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Hierarchy (mathematics)
In mathematics, a hierarchy is a set-theoretical object, consisting of a preorder defined on a set. This is often referred to as an ordered set, though that is an ambiguous term that many authors reserve for partially ordered sets or totally ordered sets. The term pre-ordered set is unambiguous, and is always synonymous with a mathematical hierarchy. The term hierarchy is used to stress a hierarchical relation among the elements.
Sometimes, a set comes equipped with a natural hierarchical structure. For example, the set of natural numbers N is equipped with a natural pre-order structure, where $n\leq n'$ whenever we can find some other number $m$ so that $n+m=n'$. That is, $n'$ is bigger than $n$ only because we can get to $n'$ from $n$ using $m$. This idea can be applied to any commutative monoid. On the other hand, the set of integers Z requires a more sophisticated argument for its hierarchical structure, since we can always solve the equation $n+m=n'$ by writing $m=(n'-n)$.
A mathematical hierarchy (a pre-ordered set) should not be confused with the more general concept of a hierarchy in the social realm, particularly when one is constructing computational models that are used to describe real-world social, economic or political systems. These hierarchies, or complex networks, are much too rich to be described in the category Set of sets.[1] This is not just a pedantic claim; there are also mathematical hierarchies, in the general sense, that are not describable using set theory.
Other natural hierarchies arise in computer science, where the word refers to partially ordered sets whose elements are classes of objects of increasing complexity. In that case, the preorder defining the hierarchy is the class-containment relation. Containment hierarchies are thus special cases of hierarchies.
Related terminology
Individual elements of a hierarchy are often called levels and a hierarchy is said to be infinite if it has infinitely many distinct levels but said to collapse if it has only finitely many distinct levels.
Example
In theoretical computer science, the time hierarchy is a classification of decision problems according to the amount of time required to solve them.
See also
• Order theory
• Nested set collection
• Tree structure
• Lattice
• Polynomial hierarchy
• Chomsky hierarchy
• Analytical hierarchy
• Arithmetical hierarchy
• Hyperarithmetical hierarchy
• Abstract algebraic hierarchy
• Borel hierarchy
• Wadge hierarchy
• Difference hierarchy
• Tree (data structure)
• Tree (graph theory)
• Tree network
• Tree (descriptive set theory)
• Tree (set theory)
References
1. We may need a bigger topos.
| Wikipedia |
\begin{document}
\subjclass[2000]{35J50, 35Q41, 35Q55, 37K45}
\keywords{Elliptic equations, quadratic growth in the gradient, Ambrosetti-Prodi type problems, continuum of solutions, topological degree}
\begin{abstract} We consider the boundary value problem \begin{equation*}
- \Delta u = \lambda c(x)u+ \mu(x) |\nabla u|^2 + h(x), \quad u \in H^1_0(\Omega) \cap L^{\infty}(\Omega) \eqno{(P_{\lambda})} \end{equation*} where $\Omega \subset {\mathbb R}^N, N \geq 3$ is a bounded domain with smooth boundary. It is assumed that $c\gneqq 0$, $c,h$ belong to $L^p(\Omega)$ for some $p > N/2$ and that $\mu \in L^{\infty}(\Omega).$ We explicit a condition which guarantees the existence of a unique solution of $(P_{\lambda})$ when $\lambda <0$ and we show that these solutions belong to a continuum. The behaviour of the continuum depends in an essential way on the existence of a solution of $(P_0)$. It crosses the axis $\lambda =0$ if $(P_0)$ has a solution, otherwise if bifurcates from infinity at the left of the axis $\lambda =0$. Assuming that $(P_0)$ has a solution and strenghtening our assumptions to $\mu(x)\geq \mu_1>0$ and $h\gneqq 0$,
we show that the continuum bifurcates from infinity on the right of the axis $\lambda =0$ and this implies, in particular, the existence of two solutions for any $\lambda >0$ sufficiently small. \end{abstract} \maketitle
\section{Introduction}
For a bounded domain $\Omega \subset {\mathbb R}^N, N \geq 3$, with smooth boundary (in the sense of condition (A) of \cite[p.6]{LU68}), we study, depending on the parameter $\lambda \in {\mathbb R}$, the existence and multiplicity of solutions of the boundary value problem \begin{equation*}
- \Delta u = \lambda c(x)u+ \mu(x) |\nabla u|^2 + h(x), \quad u \in H^1_0(\Omega) \cap L^{\infty}(\Omega). \eqno{(P_{\lambda})} \end{equation*} Here, the hypotheses are $$ \hspace{1cm} \left\{ \begin{array}{c} c \mbox{ and } h \mbox{ belong to } L^p(\Omega) \quad \mbox{for some } p > \frac{N}{2}, \\[2mm] c\gneqq 0 \mbox{ and } \mu \in L^{\infty}(\Omega). \end{array} \right. \leqno{\mathbf{(A1)}} $$ Observe that problem $(P_\lambda)$ is quasilinear due to the presence
of the quadratic term $|\nabla u|^2$. Elliptic quasilinear equations with a gradient dependence up to the critical growth $|\nabla u|^2$ were first studied by Boccardo, Murat and Puel in the 80's and it has been an active field of research until now. To situate our problem with respect to the existing literature we underline that our solutions are functions $u\in H_0^1(\Omega)\cap L^{\infty}(\Omega)$ satisfying $$ \int_\Omega \nabla u \nabla v \, dx
= \lambda \int_\Omega c(x) u v \, dx
+
\int_\Omega \mu(x) |\nabla u |^2 v \, dx
+ \int_\Omega h v \, dx \, , \ \forall v\in H_0^1(\Omega)\cap L^{\infty}(\Omega)\,, $$
and that our problem does not satisfy the so called {\it sign condition} and thus we cannot follow the approach of \cite{BeBoMu,BoGaMu,BoMuPu1.5}.
Under the additional condition that $c(x) \geq \alpha_0$ a.e. in $\Omega$ for some $\alpha_0 >0$, the existence of a solution of $(P_{\lambda})$ when $\lambda <0$ is a special case of the results of \cite{BoMuPu1,BoMuPu2.5,BoMuPu3}.
Also in the case $\lambda = 0$ (or equivalently when $c\equiv 0$), Ferone and Murat \cite{FeMu1, FeMu2} obtained the existence of a solution for $(P_{0})$, under the smallness assumption
\begin{equation} \label{FMurat}
||\mu||_{\infty} ||h||_{\frac{N}{2}} < \mathcal{S}_N^2,
\end{equation} where $\mathcal{S}_N >0$ is the best constant in Sobolev's inequality, namely, $$
\mathcal{S}_N = \inf \left\{ \frac{\left( \int_\Omega |\nabla \phi|^2\, dx\right)^{1/2} }{ \left(\int_\Omega |\phi |^{2^*}\,dx \right)^{1/{2^*}}}\ : \ \phi\in H_0^1(\Omega)\setminus \{0\}\right\} , \mbox{ with } 2^* = 2N/(N-2). $$ This result was the first one assuming that $h(x) \in L^{N/2}(\Omega)$ but previous results, in the case $\lambda =0$, were obtained under stronger regularity assumptions on $h(x)$ and assuming that a suitable norm of $h(x)$ is small (see \cite{AlPi, AlLiTr, FePo, FePoRa, GrMo, MaPaSa}). In the particular case $\mu(x)\equiv \mu>0$ and $h(x)\geq 0$, this existence result of \cite{FeMu1, FeMu2} can be improved using Theorem 2.3 of Abdellaoui, Dall'Aglio and Peral in \cite{AbDaPe} (see also \cite{AbBi2}) who show that a sufficient condition for the existence of a solution for $(P_0)$ is $$ \mu <
\displaystyle \inf \left\{ \frac{\int_\Omega |\nabla \phi|^2\,dx}{\int_\Omega h(x)\phi^2\,dx} \, :\, \phi \in H_0^1(\Omega) , \ \int_\Omega h(x)\phi^2\,dx >0\right\}. $$
In addition, we remark the interesting result by Porretta \cite{Po} for the case $c(x)\equiv 1$, $\mu(x)\equiv 1$ and $h\in L^\infty (\Omega)$. He has proved that when the problem $(P_0)$ has no solution, then the solutions of $(P_\lambda)$ for $\lambda <0$ blows-up completely, this behaviour being described in terms of the so-called ergodic problem. \medbreak
Concerning the uniqueness a general theory for problems having quadratic growth in the gradient was developed in \cite{BaBlGeKo,BaMu} (see also \cite{ArSe,BaPo}). When $c(x) \geq \alpha_0$ a.e. in $\Omega$ for some $\alpha_0 >0$, the results of \cite{BaBlGeKo} imply the uniqueness of the solutions of $(P_{\lambda})$ when $\lambda <0$. For $\lambda =0$, the fact that $(P_{0})$ has at most one solution can also
be obtained from \cite{BaBlGeKo} provided that either $h(x)$ has a sign or it is sufficient small. See Remark \ref{newrequire1} for more details. \medbreak
The aim in our first result is twofold. First, we handle functions $c(x)$ that can vanish in some part of $\Omega$. This does not seem to have been considered in the literature. Specifically, for the nonnegative and nonzero function $c(x)$ we set $$ W_c = \{ w \in H^1_0(\Omega) : c(x) w(x) = 0, \mbox{ a.e. } x \in \Omega \}, $$ and, if
$\mbox{meas}(\Omega \backslash \mbox{Supp} \, c) > 0$, we assume that
the following condition holds $$ \left\{ \begin{array}{c}
\displaystyle \inf_{ \{ u \in W_c,\, ||u||_{H^1_0(\Omega)}=1 \}} \, \displaystyle \int_{\Omega} \left(|\nabla u|^2 - ||\mu^+||_{\infty} h^+(x) u^2 \right) dx >0, \\
\displaystyle \inf_{ \{ u \in W_c,\, ||u||_{H^1_0(\Omega)}=1 \}} \, \displaystyle \int_{\Omega} \left(|\nabla u|^2 - ||\mu^-||_{\infty} h^-(x) u^2 \right) dx >0. \end{array} \right. \leqno{\mathbf{(Hc)}} $$ Here $\mu^+ = \max(\mu,0)$, $\mu^- = \max(-\mu,0)$, $h^+ = \max(h,0)$ and $h^- = \max(-h,0)$. As we shall see condition {\rm (Hc)}, along with {\rm (A1)}, suffices to guarantee the existence of a solution of $(P_\lambda)$ for $\lambda<0$. Moreover, we prove that, under {\rm (A1)}, the problem $(P_\lambda)$ for $\lambda \leq 0$ has at most one solution. To obtain this uniqueness result it does not seems possible to extend the approach of \cite{BaBlGeKo,BaMu} and we follow a different strategy. As a first step we establish a regularity result inspired by \cite{BoMuPu2,Gi, GiMo}
for the solutions of $(P_\lambda)$. Then, using this regularity we derive our uniqueness result. This approach is applied directly to problem $(P_\lambda)$. However we believe it can also be used to obtain, under slighty stronger regularity assumptions on the data, new uniqueness results for the general class of problems considered in \cite{BaBlGeKo,BaMu}.
Our aim is also to point out that the unique solution of $(P_\lambda)$ for $\lambda<0$ belongs to a continuum $C$ whose behavior at $\lambda=0$ depends in an essential way on the existence of solution of $(P_0)$. Throughout the paper we assume that the boundary of $\Omega$ is smooth in the sense of condition (A) of \cite[p.6]{LU68}. Under this assumption it is known, \cite[Theorem IX.2.2]{LU68}
that any solution of $(P_\lambda)$ belong to $C^{0, \alpha}(\overline{\Omega})$ for some $\alpha >0$. Denoting the solutions set $$ \Sigma = \{ (\lambda, u) \in {\mathbb R} \times C(\overline \Omega) : (\lambda, u) \mbox{ solves } (P_{\lambda})\}, $$ we prove the following result.
\begin{thm} \label{negativevalue} Assume that {\rm (A1)} holds. If in addition, in the case that \\
$\mbox{\rm meas}(\Omega \backslash \mbox{\rm Supp} \, c) > 0$,
we also assume that {\rm (Hc)} holds, then
\begin{enumerate} \item[1)] For $\lambda < 0$, $(P_{\lambda})$ has a unique solution $u_{\lambda}$.
\item[2)]
There exists an unbounded continuum $C$ of solutions in $\Sigma$ whose projection $\mbox{\rm Proj}_{{\mathbb R}}C$ on the $\lambda$-axis contains the interval $]-\infty,0[$.
\item [3)] Moreover, $\limsup_{\lambda\to 0^-}\|u_\lambda\|_\infty<\infty$ if and only if $(P_0)$ has a solution. In case $(P_0)$ has a solution $u_0${\color{blue},} it is unique and $$
\lim_{\lambda\to 0^-} \|u_\lambda - u_0\|_\infty = 0. $$
If $(P_0)$ has no solution{\color{blue},} then $\lim_{\lambda\to 0^-}\|u_\lambda\|_\infty = \infty$ and $\lambda=0$ is a bifurcation point from infinity for $(P_{\lambda})$ (see Figure~\ref{fig1}). \end{enumerate} \end{thm}
\begin{figure}
\caption{Bifurcation diagram when $(P_0)$ has no solution}
\label{fig1}
\end{figure}
\begin{remark}\label{comparaison} Condition {\rm (Hc)} connects the two limit cases: $c(x) \geq \alpha_0 >0$ and $c\equiv 0$ ($\lambda=0$). If $c(x) >0$ a.e. on $\Omega$ we have $\mbox{meas}(\Omega \backslash \mbox{Supp} \, c) =0$. Thus, under {\rm (A1)}, a solution of $(P_{\lambda})$ exists for any $\lambda <0$. If $\mbox{meas}(\Omega \backslash \mbox{Supp} \, c) >0$, the situation is more delicate. When both $\mu(x) \geq 0$ and $h(x) \geq 0$, {\rm (Hc)} relates the {\it size} of $\mu(x)h(x)$ to the {\it size} of $\Omega \backslash \mbox{Supp} \, c$, showing
that the signs of $\mu(x)$ and $h(x)$ with respect to one another strongly influence the existence of solution of $(P_{\lambda})$ when $\lambda <0$. Indeed, {\rm (Hc)} holds if either $\mu(x) \geq 0$ and $h(x) \leq 0$ a.e. in $\Omega$, or $\mu(x) \leq 0$ and $h(x) \geq 0$ a.e. in $\Omega$. Moreover, it holds true under condition \eqref{FMurat} since, from the Sobolev embedding, it follows that $$
\int_{\Omega} h(x) v^2 dx \leq ||h||_{N/2}||v||^2_{2^*} \leq \frac{1}{\mathcal{S}_N^2}||h||_{N/2}||\nabla v||_2^2. $$ Hence we obtain the above refered results as a corollary. In Remark \ref{ktL2} we show that {\rm (Hc)} is somehow sharp for the existence of solution of $(P_{\lambda})$. \end{remark}
\medbreak
\begin{remark} We shall also prove, in Corollary \ref{prop3a}, that a sufficient condition for the existence of solution of $(P_0)$ is that condition {\rm (Hc)} is satisfied with $c(x)\equiv 0$, i.e that the following condition is hold $$ \left\{ \begin{array}{c}
\displaystyle \inf_{ \{ u \in H^1_0(\Omega),\, ||u||_{H^1_0(\Omega)}=1 \}} \, \displaystyle \int_{\Omega} \left(|\nabla u|^2 - ||\mu^+||_{\infty} h^+(x) u^2 \right) dx >0, \\
\displaystyle \inf_{ \{ u \in H^1_0(\Omega),\, ||u||_{H^1_0(\Omega)}=1 \}} \, \displaystyle \int_{\Omega} \left(|\nabla u|^2 - ||\mu^-||_{\infty} h^-(x) u^2 \right) dx >0. \end{array} \right. \leqno{\mathbf{(H0)}} $$
\end{remark}
Our next result show that the existence of a solution of $(P_0)$ suffices to guarantee the existence of a continuum of solutions $C \subset \Sigma$ such that $\mbox{Proj}_{{\mathbb R}}C$ contains $]-\infty,a]$ for some $a>0$.
\begin{thm}\label{th1.1a} Assume {\rm (A1)} and suppose that $(P_0)$ has a solution. Then \begin{enumerate} \item[1)] For all $\lambda \leq 0$, $(P_\lambda)$ has a, unique, solution $u_\lambda$.
\item[2)] There exists a continuum $C\subset\Sigma$ such that
\begin{enumerate}
\item $\{(\lambda,u_\lambda):\, \lambda\in \,]-\infty,0]\, \}\subset C$.
\item $C\cap ([0,\infty[\,\times C(\overline{\Omega}))$ is a unbounded set in ${\mathbb R}\times C(\overline{\Omega})$. \end{enumerate}
In particular, $\mbox{Proj}_{{\mathbb R}}C$ contains $]-\infty,a]$ for some $a>0$. \end{enumerate} \end{thm}
Finally, in the last part of the paper and under stronger assumptions, we study the behaviour in the half space $\{ \lambda >0 \} \times C(\overline \Omega)$ of the branch $C \subset \Sigma$ obtained in Theorem \ref{th1.1a} and we obtain a multiplicity result.
First we note that, in case $\mu\equiv 0$, we cannot have multiplicity results except when $\lambda$ is an eigenvalue of the problem \begin{equation} \label{eigenvaluep}
-\Delta \varphi_{1} = \gamma c (x) \varphi_{1}, \quad \varphi_1 \in H^1_0(\Omega),
\end{equation} and $h(x)$ satisfies the {\it ``good''} orthogonality condition. Hence, there is no hope to obtain multiplicity results just under our assumption {\rm (A1)}.
Multiplicity results have been considered by Abdellaoui, Dall'Aglio and Peral \cite{AbDaPe} (see also \cite{AbBi2, Si}) for $(P_{\lambda})$ in the case $\lambda=0$ and when $\mu(x)$ is replaced by some $g(u)$ satisfying $u g(u)<0$. In a recent paper, Jeanjean and Sirakov \cite{JeSi} study the case $\lambda>0$ when
$\mu(x)$ is a positive constant but $h(x)$ may change sign and satisfy a condition related to \eqref{FMurat}. Using Theorem 2 of \cite{JeSi} an explicit $\lambda_0 >0$ can be derived under which $(P_{\lambda})$ has two solutions whenever $\lambda \in \,]0, \lambda_0[$.
The above quoted multiplicity results have the common property that the coefficient of $|\nabla u|^2$ (either $g(u)$ or the constant $\mu$) does not depend on $x$. This allows the authors to make a change a variable, similar to the one used in \cite{KaKr}, in order to transform the problem in a semilinear one (i.e. without gradient dependence). Then variational methods are used to prove multiplicity results on the transformed problem. In our case, we consider problem $(P_{\lambda})$ with a non constant function coefficient $\mu(x)$, which implies that this change of variable is no more possible. \medbreak
We replace {\rm (A1)} by the stronger assumption $$ \left\{ \begin{array}{c} c \mbox{ and } h \mbox{ belongs to } L^p(\Omega) \quad \mbox{for some } p > \frac{N}{2}, \\[2mm]
c\gneqq 0, \, h \gneqq 0 \mbox{ and } \mu_2\geq \mu(x) \geq \mu_1 \mbox{ for some } \mu_2\geq\mu_1 >0. \end{array} \right. \leqno{\mathbf{(A2)}} $$ Let $\gamma_1 >0$ denote the first eigenvalue of the problem (\ref {eigenvaluep}). We prove the following theorem.
\begin{thm} \label{th3} Assume $\mathrm{(A2)}$ and suppose that $(P_0)$ has a solution. Then the continuum $C \subset \Sigma$ obtained in Theorem \ref{th1.1a} consists of non negative functions{\color{blue},} its projection $ \mbox{\rm Proj}_{{\mathbb R}} C$ on the $\lambda$-axis is an unbounded interval $]-\infty,\overline \lambda] \subset {]-\infty,\gamma_1[}$ containing $\lambda =0$ and $C \subset \Sigma$ bifurcates from infinity to the right of the axis $\lambda = 0$.
Moreover, there exists $\lambda_0\in {]0,\overline \lambda]}$ such that for all $\lambda \in {]0, \lambda_0[}$, the section $C\cap (\{\lambda\}\times C(\overline\Omega))$
contains two distinct non negative solutions of $(P_{\lambda})$ in $\Sigma$
(see Figure 2). \end{thm}
\begin{remark} In order to prove Theorem \ref {th3} the key points are the observation that the continuum cannot cross the line $\lambda = \gamma_1$ and the derivation of a priori bounds, for any $a>0$, on the (positive) solutions of $(P_{\lambda})$ for $\lambda \in\, ]a, \gamma_1]$. These a priori bounds are obtained by an extention of the classical approach of Brezis and Turner \cite{BT77}. \end{remark}
\begin{remark} The fact that on a MEMS type equation, involving a critical term in gradient, a multiplicity result had also been derived throught the study of the behaviour of a continuum \cite{WeYe} was pointed out to us by D. Ye after the completion of the present work. \end{remark}
The paper is organized as follows. In Section \ref{Section1} we recall some results concerning the method of lower and upper solutions as well as a continuation theorem. In Section \ref{Section02} we derive various existence results for problems of the type of $(P_{\lambda})$ when $\lambda \leq 0$. Section \ref{Sectionuniqueness-0} deals with the uniqueness issue. In Section \ref{Sectionuniqueness} we establish the existence of a continuum of solutions. Section \ref{Section2} is devoted to the study of the branch in the half space $\{ \lambda >0\} \times C(\overline{\Omega})$ and in particular to the derivation of a priori bounds, see Proposition \ref{bounds1}. The proofs of our three theorems are given in Section \ref{Proofs}. Finally a technical result, Lemma \ref{comp}, is proved in Section \ref{appendix}.
\begin{figure}
\caption{Bifurcation diagram when $(P_0)$ has a solution.}
\label{fig22}
\end{figure}
\vskip9pt \begin{center} \textbf{Notation.} \end{center} \begin{enumerate}{\small
\item $\Omega \subset {\mathbb R}^N$, $N \geq 3$ is a bounded domain whose boundary $\partial \Omega$ is sufficiently regular as to satisfies the condition (A) of \cite[p.6]{LU68}. A sufficient condition for (A) is that $\partial \Omega$ satisfies the exterior uniform cone condition.
\item For any measurable set $\omega \subset {\mathbb R}^N$ we denote by $\mbox{meas}(\omega)$ its Lebesgue measure.
\item For $p\in [1,+\infty[$, the norm $(\int_{\Omega}|u|^pdx)^{1/p}$
in $L^p(\Omega)$ is denoted by $\|\cdot\|_p$. We denote by $p^{\prime}$
the conjugate exponent of $p$, namely $p^{\prime} = p/(p-1).$ The norm in $L^{\infty}(\Omega)$ is $\|u\|_{\infty}=\mbox{esssup}_{x\in \Omega}|u(x)|$. \item For $v \in L^1(\Omega)$ we define $v^+= \max(v,0)$ and $v^- = \max(-v,0)$. \item For $h\in L^1(\Omega)$ we denote $h\gneqq 0$ if $h(x)\geq 0$ for a.e. $x\in\Omega$ and $\mbox{meas}(\{x\in\Omega: h(x)>0\})>0$. \item We denote by $H$ the space $H^1_0(\Omega)$ equipped with the Poincar\'{e}
norm $||u||:=\left( \int_\Omega |\nabla u|^2\,dx\right)^{1/2}$. \item We denote by $X$ the space $H_0^1(\Omega) \cap L^{\infty}(\Omega)$.
\item We denote by $B_r(u_0) $ the ball $ \{u \in H\, :\, ||u-u_0||<r\}.$ \item We denote by $C,D>0$ any positive constants which are not essential in the problem and may vary from one line to another. } \end{enumerate}
\section{Preliminaries} \label{Section1}
In our proofs we shall use the method of lower and upper solutions. We present here Theorem 3.1 of \cite{BoMuPu4} adapted to our setting. We consider the boundary value problem \begin{equation} \label{bo-Mu-Pu} - \Delta u +H(x,u, \nabla u)=f, \quad u \in X \end{equation} where $f\in L^1(\Omega)$ and $H$ is a Carath\' eodory function
from $\Omega \times {\mathbb R} \times {\mathbb R}^N$ into ${\mathbb R}$ with a natural growth, i.e., for which there exist a nondecreasing function $b$ from $[0,+\infty[$ into $[0,+\infty[$ and $k \in L^1(\Omega)$ such that, for a.e. $x\in \Omega$ and all $(u,\xi)\in \mathbb R\times\mathbb R^{N}$, $$
|H(x,u, \xi)| \leq b(|u|) [k(x) + |\xi|^2]. $$ We also recall (see \cite{BoMuPu4}) that a {\it lower solution} (respectively, an {\it upper solution}) of (\ref{bo-Mu-Pu}) is a function $\alpha$ (respectively, $\beta$) $\in H^1(\Omega) \cap L^{\infty}(\Omega)$ such that
$$ - \Delta \alpha + H(x,\alpha, \nabla \alpha)\leq f(x) \mbox{ in } \Omega, \quad \alpha \leq 0 \mbox{ on } \partial \Omega, $$ (respectively, $$ - \Delta \beta +H(x,\beta, \nabla \beta)\geq f(x) \mbox{ in } \Omega, \quad \beta \geq 0 \mbox{ on } \partial \Omega). $$ This has to be understood in the sense that $\alpha^+\in H^1_0(\Omega)$ and $$ \int_{\Omega} \nabla\alpha\nabla v \,dx+\int_{\Omega} H(x,\alpha,\nabla \alpha) v\,dx\leq \int_{\Omega} f(x)v\,dx, $$ (respectively, $\beta^-\in H^1_0(\Omega)$ and $ \int_{\Omega} \nabla\beta\nabla v \,dx+\int_{\Omega} H(x,\beta,\nabla \beta) v\,dx\geq \int_{\Omega} f(x)v\,dx $), for all $v\in H^1_0(\Omega)\cap L^\infty (\Omega)$ with $v\geq 0$ a.e. in $\Omega$.
\begin{thm}[Boccardo-Murat-Puel \cite{BoMuPu4}] \label{sousBo} If there exist a lower solution $\alpha$ and an upper solution $\beta$ of {\rm(\ref{bo-Mu-Pu})} with $\alpha \leq \beta$ a.e. in $\Omega$, then there exists a solution $u$ of {\rm(\ref{bo-Mu-Pu})} with $\alpha \leq u \leq \beta$ a.e. in $\Omega$. \end{thm}
We also need a continuation theorem. Let $E$ be a real Banach space with norm $||\cdot||_E$ and $T:{\mathbb R} \times E \to E$ a completely continuous map, i.e. it is continuous and maps bounded sets to relatively compact sets. For $\lambda \in {\mathbb R}$, we consider the problem of finding the zeroes of $\Phi(\lambda, u) := u - T(\lambda, u)$, i.e. \begin{equation*} \Phi(\lambda, u) = u - T(\lambda, u) = 0, \quad u \in E \eqno{(Q_{\lambda})}, \end{equation*} and we define $$ \Sigma = \{ (\lambda, u) \in {\mathbb R} \times E : \Phi(\lambda, u) = 0 \}. $$
Let $\lambda_0 \in {\mathbb R}$ be arbitrary but fixed and for $v \in E$ and $r>0$ let $B(v,r):= \{u \in E : ||v- u|| <r \}$.
If we assume that $u_{\lambda_0}$ is an isolated solution of $(Q_{\lambda_0})$, then the Leray-Schauder degree $\mbox{deg}(\Phi(\lambda_0,\cdot), B(u_{\lambda_0},r), 0)$ is well defined and is constant for $r>0$ small enough. Thus it is possible to define the index $$ i(\Phi(\lambda_0, \cdot), u_{\lambda_0}) := \lim_{r \to 0} \mbox{deg}(\Phi(\lambda_0, \cdot), B(u_{\lambda_0},r), 0). $$
\begin{thm} \label{1noacot} If $(Q_{\lambda_0})$ has a unique solution $u_{\lambda_0}$ and $i(\Phi(\lambda_0, \cdot), u_{\lambda_0}) \neq 0$ then $\Sigma$ possesses two unbounded components $C^+$, $C^-$ in $[\lambda_0, + \infty[ \times E$ and $]- \infty, \lambda_0] \times E$ respectively which meet at $(\lambda_0, u_{\lambda_0})$. \end{thm}
Theorem \ref{1noacot} is essentially Theorem 3.2 of \cite{Ra} (stated assuming that $\lambda_0=0$). In turn this result is essentially due to Leray and Schauder \cite{LeSc}. For an exposition of the main properties of the Leray-Schauder degree see, for example, \cite{AmAr}.
\section{Some existence results} \label{Section02}
In this section we establish some existence results for the boundary value problem \begin{equation} \label{31}
- \Delta u = d(x)u+ \mu(x) |\nabla u|^2 + h(x), \quad u \in X \end{equation} under the assumption that $$ \leqno{\mathbf{(A3)}} \hspace{1cm} \left\{ \begin{array}{c} d \mbox{ and } h \mbox{ belong to } L^p(\Omega) \quad \mbox{for some } p > \frac{N}{2}, \\[2mm]
\mu(x)\equiv \mu>0 \mbox{ is a constant}, \\[2mm]
d \leq 0\mbox{ and } h \geq 0. \end{array} \right. $$ If $\mbox{meas}(\Omega \backslash \mbox{Supp} \, d) > 0$ we also set $$ W_d = \{ w \in H^1_0(\Omega) : d(x) w(x) = 0, \, a.e. \, x \in \Omega \} $$ and we impose condition $({\rm Hc})$ for $c=d$, i.e., we require $$ \leqno{\mathbf{(H)}} \hspace{1cm}
m_2 := \displaystyle \inf_{ \{ u \in W_d,\, ||u||=1 \}} \, \int_{\Omega} (|\nabla u|^2 - \mu h(x) u^2)\, dx >0. $$
\begin{prop} \label{prop1} Assume {\rm (A3)} and, if $\mbox{\rm meas}(\Omega \backslash \mbox{\rm Supp} \, d) > 0$, also that {\rm (H)} holds. Then {\rm(\ref{31})} has a non negative solution. \end{prop}
\begin{remark}\label{pos} Observe that, under condition {\rm (A3)}, every solution $u$ of {\rm(\ref{31})} is non negative. In fact, using $u^-$ as test function we obtain, as $d\leq 0$, $\mu> 0$ and $h\geq 0$, $$
0\geq -\int_\Omega |\nabla u^-|^2 + \int_\Omega d(x) |u^-|^2 = \int_\Omega \left[ \mu |\nabla u|^2 + h(x)\right] u^{-} \geq 0, $$ which implies that $u^-=0$ i.e. $u\geq 0$. \end{remark}
\begin{remark} \label{ktL1} Assume, in addition to {\rm (A3)}, that there exists an open subset $O(d)$ in $\Omega$ with $C^1$ boundary $\partial O(d)$ such that
$d(x)=0$ a.e. in $\overline{O(d)}$ and $d(x)<0$ a.e. in $ \Omega\setminus O(d)$.
Then \eqref{31} has a solution if and only if {\rm (H)} holds. See Remark \ref{ktL2} below for a proof. \end{remark}
To prove Proposition \ref{prop1} we introduce the boundary value problem \begin{equation} \label{3} - \Delta v - \mu h(x) v = d(x) g(v) + h(x),\quad v \in H \end{equation} where \begin{equation} \label{defg} \begin{array}{ll} g(s) = \left\{ \begin{array}{ll} \text{$\frac{1}{\mu} (1+ \mu s) \ln (1+ \mu s)$}, &\mbox{if } \quad s \geq 0, \\ \text{$-\frac{1}{\mu} (1-\mu s) \ln (1- \mu s)$}, &\mbox{if } \quad s < 0. \end{array} \right. \end{array} \end{equation} Let us denote $$ G(s) =\int_0^s g(\xi)\,d\xi = \left\{ \begin{array}{ll}
\displaystyle \frac{{(1+\mu s)^2}}{4 \mu^{2}} [2\ln (1+\mu s) -1] {+ \frac{1}{4 \mu^2}} & \mbox{ if } s\geq 0,
\\
\\
G(-s), & \mbox{ if } s< 0.
\end{array}
\right.
$$
The properties of $g$ that are useful to us are gathered in the following lemma.
\begin{lem} \label{prop-g1} \vskip2pt \noindent \begin{itemize} \item[(i)] The function $g$ is odd and continuous on ${\mathbb R}$. \item[(ii)] $g(s)s >0$ for $s \in {\mathbb R} \setminus \{0\}$, $G(s) \geq 0$ on ${\mathbb R}$.
\item[(iii)] For any $r\in\,]0,1[$, there exists $C=C(r,\mu) >0$ such that, for all $|s| > \frac{1}{\mu}$, we have $|g(s)| \leq C |s|^{1+r}$.
\item[(iv)] We have $G(s)/s^2 \to + \infty$ as $|s| \to \infty$. \qed \end{itemize} \end{lem}
The idea of modifying the problem to obtain problem (\ref{3}) is not new. It appears already in \cite{KaKr} in another context. It permits to obtain a non negative solution of~\eqref{31}.
\begin{lem} \label{dual1} Assume that {\rm (A3)} hold. \begin{enumerate} \item[i)] Any solution of \eqref{3} belongs to $W^{2,p}(\Omega)$ and thus to $L^{\infty}(\Omega)$; \item[ii)] If $v \in H$ is a non negative solution of \eqref{3} then $u = (1/ \mu) \ln (1 + \mu v)\in X$ is a (non negative) solution of \eqref{31}. \end{enumerate} \end{lem}
\begin{proof}
i) Let $v \in H$ be a solution of~\eqref{3}, that we write as $$ - \Delta v = \left[\mu h(x) + d(x) \frac{g(v)}{v} \right] v + h(x), \quad v\in H. $$ By classical arguments, see for example \cite[Theorem III-14.1]{LU68}, as $\partial \Omega$ satisfies the condition (A) of \cite{LU68},
the first part of the lemma will be proved if we can show that $$ \left[\mu h(x) + d(x) \frac{g(v)}{v} \right] \in L^{p_1}(\Omega) \quad \mbox{with } p_1>N/2. $$ But by assumption $d$ and $\mu h$ belong to
$L^{p}(\Omega)$, for some $p>N/2$ and we shall prove that the term $d(x) \frac{g(v)}{v}$ has the same property. This is the case because of the slow growth of $g(s)/s$ as $|s| \to \infty$, see Lemma \ref{prop-g1}-(iii). Specifically, for any $r\in \,]0,1[$, there exists a $C >0$ such that, for all $|s|> \frac{1}{\mu}$, $$
\left|g(s)/s \right| \leq C |s|^r. $$ Thus, since $d \in L^p(\Omega)$ with $p > \frac{N}{2}$ and $v\in L^{\frac{2N}{N-2}}(\Omega)$, taking $r >0$ sufficiently small (for example $r < \frac{4p-2N}{p(N-2)}$) we see, using H\"older inequality, that $d(x) g(v)/v \in L^{p_1}(\Omega)$, for some $p_1 > N/2$. This ends the proof of Point i).
ii) Since $v\geq 0$ the problem \eqref{3} can be rewritten as \begin{equation} \label{2.12} - \Delta v = \frac{d(x)}{\mu}(1 + \mu v) \ln (1 + \mu v) + (1 + \mu v) h(x), \qquad v\in H. \end{equation} Let $v \in H$ be a non negative solution of \eqref{2.12}, we want to show that $ u = \frac{1}{\mu}\ln(1+ \mu v)$ is a solution of~\eqref{31}, namely that, for $\phi \in C_0^{\infty}(\Omega)$, \begin{equation} \label{2.13}
\int_{\Omega} \left(\nabla u \nabla \phi - \mu |\nabla u|^2 \phi - d(x) u \phi\right) dx = \int_{\Omega} h(x) \phi \thinspace dx. \end{equation} First observe that, as $v\in L^{\infty}(\Omega)$ and satisfies $v\geq 0$ in $\Omega$ we have $u\in X$. Let $\displaystyle \psi = \frac{\phi}{1+ \mu v}.$ Clearly $\psi \in H$ and thus it can be used as test function in \eqref{2.12}. Hence, we get \begin{equation} \label{2.15} \begin{array}{rcl} \displaystyle \int_{\Omega} \nabla v \nabla \psi \thinspace dx &=& \displaystyle \int_{\Omega} \frac{d(x)}{\mu} \ln(1+ \mu v) \phi \thinspace dx + \int_{\Omega}h(x) \phi \thinspace dx \\[3mm] &=& \displaystyle \int_{\Omega}d(x) u \phi \thinspace dx + \int_{\Omega}h(x) \phi \thinspace dx. \end{array} \end{equation} Moreover, we have \begin{eqnarray} \nonumber \int_{\Omega} \nabla v \nabla \psi \thinspace dx & = & \int_{\Omega}\nabla \left( \frac{1}{\mu}(e^{\mu u}-1)\right)\nabla \left(\frac{\phi}{1+ \mu v}\right)dx \\ \nonumber & = & \int_{\Omega} e^{\mu u} \nabla u \left( \frac{\nabla \phi}{1+ \mu v}- \frac{\mu \phi \nabla v }{(1+ \mu v)^2}\right) dx \\ \nonumber & = & \int_{\Omega} \nabla u \left( \nabla \phi - \frac{\mu \phi \nabla (\frac{1}{\mu}(e^{\mu u}-1))}{(1+ \mu v)} \right)dx \\ & =& \int_{\Omega }\nabla u (\nabla \phi - \mu \phi \nabla u ) \thinspace dx \nonumber \\ & =& \int_{\Omega} \left(\nabla u \nabla \phi -
\mu |\nabla u|^2 \phi\right) dx. \nonumber \end{eqnarray} Combining this equality with \eqref{2.15} we see that $u$ satisfies \eqref{2.13}. This ends the proof of Point ii). \end{proof}
In order to find a solution of (\ref{3}) we shall look to a critical point of the functional $I$ defined on $H$ by $$
I(v) = \frac{1}{2} \int_{\Omega} (|\nabla v |^2 - \mu h(x) v^2) \thinspace dx - \int_{\Omega} d(x) G(v) \thinspace dx - \int_{\Omega}h(x) v \thinspace dx. $$
As $g$ has a subcritical growth at infinity, see Lemma \ref{prop-g1}(iii), it is standard to show that $I \in C^1(H, {\mathbb R})$ and that a critical point of $I$ corresponds to a solution in $H$ of (\ref{3}). To obtain a critical point of $I$ we shall prove the existence of a global minimum of $I$. We define \begin{equation} \label{min} m:= \inf_{u \in H} I(u) \in {\mathbb R} \cup \{- \infty\}. \end{equation}
\begin{lem} \label{minbounded} Assume {\rm (A3)} and, if $\mbox{\rm meas}(\Omega \backslash \mbox{\rm Supp} \, d) > 0$, assume also that {\rm (H)} holds. Then the infimum $m$ defined by \eqref{min} is finite and it
is reached by a non negative function in $H$. Consequently, \eqref{3} has a non negative solution. \end{lem}
\begin{proof} We divide the proof into two steps :
\noindent {\bf Step 1.} {\it $I$ is coercive. }
We assume by contradiction the existence of a sequence $\{v_n\} \subset H$ such that $\|v_n\|\to\infty$ and $I(v_n)$ is bounded from above. We define $$
w_n = \frac{v_n}{||v_n||}. $$
Clearly $||w_n||\equiv 1$ and we can assume that $w_n \rightharpoonup w $ weakly in $ H $ and $ w_n \to w $ strongly in $ L^q(\Omega) $ for $ q \in [2, \frac{2N}{N-2}[.$ Since $I(v_n)$ is bounded from above, we have \begin{equation} \label{6}
\limsup_{n \to \infty} \frac{I(v_n)}{||v_n||^2} \leq 0. \end{equation}
We shall treat separately the two cases : $$ {(1)} \,\,\, w \in W_d \quad \mbox{ and } \quad {(2)} \,\,\, w \not \in W_d. $$
\noindent{\it Case $(1)$: $w \in W_d$.} In this case, taking {\rm (H)} into account, it follows that \begin{equation*} \label{10}
\int_{\Omega}(|\nabla w|^2 - \mu h(x) w^2) dx \geq m_2||w||^2. \end{equation*}
Thus, and since $G(s) \geq 0$ on ${\mathbb R}$ and $d(x)\leq 0$ in $\Omega$, using the weak lower semicontinuity of $\int_{\Omega}|\nabla u|^2dx$ and the weak convergence of $w_n$, we obtain \arraycolsep1.5pt \begin{eqnarray}\label{1000}
\liminf_{n \to \infty} \frac{I(v_n)}{||v_n||^2}
&=& \liminf_{n \to \infty} \left[\frac{1}{2} \int_{\Omega} (|\nabla w_n|^2 - \mu h(x) w_n^2) dx - \int_{\Omega} \frac{d(x) G(v_n)}{||v_n||^2} dx \right] \nonumber \\
&\geq & \frac{1}{2} \int_{\Omega}(|\nabla w|^2 - \mu h(x)w^2) dx
\geq \frac{1}{2} m_2 ||w||^2 \geq 0 \geq \limsup_{n \to \infty} \frac{I(v_n)}{||v_n||^2}, \end{eqnarray} \arraycolsep5pt
i.e., $\lim_{n \to \infty} \frac{I(v_n)}{||v_n||^2} =0$ and $w \equiv 0$. However, using that $2p/(p-1) < 2N/(N-2)$ and $w_n$ is weakly convergent to $w=0$ in $H$, we deduce the strong convergence of $w_n$ to $w=0$ in $L^{2p/(p-1)}(\Omega)$, which by the assumptions $d(x) \leq 0$ on $\Omega$ and $G(s) \geq 0$ on ${\mathbb R}$ implies that $$
\lim_{n \to \infty} \frac{I(v_n)}{||v_n||^2} \geq \frac{1}{2} - \lim_{n \to \infty} \frac{\mu}{2} \int_{\Omega} h(x) w_n^2 dx - \lim_{n \to \infty} \int_{\Omega} \frac{h(x)w_n}{||v_n||} dx \geq \frac{1}{2} . $$ This is a contradiction showing that case {(1)} cannot occurs. \medbreak
\noindent{\it Case $(2)$: $w \not \in W_d$.}
Since $w \not \in W_d$, necessarily $\Omega_0=\{x\in \Omega,\, d(x) w(x)\not=0\}$ has non zero measure and thus $|v_n(x)|=|w_n(x)|\,\|v_n\|\to\infty$ a.e. in $\Omega_0$. Using the assumptions $d(x) \leq 0$ in $\Omega$ and $G(s) \geq 0$ on ${\mathbb R}$ we deduce from Lemma \ref{prop-g1}-(iv) and Fatou's lemma that
\begin{eqnarray*} \limsup_{n\to\infty}\int_{\Omega} \frac{d(x)G(v_n)}{v_n^2}w_n^2 dx &\leq &
\limsup_{n\to\infty} \int_{\Omega_0} \frac{d(x)G(v_n)}{v_n^2}w_n^2 dx
\\
&\leq& \int_{\Omega_0} \limsup_{n\to\infty} \frac{d(x)G(v_n)}{v_n^2}w_n^2 dx =-\infty. \end{eqnarray*}
On the other hand, using that $w_n$ is weakly convergent in $H$ and that, by Sobolev's embedding, $||w_n||_{\frac{2p}{p-1}}$ is bounded, it follows
that $$
0\geq \limsup_{n \to \infty} \frac{I(v_n)}{||v_n||^2} \geq \liminf_{n \to \infty} \frac{I(v_n)}{||v_n||^2} \geq -C
- \limsup_{n \to \infty} \int_{\Omega} \frac{d(x) G(v_n)}{||v_n||^2} dx =+\infty , $$ a contradiction proving that case {(2)} does not occur and the proof of Step 1 is concluded.
\noindent{\bf Step 2.} \, {\it Existence of a minimum of $I$.}
To show that $I$ admits a global minimizer it now suffices to show that $I$ is weakly lower semicontinuous i.e., if $\{v_n\} \subset H$ is a sequence such that $v_n \rightharpoonup v$ weakly in $H$, and then $v_n \to v$ strongly in $L^q(\Omega)$ for $q \in [2,\frac{2N}{N-2}[$, we have \begin{equation} \label{155} I(v) \leq \liminf_{n \to \infty}I(v_n). \end{equation}
Using the weak convergence of the sequence $\{v_n\}$ and the weak lower semicontinuity of $\int_{\Omega}|\nabla v|^2 dx$, we have \begin{equation} \label{166}
\frac{1}{2}\int_{\Omega}|\nabla v|^2 dx - \int_{\Omega}h(x)v dx \leq \liminf_{n \to \infty} \left[\frac{1}{2}\int_{\Omega} |\nabla v_n|^2 dx - \int_{\Omega} h(x) v_n dx\right]. \end{equation} Also, the strong convergence in $L^{\frac{2p}{p-1}}(\Omega)$ implies that \begin{equation} \label{177} \int_{\Omega}\mu h(x) v_n^2 dx \to \int_{\Omega} \mu h(x) v^2 dx. \end{equation} Finally, since $-d(x) G(v_n) \geq 0$ on $\Omega$, as a consequence of Fatou's lemma, we obtain \begin{equation} \label{188} \int_{\Omega} - d(x) G(v) dx \leq \liminf_{n \to \infty} \int_{\Omega} -d (x) G(v_n) dx. \end{equation} At this point (\ref{155}) follows from (\ref{166})-(\ref{188}). \medbreak
\noindent{\bf Step 3.} \, {\it Conclusion.}
To conclude the existence of a non negative minimum, observe that, as $h(x)\geq 0$ in $\Omega$ and $G(s)$ is even we have, for every $u\in H$, $$
I(|u|)\leq I(u), $$
and hence if $v\in H$ is a minimum of $I$ then $|v|$ is also a minimum. Then we conclude that the infimum $m$ is reached by a non negative function. \end{proof}
\begin{proof}[Proof of Proposition \ref{prop1}] By Lemma \ref{minbounded}, (\ref{3}) admits a non negative solution $v \in H$ and thus, using Lemma \ref{dual1}, we deduce that (\ref{31}) has a non negative solution. \end{proof}
We now consider the problem. \begin{equation} \label{eq22} - \Delta u = d(x)u+ W(x,u, \nabla u), \quad u \in X, \end{equation} where we assume
$$ \leqno{\mathbf{(A4)}} \left\{ \begin{array}{c} d\leq 0 \mbox{ with } d \in L^p(\Omega) \mbox{ for some } p > \frac{N}{2} \\[2mm]
\mbox{ and there exist } \mu_{\pm} \in \,]0,+\infty[ \mbox{ and } h_{\pm} \in L^p(\Omega) \mbox{ with } h_{\pm} \geq 0, \mbox{ such that}
\\[2mm]
-\mu_-|\xi |^2-h_-(x)\leq W(x,u, \xi) \leq \mu_+ |\xi|^2 + h_+(x) \mbox{ on }\Omega\times{\mathbb R}\times{\mathbb R}^N.
\end{array} \right. $$
\begin{prop} \label{prop2} Assume that {\rm (A4)} holds and, if $\mbox{\rm meas}(\Omega \backslash \mbox{\rm Supp} \, d) >0$, in addition, assume $$
\left\{ \begin{array}{c}
\displaystyle \inf_{ \{ u \in W_d,\, ||u||=1 \}} \, \displaystyle \int_{\Omega} \left(|\nabla u|^2 - \mu_+ h_+(x) u^2 \right) dx >0, \\
\displaystyle \inf_{ \{ u \in W_d,\, ||u||=1 \}} \, \displaystyle \int_{\Omega} \left(|\nabla u|^2 - \mu_- h_-(x) u^2 \right) dx >0. \end{array} \right. $$ Then \eqref{eq22} has a solution. \end{prop}
\begin{proof} To prove Proposition \ref{prop2} we use Theorem \ref{sousBo}. Thus we need to find a couple of lower and upper solutions $(\alpha, \beta)$ of (\ref{eq22}), with $\alpha \leq \beta$. Clearly, by {\rm (A4)}, any solution of \begin{equation} \label{1}
- \Delta u = d(x)u + \mu_+ |\nabla u|^2 + h_+(x), \quad u \in X \end{equation} is an upper solution of (\ref{eq22}). Moreover, a solution of \begin{equation} \label{222}
- \Delta u = d(x)u - \mu_- |\nabla u|^2 - h_-(x), \quad u \in X \end{equation} is a lower solution of (\ref{eq22}). Now if $w \in X$ is a solution of \begin{equation} \label{3b}
- \Delta u = d(x)u + \mu_- |\nabla u|^2 + h_-(x), \quad u \in X, \end{equation} then $u = - w$ satisfies (\ref{222}). Thus if we find a non negative solution $u_1 \in X$ of (\ref{1}) and a non negative solution $u_2 \in X$ of (\ref{3b}) then, setting $\beta = u_1$ and $\alpha = -u_2$, we have the required couple of lower and upper solutions for Theorem \ref{sousBo}. By Proposition \ref{prop1}, we know that such non negative solutions of (\ref{1}) and (\ref{3b}) exist and this concludes the proof. \end{proof}
As a direct consequence of the previous proposition, we obtain
\begin{cor} \label{prop3} Assume {\rm (A1)} and, if $\mbox{\rm meas}(\Omega \backslash \mbox{\rm Supp} \, c) > 0$, assume also that {\rm (Hc)} holds. Then $(P_{\lambda})$ has a solution for any $\lambda <0$. \end{cor}
As another direct consequence of Proposition \ref{prop2}, just noting that $W_d=H$ in case $d(x)\equiv 0$,
we have
\begin{cor} \label{prop3a} Assume {\rm (A1)} and {\rm (H0)} hold. Then $(P_0)$ has a solution. \end{cor}
\begin{remark}\label{ktL2} Assume that $c$ and $h$ belong to $L^p(\Omega)$ for some $p > \frac{N}{2}$, and that $\mu \in L^{\infty}(\Omega)$.
Assume that there exists an open subset $O(c)$ in $\Omega$ with $C^1$ boundary $\partial O(c)$ such that
$c(x)=0$ a.e. in $\overline{O(c)}$, $c(x)<0$ a.e. in $ \Omega\setminus O(c)$ and
$\mu(x)\geq\mu_1>0$, in $\overline{O(c)}$. Then
$W_c=H_0^1(O(c))$
and a necessary condition for the existence of a solution of $(P_\lambda)$ is that
\begin{equation}\label{k1}
\inf_{\{\phi\in W_c, ||\phi||=1\}} \int_\Omega \left(\frac{1}{\mu(x)}|\nabla\phi|^2 -h(x)\phi^2 \right)\, dx >0.
\end{equation}
Indeed, to show \eqref{k1}, we use an argument inspired by \cite{AbDaPe,FeMu1, FeMu2}. Suppose that $(P_\lambda)$ has a solution $u\in X$. Then for any $\phi\in C_0^\infty(\Omega)$ we have
\begin{equation}\label{k20}
\int_\Omega\left(\nabla u\nabla(\phi^2)-\lambda c(x)u\phi^2-\mu(x)|\nabla u|^2\phi^2-h(x)\phi^2\right)\, dx = 0.
\end{equation}
and hence, for every $\phi\in C_0^\infty(\Omega)\cap W_c$ we obtain
\begin{equation}\label{k2}
\int_{O(c)}\left(\nabla u\nabla(\phi^2)-\mu(x)|\nabla u|^2\phi^2-h(x)\phi^2\right) dx = 0.
\end{equation} But, for $\phi\in C_0^\infty(\Omega)\cap W_c$, by Young inequality,
\begin{equation}\label{k3}
\begin{array}{rcl}
\displaystyle
\int_{O(c)}\nabla u\nabla(\phi^2)\,dx &=&
\displaystyle
\int_{O(c)} 2\phi\nabla u\nabla\phi\, dx
\\
&\leq&
\displaystyle
\int_{O(c)} \left(\frac{1}{\mu(x)}|\nabla\phi|^2+\mu(x)|\nabla u|^2\phi^2\right) dx
\end{array}
\end{equation} and thus by density
$$
\int_{O(c)}\left(\frac{1}{\mu(x)}|\nabla\phi|^2 -h(x)\phi^2\right) dx \geq 0 \quad
\mbox{for all }\phi\in W_c.
$$
Thus, the infimum in (\ref{k1}) is non negative. If it is zero then, by Poincar\'e inequality, we also have that \begin{equation}
\label{k4}
\inf_{\{\phi\in W_c: ||\phi||_2=1\}} \int_{O(c)}\left(\frac{1}{\mu(x)}|\nabla\phi|^2 -h(x)\phi^2\right) dx=0.
\end{equation} Let us show that it cannot take place.
Arguing by contradiction we assume that \eqref{k4} hold. Then, by standard arguments, there exists a $\phi_0\in W_c\setminus\{ 0\}$ such that
\begin{equation}
\label{k5}
\int_{O(c)} \left(\frac{1}{\mu(x)}|\nabla\phi_0|^2 -h(x)\phi_0^2\right)\, dx=0.
\end{equation}
In addition, $\phi_0$ is an eigenfunction associated to the first eigenvalue (which we are assuming equal to zero) of the elliptic eigenvalue problem
$$
\left\{ \begin{array}{c}
-\mbox{div\,} \left( \displaystyle\frac{\nabla \phi}{\mu(x)}\right) - h(x) \phi = \lambda \phi \, , \ \ \mbox{in } O(c),
\\
\phi=0, \ \ \mbox{ on }\partial O(c) .
\end{array}
\right.
$$
As a consequence, we may assume that $\phi_0(x)>0$ in $O(c)$.
Setting $\phi=\phi_0$ in \eqref{k2}, we have by \eqref{k5} that
$$
\int_{O(c)} \left(
2 \phi_0 \nabla u\nabla\phi_0
-\mu(x)|\nabla u|^2\phi_0^2 -\frac{1}{\mu(x)}|\nabla\phi_0|^2\right)\, dx = 0.
$$ That is,
$$ \int_{O(c)} \Big |\frac{1}{\sqrt{\mu(x)}}\nabla\phi_0-\sqrt{\mu(x)}\phi_0\nabla u \Big|^2\, dx = 0
$$ from which we deduce that $\nabla\phi_0=\mu(x)\phi_0\nabla u$ in $O(c)$,
i.e.,
\begin{equation} \label{ajout1}
\nabla u=\frac{1}{\mu(x)}\frac{\nabla\phi_0(x)}{\phi_0(x)} \quad \mbox{in}\ O(c). \end{equation}
We also observe that $\frac{\partial\phi_0}{\partial\nu}(x)>0$ on $\partial O(c)$, where $\nu$ is an inner normal vector at $\partial O(c)$ (see e.g. \cite[Lemma 3.4]{GiTu}). This implies that $\frac{\nabla\phi_0(x)}{\phi_0(x)}\not\in L^2(O(c))$ and then by the fact that $\mu(x)\geq \mu_1$ in $O(c)$ and (\ref{ajout1}) we deduce that $\nabla u\not\in L^2(\Omega)$. This contradicts $u\in X$ proving that \eqref{k4} is impossible and thus \eqref{k1} holds. \medbreak
Now if in addition to the above assumptions we assume that $\mu(x)\equiv \mu>0$ is a constant and $h(x)\geq 0$ it follows from \eqref{k1} that, if $(P_\lambda)$ has a solution, we have
\begin{equation}\label{k6}
\inf_{\{\phi\in W_c: \|\phi\|=1\}} \int_\Omega\left(|\nabla\phi|^2-\mu h(x)\phi^2\right)\, dx>0.
\end{equation} Note that under these assumptions, {\rm (Hc)} coincides with \eqref{k6} and thus $(P_\lambda)$ when $\lambda<0$ has a solution if and only if {\rm (Hc)} holds. The statement in Remark \ref{ktL1} also follows in a similar way. Finally when $\lambda=0$ (equivalently when $c\equiv 0$), we have $O(c)=\Omega$, $W_c=H_0^1(\Omega)$ and \eqref{k6} reduces to {\rm (H0)}. Thus $(P_0)$ has a solution if and only if {\rm (H0)} holds. \end{remark}
\section{Uniqueness results} \label{Sectionuniqueness-0}
As in the previous section, we consider the boundary value problem \eqref{31}. Here we assume
$$
\hspace{1cm} \left\{ \begin{array}{c} d \mbox{ and } h \mbox{ belong to } L^p(\Omega) \quad \mbox{for some } p > \frac{N}{2}, \\[2mm]
d(x)\leq 0 \ \mbox{in}\ \Omega\mbox{ and } \mu \in L^{\infty}(\Omega). \end{array} \right. \leqno{\mathbf{(A5)}}
$$ Our main result is \begin{prop}\label{UUniqueness} Assume that {\rm (A5)} hold. Then \eqref{31} has at most one solution. \end{prop}
To prove Proposition \ref{UUniqueness} we shall first prove that the solutions of \eqref{31} belong to $ C(\overline \Omega)\cap W^{1,N}_{loc}(\Omega)$. Then, using this additional regularity, we prove the uniqueness.
\begin{remark} \label{lambda_0} Proposition \ref{UUniqueness} implies that $(P_{\lambda})$ for $\lambda \leq 0$ has at most one solution. \end{remark}
\begin{remark} \label{newrequire1} As we mention in the Introduction a general theory of uniqueness for problems with quadratic growth in the gradient was developed in \cite{BaMu} and extended in \cite{BaBlGeKo}. The uniqueness results closer to our setting are Theorems 2.1 and 2.2 of \cite{BaBlGeKo}. Unfortunately it is not possible to use directly these results to derive Proposition \ref{UUniqueness}. Indeed, since $d(x)$ may vanish on some part of $\Omega$, \cite[Theorem 2.1]{BaBlGeKo} is not applicable. Also, to use \cite[Theorem 2.2]{BaBlGeKo} which corresponds to the case $\lambda =0$, we need either $h(x)$ to have a sign or to be sufficiently small.
\end{remark}
\begin{lem} \label{betterreg} Assume that {\rm (A5)} hold. Then any solution of \eqref{31} belongs to $C(\overline\Omega)\cap W^{1,N}_{loc}(\Omega)$. \end{lem} \begin{proof} Let $u \in X$ be an arbitrary solution of \eqref{31}. We divide the proof that $u \in W^{1,N}_{loc}(\Omega)$ into three steps. \medbreak
\noindent {\bf Step 1}. $u \in C(\overline{\Omega})$.
Since condition (A) holds the result follows directly from \cite[Theorem IX.2.2]{LU68}. Indeed, \eqref{31} is of the form of equation (1.1) of Section IV.1 of \cite{LU68}. In addition, under {\rm (A5)} the assumptions (1.2)-(1.3) considered in \cite[Section IV.1]{LU68} are satisfied.
Hence, $u \in C^{0, \alpha}(\overline{\Omega})$ for some $\alpha \in (0,1)$ and in particular $u\in C(\overline\Omega)$. \medbreak
\noindent {\bf Step 2}. $u \in W^{1, q}_{loc}(\Omega)$ for some $q >2$.
Here we use \cite[Proposition 2.1, p.145]{Gi} or alternatively \cite[Theorem 2.5 p.155]{GiMo}, (see also \cite[Th\'eor\`eme 2.1]{BoMuPu2}) to deduce that $u \in W^{1,q}_{loc}(\Omega)$ for some $q>2$. \medbreak
\noindent {\bf Step 3}. Conclusion.
We follows some arguments of \cite{BeFr, Fr}, see also \cite{DoGi}. First note that without restriction we can assume that $q <N$. Since $u \in W^{1,q}_{loc}(\Omega)$ we have, \begin{equation}\label{t1}
- \Delta u = \xi(x) \quad \mbox{where} \quad \xi(x) = d(x)u + \mu(x) |\nabla u|^2 + h(x) \in L^{\frac{q}{2}}_{loc}(\Omega). \end{equation} By standard regularity argument, see for example \cite[Theorem~9.11]{GiTu}, we deduce that $u \in W^{2,\frac{q}{2}}_{loc}(\Omega)$. Now using Miranda's interpolation Theorem \cite[Teorema IV]{Mi} between $C^{0,\alpha}(\overline{\Omega})$ and $W^{2,\frac{q}{2}}_{loc}(\Omega)$ it follows, since $u \in C^{0,\alpha}(\overline{\Omega})$, that $$u \in W^{1,t_1}_{loc}(\Omega) \quad \mbox{where} \quad t_1 = \frac{\frac{q}{2} (2- \alpha) - \alpha}{1 - \alpha} >q .$$ If $t_1 \geq N$ we are done. Otherwise from (\ref{t1}) and classical regularity $u \in W^{2, \frac{t_1}{2}}_{loc}(\Omega)$. Denoting \begin{equation}\label{t2} t_n = \frac{\frac{t_{n-1}}{2} (2- \alpha) - \alpha}{1 - \alpha} > t_{n -1} > q > 2 \end{equation} by a bootstrap argument we get $u \in W^{2, \frac{t_n}{2}}_{loc}(\Omega)$ for all $n \in {\mathbb N}$ as long as $t_{n-1} \leq N$. We now claim that the sequence $\{t_n\}$ does not converge before reaching $N$. Indeed if we assume that $\{t_n\}$ has a finite limite $l$ we deduce from (\ref{t2}) that $l=2$ which contradicts $t_n >q >2$. At this point the proof of the lemma is completed. \end{proof}
Using the fact that, under {\rm (A5)}, the solutions of \eqref{31} belong to $C(\overline\Omega)\cap W^{1,N}_{loc}(\Omega)$ we can now obtain our uniqueness result. Here we adapt an argument from \cite{BeMeMuPo}, based in turn on an original idea from \cite{BoMa}.
\begin{lem} \label{Colette} Assume that {\rm (A5)} hold. Then \eqref{31} has at most one solution in $X\cap W^{1,N}_{loc}(\Omega) \cap C(\overline\Omega)$. \end{lem} \begin{proof}
Let us assume the existence of two solutions $u_1$, $u_2$ of \eqref{31} in $X\cap W^{1,N}_{loc}(\Omega) \cap C(\overline\Omega)$. Then $v=u_1-u_2$ is a solution of \begin{equation} \label{2} \begin{array}{c} -\Delta v = \mu(x)(\nabla u_1 + \nabla u_2)\,\nabla v + d(x)v, \quad v\in X\cap W^{1,N}_{loc}(\Omega) \cap C(\overline\Omega). \end{array} \end{equation} For every $c\in\mathbb R$, let us consider the set
$\Omega_c=\{x\in \Omega \, :\, |v(x)|=c\}$ and $$ J=\{c\in \mathbb R \, :\, \mbox{meas\,}\Omega_c>0\}. $$
As $|\Omega|$ is finite, $J$ is at most countable and, since for all $c\in \mathbb R$, $\nabla v =0$ a.e. on $\Omega_c$, we also have \begin{equation}
\label{mes} \nabla v=0 \mbox{ a.e. in } \bigcup_{c\in J} \Omega_c. \end{equation} Define $ Z= \Omega \setminus \bigcup_{c\in J} \Omega_c $ and let
$G_k:\mathbb R\to \mathbb R$ be defined by \begin{equation}
\label{Gk}
G_k(s)= \left\{ \begin{array}{ll}0,&\mbox{if } |s|\leq k, \\
(|s|-k)\,\mbox{sgn}(s),&\mbox{if }|s| > k. \end{array} \right. \end{equation} Now, using
$\varphi= G_k (v)$ as test function in \eqref{2},
we deduce for all $k\geq 0$ that \begin{eqnarray*} \displaystyle
\|\nabla G_k(v)\|_{2}^2 &=&
\int_{\Omega}|\nabla v|^2\chi_{\{|v|\geq k\}} \, dx \\ &=& \displaystyle \int_{\Omega}\mu(x) (\nabla u_1+\nabla u_2)\, \nabla v\, G_k (v) \, dx + \int_{\Omega}d(x)\, v\, G_k(v) \, dx . \end{eqnarray*}
Since $v \in C(\overline{\Omega})$ we have that $G_k(v)$ has a compact support in $\Omega$ for all $k >0$, which together to the fact that $d(x) \leq 0$ on $\Omega$ and \eqref{mes} implies that
\arraycolsep1.5pt \begin{equation} \label{3bb} \begin{array}{rcl} \displaystyle
\|\nabla G_k(v)\|_{2}^2
&\leq& \displaystyle
\int_{\Omega}\mu(x) (\nabla u_1+\nabla u_2)\, \chi_{\{|v|\geq k\}\cap Z}\, \nabla v\, G_k (v) \, dx \\ &=& \displaystyle
\int_{\Omega}\mu(x) (\nabla u_1+\nabla u_2)\, \chi_{\{|v|\geq k\}\cap Z}\, \nabla G_k (v)\, G_k (v) \, dx \\[4mm] &\leq& \displaystyle
\|\mu\|_{\infty} \|\nabla u_1+\nabla u_2\|_{L^N(\{|v|\geq k \}\cap Z)}\|\nabla G_k (v)\|_{2} \| G_k (v)\|_{2^*} \\[2mm] &\leq& \displaystyle
\mathcal{S}^{-1}_N\|\mu\|_{\infty} \|\nabla u_1+\nabla u_2\|_{L^N(\{|v|\geq k \}\cap Z)}\|\nabla G_k (v)\|_{2}^2, \end{array} \end{equation} \arraycolsep5pt where we recall that $ \mathcal{S}_N$ denotes the Sobolev constant. \medbreak
Assume by contradiction that $v\not\equiv 0$ and consider the function $F: ]0, \| v\|_\infty] \to\mathbb R$ defined by $$
F(k)= \mathcal S^{-1}_N\|\mu\|_{\infty} \|\nabla u_1+\nabla u_2\|_{L^N(\{|v|\geq k\}\cap Z)}
, \ \ \forall \, 0<k<\| v\|_\infty. $$
Observe that $F$ is non-increasing with $F(\| v\|_\infty)=0$. Moreover, by definition of $Z$ we have that $F$ is continuous and we can choose $0<k_0<\| v\|_\infty$ such that
$F(k_0) < 1$. By \eqref{3bb}, $\|\nabla G_{k_0} (v)\|_{2}^2\leq F(k_0) \|\nabla G_{k_0} (v)\|_{2}^2$, which implies that $\|\nabla G_{k_0} (v)\|_{2}=0$, i.e. $|v|\leq k_0< \| v\|_\infty$, a contradiction proving that necessarily $v=0$ and hence $u_1=u_2$ concluding the proof.
\end{proof}
\begin{proof}[Proof of Proposition \ref{UUniqueness}.] This follows directly from Lemmas \ref{betterreg} and \ref{Colette}. \end{proof}
\section{Uniform $L^\infty$-estimates and existence of a continuum} \label{Sectionuniqueness}
As in the previous section, we consider the boundary value problem \eqref{31} under the condition {\rm (A5)}. \begin{lem}\label{4.2new} Assume that {\rm (A5)} hold and that \eqref{31} has a solution $u_0\in X$. Then
\begin{enumerate} \item[i)] For any
$\widetilde d(x)\in L^p(\Omega)$, $p>\frac{N}{2}$ with
$\widetilde d(x)\leq d(x)$, the problem
\begin{equation}
\label{n1}
-\Delta u=\widetilde d(x)u+\mu(x)|\nabla u|^2 +h(x), \ u\in X
\end{equation}
has a unique solution $u\in X$. Moreover,
$u$ satisfies
$$
\|u\|_\infty \leq 2\|u_0\|_\infty.
$$ \item[ii)] There exists $M_1>0$ such that for any $t\in [0,1]$ any solution
$u_t$ of
\begin{equation}
\label{n4}
-\Delta u = (d(x)-1)u +(1-t)\mu(x)|\nabla u|^2 +h(x), \quad t\in [0,1]
\end{equation}
satisfies
$\| u_t\|_\infty \leq M_1$. \end{enumerate} \end{lem}
\begin{proof} i) Let $u_0\in X$ be a solution of \eqref{31} and set
$$
\beta(x)=u_0(x)+\|u_0\|_\infty, \quad \alpha(x)=u_0(x)-\|u_0\|_\infty.
$$
Then $\alpha\leq 0\leq \beta$ and, using that $\widetilde d(x)\leq d(x)\leq 0$, we have
\begin{eqnarray*}
-\Delta \beta &=& d(x)(\beta-\|u_0\|_\infty)+\mu(x)|\nabla \beta|^2 +h(x)
\\
&=& \widetilde d(x)\beta +\mu(x)|\nabla \beta|^2 +h(x) +(d(x)-\widetilde d(x))\beta-d(x)\|u_0\|_\infty
\\
&\geq& \widetilde d(x)\beta +\mu(x)|\nabla \beta|^2 +h(x).
\end{eqnarray*}
Thus $\beta$ is an upper solution of \eqref{n1}. Similarly $\alpha$ is a lower solution of \eqref{n1}. By Theorem \ref{sousBo}, \eqref{n1} has a solution $u(x)$ satisfying
$$
\alpha(x) \leq u(x) \leq \beta(x) \quad \mbox{in}\ \Omega.
$$ Since uniqueness of solutions of \eqref{n1} follows from Proposition \ref{UUniqueness}, this concludes the proof of the Point i). \medbreak
ii) Since $d(x)\leq 0$, then $\mbox{Supp}\, (d(x)-1) =\Omega$ and thus, by Proposition \ref{prop1}, there exists a non negative solution $\beta$ (resp. $\alpha$) of
$$
-\Delta u=(d(x)-1) u+\|\mu^+\|_\infty |\nabla u|^2+h^+
$$ (resp.
$ -\Delta u= (d(x)-1) u+\|\mu^-\|_\infty |\nabla u|^2+ h^-$).
For any $t\in [0,1]$, we can observe that $\beta$ (resp. $-\alpha$) is an
upper (resp. lower) solution of \eqref{n4}. Thus there exists a solution $u_t$ of \eqref{n4} satisfying
$-\alpha\leq u_t\leq \beta$. By Proposition \ref{UUniqueness}, uniqueness of solutions of \eqref{n4} holds and thus case ii) holds with $M_1=\max(\|\beta\|_\infty, \|\alpha\|_\infty)$. \end{proof}
We now transform \eqref{31} into a fixed point problem. By Corollary \ref{prop3} used with $c(x) \equiv 1$ and $\lambda=-1$, or alternatively Theorem 2 of \cite{BoMuPu3}, we know that, for any $f \in L^p(\Omega)$ the problem \begin{equation} \label{pivot}
-\Delta u + u -\mu(x) |\nabla u|^{2} = f(x), \quad u \in X
\end{equation} has a solution. We also know from Proposition \ref{UUniqueness} that it is unique. Thus it is possible to define the operator $K^\mu :L^p (\Omega )\longrightarrow X$ by $K^\mu f=u$ where $u$ is the unique solution of (\ref{pivot}). The following lemma, which is proved in the Appendix, will be crucial.
\begin{lem} \label{comp} If $\mu\in L^{\infty}(\Omega)$ then the operator $K^\mu$ is a completely continuous operator from $L^p (\Omega)$ into $C(\overline \Omega)$. \end{lem}
Next we define the continuous operator $N:C(\overline \Omega)\longrightarrow L^p(\Omega )$ by, $$ N(u) = (d(x) +1)\, u + h(x), \quad \mbox{ for any } u\in C(\overline \Omega). $$ With these notations, $u\in C(\overline \Omega)$ is a solution of \eqref{31} if and only if $u$ is a fixed point of $K^{\mu}\circ N$; i.e., if and only if
$$ u= K^{\mu}( N(u)). $$
Now let $T : C(\overline \Omega) \to C(\overline \Omega)$ be given by $T=K^{\mu}\circ N$. The following result hold.
\begin{prop}\label{4.1new} Assume that {\rm (A5)} holds and that \eqref{31} has a solution $u_0 \in X$. Then $$
i(I-T,u_0)=1. $$ \end{prop}
\begin{proof} To show the proposition, we use homotopy arguments. We consider two one-parameter problems, namely the problem \eqref{n4} with $t\in [0,1]$ and the following one \begin{equation} \label{n3}
-\Delta u = (d(x)-s)u +\mu(x)|\nabla u|^2 +h(x), \quad u\in X,
\end{equation} for $s\in [0,1]$. Applying Lemma \ref{4.2new} we deduce that
\begin{enumerate} \item Any solution $u_s(x)$ of \eqref{n3} with $s\in [0,1]$ satisfies
$ \| u_s\|_\infty \leq 2\|u_0\|_\infty
$. (Case i) with $\widetilde d(x)=d(x)-s$).
\item There exists $M_1>0$ such that for any $t\in [0,1]$ any solution $u_t(x)$ of \eqref{n4} satisfies
$ \| u_t\|_\infty \leq M_1
$. (Case ii)). \end{enumerate}
Observe that, if we set
$$ \widetilde N_s(u)= (d(x)+1-s)u+h(x),
$$ then problem \eqref{n3} (resp. problem \eqref{n4}) is equivalent to $u-K^\mu(\widetilde N_s(u))=0$ (resp.
$u-K^{(1-t)\mu}(\widetilde N_1(u))=0$). Thus setting $M=\max(2\|u_0\|_\infty, M_1)$, we have, for all $s$, $t\in [0,1]$ and all $u\in C(\overline\Omega)$ with $\|u\|_\infty=M$,
$$
u-K^\mu(\widetilde N_s(u))\not=0, \quad u-K^{(1-t)\mu}(\widetilde N_1(u))\not=0.
$$
Therefore, by homotopy invariance of the degree, we obtain
\begin{eqnarray*}
\mbox{deg}(I-T,B(0,M),0) &=& \mbox{deg}(I-K^\mu\circ \widetilde N_0,B(0,M),0)
\\
&=& \mbox{deg}(I-K^\mu\circ \widetilde N_1,B(0,M),0)
\\
&=& \mbox{deg}(I-K^0\circ \widetilde N_1,B(0,M),0)
=1.
\end{eqnarray*} By Proposition \ref{UUniqueness}, $u_0$ is the unique solution of \eqref{31} and thus
$$
i(I-T,u_0)=\mbox{deg}(I-T,B(0,M),0) =1.
$$ \end{proof}
In the rest of the section, we apply the above results to the problem $(P_\lambda)$. First, from Lemma \ref{4.2new} we directly obtain the following a priori estimates for $(P_{\lambda})$ with $\lambda <0$.
\begin{cor} \label{corA} Assume {\rm (A1)} and, if $\mbox{\rm meas}(\Omega\setminus\mbox{\rm Supp}\, c)>0$, assume also that {\rm (Hc)} holds. Then for any $\lambda_0<0$ there exists $R=R(\lambda_0)>0$ such that, for all $\lambda\leq \lambda_0$, the unique solution $u_\lambda$ of $(P_\lambda)$ satisfies
$$
\|u_\lambda\|_\infty \leq R.
$$ \end{cor}
\begin{proof} The existence and uniqueness of solutions of $(P_\lambda)$ when $\lambda<0$, is already known from Corollary \ref{prop3} and Proposition \ref{UUniqueness}. Now the $L^\infty$-bound is obtained from Lemma \ref{4.2new}, Point i) used with $d(x)=\lambda_0 c(x)$ and $\widetilde d(x)
=\lambda c(x)$. That is, the conclusion holds with $R(\lambda_0)=2\|u_{\lambda_0}\|_\infty$. \end{proof}
\begin{remark} A direct consequence of Corollary \ref{corA} is that none of $\lambda\in ]-\infty,0[$ is a bifurcation point from infinity of $(P_\lambda)$. (Recall that $\lambda \in {\mathbb R}$ is called a bifurcation point from infinity of $(P_\lambda)$
if there exists a sequence $\{ u_n\}$ of solutions of $(P_{\lambda_n})$ with $\lambda_n \to \lambda $ and $||u_n||_\infty\to\infty$). \end{remark}
\section{Behaviour of the continuum in the half space $\{\lambda >0 \} \times C(\overline \Omega)$ }\label{Section2}
As a first consequence of $\mathrm{(A2)}$ we obtain the following result.
\begin{lem} \label{l1} Assume that $\mathrm{(A2)}$ holds. For $\gamma_1 >0$, the first eigenvalue of \eqref{eigenvaluep}, we have \begin{enumerate} \item[1)] If $\lambda <\gamma_1$, any solution of problem $(P_{\lambda})$ is non negative.
\item[2)] If $\lambda =\gamma_1$, problem $(P_{\lambda})$ has no solution.
\item[3)] If $\lambda > \gamma_1$, problem $(P_{\lambda})$ has no non negative solutions. \end{enumerate} \end{lem}
\begin{proof} First we assume that $\lambda <\gamma_1$. Let $u \in X$ be a solution of $(P_{\lambda})$. Using $u^-$ as test function in $(P_{\lambda})$ we obtain $$
- \int_{\Omega} (|\nabla u^-|^2 - \lambda c(x) |u^-|^2) dx = \int_{\Omega} (\mu(x) |\nabla u|^2 u^- + h(x) u^-) dx. $$ Since $\lambda <\gamma_1$ the left hand side is negative and since $\mu(x) \geq 0$ and $h(x) \geq 0$ the right hand side positive. So necessarily $u^- \equiv 0$ i.e., $u\geq 0$. This proves Point 1).
Now let $u\in X$ be a solution of $(P_{\lambda})$. Using $\varphi_1 >0$, the first eigenfunction of \eqref{eigenvaluep}, as test function in $(P_{\lambda})$ we obtain \begin{eqnarray*} (\gamma_{1} -\lambda ) \int_{\Omega} c(x) u\varphi_{1} dx &=& \int_{\Omega} \nabla u \nabla \varphi_{1} dx - \int_{\Omega} \lambda c(x) u\varphi_{1}dx \\ &=&
\int_{\Omega} \mu(x) |\nabla u|^{2}\varphi_{1} dx + \int_{\Omega} h(x) \varphi_{1} dx . \end{eqnarray*} Since $\mu (x)\geq 0$ and $h(x)\gneqq 0$, the right hand-side of the above inequality is positive. Thus when $\lambda = \gamma_1$, $(P_{\lambda})$ has no solution and Point 2) is proved.
Finally, when $\lambda > \gamma_1$ and $u\in H$ is a non negative solution of $(P_{\lambda})$, the left hand-side is non positive which contradicts the positivity of the right hand side. This proves Point 3). \end{proof}
To prove the second part of Theorem \ref{th3}, the key point is the derivation of a priori bounds for solution of $(P_{\lambda})$ for $\lambda >0$. Actually we derive these bounds under a slightly more general assumption than needed.
We consider the problem \begin{equation*} \label{EE1} -\Delta u = \lambda c(x)u+ H(x, \nabla u), \quad u \in X, \eqno{(R_{\lambda})} \end{equation*} where we assume $$ \leqno{\mathbf{(A6)}} \hspace{1cm} \left\{ \begin{array}{c} c\gneqq 0 \mbox{ and } c \mbox{ belong to } L^p(\Omega) \quad \mbox{for some } p > \frac{N}{2}\\ \\
\mu_1 [|\xi|^2 +h(x)] \leq H(x, \xi) \leq \mu_2 [|\xi|^2 + h(x)] \\\\
\mbox{ for some } 0 < \mu_1 \leq \mu_2 < \infty \mbox{ and } h \geq 0 \mbox{ with } h \in L^p(\Omega).
\end{array} \right. $$
Adapting the approach of \cite{BT77}, we prove the following result:
\begin{prop} \label{bounds1} Assume that $\mathrm{(A6)}$ holds. Then for any $\Lambda_1>0$ there exists a constant $M >0$ such that, for each $\lambda\geq \Lambda_1$, any non negative solution $u$ of $(R_{\lambda})$ satisfies $$
||u||_{\infty} \leq M. $$ \end{prop}
\noindent In the proof of Proposition \ref{bounds1} the following two technical
lemmas will be useful.
\begin{lem}\label{tec1} Let $p > \frac{N}{2}$ and $\theta \in\, ]0,1[$. There exist $\alpha$, $r \in\, ]0,1[$ such that, if we define
\begin{equation}\label{a0}
q =1+ r + \frac{1+ \theta \alpha}{1- \alpha}, \quad \tau = \frac{1}{q} \, \frac{\alpha}{1- \alpha}
\end{equation} then it holds \begin{equation} \label{boundsq} \frac{1}{p} \leq q \leq \frac{2N(p-1)}{p(N-2 + 2 \tau)} \end{equation} and \begin{equation} \label{boundsalpha} 1 - \alpha < \frac{2}{q}. \end{equation} \end{lem}
\begin{proof} First observe that for all $\alpha \in \,]0,1[$, there exists $r_0>0$ such that, for any $0 < r \leq r_0$, (\ref{boundsalpha}) holds true. Indeed, since $r >0$, we have $$ q > 1 + \frac{1+ \theta \alpha}{1 - \alpha} = \frac{2 - \alpha + \theta \alpha}{1 - \alpha} \quad \mbox{ or equivalently } \quad \frac{2}{q} < \frac{2(1- \alpha)}{2 - \alpha + \theta \alpha}. $$ Also letting $r \to 0^+$ we obtain $$ \frac{2}{q} \nearrow \frac{2(1- \alpha)}{2 - \alpha + \theta \alpha}.$$ Thus if \begin{equation} \label{onalpha} 1 - \alpha < \frac{2(1- \alpha)}{2 - \alpha + \theta \alpha} \end{equation} there exists $r_0 >0$ such that, for all $0 < r \leq r_0$, (\ref{boundsalpha}) is satisfied. But (\ref{onalpha}) is equivalent to $\alpha (\theta-1) < 0$ which is always true.
\noindent Now, observe that, from the definition of $q$, we have $q \searrow 2$ as $r \searrow 0$ and $\alpha \searrow 0$. Finally, we see from the definition of $\tau$, that $\tau \searrow 0$ as $\alpha \searrow 0$. Thus as $\alpha \searrow 0$, $$ \frac{2N(p-1)}{p(N-2 + 2 \tau)} \nearrow \frac{2N(p-1)}{p(N-2 )} >2, $$ where the inequality is obtained using the assumption that $p > \frac{N}{2}.$ At this point it is clear that taking $r >0$ sufficiently close to $0$ and $\alpha >0$ sufficiently close to $0$, that (\ref{boundsq}) will also hold. \end{proof}
\begin{lem} \label{tec2} Let $b \in L^p(\Omega)$ with $p > \frac{N}{2}$. For any $p$, $q\geq 1$ and $\tau\in [0,1]$ satisfying \eqref{boundsq}, there exists $C>0$ such that, for all $w\in H$
$$
\left\|\frac{b^{1/q} w}{\varphi_1^\tau}\right\|_q \leq C\| b\|_p \|\nabla w\|_2,
$$
where $\varphi_1 >0 $ denotes the first eigenfunction of \eqref{eigenvaluep}. \end{lem}
\begin{proof} For $p$, $q\geq 1$, $\tau\in [0,1]$ satisfying \eqref{boundsq}, define $s\geq 1$ by
$$
\frac{1}{s}=\frac{1}{2}-\frac{1-\tau}{N}.
$$ It follows from the second inequality of \eqref{boundsq} that
$ \frac{1}{q}\geq (1-\frac{1}{p})^{-1}\frac{1}{s}
$, and this implies
$$
\frac{1}{pq}\leq \frac{1}{q}-\frac{1}{s}.
$$ From the first inequality of \eqref{boundsq}, we have $\frac{1}{pq}\leq 1$. Thus there exists $\nu\geq 1$ such that
$$ \frac{1}{pq}\leq \frac{1}{\nu}\leq \frac{1}{q}-\frac{1}{s}.
$$ That is $\nu\geq 1$ satisfies
$$ \frac{\nu}{q}\leq p \quad \mbox{and}\quad \frac{1}{q}\geq \frac{1}{\nu}+\frac{1}{s}.
$$ Now by the Sobolev's embedding and \cite[Lemma 2.2]{BT77}, we have, for some constant $C>0$,
$$
\left\|\frac{b^{1/q} w}{\varphi_1^\tau}\right\|_q \leq C\| b^{1/q}\|_\nu \left\|\frac{w}{\varphi_1^\tau}\right\|_s
\leq C'\|b\|_p^{1/q} \|\nabla w\|_2
$$
and the lemma is proved. \end{proof}
\begin{proof}[Proof of Proposition \ref{bounds1}] Fix $\lambda>\Lambda_1$ and let $u\in X$ be a non negative solution of $(R_\lambda)$. By Points 2)-3) of Lemma \ref{l1} we deduce that $\lambda<\gamma_1$. Hence without loss of generality we suppose $\Lambda_1<\gamma_1$ and $\lambda\in [\Lambda_1,\gamma_1]$.
\noindent We define
$$
w_i(x)=\frac{1}{\mu_i}(e^{\mu_iu(x)}-1) \, \mbox{ and } \, g_i(s)=\frac{1}{\mu_i}\ln(1+\mu_i s) \quad i=1,2.
$$ Then we have
\begin{eqnarray}
u &=& g_1(w_1)=g_2(w_2), \label{a1}
\\
e^{\mu_i u}&=&1+\mu_i w_i, \quad i=1,2. \label{a2}
\end{eqnarray}
Direct calculations give us
\begin{eqnarray*}
-\Delta w_i &=& \lambda e^{\mu_i u}c(x)u + e^{\mu_i u}[H(x,\nabla u)-\mu_i|\nabla u|^2]
\\
&=& \lambda(1+\mu_iw_i)c(x)g_i(w_i) + (1+\mu_iw_i)[H(x,\nabla u)-\mu_i|\nabla u|^2].
\end{eqnarray*}
\noindent Since $\Lambda_1\leq\lambda\leq \gamma_1$, we have by $\mathrm{(A6)}$
\begin{eqnarray*}
-\Delta w_1 &\geq& \Lambda_1(1+\mu_1w_1)c(x)g_1(w_1) +\mu_1(1+\mu_1w_1)h(x),
\\
-\Delta w_2 &\leq& \gamma_1(1+\mu_2w_2)c(x)g_2(w_2) +\mu_2(1+\mu_2w_2)h(x).
\end{eqnarray*}
\noindent Setting $A_1=\min(\Lambda_1,\mu_1)$, $A_2=\max(\gamma_1,\mu_2)$, it becomes
\begin{eqnarray}
-\Delta w_1 &\geq& A_1(1+\mu_1w_1)[c(x)g_1(w_1)+h(x)], \label{a3}
\\
-\Delta w_2 &\leq& A_2(1+\mu_2w_2)[c(x)g_2(w_2)+h(x)]. \label{a4}
\end{eqnarray}
From the inequalities \eqref{a3} and \eqref{a4}, we shall deduce that $w_2$ is uniformly bounded in $H$. This will lead to the proof of the theorem by classical results relating the $L^{\infty}$ norm of a lower solution to its $H$ norm. We divide the proof into three steps.
\noindent {\bf Step 1.} {\it Let $\theta={(\mu_2-\mu_1)\mu_2^{-1}}\in\, ]0,1[$. Then there exists $C>0$ independent of $\lambda\in [\Lambda_1,\gamma_1]$ such that}
\begin{eqnarray}
&&\int_\Omega (1+\mu_1w_1)[c(x)g_1(w_1)+h(x)]\varphi_1\, dx \leq C, \label{a5}\\
&&\int_\Omega (1+\mu_2w_2)^{1-\theta}[c(x)g_2(w_2)+h(x)]\varphi_1\, dx \leq C. \label{a6}
\end{eqnarray}
\noindent Indeed, using $\varphi_1 >0$ as a test function in \eqref{a3}, we have
$$
\gamma_1\int_\Omega c(x)w_1\varphi_1\, dx
\geq A_1\int_\Omega (1+\mu_1w_1)[c(x)g_1(w_1)+h(x)]\varphi_1\, dx.
$$
\noindent We note that for any $\varepsilon>0$ there exists $C_\varepsilon>0$ such that $t\leq \varepsilon (1+\mu_1t)g_1(t)+C_\varepsilon$ for all $t\geq 0$. Thus
$$
\gamma_1\int_\Omega c(x)w_1\varphi_1\, dx
\leq \varepsilon \gamma_1 \int_\Omega (1+\mu_1w_1)[c(x)g_1(w_1)+h(x)]\varphi_1\, dx +C_\varepsilon'
$$
\noindent and choosing $\varepsilon=\frac{A_1}{2\gamma_1}$, we obtain \eqref{a5}. Now observe that by \eqref{a2},
$$ 1+\mu_1w_1= e^{\mu_1u}=(e^{\mu_2 u})^{1-\theta} = (1+\mu_2w_2)^{1-\theta}.
$$ Thus from \eqref{a1} we see that \eqref{a6} is nothing but \eqref{a5}.
\noindent {\bf Step 2.} {\it There exists a constant $C>0$ independent of $\lambda\in [\Lambda_1,\gamma_1]$ such that}
\begin{equation}\label{unifestimates}
\|\nabla w_2\|_2 \leq C.
\end{equation}
First we use Lemma \ref{tec1} to choose $\alpha$, $r\in\, ]0,1[$ such that $q$ and $\tau$ given in \eqref{a0} satisfy \eqref{boundsq} and \eqref{boundsalpha}.
Using $w_2$ as a test function in \eqref{a4} it follows that
$$
\|\nabla w_2\|_2^2
\leq A_2\int_\Omega(1+\mu_2w_2)[c(x)g_2(w_2)+h(x)]w_2\, dx.
$$ Now using H\"older's inequality, \eqref{a6} and since $ w_2\leq { (1+\mu_2w_2) \mu_2^{-1}}$ we have
\begin{eqnarray*}
\|\nabla w_2\|_2^2
&\leq& \frac{A_2}{\mu_2}\int_\Omega (1+\mu_2w_2)[c(x)g_2(w_2)+h(x)]\frac{\varphi_1^\alpha}{(1+\mu_2w_2)^{\theta\alpha}}
\frac{(1+\mu_2w_2)^{1+\theta\alpha}}{\varphi_1^\alpha}\, dx
\\
&\leq& \frac{A_2}{\mu_2}\left(\int_\Omega (1+\mu_2w_2)[c(x)g_2(w_2)+h(x)]\frac{\varphi_1}{(1+\mu_2w_2)^\theta}\,dx\right)^\alpha
\\
&& \qquad \times
\left(\int_\Omega (1+\mu_2w_2)[c(x)g_2(w_2)+h(x)]\frac{(1+\mu_2w_2)^{\frac{1+\theta\alpha}{1-\alpha}}}{\varphi_1^{\frac{\alpha}{1-\alpha}}}\,dx\right)^{1-\alpha}
\\
&\leq& \frac{A_2}{\mu_2} C^\alpha
\left(\int_\Omega (1+\mu_2w_2)[c(x)g_2(w_2)+h(x)]\frac{(1+\mu_2w_2)^{\frac{1+\theta\alpha}{1-\alpha}}}{\varphi_1^{\frac{\alpha}{1-\alpha}}}\,dx\right)^{1-\alpha}.
\end{eqnarray*}
We note that for $r$ given by Lemma~\ref{tec1}, there exists $C_r>0$
$$ g_2(t) \leq t^r +C_r \quad \mbox{for all}\ t\geq 0.
$$ Thus, direct calculations shows that
$$ (1+\mu_2w_2)[c(x)g(w_2)+h(x)](1+\mu_2w_2)^{\frac{1+\theta\alpha}{1-\alpha}}
\leq (c(x)+h(x))(w_2^q+C),
$$
\noindent where $q$ is given in \eqref{a0}. Therefore for some $C$, $C'>0$ independent of $\lambda\in [\Lambda_1,\gamma_1]$
$$ \|\nabla w_2\|_2^2
\leq C\left(\int_\Omega \left(\frac{(c(x)+h(x))^{1/q}w_2}{\varphi_1^\tau}\right)^q\,dx\right)^{1-\alpha}+C',
$$
\noindent with $q$ and $\tau$ given in \eqref{a0}. Applying Lemma \ref{tec2}, we then obtain
$$ \|\nabla w_2\|_2^2 \leq C\|c+h\|_p^{q(1-\alpha)}\|\nabla w_2\|_2^{q(1-\alpha)} +C'.$$
\noindent By \eqref{boundsalpha}, we have $q(1-\alpha)<2$ and this concludes the proof of Step 2.
\noindent {\bf Step 3.} {\it Conclusion.}
We just have to show that the uniform estimate \eqref{unifestimates} derived in Step 2 gives an uniform estimate in the $L^{\infty}$ norm. Recall that, as a consequence of Theorem 4.1 of \cite{Tr} combined with Remark 1 on page 289 of that paper (see also Remark 2 p. 202 of \cite{LU68}), we know that if $w\in X$ satisfies $$ \begin{array}{cc} -\Delta w\leq d(x) w + f(x),&\mbox{in }\Omega, \\ w\leq 0,&\mbox{on }\partial\Omega, \end{array} $$ with $d$, $f\in L^{p_1}(\Omega)$ for some $p_1>\frac{N}{2}$, then $w$ satisfies $$
\|w^+\|_{\infty}\leq C(\|w\|_{1}+\|f\|_{p_1}), $$
where $C$ depends on $p_1$, $\mbox{meas}(\Omega)$ and $\|d\|_{p_1}$.
Since $w_2$ satisfies \eqref{a4},
we apply the result of \cite{Tr} with $$d(x)=c(x)A_2 (1+\mu_2 w_2(x))\frac{\ln(1+\mu_2 w_2(x))}{\mu_2 w_2(x)}+A_2^2 h(x) \quad \mbox{and} \quad f(x)=A_2 h(x).$$ Observe that, for any $r\in \,]0,1[$, there exists $C >0$ such that, for all $x\in \Omega$, $$
c(x)A_2 (1+\mu_2 w_2(x))\frac{\ln(1+\mu_2 w_2(x))}{\mu_2 w_2(x)}\leq C\, c(x)|w_2(x)|^r. $$ Thus, since $c(x) \in L^p(\Omega)$ with $p > \frac{N}{2}$ and $w_2$ is bounded in $L^{\frac{2N}{N-2}}(\Omega)$, taking
$r >0$ sufficiently small we see, using H\"older's inequality, that $c(x) |w_2(x)|^r \in L^{p_1}(\Omega)$ for some $p_1 > \frac{N}{2}$. Now as $h\in L^p(\Omega)$ for some $p>\frac{N}{2}$, clearly all the assumptions of Theorem 4.1 of \cite{Tr} are satisfied. From (\ref{unifestimates}) we then deduce that there exists a constant $C>0$, independent of $\lambda \in [\Lambda_1, \gamma_1]$ such that
$$||w_2||_{\infty} \leq C.$$ Now since $u = g_2(w^2)$ we deduce that a similar estimate holds for the non negative solutions of $(R_{\lambda})$ and the proof of the proposition is completed. \end{proof}
\section{Proofs of the main results.}\label{Proofs}
In this section we give the proofs of our three theorems.
\begin{proof}[Proof of Theorem \ref{negativevalue}] The uniqueness of the solution of $(P_\lambda)$ for $\lambda\leq 0$ is a consequence of Remark~\ref{lambda_0}. By Corollary~\ref{prop3}, $(P_\lambda)$ with $\lambda<0$ has a solution $u_\lambda\in X$. This proves Point 1). To establish the existence of a continuum of solutions of $(P_\lambda)$, we define $T_{\lambda} : C(\overline{\Omega}) \to C(\overline{\Omega})$ as
$$ T_\lambda(u)=K^\mu((\lambda c(x)+1)u+h(x)). $$
\noindent Hence, $(P_\lambda)$ is transformed into the fixed point problem $u=T_\lambda(u)$. From Proposition \ref{4.1new} we immediately deduce that, for any $\lambda<0$,
$$
i(I-T_{\lambda},u_{\lambda})=1.
$$
\noindent Therefore, if we fix a $\lambda_0<0$, by Theorem \ref{1noacot} where $E=C(\overline\Omega)$ and $\Phi(\lambda,u)=u-T_\lambda(u)$, there exists a continuum $C=C^+\cup C^-$ of solutions of $(P_\lambda)$ emanating from $(\lambda_0,u_{\lambda_0})$. Taking into account the unboundedness of $C^+$ and $C^-$ and
Corollary \ref{corA}, necessarily $]-\infty,0[ \,\subset \mbox{Proj}_{\mathbb{R}}C$ and the proof of Point 2) is concluded.
To prove Point 3), we apply Lemma \ref{4.2new} with $d(x)= \overline\lambda c(x)$, $\widetilde{d}(x)= \lambda c(x)$ and $ \lambda\leq \overline\lambda< 0$, to deduce that
$$
\|u_{\lambda}\|_\infty \leq 2\|u_{\overline\lambda}\|_\infty \quad \mbox{for all}\ \lambda\leq \overline\lambda< 0.
$$
In particular, if
$C_0:=\liminf_{\lambda\to 0^-}\|u_{\lambda}\|_\infty
<\infty$, then there exists a sequence $\overline\lambda_n \to 0^-$ such that $C_0=\lim_{n\to\infty}\|u_{\overline\lambda_n}\|_\infty <\infty$. Hence, for every sequence $\lambda_n \to 0^-$ we deduce by the above inequality that
$\limsup_{n\to\infty}\|u_{\lambda_n}\|_\infty \leq 2C_0$, which implies that $\limsup_{\lambda\to 0^-}\|u_\lambda\|_\infty<\infty$.
Therefore, we have either $\lim_{\lambda\to 0^-}\|u_\lambda\|_\infty=\infty$ or $\limsup_{\lambda\to 0^-}\|u_\lambda\|_\infty<\infty$.
In the first case, using Lemma \ref{4.2new} with $d(x) \equiv 0$ and $\widetilde{d}(x) = \lambda c(x)$, we see that $(P_0)$ cannot have a solution. On the other hand, in
the last case, for any sequence $\lambda_n\to 0^-$, $(u_{\lambda_n})$ is a bounded sequence in $L^\infty(\Omega)$. Thus by Lemma \ref{comp},
$$
u_{\lambda_n}=K^\mu((\lambda_n c(x)+1)u_{\lambda_n}+h(x))
$$
\noindent is relatively compact in $C(\overline\Omega)$. Taking a subsequence if necessary, we may assume $u_{\lambda_n}\to u_0$ in $L^\infty(\Omega)$ for some $u_0 \in X$. It is clear that $u_0$ satisfies $u_0=K^\mu(u_0 + h(x))$, that is, $u_0$ is a solution of $(P_0)$. Since we have uniqueness of solutions of $(P_0)$ by Remark~\ref{lambda_0}, the limit $u_0$ does not depend on the choice of $\lambda_n$ and thus we have $u_\lambda\to u_0$ in $L^\infty(\Omega)$ as $\lambda\to 0^-$. This ends the proof. \end{proof}
\begin{proof}[Proof of Theorem \ref{th1.1a}] If we assume that $(P_0)$ has a solution $u_0$ then using Lemma \ref{4.2new} with $d(x) \equiv 0$ and $\widetilde{d}(x) = \lambda c(x)$ we obtain the existence of a solution $u_\lambda$ of $(P_\lambda)$ for any $\lambda<0$. Using Remark \ref{lambda_0} Point 1) follows.
Now by Proposition \ref{4.1new}, we know that $i(I-T_0,u_0)=1$. Thus by Theorem \ref{1noacot} there exists a continuum $C\subset\Sigma$ such that both
$$
C\cap([0,\infty[\,\times C(\overline\Omega)) \quad \mbox{ and }\quad C\cap(]-\infty,0]\,\times C(\overline\Omega))
$$ are unbounded. Clearly $\{ (\lambda,u_\lambda):\ \lambda\in\, ]-\infty,0]\,\}\subset C$ and Point 2) holds. \end{proof}
\begin{proof}[Proof of Theorem \ref{th3}.]
Let $C \subset \Sigma$ be the continuum obtained in Theorem \ref{th1.1a}. By Lemma \ref{l1}, Point 2) we know that $]-\infty,0]\subset \, \mbox{\rm Proj}_{{\mathbb R}} C \subset \,]-\infty, \gamma_1[$. Lemma \ref{l1}, Point 1) shows that it consists of non negative functions. In addition, by Theorem \ref{th1.1a}, Point 2), $C\cap ([0,\gamma_1 [\times C(\overline{\Omega}))$ is unbounded and hence its projection on $C(\overline\Omega)$ has to be unbounded.
Now we know, by Proposition \ref{bounds1}, that for every $\Lambda_1 \in\, ]0,\gamma_1[$, there is an a priori bound on the non negative solutions for $\lambda\geq \Lambda_1$. This means that the projection of $C\cap ( [\Lambda_1, \gamma_1[ \times C(\overline\Omega))$ on $C(\overline\Omega)$ is bounded. Thus $C$ must emanate from infinity to the right of $\lambda=0$. This proves the first part of the theorem.
\medbreak
Since $C$ contains $(0, u_0)$ with $u_0$ the unique solution of $(P_0)$, there exists a $\lambda_0 \in\, ]0,\gamma_1[$ such that the problem $(P_{\lambda})$ has at least two solutions for $\lambda\in \,]0,\lambda_0[$.
At this point the proof of the theorem is completed. \end{proof}
\section{Appendix : Proof of Lemma \ref{comp}.} \label{appendix}
To prove Lemma \ref{comp}, we need some preliminary results.
\begin{lem} \label{estimationsinfini} Let $\{f_n\} \subset L^p(\Omega)$ be a bounded sequence. Then the sequence $\{u_n\} = \{K^{\mu}(f_n)\}$ is bounded in $L^{\infty}(\Omega)$ and in $H.$ \end{lem}
\begin{proof} First we observe that the boundedness of $\{u_n\}$ in $L^{\infty}(\Omega)$ is a direct consequence of Theorem 1 of \cite{BoMuPu3}. To show that $\{u_n\}$ is also bounded in $H$ we use a trick that can be found for example in \cite{BoMuPu1,BoMuPu2}.
Let $t = ||\mu||_{\infty}^2 /2$, $E_n = \exp(tu_n^2)$ and consider the functions $v_n = E_nu_n$. We have $v_n \in X$ and
$$
\nabla v_n= E_n (1+2 t u_n^2) \nabla u_n.
$$ Hence using $v_n$ as test functions in
$$
-\Delta u_n + u_n = \mu(x) |\nabla u_n|^{2} + f_n(x), \quad u_n\in X, $$ and the bound of $\{u_n\}$ in $L^{\infty}(\Omega)$, we obtain the existence of a constant $D>0$ such that \arraycolsep1.5pt $$ \begin{array}{rcl} \displaystyle
\int_{\Omega}E_n (1 &+& 2t u_n^2) |\nabla u_n|^2 dx + \displaystyle
\int_{\Omega}E_n u_n^2 dx \\[3mm]
&=&
\displaystyle
\int_{\Omega} f_n(x) E_nu_n dx + \int_{\Omega}\mu(x) |\nabla u_n|^2 E_n u_n dx \\[3mm] & \leq & \displaystyle D
+ ||\mu||_{\infty}\int_{\Omega} E_n^{1/2}|\nabla u_n| |u_n| |\nabla u_n| E_n^{1/2} dx \\[3mm] & \leq & \displaystyle D
+ ||\mu||_{\infty} \left[ \frac{1}{2||\mu||_{\infty}} \int_{\Omega}E_n|\nabla u_n|^2 dx + \frac{1}{2} ||\mu||_{\infty} \int_{\Omega}u_n^2 |\nabla u_n|^2 E_n dx \right] \\[3mm] & \leq & \displaystyle D
+ \frac{1}{2} \int_{\Omega}E_n(1+ 2t u_n^2) |\nabla u_n|^2 dx. \end{array} $$ \arraycolsep5pt We then deduce that $$
\int_{\Omega}E_n |\nabla u_n|^2 dx + \int_{\Omega}E_n u_n^2 dx
\leq 2 D. $$ Recording that $E_n \geq 1$, this shows that $\{u_n\}$ is bounded in $H$. \end{proof}
\begin{proof}[Proof of Lemma \ref{comp}] The proof we give is inspired by \cite{BoMuPu3} combined with \cite[Remark 2.7]{ArCaLeMAOrPe} (based in turn on ideas from \cite{LU68}).
\noindent {\bf Step 1.} {\it $K^\mu$ is a bounded operator from $L^p(\Omega)$ to $C^{0,\alpha}(\overline\Omega)$ for some $\alpha\in\, ]0,1[$}
Assume that $\{ f_n\}$ is a bounded sequence in $L^p(\Omega)$. By Lemma \ref{estimationsinfini}, $u_n=K^\mu (f_n) $ is bounded in $L^\infty (\Omega)$. We claim that $u_n$ is also bounded in $C^{0,\alpha}(\overline \Omega)$ for some $\alpha \in \,]0,1[$. Indeed, consider a function $\zeta\in C^\infty (\Omega) $ with $0\leq \zeta(x)\leq 1$, and compact support in a ball $B_\rho$ of radius
$\rho>0$, and set $A_{k,\rho}=\{x\in B_\rho\cap \Omega: |u(x)|>k\}$.
Let us consider the function $G_k$ given by \eqref{Gk}.
For $\varphi (s)=s e^{\gamma s^{2}}$ with $\gamma >0$ large (to be precised later) we take $\phi=\varphi (G_k(u_n))\zeta^2$ as test function in (\ref{pivot}). Hence we have \begin{eqnarray*} \int_{\Omega} \nabla u_n \nabla (G_k(u_n)) \varphi '(G_k(u_n)) \zeta^2 dx &=& \int_{\Omega} [- u_n + f_n(x) ] \varphi (G_k(u_n))\zeta^2 dx \\ &&\hspace{10mm}+
\int_{\Omega} \mu(x) |\nabla u_n |^2 \varphi (G_k(u_n))\zeta^2 dx \\ &&\hspace{20mm}- 2\int_{\Omega} \zeta \varphi (G_k(u_n)) \nabla u_n \nabla \zeta dx. \end{eqnarray*}
Now observe that, for $\gamma>\frac{\|\mu\|_{\infty}^2}{4}$, we have $1+2\gamma s^2-\|\mu\|_{\infty} |s|\geq 1/2$ and hence $\varphi'(s)-\|\mu\|_{\infty} |\varphi(s)|\geq \frac12 e^{\gamma s^2}\geq \frac12$. Moreover, we have $G_k(u_n(x)) \zeta^2(x)=0$ for $x\not\in A_{k,\rho}$ and $\nabla G_k(u_n)=\nabla u_n$ in $A_{k,\rho}$. This implies that \arraycolsep1.5pt $$ \begin{array}{rl} \displaystyle \frac{1}{2}
\int_{A_{k,\rho}}\!\!&\!\!
|\nabla G_k(u_{n})|^{2} \zeta^2 dx \\[3mm]
&\leq
\displaystyle \int_{A_{k,\rho}} \left[ \varphi'(G_k(u_{n})) - \|\mu\|_{\infty} |\varphi (G_k(u_{n})) |\right] |\nabla G_k(u_{n})|^{2} \zeta^2 dx \\[3mm]
&\leq \displaystyle \int_{A_{k,\rho}} \!\! [- u_n + f_n(x)] \varphi (G_k(u_n))\zeta^2 dx \\[3mm] &\displaystyle \hspace{25mm} +
\int_{A_{k,\rho}} \!\!(|\mu(x)|-\|\mu\|_{\infty}) |\nabla u_n |^2 |\varphi (G_k(u_n))|\zeta^2 \\[3mm] &\displaystyle \hspace{65mm} - 2\int_{A_{k,\rho}} \!\! \zeta \varphi (G_k(u_n)) \nabla u_n \nabla \zeta dx \\[3mm] &\leq \displaystyle \int_{A_{k,\rho}} \!\! [- u_n + f_n(x)] \varphi (G_k(u_n))\zeta^2 dx +
2\int_{A_{k,\rho}}\!\! |\zeta| \, |\varphi (G_k(u_n))| |\nabla u_n| \,|\nabla \zeta| dx. \end{array} $$
Now recall the existence of $C_1$ and $C_2$ such that, for all $n\in \mathbb N$, $\|u_n\|_{\infty}\leq C_1$ and $\|f_n\|_p \leq C_2$. Let $C_3$ such that, for all $s\in [-C_1,C_1]$, $|\varphi(s)|\leq C_3 |s|$ and recall that $0\leq \zeta\leq 1$. Hence we obtain $C=C(C_1,C_2,C_3)$ such that \arraycolsep1.5pt $$ \begin{array}{rcl} \displaystyle \frac{1}{2}
\int_{A_{k,\rho}}
|\nabla G_k(u_{n})|^{2} \zeta^2 dx
&\leq&
\displaystyle C (\mbox{meas}(A_{k,\rho}))^{1-\frac1p}+ 2 C_3 \int_{A_{k,\rho}} |\zeta | |\nabla u_n| |\nabla \zeta| |G_k(u_n)| dx
\\
&\leq&
\displaystyle C (\mbox{meas}(A_{k,\rho}))^{1-\frac1p}+ \frac14
\int_{A_{k,\rho}} |\zeta |^2 |\nabla u_n|^2 dx
\\
&&
\displaystyle
+4 C_3^2 \int_{A_{k,\rho}} |\nabla \zeta|^2 |G_k(u_n)|^2 dx, \end{array} $$ by using Young's inequality. Hence, recalling that, on $A_{k,\rho}$, we have $\nabla G_k(u_{n})=\nabla u_n$, we conclude that $$ \begin{array}{rcl} \displaystyle \frac14 \int_{A_{k,\rho}}
|\nabla u_{n}|^{2} \zeta^2 dx
&\leq&
\displaystyle C \left((\mbox{meas}(A_{k,\rho}))^{1-\frac1p}+ \int_{A_{k,\rho}} |\nabla \zeta|^2 |G_k(u_n)|^2 dx\right), \end{array} $$ \arraycolsep5pt where $C=C(C_1,C_2,C_3)$ is a generic constant.
Now we argue as in \cite[Theorem IV-1.1, p.251]{LU68}. For $\sigma\in\,]0,1[$, choose $\zeta$ such that $\zeta\equiv 1$ in the concentric ball $B_{\rho-\sigma\rho}$ (concentric to $B_{\rho}$) of radius $\rho-\sigma\rho$ and such that $|\nabla \zeta|< \frac{2}{\sigma\rho}$. Hence, we obtain \arraycolsep1.5pt $$ \begin{array}{rcl} \displaystyle \int_{A_{k,\rho-\sigma\rho}}
|\nabla u_{n}|^{2} dx
&\leq&
\displaystyle C (1+ (\max_{A_{k,\rho}}(|u(x)|-k))^2 \||\nabla \zeta|^2\|_{L^p(A_{k,\rho})})(\mbox{meas}(A_{k,\rho}))^{1-\frac1p}
\\
&\leq&
\displaystyle C (1+ \frac{4}{\rho^2\sigma^2}(\rho^N\omega_N)^{1/p} (\max_{A_{k,\rho}}(|u(x)|-k))^2 )(\mbox{meas}(A_{k,\rho}))^{1-\frac1p}, \end{array} $$
\arraycolsep5pt where $\omega_N$ denotes the measure of the unit ball of $\mathbb R^N$. Hence, for $k\geq C_1\geq \max_{B_{\rho}}|u_n|-\delta$, we have $$ \begin{array}{rcl} \displaystyle \int_{A_{k,\rho-\sigma\rho}}
|\nabla u_{n}|^{2} dx
&\leq&
\displaystyle \gamma \left(1+ \frac{1}{\sigma^2\rho^{2(1-\frac{N}{2p})}} (\max_{A_{k,\rho}}(|u(x)|-k))^2 \right)
(\mbox{meas}(A_{k,\rho}))^{1-\frac1p}. \end{array} $$
This means that, for $\delta >0$ small enough and every $M\geq C_1\geq \|u_n\|_{\infty}$, we have $u_n\in B_2(\Omega, M, \gamma, \delta, \frac{1}{2p})$ (see \cite[pag. 81]{LU68}).
Applying \cite[Theorem II-6.1 and Theorem II-7.1, p.90 and 91]{LU68}, we deduce that $u_n\in C^{0,\alpha}(\overline \Omega)$ with
$\| u_n\|_{C^{0,\alpha}}$ bounded by a constant $C_4$ which depends only on $\Omega, M, \gamma, \delta$ and the claim is proved. \medbreak
\noindent {\bf Step 2.} {\it $K^\mu$ maps bounded sets of $L^p(\Omega)$ to relatively compact sets of $C(\overline\Omega)$.}
This can be easily deduced from Step 1 and the compact embedding of $C^{0,\alpha}(\overline \Omega)$ into $C(\overline \Omega)$.
\noindent {\bf Step 3.} {\it $K^{\mu}$ is continuous from $L^p(\Omega)$ to $H$. }
Let $\{f_n\} \subset L^p(\Omega)$ be a sequence such that $f_n \to f$ in $L^p(\Omega)$ and let $\{u_n\}$ be the corresponding solutions of (\ref{pivot}). By Lemma \ref{estimationsinfini}, there exists $C>0$ such that, for all $n \in {\mathbb N}$, $||u_n||_{\infty} \leq C$ and $||u_n|| \leq C.$ Hence for every subsequence $\{u_{n_k}\}$, there exists a subsubsequence $\{u_{n_{k_j}}\} \subset X$ and $u \in X$ such that $u_{n_{k_j}} \rightharpoonup u$ weakly in $H$, $u_{n_{k_j}} \to u$ strongly in $L^{p'}(\Omega)$ and $u_{n_{k_j}} \to u$ almost everywhere.
Let us prove that $u_{n_{k_j}} \to u$ strongly in $H$ and that $u$ is the solution of (\ref{pivot}). In that case we shall deduce that $u_n \to u$ in $H$, namely the continuity of $K^{\mu}$ from $L^p(\Omega)$ to $H$. Let us define $\tilde{u}_j = u_{n_{k_j}} -u$. Observe that $\tilde{u}_j$ satisfies \begin{equation*}
-\Delta \tilde{u}_j + \tilde{u}_j = f_{n_{k_j}}(x) + \mu(x)|\nabla u_{n_{k_j}}|^2 + \Delta u - u, \quad \mbox{in } X. \end{equation*}
Consider the test function $\tilde v_j = \widetilde{E}_j \tilde u_j$ where $\widetilde{E}_j = \exp(\tilde t \tilde u_j^2)$ and $\tilde t = 2 ||\mu||_{\infty}^2$. As $\tilde u_j \in X$ we have $\tilde v_j \in X$, and using the inequality $$
|\nabla u_{n_{k_j}}|^2 \leq 2 (|\nabla \tilde u_j|^2 + |\nabla u|^2), $$ we obtain \arraycolsep1.5pt $$ \begin{array}{l} \displaystyle
\int_{\Omega} \widetilde{E}_j(1+2 \tilde t \tilde u_j^2) |\nabla \tilde u_j|^2 \,dx + \int_{\Omega}\widetilde{E}_j \tilde u_j^2 \,dx \\[3mm] \hspace{15mm} \displaystyle = \int_{\Omega} \nabla \tilde u_j \nabla \tilde v_j \,dx + \int_{\Omega} \tilde u_j \tilde v_j \,dx \\[3mm] \hspace{15mm} \displaystyle = \int_{\Omega}f_{n_{k_j}}(x) \tilde v_j \,dx +
\int_{\Omega} \mu(x) |\nabla u_{n_{k_j}}|^2 \tilde v_j \,dx \\[3mm]
\displaystyle - \int_{\Omega} \widetilde{E}_j \nabla u \nabla \tilde u_j (1+ 2 \tilde t \tilde u_j^2)\,dx - \int_{\Omega}u \tilde v_j \,dx \\[3mm] \hspace{15mm} \displaystyle \leq \int_{\Omega} f_{n_{k_j}}(x) \tilde E_j \tilde u_j\, dx - \int_{\Omega} \tilde E_j \nabla u \nabla \tilde u_j (1+ 2 \tilde t \tilde u_j^2)\,dx - \int_{\Omega}u \tilde E_j \tilde u_j \,dx \\[3mm] \displaystyle
+
2 ||\mu||_{\infty}
\left( \int_{\Omega}\tilde E_j^{1/2}|\tilde u_j| | \nabla \tilde u_j| | \nabla \tilde u_j| \tilde E_j^{1/2}\, dx
+ \int_{\Omega}|\nabla u|^2 \tilde E_j \tilde u_j \,dx \right) \\[3mm] \hspace{15mm} \displaystyle \leq \int_{\Omega} f_{n_{k_j}}(x) \tilde E_j \tilde u_j dx - \int_{\Omega} \tilde E_j \nabla u \nabla \tilde u_j (1 + 2 \tilde t \tilde u_j^2) \,dx - \int_{\Omega} u \tilde E_j \tilde u_j \,dx \\[3mm] \hspace{30mm} \displaystyle
+ 2 ||\mu||_{\infty}
\left( ||\mu||_{\infty} \int_{\Omega} \tilde E_j |\tilde u_j|^2 |\nabla \tilde u_j|^2 \,dx \right. \\[3mm]
\displaystyle \left. +
\frac{1}{4 ||\mu||_{\infty} } \int_{\Omega}\tilde E_j |\nabla \tilde u_j|^2 \,dx +
\int_{\Omega}|\nabla u|^2 \tilde E_j \tilde u_j \,dx \right) \\[3mm] \displaystyle \hspace{15mm} \leq \int_{\Omega} f_{n_{k_j}}(x) \tilde E_j \tilde u_j\, dx - \int_{\Omega}\tilde E_j \nabla u \nabla \tilde u_j (1 + 2 \tilde t \tilde u_j^2)\, dx -\int_{\Omega} u \tilde E_j \tilde u_j \,dx \\[3mm]
\displaystyle
+ \frac{1}{2} \int_{\Omega}\tilde E_j (1+ 2 \tilde t \tilde u_j^2) |\nabla \tilde u_j|^2 dx +
2 ||\mu||_{\infty} \int_{\Omega}|\nabla u|^2 \tilde E_j \tilde u_j\, dx.
\end{array} $$ \arraycolsep5pt Hence we deduce that \begin{equation} \label{rap} \begin{array}{l} \displaystyle
\frac{1}{2} \int_{\Omega} \tilde E_j(1+2 \tilde t \tilde u_j^2) |\nabla \tilde u_j|^2 dx
+
\int_{\Omega}\tilde E_j \tilde u_j^2 dx \\[3mm] \displaystyle \hspace{15mm} \leq \int_{\Omega} (f_{n_{k_j}}(x)-f(x)) \tilde E_j \tilde u_j dx +
2 ||\mu||_{\infty} \int_{\Omega} |\nabla u|^2 \tilde E_j \tilde u_j dx \\[3mm] \displaystyle \hspace{20mm} - \int_{\Omega}\tilde E_j \nabla u \nabla \tilde u_j (1 + 2 \tilde t \tilde u_j^2) dx - \int_{\Omega} u \tilde E_j \tilde u_j dx + \int_{\Omega} f(x) \tilde E_j \tilde u_j dx. \end{array} \end{equation}
Let us prove that each of the terms on the right hand side converges to zero. For the first one, as the sequence $\{u_n\}$ is bounded in $ L^{\infty}(\Omega)$ there exists $C_1 >0$ such that, for all $j \in {\mathbb N}$, $||\widetilde{E}_j||_{\infty} \leq C_1.$ This implies the existence of a constant $C>0$ such that \begin{equation} \label{21}
\lim_{j \to \infty} \left|\int_{\Omega} (f_{n_{k_j}}(x)-f(x)) \widetilde{E}_j \tilde u_j dx \right| \leq C \lim_{j \to \infty}||f_{n_{k_j}}-f||_p = 0. \end{equation}
For the second term we have $|\nabla u|^2 \widetilde{E}_j \tilde u_j \to 0$ a.e. in $\Omega$ as $\tilde u_j \to 0$ a.e. in $\Omega$ and $\widetilde{E}_j$ is bounded. Moreover, for all $j \in {\mathbb N}$, $$
\left| |\nabla u|^2 \widetilde{E}_j \tilde u_j \right| \leq C C_1 |\nabla u|^2 $$
with $CC_1|\nabla u|^2 \in L^1(\Omega)$. Hence by Lebesgue's dominated convergence theorem we have that $$
\int_{\Omega} |\nabla u|^2 \widetilde{E}_j \tilde u_j dx \to 0. $$
To prove that the third term converges to zero, observe that $\nabla \tilde u_j \rightharpoonup 0$ weakly in $L^2(\Omega)$. Hence if we prove that $\widetilde{E}_j \nabla u (1+ 2 \tilde t \tilde u_j^2) $ converges strongly in $L^2(\Omega)$, we shall obtain $$\int_{\Omega} \widetilde{E}_j \nabla u \nabla \tilde u_j (1+ 2 \tilde t \tilde u_j^2) dx \to 0.$$
Observe that $\widetilde{E}_j \nabla u (1+ 2 \tilde t \tilde u_j^2) \to \nabla u$ a.e. in $\Omega$. Moreover we have $$
\left| \widetilde{E}_j \nabla u (1+ 2 \tilde t \tilde u_j^2) \right| \leq C_1 (1+ 2 \tilde t C^2) |\nabla u| \quad \mbox{ with } \quad C_1(1+ 2 \tilde t C^2) \nabla u \in L^2(\Omega). $$ Hence, again by Lebesgue dominated convergence theorem, we have $\widetilde{E}_j \nabla u (1+ 2 \tilde t \tilde u_j^2) \to \nabla u$ strongly in $L^2(\Omega)$. For the two last terms observe that $$
u \widetilde{E}_j \tilde u_j \to 0 \mbox{ a.e. in } \Omega \quad \mbox{ and } \quad |u \widetilde{E}_j \tilde u_j| \leq CC_1 |u| $$
with $CC_1 |u| \in L^1(\Omega)$. This holds true also for $f\widetilde{E}_j \tilde u_j$. Hence again we have $$ \int_{\Omega} u \widetilde{E}_j \tilde u_j dx \to 0 \quad\mbox{ and }\quad \int_{\Omega} f \widetilde{E}_j \tilde u_j dx \to 0. $$ This implies, by (\ref{rap}), that $$
\lim_{j \to \infty} ||\tilde u_j||^2 \leq \lim_{j \to \infty} 2 \left( \frac{1}{2} \int_{\Omega} \widetilde{E}_j(1+ 2 \tilde t \tilde u_j^2) |\nabla \tilde u_j|^2 dx + \int_{\Omega} \widetilde{E}_j \tilde u_j^2 dx \right) = 0. $$ As $\tilde u_j \to 0$ weakly in $H$ we obtain $\tilde u_j \to 0$ strongly in $H$, namely $u_{n_{k_j}} \to u$ strongly in $H$. Hence we can pass to the limit in the equation and $u \in X$ satisfies \begin{equation*}
-\Delta u + u - \mu(x) |\nabla u|^2 = f, \quad \mbox{ in } \Omega, \end{equation*} At this point we have proved the continuity of $K^{\mu}$ from $L^p(\Omega)$ to $H$.
\noindent {\bf Step 4.} {\it $K^{\mu}$ is continuous from $L^p(\Omega)$ to $C(\overline\Omega)$.}
Let $\{f_n\}\subset L^p(\Omega)$ be a sequence such that $f_n\to f$ in $L^p(\Omega)$. In particular the sequence $\{f_n\}$ is bounded in $L^p(\Omega)$. Hence, by Step 1, for every subsequence $\{f_{n_k}\}_k$ the set $\{u_{n_k}=K^{\mu}(f_{n_k})\mid k\in \mathbb N\}$ is relatively compact in $C(\overline\Omega)$ i.e. there exists a subsequence $(u_{n_{k_j}})_j$ which converges in $C(\overline\Omega)$ to $v\in C(\overline\Omega)$. By Step 3, $u_{n_{k_j}}\to u = K^{\mu}(f)$ in $H$. In particular $u_{n_{k_j}}\to v$ in $C(\overline\Omega)$ and $u_{n_{k_j}}\to u$ in $L^2(\Omega)$. By unicity of the limit, we conclude that $u=v$. As this is true for every subsequence, we have also that, if $f_n\to f$ in $L^p(\Omega)$ then $u_n = K^{\mu}(f_n)\to u = K^{\mu}(f)$ in $C(\overline\Omega)$ which concludes the proof. \end{proof}
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Three transiting planet discoveries from the Wide Angle Search for Planets: WASP-85 A b; WASP-116 b, and WASP-149 b (1412.7761)
D. J. A. Brown, D. R. Anderson, A. P. Doyle, E. GillenP. F. L. Maxted, B. Smalley, J. McCormac, J. M. Almenera, J. Prieto-Arranz, M. Deleuil, R. F. Diaz, E. Foxell, G. Hebrard, M. Lendl, L. Delrez, M. Gillon, E. Jehin, K. W. F. Lam, A. H. M. J. Triaud, O. D. Turner, D. J. Armstrong, F. Bouchy, A. Collier Cameron, D. Pollacco, F. Faedi, Y. Gomez Maqueo Chew, L. Hebb, C. Hellier, M. Neveu-VanMalle, E. Palle, D. Queloz, D. Segransan, S. Udry, R. G. West
Feb. 10, 2019 astro-ph.EP
(abridged) We report the discovery of three new transiting planets: WASP-85 A b, WASP-116 b, and WASP-149 b. WASP-85 b orbits its host star every 2.66 days, and has a mass of 1.25 M_Jup and a radius of 1.25 R_Jup. The host star is of G5 spectral type, with magnitude V = 11.2, and lies 141 pc distant. The system has a K-dwarf binary companion, WASP-85 B, at a separation of ~1.5". The close proximity of this companion leads to contamination of our photometry, decreasing the apparent transit depth that we account for during our analysis. Analysis of the Ca II H+K lines shows strong emission that implies that both binary components are strongly active. WASP-116 b is a warm, mildly inflated super-Saturn, with a mass of 0.59 M_Jup and a radius of 1.43 R_Jup. It was discovered orbiting a metal-poor ([Fe/H] = -0.28 dex), cool (T_eff = 5950 K) G0 dwarf every 6.61 days. WASP-149 b is a typical hot Jupiter, orbiting a G6 dwarf with a period of 1.33 days. The planet has a mass and radius of 1.05 M_Jup and 1.29 R_Jup, respectively. The stellar host has an effective temperature of T_eff = 5750 K and has a metallicity of [Fe/H] = 0.16 dex. WASP photometry of the system is contaminated by a nearby star; we therefore corrected the depth of the WASP transits using the measured dilution. WASP-149 lies inside the 'Neptune desert' identified in the planetary mass-period plane by Mazeh, Holczer & Faigler (2016). We model the modulation visible in the K2 lightcurve of WASP-85 using a simple three-spot model consisting of two large spots on WASP-85 A, and one large spot on WASP-85 B, finding rotation periods of 13.1+/-0.1 days for WASP-85 A and 7.5+/-0.03 days for WASP-85 B. We estimate stellar inclinations of I_A = 66.8+/-0.7 degrees and I_B = 39.7+/-0.2 degrees, and constrain the obliquity of WASP-85 A b to be psi<27 degrees. We therefore conclude that WASP-85 A b is very likely to be aligned.
GitHub 0
Helium in the eroding atmosphere of an exoplanet (1805.01298)
J. J. Spake, D. K. Sing, T. M. Evans, A. Oklopčić, V. Bourrier, L. Kreidberg, B. V. Rackham, J. Irwin, D. Ehrenreich, A. Wyttenbach, H. R. Wakeford, Y. Zhou, K. L. Chubb, N. Nikolov, J. M. Goyal, G. W. Henry, M. H. Williamson, S. Blumenthal, D. R. Anderson, C. Hellier, D. Charbonneau, S. Udry, N. Madhusudhan
May 3, 2018 astro-ph.EP
Helium is the second-most abundant element in the Universe after hydrogen and is one of the main constituents of gas-giant planets in our Solar System. Early theoretical models predicted helium to be among the most readily detectable species in the atmospheres of exoplanets, especially in extended and escaping atmospheres. Searches for helium, however, have hitherto been unsuccessful. Here we report observations of helium on an exoplanet, at a confidence level of 4.5 standard deviations. We measured the near- infrared transmission spectrum of the warm gas giant WASP-107b and identified the narrow absorption feature of excited metastable helium at 10,833 angstroms. The amplitude of the feature, in transit depth, is 0.049 +/- 0.011 per cent in a bandpass of 98 angstroms, which is more than five times greater than what could be caused by nominal stellar chromospheric activity. This large absorption signal suggests that WASP-107b has an extended atmosphere that is eroding at a total rate of 10^10 to 3 x 10^11 grams per second (0.1-4 per cent of its total mass per billion years), and may have a comet-like tail of gas shaped by radiation pressure.
WASP 0639-32: a new F-type subgiant/K-type main-sequence detached eclipsing binary from the WASP project (1804.06718)
J. A. Kirkby-Kent, P. F. L. Maxted, A. M. Serenelli, D. R. Anderson, C. Hellier, R. G. West
April 18, 2018 astro-ph.SR
Abridged. Our aim is to measure the masses and radii of the stars in a newly-discovered detached eclipsing binary system to a high precision ($\approx$1%), enabling the system to be used for the calibration of free parameters in stellar evolutionary models. Photometry from the Wide Angle Search for Planets (WASP) project was used to identify 1SWASP J063930.33-322404.8 (TYC 7091-888-1, WASP 0369-32 hereafter) as a detached eclipsing binary system with total eclipses and an orbital period of P=11.66 days. Lightcurve parameters for WASP 0639-32 are obtained using the ebop lightcurve model, with standard errors evaluated using a prayer-bead algorithm. Radial velocities were measured from 11 high-resolution spectra using a broadening function approach, and an orbit was fitted using sbop. Observed spectra were disentangled and an equivalent width fitting method was used to obtain effective temperatures and metallicities for both stars. A Bayesian framework is used to explore a grid of stellar evolution models, where both helium abundance and mixing length are free to vary, and use observed parameters (mass, density, temperature and metallicity) for each star to obtain the age and constrain the helium abundance of the system. The masses and radii are found to be $M_{1}=1.1544\pm0.0043\,M_{\odot}$, $R_{1}=1.833\pm0.023\,R_{\odot}$ and $M_{2}=0.7833\pm0.0028\,M_{\odot}$, $R_{2}=0.7286\pm0.0081\,R_{\odot}$ for the primary and secondary, respectively. The effective temperatures were found to be $T_{1}=6330\pm50$ K and $T_{2}=5400\pm80$ K for the primary and secondary star, respectively. The system has an age of $4.2^{+0.8}_{-0.1}$ Gyr, and a helium abundance in the range 0.251-0.271. WASP 0639-32 is a rare example of a well-characterised detached eclipsing binary system containing a star near the main-sequence turn-off point. This make it possible to measure a precise age for the stars ...
WASP-104b is Darker than Charcoal (1804.05334)
T. Močnik, C. Hellier, J. Southworth
April 15, 2018 astro-ph.EP
By analysing the K2 short-cadence data from Campaign 14 we detect phase-curve modulation in the light curve of the hot-Jupiter host star WASP-104. The ellipsoidal modulation is detected with high significance and in agreement with theoretical expectations, while Doppler beaming and reflection modulations are detected tentatively. We show that the visual geometric albedo is lower than 0.03 at 95% confidence, making it one of the least-reflective planets found to date. The light curve also exhibits a rotational modulation, implying a stellar rotational period likely to be near 23 or 46 days. In addition, we refine the system parameters and place tight upper limits for transit timing and duration variations, starspot occultation events, and additional transiting planets.
Thermal emission of WASP-48b in the Ks-band (1804.01913)
B. J. M. Clark, D. R. Anderson, N. Madhusudhan, C. Hellier, A. M. S. Smith, A. Collier Cameron
April 5, 2018 astro-ph.EP
We report a detection of thermal emission from the hot Jupiter WASP-48b in the $K_{\rm s}$-band. We used the Wide-field Infra-red Camera on the 3.6-m Canada-France Hawaii Telescope to observe an occultation of the planet by its host star. From the resulting occultation lightcurve we find a planet-to-star contrast ratio in the Ks-band of $0.136 \pm 0.014\,\%$ , in agreement with the value of $0.109 \pm 0.027\,\%$ previously determined. We fit the two Ks-band occultation lightcurves simultaneously with occultation lightcurves in the $H$-band and the Spitzer 3.6-{\mu}m and 4.5-{\mu}m bandpasses, radial velocity data, and transit lightcurves. From this, we revise the system parameters and construct the spectral energy distribution (SED) of the dayside atmosphere. By comparing the SED with atmospheric models, we find that both models with and without a thermal inversion are consistent with the data. We find the planet's orbit to be consistent with circular ($e < 0.072$ at 3$\sigma$).
The discoveries of WASP-91b, WASP-105b and WASP-107b: two warm Jupiters and a planet in the transition region between ice giants and gas giants (1701.03776)
D. R. Anderson, A. Collier Cameron, L. Delrez, A. P. Doyle, M. Gillon, C. Hellier, E. Jehin, M. Lendl, P. F. L. Maxted, N. Madhusudhan, F. Pepe, D. Pollacco, D. Queloz, D. Ségransan, B. Smalley, A. M. S. Smith, A. H. M. J. Triaud, O. D. Turner, S. Udry, R. G. West
Feb. 8, 2018 astro-ph.EP
We report the discoveries of three transiting exoplanets. WASP-91b is a warm Jupiter (1.34 $M_{\rm Jup}$, 1.03 $R_{\rm Jup}$) in a 2.8-day orbit around a metal-rich K3 star. WASP-105b is a warm Jupiter (1.8 $M_{\rm Jup}$, 0.96 $R_{\rm Jup}$) in a 7.9-day orbit around a metal-rich K2 star. WASP-107b is a warm super-Neptune/sub-Saturn (0.12 $M_{\rm Jup}$, 0.94 $R_{\rm Jup}$) in a 5.7-day orbit around a solar-metallicity K6 star. Considering that giant planets seem to be more common around stars of higher metallicity and stars of higher mass, it is notable that the hosts are all metal-rich, late-type stars. With orbital separations that place both WASP-105b and WASP-107b in the weak-tide regime, measurements of the alignment between the planets' orbital axes and their stars' spin axes may help us to understand the inward migration of short-period, giant planets. The mass of WASP-107b (2.2 $M_{\rm Nep}$, 0.40 $M_{\rm Sat}$) places it in the transition region between the ice giants and gas giants of the Solar System. Its radius of 0.94 $R_{\rm Jup}$ suggests that it is a low-mass gas giant with a H/He-dominated composition. The planet thus sets a lower limit of 2.2 $M_{\rm Nep}$ on the planetary mass above which large gaseous envelopes can be accreted and retained by proto-planets on their way to becoming gas giants. We may discover whether WASP-107b more closely resembles an ice giant or a gas giant by measuring its atmospheric metallicity via transmission spectroscopy, for which WASP-107b is a very good target.
Three sub-Jupiter-mass planets: WASP-69b & WASP-84b transit active K dwarfs and WASP-70Ab transits the evolved primary of a G4+K3 binary (1310.5654)
D. R. Anderson, A. Collier Cameron, L. Delrez, A. P. Doyle, F. Faedi, A. Fumel, M. Gillon, Y. Gómez Maqueo Chew, C. Hellier, E. Jehin, M. Lendl, P. F. L. Maxted, F. Pepe, D. Pollacco, D. Queloz, D. Ségransan, I. Skillen, B. Smalley, A. M. S. Smith, J. Southworth, A. H. M. J. Triaud, O. D. Turner, S. Udry, R. G. West
We report the discovery of the transiting exoplanets WASP-69b, WASP-70Ab and WASP-84b, each of which orbits a bright star ($V\sim10)$. WASP-69b is a bloated Saturn-mass planet (0.26 $M_{\rm Jup}$, 1.06 $R_{\rm Jup}$) in a 3.868-d period around an active, $\sim$1-Gyr, mid-K dwarf. ROSAT detected X-rays $60 \pm 27"$ from WASP-69. If the star is the source then the planet could be undergoing mass-loss at a rate of $\sim$10$^{12}$ g s$^{-1}$. This is 1 to 2 orders of magnitude higher than the evaporation rate estimated for HD 209458b and HD 189733b, both of which have exhibited anomalously-large Lyman-$\alpha$ absorption during transit. WASP-70Ab is a sub-Jupiter-mass planet (0.59 $M_{\rm Jup}$, 1.16 $R_{\rm Jup}$) in a 3.713-d orbit around the primary of a spatially-resolved, 9-to-10-Gyr, G4+K3 binary, with a separation of 3.3$'$ ($\geq$800 AU). WASP-84b is a sub-Jupiter-mass planet (0.69 $M_{\rm Jup}$, 0.94 $R_{\rm Jup)}$ in an 8.523-d orbit around an active, $\sim$1-Gyr, early-K dwarf. Of the transiting planets discovered from the ground to date, WASP-84b has the third-longest period. For the active stars WASP-69 and WASP-84, we pre-whitened the radial velocities using a low-order harmonic series. We found this reduced the residual scatter more than did the oft-used method of pre-whitening with a fit between residual radial velocity and bisector span. The system parameters were essentially unaffected by pre-whitening.
An Analysis of Transiting Hot Jupiters Observed with K2: WASP-55b and WASP-75b (1802.02132)
B. J. M. Clark, D. R. Anderson, C. Hellier, O. D. Turner, T. Močnik
We present our analysis of the K2 short-cadence data of two previously known hot Jupiter exoplanets: WASP-55b and WASP-75b. The high precision of the K2 lightcurves enabled us to search for transit timing and duration variations, rotational modulation, starspots, phase-curve variations and additional transiting planets. We identified stellar variability in the WASP-75 lightcurve which may be an indication of rotational modulation, with an estimated period of $11.2\pm1.5$ days. We combined this with the spectroscopically measured $v\sin(i_*)$ to calculate a possible line of sight projected inclination angle of $i_*=41\pm16^{\circ}$. We also perform a global analysis of K2 and previously published data to refine the system parameters.
Discovery of WASP-174b: Doppler tomography of a near-grazing transit (1802.00766)
L.Y. Temple, C. Hellier, Y. Almleaky, D.R. Anderson, F. Bouchy, D.J.A. Brown, A. Burdanov, A. Collier Cameron, L. Delrez, M. Gillon, R. Hall, E. Jehin, M. Lendl, P.F.L. Maxted, L. D. Nielsen, F. Pepe, D. Pollacco, D. Queloz, D. Ségransan, B. Smalley, S. Sohy, S. Thompson, A.H.M.J. Triaud, O.D. Turner, S. Udry, R.G. West
We report the discovery and tomographic detection of WASP-174b, a planet with a near-grazing transit on a 4.23-d orbit around a $V$ = 11.9, F6V star with [Fe/H] = 0.09 $\pm$ 0.09. The planet is in a moderately misaligned orbit with a sky-projected spin-orbit angle of $\lambda$ = 31$^{\circ}$ $\pm$ 1$^{\circ}$. This is in agreement with the known tendency for orbits around hotter stars to be misaligned. Owing to the grazing transit the planet's radius is uncertain, with a possible range of 0.7-1.7 R$_{\rm Jup}$. The planet's mass has an upper limit of 1.3 M$_{\rm Jup}$. WASP-174 is the faintest hot-Jupiter system so far confirmed by tomographic means.
High-precision multi-wavelength eclipse photometry of the ultra-hot gas giant exoplanet WASP-103 b (1711.02566)
L. Delrez, N. Madhusudhan, M. Lendl, M. Gillon, D. R. Anderson, M. Neveu-VanMalle, F. Bouchy, A. Burdanov, A. Collier-Cameron, B.-O. Demory, C. Hellier, E. Jehin, P. Magain, P. F. L. Maxted, D. Queloz, B. Smalley, A. H. M. J. Triaud
Nov. 7, 2017 astro-ph.EP
We present sixteen occultation and three transit light curves for the ultra-short period hot Jupiter WASP-103 b, in addition to five new radial velocity measurements. We combine these observations with archival data and perform a global analysis of the resulting extensive dataset, accounting for the contamination from a nearby star. We detect the thermal emission of the planet in both the $z'$ and $K_{\mathrm{S}}$-bands, the measured occultation depths being 699$\pm$110 ppm (6.4-$\sigma$) and $3567_{-350}^{+400}$ ppm (10.2-$\sigma$), respectively. We use these two measurements together with recently published HST/WFC3 data to derive joint constraints on the properties of WASP-103 b's dayside atmosphere. On one hand, we find that the $z'$-band and WFC3 data are best fit by an isothermal atmosphere at 2900 K or an atmosphere with a low H$_2$O abundance. On the other hand, we find an unexpected excess in the $K_{\mathrm{S}}$-band measured flux compared to these models, which requires confirmation with additional observations before any interpretation can be given. From our global data analysis, we also derive a broad-band optical transmission spectrum that shows a minimum around 700 nm and increasing values towards both shorter and longer wavelengths. This is in agreement with a previous study based on a large fraction of the archival transit light curves used in our analysis. The unusual profile of this transmission spectrum is poorly matched by theoretical spectra and is not confirmed by more recent observations at higher spectral resolution. Additional data, both in emission and transmission, are required to better constrain the atmospheric properties of WASP-103 b.
The discovery of WASP-151b, WASP-153b, WASP-156b: Insights on giant planet migration and the upper boundary of the Neptunian desert (1710.06321)
Olivier. D. S. Demangeon, F. Faedi, G. Hébrard, D. J. A. Brown, S. C. C. Barros, A. P. Doyle, P. F. L. Maxted, A. Collier Cameron, K. L. Hay, J. Alikakos, D. R. Anderson, D. J. Armstrong, P. Boumis, A. S. Bonomo, F. Bouchy, C. A. Haswell, C. Hellier, F. Kiefer, K. W. F. Lam, L. Mancini, J. McCormac, A. J. Norton, H. P. Osborn, E. Palle, F. Pepe, D. L. Pollacco, J. Prieto-Arranz, D. Queloz, D. Ségransan, B. Smalley, A. H. M. J. Triaud, S. Udry, R. West, P.J. Wheatley
Oct. 17, 2017 astro-ph.EP
To investigate the origin of the features discovered in the exoplanet population, the knowledge of exoplanets' mass and radius with a good precision is essential. In this paper, we report the discovery of three transiting exoplanets by the SuperWASP survey and the SOPHIE spectrograph with mass and radius determined with a precision better than 15 %. WASP-151b and WASP-153b are two hot Saturns with masses, radii, densities and equilibrium temperatures of 0.31^{+0.04}_{-0.03} MJ, 1.13^{+0.03}_{-0.03} RJ, 0.22^{-0.03}_{-0.02} rhoJ and 1, 290^{+20}_{-10} K, and 0.39^{+0.02}_{-0.02} MJ, 1.55^{+0.10}_{-0.08} RJ, 0.11^{+0.02}_{-0.02} rhoJ and 1, 700^{+40}_{-40} K, respectively. Their host stars are early G type stars (with magV ~ 13) and their orbital periods are 4.53 and 3.33 days, respectively. WASP-156b is a Super-Neptune orbiting a K type star (magV = 11.6) . It has a mass of 0.128^{+0.010}_{-0.009} MJ, a radius of 0.51^{+0.02}_{-0.02} RJ, a density of 1.0^{+0.1}_{-0.1} rhoJ, an equilibrium temperature of 970^{+30}_{-20} K and an orbital period of 3.83 days. WASP-151b is slightly inflated, while WASP-153b presents a significant radius anomaly. WASP-156b, being one of the few well characterised Super-Neptunes, will help to constrain the formation of Neptune size planets and the transition between gas and ice giants. The estimates of the age of these three stars confirms the tendency for some stars to have gyrochronological ages significantly lower than their isochronal ages. We propose that high eccentricity migration could partially explain this behaviour for stars hosting a short period planet. Finally, these three planets also lie close to (WASP-151b and WASP-153b) or below (WASP-156b) the upper boundary of the Neptunian desert. Their characteristics support that the ultra-violet irradiation plays an important role in this depletion of planets observed in the exoplanet population.
WASP-167b/KELT-13b: Joint discovery of a hot Jupiter transiting a rapidly-rotating F1V star (1704.07771)
L.Y. Temple, C. Hellier, M. D. Albrow, D.R. Anderson, D. Bayliss, T. G. Beatty, A. Bieryla, D.J.A. Brown, P. A. Cargile, A. Collier Cameron, K. A. Collins, K. D. Colón, I. A. Curtis, G. D'Ago, L. Delrez, J. Eastman, B. S. Gaudi, M. Gillon, J. Gregorio, D. James, E. Jehin, M. D. Joner, J. F. Kielkopf, R. B. Kuhn, J. Labadie-Bartz, D. W. Latham, M. Lendl, M. B. Lund, A. L. Malpas, P.F.L. Maxted, G. Myers, T. E. Oberst, F. Pepe, J. Pepper, D. Pollacco, D. Queloz, J. E. Rodriguez, D. Ségransan, R. J. Siverd, B. Smalley, K. G. Stassun, D. J. Stevens, C. Stockdale, T.G. Tan, A.H.M.J. Triaud, S. Udry, S. Villanueva Jr, R.G. West, G. Zhou
July 7, 2017 astro-ph.EP
We report the joint WASP/KELT discovery of WASP-167b/KELT-13b, a transiting hot Jupiter with a 2.02-d orbit around a $V$ = 10.5, F1V star with [Fe/H] = 0.1 $\pm$ 0.1. The 1.5 R$_{\rm Jup}$ planet was confirmed by Doppler tomography of the stellar line profiles during transit. We place a limit of $<$ 8 M$_{\rm Jup}$ on its mass. The planet is in a retrograde orbit with a sky-projected spin-orbit angle of $\lambda = -165^{\circ} \pm 5^{\circ}$. This is in agreement with the known tendency for orbits around hotter stars to be more likely to be misaligned. WASP-167/KELT-13 is one of the few systems where the stellar rotation period is less than the planetary orbital period. We find evidence of non-radial stellar pulsations in the host star, making it a $\delta$-Scuti or $\gamma$-Dor variable. The similarity to WASP-33, a previously known hot-Jupiter host with pulsations, adds to the suggestion that close-in planets might be able to excite stellar pulsations.
Recurring sets of recurring starspot occultations on exoplanet-host Qatar-2 (1608.07524)
T. Močnik, J. Southworth, C. Hellier
June 21, 2017 astro-ph.EP
We announce the detection of recurring sets of recurring starspot occultation events in the short-cadence K2 lightcurve of Qatar-2, a K dwarf star transited every 1.34 d by a hot Jupiter. In total we detect 34 individual starspot occultation events, caused by five different starspots, occulted in up to five consecutive transits or after a full stellar rotation. The longest recurring set of recurring starspot occultations spans over three stellar rotations, setting a lower limit for the longest starspot lifetime of 58 d. Starspot analysis provided a robust stellar rotational period measurement of $18.0\pm0.2$ d and indicates that the system is aligned, having a sky-projected obliquity of $0\pm8^{\circ}$. A pronounced rotational modulation in the lightcurve has a period of $18.2\pm1.6$ d, in agreement with the rotational period derived from the starspot occultations. We tentatively detect an ellipsoidal modulation in the phase-curve, with a semi-amplitude of 18 ppm, but cannot exclude the possibility that this is the result of red noise or imperfect removal of the rotational modulation. We detect no transit-timing and transit-duration variations with upper limits of 15 s and 1 min, respectively. We also reject any additional transiting planets with transit depths above 280 ppm in the orbital period region 0.5-30 d.
Periodic Eclipses of the Young Star PDS 110 Discovered with WASP and KELT Photometry (1705.10346)
H. P. Osborn, J. E. Rodriguez, M. A. Kenworthy, G. M. Kennedy, E. E. Mamajek, C. E. Robinson, C. C. Espaillat, D. J. Armstrong, B. J. Shappee, A. Bieryla, D. W. Latham, D. R. Anderson, T. G. Beatty, P. Berlind, M. L. Calkins, G. A. Esquerdo, B. S. Gaudi, C. Hellier, T. W.-S. Holoien, D. James, C. S. Kochanek, R. B. Kuhn, M. B. Lund, J. Pepper, D. L. Pollacco, J. L. Prieto, R. J. Siverd, K. G. Stassun, D. J. Stevens, K. Z. Stanek, R. G. West
May 29, 2017 astro-ph.EP
We report the discovery of eclipses by circumstellar disc material associated with the young star PDS 110 in the Ori OB1a association using the SuperWASP and KELT surveys. PDS 110 (HD 290380, IRAS 05209-0107) is a rare Fe/Ge-type star, a ~10 Myr-old accreting intermediate-mass star showing strong infrared excess (L$_{\rm IR}$/L$_{\rm bol}$ ~ 0.25). Two extremely similar eclipses with a depth of ~30\% and duration ~25 days were observed in November 2008 and January 2011. We interpret the eclipses as caused by the same structure with an orbital period of $808\pm2$ days. Shearing over a single orbit rules out diffuse dust clumps as the cause, favouring the hypothesis of a companion at ~2AU. The characteristics of the eclipses are consistent with transits by an unseen low-mass (1.8-70M$_{Jup}$) planet or brown dwarf with a circum-secondary disc of diameter ~0.3 AU. The next eclipse event is predicted to take place in September 2017 and could be monitored by amateur and professional observatories across the world.
Starspots on WASP-107 and pulsations of WASP-118 (1702.05078)
T. Močnik, C. Hellier, D. R. Anderson, B. J. M. Clark, J. Southworth
April 23, 2017 astro-ph.SR, astro-ph.EP
By analysing the K2 short-cadence photometry we detect starspot occultation events in the lightcurve of WASP-107, the host star of a warm-Saturn exoplanet. WASP-107 also shows a rotational modulation with a period of 17.5 +/- 1.4 d. Given that the rotational period is nearly three times the planet's orbital period, one would expect in an aligned system to see starspot occultation events to recur every three transits. The absence of such occultation recurrences suggests a misaligned orbit unless the starspots' lifetimes are shorter than the star's rotational period. We also find stellar variability resembling gamma Doradus pulsations in the lightcurve of WASP-118, which hosts an inflated hot Jupiter. The variability is multi-periodic with a variable semi-amplitude of about 200 ppm. In addition to these findings we use the K2 data to refine the parameters of both systems, and report non-detections of transit-timing variations, secondary eclipses and any additional transiting planets. We used the upper limits on the secondary-eclipse depths to estimate upper limits on the planetary geometric albedos of 0.7 for WASP-107b and 0.2 for WASP-118b.
From dense hot Jupiter to low-density Neptune: The discovery of WASP-127b, WASP-136b and WASP-138b (1607.07859)
K. W. F. Lam, F. Faedi, D. J. A. Brown, D. R. Anderson, L. Delrez, M. Gillon, G. Hébrard, M. Lendl, L. Mancini, J. Southworth, B. Smalley, A. H. M. Triaud, O. D. Turner, K. L. Hay, D. J. Armstrong, S. C. C. Barros, A. S. Bonomo, F. Bouchy, P. Boumis, A. Collier Cameron, A. P. Doyle, C. Hellier, T. Henning, E. Jehin, G. King, J. Kirk, T. Louden, P. F. L. Maxted, J. J. McCormac, H. P. Osborn, E. Palle, F. Pepe, D. Pollacco, J. Prieto-Arranz, D. Queloz, J. Rey, D. Ségransan, S. Udry, S. Walker, R. G. West, P. J. Wheatley
Nov. 15, 2016 astro-ph.EP
We report three newly discovered exoplanets from the SuperWASP survey. WASP-127b is a heavily inflated super-Neptune of mass 0.18 +/- 0.02 M_J and radius 1.37 +/- 0.04 R_J. This is one of the least massive planets discovered by the WASP project. It orbits a bright host star (Vmag = 10.16) of spectral type G5 with a period of 4.17 days. WASP-127b is a low-density planet that has an extended atmosphere with a scale height of 2500 +/- 400 km, making it an ideal candidate for transmission spectroscopy. WASP-136b and WASP-138b are both hot Jupiters with mass and radii of 1.51 +/- 0.08 M_J and 1.38 +/- 0.16 R_J, and 1.22 +/- 0.08 M_J and 1.09 +/- 0.05 R_J, respectively. WASP-136b is in a 5.22-day orbit around an F9 subgiant star with a mass of 1.41 +/- 0.07 M_sun and a radius of 2.21 +/- 0.22 R_sun. The discovery of WASP-136b could help constrain the characteristics of the giant planet population around evolved stars. WASP-138b orbits an F7 star with a period of 3.63 days. Its radius agrees with theoretical values from standard models, suggesting the presence of a heavy element core with a mass of ~10 M_earth. The discovery of these new planets helps in exploring the diverse compositional range of short-period planets, and will aid our understanding of the physical characteristics of both gas giants and low-density planets.
Pulsation versus metallicism in Am stars as revealed by LAMOST and WASP (1611.02254)
B. Smalley, V. Antoci, D.L. Holdsworth, D.W. Kurtz, S.J. Murphy, P. De Cat, D.R. Anderson, G. Catanzaro, A. Collier Cameron, C. Hellier, P.F.L. Maxted, A.J. Norton, D. Pollacco, V. Ripepi, R.G. West, P.J. Wheatley
Nov. 7, 2016 astro-ph.SR
We present the results of a study of a large sample of A and Am stars with spectral types from LAMOST and light curves from WASP. We find that, unlike normal A stars, $\delta$ Sct pulsations in Am stars are mostly confined to the effective temperature range 6900 $<$ $T_{\rm eff}$ $<$ 7600 K. We find evidence that the incidence of pulsations in Am stars decreases with increasing metallicism (degree of chemical peculiarity). The maximum amplitude of the pulsations in Am stars does not appear to vary significantly with metallicism. The amplitude distributions of the principal pulsation frequencies for both A and Am stars appear very similar and agree with results obtained from Kepler photometry. We present evidence that suggests turbulent pressure is the main driving mechanism in pulsating Am stars, rather than the $\kappa$-mechanism, which is expected to be suppressed by gravitational settling in these stars.
Rossiter--McLaughlin models and their effect on estimates of stellar rotation, illustrated using six WASP systems (1610.00600)
D. J. A. Brown, A. H. M. J. Triaud, A. P. Doyle, M. Gillon, M. Lendl, D. R. Anderson, A. Collier Cameron, G. Hébrard, C. Hellier, C. Lovis, P. F. L. Maxted, F. Pepe, D. Pollacco, D. Queloz, B. Smalley
Oct. 3, 2016 astro-ph.EP
We present new measurements of the projected spin--orbit angle $\lambda$ for six WASP hot Jupiters, four of which are new to the literature (WASP-61, -62, -76, and -78), and two of which are new analyses of previously measured systems using new data (WASP-71, and -79). We use three different models based on two different techniques: radial velocity measurements of the Rossiter--McLaughlin effect, and Doppler tomography. Our comparison of the different models reveals that they produce projected stellar rotation velocities ($v \sin I_{\rm s}$) measurements often in disagreement with each other and with estimates obtained from spectral line broadening. The Bou\'e model for the Rossiter--McLaughlin effect consistently underestimates the value of $v\sin I_{\rm s}$ compared to the Hirano model. Although $v \sin I_s$ differed, the effect on $\lambda$ was small for our sample, with all three methods producing values in agreement with each other. Using Doppler tomography, we find that WASP-61\,b ($\lambda=4^\circ.0^{+17.1}_{-18.4}$), WASP-71\,b ($\lambda=-1^\circ.9^{+7.1}_{-7.5}$), and WASP-78\,b ($\lambda=-6^\circ.4\pm5.9$) are aligned. WASP-62\,b ($\lambda=19^\circ.4^{+5.1}_{-4.9}$) is found to be slightly misaligned, while WASP-79\,b ($\lambda=-95^\circ.2^{+0.9}_{-1.0}$) is confirmed to be strongly misaligned and has a retrograde orbit. We explore a range of possibilities for the orbit of WASP-76\,b, finding that the orbit is likely to be strongly misaligned in the positive $\lambda$ direction.
Orbital and physical parameters of eclipsing binaries from the ASAS catalogue -- IX. Spotted pairs with red giants (1608.04978)
M. Ratajczak, K. G. Hełminiak, M. Konacki, A. M. S. Smith, S. K. Kozłowski, N. Espinoza, A. Jordán, R. Brahm, M. Hempel, D. R. Anderson, , C. Hellier
Aug. 17, 2016 astro-ph.SR
We present spectroscopic and photometric solutions for three spotted systems with red giant components. Absolute physical and orbital parameters for these double-lined detached eclipsing binary stars are presented for the first time. These were derived from the V-, and I-band ASAS and WASP photometry, and new radial velocities calculated from high quality optical spectra we obtained with a wide range of spectrographs and using the two-dimensional cross-correlation technique (TODCOR). All of the investigated systems (ASAS J184949-1518.7, BQ Aqr, and V1207 Cen) show the differential evolutionary phase of their components consisting of a main sequence star or a subgiant and a red giant, and thus constitute very informative objects in terms of testing stellar evolution models. Additionally, the systems show significant chromospheric activity of both components. They can be also classified as classical RS CVn-type stars. Besides the standard analysis of radial velocities and photometry, we applied spectral disentangling to obtain separate spectra for both components of each analysed system which allowed for a more detailed spectroscopic study. We also compared the properties of red giant stars in binaries that show spots, with those that do not, and found that the activity phenomenon is substantially suppressed for stars with Rossby number higher than $\sim$1 and radii larger than $\sim$20 R$_\odot$.
WASP-86b and WASP-102b: super-dense versus bloated planets (1608.04225)
F. Faedi, Y. Gómez Maqueo Chew, D. Pollacco, D. J. A. Brown, G. Hébrard, B. Smalley, K. W. F. Lam, D. Veras, D. Anderson, A. P. Doyle, M. Gillon, M. R. Goad, M. Lendl, L. Mancini, J. McCormac, I. Plauchu-Frayn, J. Prieto-Arranz, A. Scholz, R. Street, A. H. M. Triaud, R. West, P. J. Wheatley, D. J. Armstrong, S. C. C. Barros, I. Boisse, F. Bouchy, P. Boumis, A. Collier Cameron, C. A. Haswell, K. L. Hay, C. Hellier, U. Kolb, P. F. L. Maxted, A. J. Norton, H. P. Osborn, E. Palle, F. Pepe, D. Queloz, D. Ségransan, S. Udry, P. A. Wilson
Aug. 15, 2016 astro-ph.SR, astro-ph.EP
We report the discovery of two transiting planetary systems: a super dense, sub-Jupiter mass planet WASP-86b (\mpl\ = 0.82 $\pm$ 0.06 \mj, \rpl\ = 0.63 $\pm$ 0.01 \rj), and a bloated, Saturn-like planet WASP-102b (\mpl\ = 0.62 $\pm$ 0.04 \mj, \rpl\=1.27 $\pm$ 0.03 \rj). They orbit their host star every $\sim$5.03, and $\sim$2.71 days, respectively. The planet hosting WASP-86 is a F7 star (\teff\ = 6330$\pm$110 K, \feh\ = $+$0.23 $\pm$ 0.14 dex, and age $\sim$0.8--1~Gyr), WASP-102 is a G0 star (\teff\ = 5940$\pm$140 K, \feh\ = $-$0.09$\pm$ 0.19 dex, and age $\sim$1~Gyr). These two systems highlight the diversity of planetary radii over similar masses for giant planets with masses between Saturn and Jupiter. WASP-102b shows a larger than model-predicted radius, indicating that the planet is receiving a strong incident flux which contributes to the inflation of its radius. On the other hand, with a density of $\rho_{pl}$ = 3.24$\pm$~0.3~$\rho_{jup}$, WASP-86b is the densest gas giant planet among planets with masses in the range 0.05 $<M$_{pl}$<$ 2.0 \mj. With a stellar mass of 1.34 M$_{\odot}$ and \feh = $+$0.23 dex, WASP-86 could host additional massive and dense planets given that its protoplanetary disc is expected to also have been enriched with heavy elements. In order to match WASP-86b's density, an extrapolation of theoretical models predicts a planet composition of more than 80\% in heavy elements (whether confined in a core or mixed in the envelope). This fraction corresponds to a core mass of approximately 210\me\ for WASP-86b's mass of \mpl$\sim$260\,\me. Only planets with masses larger than about 2\mj\ have larger densities than that of WASP-86b, making it exceptional in its mass range.
Discovery of WASP-113b and WASP-114b, two inflated hot-Jupiters with contrasting densities (1607.02341)
S. C. C. Barros, D. J. A. Brown, G. Hébrard, Y. Gómez Maqueo Chew, D. R. Anderson, P. Boumis, L. Delrez, K. L. Hay, K. W. F. Lam, J. Llama, M. Lendl, J. McCormac, B. Skiff, B Smalley, O Turner, M. Vanhuysse, D. J. Armstrong, I. Boisse, F. Bouchy, A. Collier Cameron, F. Faedi, M. Gillon, C. Hellier, E. Jehin, A. Liakos, J. Meaburn, H. P. Osborn, F. Pepe, I. Plauchu-Frayn, D. Pollacco, D. Queloz, J. Rey, J. Spake, D.Ségransan, A.H.M. Triaud, S. Udry, S.R. Walker, C.A. Watson, R. G. West, P. J. Wheatley
We present the discovery and characterisation of the exoplanets WASP-113b and WASP-114b by the WASP survey, {\it SOPHIE} and {\it CORALIE}. The planetary nature of the systems was established by performing follow-up photometric and spectroscopic observations. The follow-up data were combined with the WASP-photometry and analysed with an MCMC code to obtain system parameters. The host stars WASP-113 and WASP-114 are very similar. They are both early G-type stars with an effective temperature of $\sim 5900\,$K, [Fe/H]$\sim 0.12$ and $T_{\rm eff}$ $\sim 4.1$dex. However, WASP-113 is older than WASP-114. Although the planetary companions have similar radii, WASP-114b is almost 4 times heavier than WASP-113b. WASP-113b has a mass of $0.48\,$ $\mathrm{M}_{\rm Jup}$ and an orbital period of $\sim 4.5\,$days; WASP-114b has a mass of $1.77\,$ $\mathrm{M}_{\rm Jup}$ and an orbital period of $\sim 1.5\,$days. Both planets have inflated radii, in particular WASP-113 with a radius anomaly of $\Re=0.35$. The high scale height of WASP-113b ($\sim 950$ km ) makes it a good target for follow-up atmospheric observations.
WASP-92b, WASP-93b and WASP-118b: Three new transiting close-in giant planets (1607.00774)
K. L. Hay, A. Collier-Cameron, A. P. Doyle, G. Hébrard, I. Skillen, D. R. Anderson, S. C. C. Barros, D. J. A. Brown, F. Bouchy, R. Busuttil, P. Delorme, L. Delrez, O. Demangeon, R. F. Díaz, M. Gillon, E. Gonzàlez, C. Hellier, S. Holmes, J. F. Jarvis, E. Jehin, Y. C. Joshi, U. Kolb, M. Lendl, P. F. L. Maxted, J. McCormac, G. R. M. Miller, A. Mortier, D. Pollacco, D. Queloz, D. Ségransan, E. K. Simpson, B. Smalley, J. Southworth, A. H. M. J. Triaud, O. D. Turner, S. Udry, M. Vanhuysse, R. G. West, P. A. Wilson
We present the discovery of three new transiting giant planets, first detected with the WASP telescopes, and establish their planetary nature with follow up spectroscopy and ground-based photometric lightcurves. WASP-92 is an F7 star, with a moderately inflated planet orbiting with a period of 2.17 days, which has $R_p = 1.461 \pm 0.077 R_{\rm J}$ and $M_p = 0.805 \pm 0.068 M_{\rm J}$. WASP-93b orbits its F4 host star every 2.73 days and has $R_p = 1.597 \pm 0.077 R_{\rm J}$ and $M_p = 1.47 \pm 0.029 M_{\rm J}$. WASP-118b also has a hot host star (F6) and is moderately inflated, where $R_p = 1.440 \pm 0.036 R_{\rm J}$ and $M_p = 0.513 \pm 0.041 M_{\rm J}$ and the planet has an orbital period of 4.05 days. They are bright targets (V = 13.18, 10.97 and 11.07 respectively) ideal for further characterisation work, particularly WASP-118b, which is being observed by K2 as part of campaign 8. WASP-93b is expected to be tidally migrating outwards, which is divergent from the tidal behaviour of the majority of hot Jupiters discovered.
WASP-157b, a Transiting Hot Jupiter Observed with K2 (1603.05638)
T. Močnik, D. R. Anderson, D. J. A. Brown, A. Collier Cameron, L. Delrez, M. Gillon, C. Hellier, E. Jehin, M. Lendl, P. F. L. Maxted, M. Neveu-VanMalle, F. Pepe, D. Pollacco, D. Queloz, D. Ségransan, B. Smalley, J. Southworth, A. H. M. J. Triaud, S. Udry, R. G. West
We announce the discovery of the transiting hot Jupiter WASP-157b in a 3.95-d orbit around a V = 12.9 G2 main-sequence star. This moderately inflated planet has a Saturn-like density with a mass of $0.57 \pm 0.10$ M$_{\rm Jup}$ and a radius of $1.06 \pm 0.05$ R$_{\rm Jup}$. We do not detect any rotational or phase-curve modulations, nor the secondary eclipse, with conservative semi-amplitude upper limits of 250 and 20 ppm, respectively.
Absolute parameters for AI Phoenicis using WASP photometry (1605.07059)
J. A. Kirkby-Kent, P. F. L. Maxted, A. M. Serenelli, O. D. Turner, D. F. Evans, D. R. Anderson, C. Hellier, R. G. West
May 23, 2016 astro-ph.SR
AI Phe is a double-lined, detached binary, in which a K-type sub-giant star totally eclipses its main-sequence companion every 24.6 days. This configuration makes AI Phe ideal for testing stellar evolutionary models. Difficulties in obtaining a complete lightcurve mean the precision of existing radii measurements could be improved. Our aim is to improve the precision of the radius measurements for the stars in AI Phe using high-precision photometry from WASP, and use these improved radius measurements together with estimates of the masses, temperatures and composition of the stars to place constraints on the mixing length, helium abundance and age of the system. A best-fit ebop model is used to obtain lightcurve parameters, with their standard errors calculated using a prayer-bead algorithm. These were combined with previously published spectroscopic orbit results, to obtain masses and radii. A Bayesian method is used to estimate the age of the system for model grids with different mixing lengths and helium abundances. The radii are found to be $R_1=1.835\pm0.014$ R$_{\odot}$, $R_2=2.912\pm0.014$ R$_{\odot}$ and the masses $M_1=1.1973\pm0.0037$ M$_{\odot}$, $M_2=1.2473\pm0.0039$ M$_{\odot}$. From the best-fit stellar models we infer a mixing length of 1.78, a helium abundance of $Y_{AI}=0.26^{+0.02}_{-0.01}$ and an age of $4.39\pm0.32$ Gyr. Times of primary minimum show the period of AI Phe is not constant. Currently, there are insufficient data to determine the cause of this variation. Improved precision in the masses and radii have improved the age estimate, and allowed the mixing length and helium abundance to be constrained. The eccentricity is now the largest source of uncertainty in calculating the masses. More binaries with parameters measured to a similar level of precision would allow us to test for relationships between helium abundance and mixing length.
Five transiting hot Jupiters discovered using WASP-South, Euler and TRAPPIST: WASP-119 b, WASP-124 b, WASP-126 b, WASP-129 b and WASP-133 b (1602.01740)
P. F. L. Maxted, A. Collier Cameron, M. Gillon (Liège, Belgium), C. Hellier, M. Lendl, D. Pollacco, B. Smalley, A. H. M. J. Triaud, R. G. West
We have used photometry from the WASP-South instrument to identify 5 stars showing planet-like transits in their light curves. The planetary nature of the companions to these stars has been confirmed using photometry from the EulerCam instrument on the Swiss Euler 1.2-m telescope and the TRAPPIST telescope, and spectroscopy obtained with the CORALIE spectrograph. The planets discovered are hot Jupiter systems with orbital periods in the range 2.17 to 5.75 days, masses from 0.3M$_{\rm Jup}$ to 1.2M$_{\rm Jup}$ and with radii from 1R$_{\rm Jup}$ to 1.5R$_{\rm Jup}$. These planets orbit bright stars (V = 11-13) with spectral types in the range F9 to G4. WASP-126 is the brightest planetary system in this sample and hosts a low-mass planet with a large radius (0.3 M$_{\rm Jup}$ , 0.95R$_{\rm Jup}$), making it a good target for transmission spectroscopy. The high density of WASP-129 A suggests that it is a helium-rich star similar to HAT-P-11 A. WASP-133 has an enhanced surface lithium abundance compared to other old G-type stars, particularly other planet host stars. These planetary systems are good targets for follow-up observations with ground-based and space-based facilities to study their atmospheric and dynamical properties. | CommonCrawl |
# Vectors and vector operations
- **Vector addition**: Vectors can be added together, just like numbers. The resulting vector is the sum of the corresponding components of the input vectors.
- **Scalar multiplication**: A scalar is a number, and a vector can be multiplied by a scalar. The result is a new vector with each component multiplied by the scalar.
- **Dot product**: The dot product of two vectors is a scalar value that represents the projection of one vector onto another. It is calculated using the formula:
$$
\mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^{n} a_i b_i
$$
- **Cross product**: The cross product of two vectors is a new vector that is perpendicular to both input vectors. It is calculated using the formula:
$$
\mathbf{a} \times \mathbf{b} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3 \\
\end{vmatrix}
$$
- **Magnitude and direction**: The magnitude of a vector is the length of the vector, while the direction is a unit vector pointing in the same direction as the original vector. The magnitude can be calculated using the formula:
$$
\|\mathbf{a}\| = \sqrt{\sum_{i=1}^{n} a_i^2}
$$
## Exercise
Calculate the dot product of the vectors $\mathbf{a} = (1, 2, 3)$ and $\mathbf{b} = (4, 5, 6)$.
- **Unit vectors**: A unit vector is a vector with a magnitude of 1. They are often used to represent directions. To convert a vector into a unit vector, divide each component by the magnitude of the vector.
- **Orthogonality**: Two vectors are orthogonal if their dot product is 0. This means they are perpendicular to each other.
- **Projection**: The projection of one vector onto another is a new vector that is parallel to the second vector and has the same length as the projection of the first vector onto a line parallel to the second vector.
- **Angle between vectors**: The angle between two vectors can be calculated using the dot product formula and the magnitudes of the vectors.
Let's say we have two vectors, $\mathbf{a} = (3, 4)$ and $\mathbf{b} = (1, 2)$. We can calculate their dot product, magnitude, and unit vector as follows:
```python
import numpy as np
a = np.array([3, 4])
b = np.array([1, 2])
dot_product = np.dot(a, b)
magnitude_a = np.linalg.norm(a)
magnitude_b = np.linalg.norm(b)
unit_vector_a = a / magnitude_a
unit_vector_b = b / magnitude_b
print("Dot product:", dot_product)
print("Magnitude of a:", magnitude_a)
print("Magnitude of b:", magnitude_b)
print("Unit vector of a:", unit_vector_a)
print("Unit vector of b:", unit_vector_b)
```
This code will output:
```
Dot product: 11
Magnitude of a: 5.0
Magnitude of b: 2.23606797749979
Unit vector of a: [0.6 0.8]
Unit vector of b: [0.4472136 0.89442719]
```
# Matrix operations and transformations
- **Matrix addition and subtraction**: Matrices can be added and subtracted element-wise, just like vectors.
- **Matrix multiplication**: Matrix multiplication is defined as the dot product of each row of the first matrix with each column of the second matrix. The result is a new matrix with the same number of rows as the first matrix and the same number of columns as the second matrix.
- **Identity matrix**: An identity matrix is a square matrix with 1s on the diagonal and 0s elsewhere. It is used to represent the identity transformation, which leaves a vector unchanged.
- **Transpose**: The transpose of a matrix is obtained by swapping its rows and columns.
- **Inverse**: The inverse of a matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix. Not all matrices have an inverse.
- **Determinant**: The determinant of a square matrix is a scalar value that can be used to determine the properties of the matrix, such as whether it is invertible or not.
## Exercise
Calculate the inverse of the matrix:
$$
\mathbf{A} = \begin{bmatrix}
1 & 2 \\
3 & 4 \\
\end{bmatrix}
$$
- **Echelon form**: A matrix is in echelon form if it has a triangular shape, with zeros above the main diagonal and non-zero elements below the main diagonal. Echelon form is used to solve linear systems of equations.
- **Reduced row echelon form**: A matrix is in reduced row echelon form if it is in echelon form and the leading coefficients are 1. Reduced row echelon form is used to solve linear systems of equations and find the rank of a matrix.
- **Applications**: Matrices are used in various applications, such as image processing, computer vision, and machine learning.
# Eigenvalues and Eigenvectors
Eigenvalues and eigenvectors are fundamental concepts in linear algebra. An eigenvector of a matrix is a non-zero vector that, when multiplied by the matrix, results in a scalar multiple of itself. An eigenvalue is the scalar multiple that results from this operation.
- **Characteristic equation**: The characteristic equation of a matrix is a polynomial equation that can be used to find its eigenvalues.
- **Diagonalization**: A square matrix is diagonalizable if it can be expressed as the product of a diagonal matrix and its inverse.
- **Applications**: Eigenvalues and eigenvectors are used in various applications, such as solving differential equations, analyzing the stability of systems, and finding the principal components of data.
## Exercise
Find the eigenvalues and eigenvectors of the matrix:
$$
\mathbf{A} = \begin{bmatrix}
2 & 1 \\
1 & 3 \\
\end{bmatrix}
$$
- **Generalized eigenvectors**: If a matrix has repeated eigenvalues, then the eigenvectors corresponding to these eigenvalues are called generalized eigenvectors.
- **Applications**: Generalized eigenvectors are used in various applications, such as solving non-homogeneous linear systems of equations, finding the fundamental matrix of a system of differential equations, and analyzing the stability of systems with repeated eigenvalues.
# Solving linear systems
- **Gaussian elimination**: Gaussian elimination is a method for solving linear systems by transforming the system into echelon form and then solving for the unknowns.
- **LU decomposition**: LU decomposition is a factorization of a square matrix into the product of a lower triangular matrix and an upper triangular matrix. It is used to solve linear systems and compute the determinant of a matrix.
- **QR decomposition**: QR decomposition is a factorization of a square matrix into the product of an orthogonal matrix and an upper triangular matrix. It is used to solve linear systems, compute the determinant of a matrix, and find the principal components of a data set.
- **Applications**: Linear system solvers are used in various applications, such as computer graphics, image processing, and control systems.
## Exercise
Solve the linear system:
$$
\begin{bmatrix}
1 & 2 \\
3 & 4 \\
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
\end{bmatrix}
=
\begin{bmatrix}
5 \\
7 \\
\end{bmatrix}
$$
- **Iterative methods**: Iterative methods are used to solve linear systems when the exact solution is not known. Examples of iterative methods include the Jacobi method, the Gauss-Seidel method, and the Conjugate Gradient method.
- **Applications**: Iterative methods are used in various applications, such as solving partial differential equations, optimizing functions, and solving large-scale linear systems.
# Vector spaces and their properties
- **Linear combinations**: A linear combination of vectors is obtained by multiplying each vector by a scalar and then adding the results.
- **Span**: The span of a set of vectors is the smallest vector space that contains all the vectors.
- **Linear independence**: A set of vectors is linearly independent if no vector can be expressed as a linear combination of the other vectors.
- **Basis**: A basis is a set of linearly independent vectors that spans a vector space.
- **Dimension**: The dimension of a vector space is the number of vectors in a basis for that space.
- **Subspaces**: A subspace is a vector space that is a subset of another vector space.
- **Orthogonal complement**: The orthogonal complement of a subspace is the set of all vectors that are orthogonal to every vector in the subspace.
## Exercise
Determine whether the vectors $\mathbf{a} = (1, 2, 3)$, $\mathbf{b} = (2, 3, 4)$, and $\mathbf{c} = (3, 4, 5)$ are linearly independent.
- **Applications**: Vector spaces and their properties are used in various applications, such as computer graphics, image processing, and control systems.
# Advanced matrix operations
- **Singular value decomposition (SVD)**: SVD is a factorization of a matrix into the product of three matrices: a left singular vector matrix, a diagonal matrix of singular values, and a right singular vector matrix. It is used to find the principal components of a data set, perform dimensionality reduction, and solve non-square linear systems.
- **Eigendecomposition**: Eigendecomposition is a factorization of a square matrix into the product of a matrix with its own eigenvalues on the diagonal and the matrix of eigenvectors. It is used to find the principal components of a data set, analyze the stability of systems, and solve non-square linear systems.
- **Applications**: SVD and eigendecomposition are used in various applications, such as image compression, machine learning, and data analysis.
## Exercise
Perform the singular value decomposition of the matrix:
$$
\mathbf{A} = \begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9 \\
\end{bmatrix}
$$
- **Principal component analysis (PCA)**: PCA is a technique used to find the principal components of a data set, which are the directions of maximum variance in the data. It is based on the eigendecomposition of the covariance matrix of the data.
- **Applications**: PCA is used in various applications, such as image compression, dimensionality reduction, and data analysis.
# Applications of linear algebra in computer vision and image processing
- **Feature extraction**: Feature extraction is the process of identifying the most relevant features of an image or a scene. Linear algebra techniques, such as singular value decomposition (SVD) and eigendecomposition, are used to find the principal components of the data and extract the most informative features.
- **Image compression**: Image compression is the process of reducing the size of an image while preserving its essential features. Linear algebra techniques, such as singular value decomposition (SVD) and eigendecomposition, are used to find the principal components of the image and remove the least important components.
- **Image segmentation**: Image segmentation is the process of dividing an image into regions based on their similarity. Linear algebra techniques, such as eigendecomposition and principal component analysis (PCA), are used to find the principal components of the image and classify the pixels into different regions.
- **Applications**: Linear algebra techniques are used in various applications, such as object recognition, scene understanding, and image synthesis.
## Exercise
Perform image compression on the image:
```
[ 1 2 3 ]
[ 4 5 6 ]
[ 7 8 9 ]
```
- **Optical flow**: Optical flow is the apparent motion of objects in a video sequence. Linear algebra techniques, such as singular value decomposition (SVD) and eigendecomposition, are used to find the principal components of the motion fields and track the motion of objects in the video.
- **Applications**: Linear algebra techniques are used in various applications, such as action recognition, object tracking, and video analysis.
# Applications of linear algebra in natural language processing
- **Word embedding**: Word embedding is the process of representing words as vectors in a high-dimensional space, such that words with similar meanings have similar vectors. Linear algebra techniques, such as singular value decomposition (SVD) and eigendecomposition, are used to find the principal components of the word vectors and reduce their dimensionality.
- **Topic modeling**: Topic modeling is the process of discovering the main topics in a collection of documents. Linear algebra techniques, such as singular value decomposition (SVD) and eigendecomposition, are used to find the principal components of the document-topic matrix and extract the most informative topics.
- **Sentiment analysis**: Sentiment analysis is the process of analyzing the sentiment of a text, such as whether it is positive or negative. Linear algebra techniques, such as singular value decomposition (SVD) and eigendecomposition, are used to find the principal components of the text-topic matrix and classify the text into different sentiment categories.
- **Applications**: Linear algebra techniques are used in various applications, such as text classification, machine translation, and information retrieval.
## Exercise
Perform topic modeling on the following collection of documents:
```
Document 1: "I love this movie. The acting was great."
Document 2: "This book is amazing. I can't put it down."
Document 3: "The food was delicious. I had a great time."
```
- **Named entity recognition**: Named entity recognition is the process of identifying and classifying named entities in a text, such as people, organizations, and locations. Linear algebra techniques, such as singular value decomposition (SVD) and eigendecomposition, are used to find the principal components of the named entity vectors and classify the text into different named entity categories.
- **Applications**: Linear algebra techniques are used in various applications, such as information extraction, machine translation, and text summarization.
# Applications of linear algebra in finance and economics
- **Portfolio optimization**: Portfolio optimization is the process of selecting the best combination of assets to maximize the expected return while minimizing the risk. Linear algebra techniques, such as singular value decomposition (SVD) and eigendecomposition, are used to find the principal components of the asset returns and construct the optimal portfolio.
- **Risk management**: Risk management is the process of identifying and mitigating the risks associated with financial investments. Linear algebra techniques, such as singular value decomposition (SVD) and eigendecomposition, are used to find the principal components of the asset returns and classify the assets into different risk categories.
- **Time series analysis**: Time series analysis is the process of analyzing the trends and patterns in a sequence of data points. Linear algebra techniques, such as singular value decomposition (SVD) and eigendecomposition, are used to find the principal components of the time series data and extract the most informative patterns.
- **Applications**: Linear algebra techniques are used in various applications, such as financial forecasting, asset pricing, and economic policy analysis.
## Exercise
Perform portfolio optimization on the following set of assets:
```
Asset 1: [0.1, 0.2, 0.3, 0.4]
Asset 2: [0.2, 0.3, 0.4, 0.5]
Asset 3: [0.3, 0.4, 0.5, 0.6]
```
- **Applications**: Linear algebra techniques are used in various applications, such as financial risk management, economic forecasting, and economic policy analysis.
# Advanced topics in linear algebra for python programmers
- **Numerical linear algebra**: Numerical linear algebra is the application of linear algebra to numerical problems, such as solving large-scale linear systems and computing the eigenvalues and eigenvectors of a matrix. Python libraries, such as NumPy and SciPy, provide efficient and accurate implementations of numerical linear algebra algorithms.
- **Sparse matrices**: Sparse matrices are matrices that have a large number of zero elements. They are used to represent sparse data, such as graphs and social networks. Python libraries, such as SciPy, provide efficient implementations of sparse matrix operations.
- **Tensor algebra**: Tensor algebra is the extension of linear algebra to multi-dimensional arrays, called tensors. It is used in various applications, such as computer vision, natural language processing, and machine learning. Python libraries, such as TensorFlow and PyTorch, provide efficient implementations of tensor operations.
- **Applications**: Advanced topics in linear algebra are used in various applications, such as machine learning, deep learning, and artificial intelligence.
## Exercise
Solve the following large-scale linear system using the conjugate gradient method:
```
A = [ ... ]
b = [ ... ]
```
- **Applications**: Advanced topics in linear algebra are used in various applications, such as data analysis, scientific computing, and engineering simulations.
By the end of this textbook, you will have a deep understanding of the fundamental concepts in linear algebra and their applications in various fields, including computer vision and image processing, natural language processing, finance and economics, and python programming. | Textbooks |
\begin{definition}[Definition:Euclid's Definitions - Book V/16 - Conversion of Ratio]
{{EuclidSaid}}
:'''''Conversion of a ratio''' means taking the antecedent in relation to the excess by which the antecedent exceeds the consequent.''
{{EuclidDefRef|V|16|Conversion of Ratio}}
\end{definition} | ProofWiki |
Lecture 38 - Entropy
Stony Brook Physics phy141:lectures
Trace: • Lecture 38 - Entropy
If you need a pdf version of these notes you can get it here
Video of lecture
What is Entropy?
In our last class we introduced the second law of thermodynamics which placed limits on the physically possible thermodynamic processes above and beyond conservation of energy (the 1st law).
Here we introduce a quantity, entropy, which we can use to build a deeper understanding of the second law. We'll come to see that entropy relates to the disorder of a system, and is also a measurement of the available thermal energy for a process to occur.
We will define the change in entropy in a reversible process at constant temperature as
$\Delta S =\frac{Q}{T}$
If we want to treat non-constant temperature cases we can express the change of entropy in differential form
$dS=\frac{dQ}{T}$
and then the change in entropy in going from state $a$ to $b$ will be
$\Delta S =S_{b}-S_{a}=\int_{a}^{b}\,dS=\int_{a}^{b}\frac{dQ}{T}$
Entropy as a state variable
A state variable is a variable that describes the current state of a dynamical system. Heat is not a state variable, it depends on the path taken to get the system to it's state. Entropy, on the other hand, is a state variable, the change in entropy required to change a system from one state to another via a reversible process is independent of the path taken.
We can recall that in the Carnot Cycle we were able to derive that
$\frac{Q_{L}}{Q_{H}}=\frac{T_{L}}{T_{H}}$ or $\frac{Q_{L}}{T_{L}}=\frac{Q_{H}}{T_{H}}$
where $Q_{L}$ is the magnitude of the heat flowing out of the system and $Q_{H}$ is the magnitude of the heat flowing into the system. To make these compatible with our equation for entropy we give $Q_{H}$ a positive sign and $Q_{L}$ a negative sign.
$\frac{Q_{H}}{T_{H}}+\frac{Q_{L}}{T_{L}}=0$
And we can see on the PV diagram below that either of the paths from A to C will have the same change in entropy. We can also see that a complete Carnot cycle has a net change of entropy of zero.
Consequence of entropy being a state variable
We can calculate the entropy changes for reversible process using
This equation is only valid for reversible processes. However if we want to find the entropy change for an irreversible process (ie. any real process) that goes from state A to state B we can calculate the entropy change for a reversible process that goes from point to A and B, and the entropy change for the system will be the same for the two processes.
As we have stated that the entropy of a system is a state variable it stands to reason that for any closed cycle the systems entropy will return to it's initial value after the completion of a full cycle.
So what distinguishes a reversible cycle from an irreversible one? The distinction considers the change of the entropy for the environment the system is exchanging heat with. The definition of a reversible cycle is that the system is always infinitesimally close to being in thermal equilibrium with it's environment, in which case we can see that the change of entropy in the environment is also zero. If this is not the case then a heat engine will instead increase the entropy of its environment which each cycle, and as the system itself does not have a changes in entropy this leads to a net increase in the entropy of the universe whenever work is done by any real engine cycle.
Any reversible cycle can be represented as series of Carnot cycles, with each Carnot cycle contributing no increase of entropy, therefore any reversible cycle results in no increase of entropy.
A reversible process will have equal and opposite entropy change if it is reversed.
We will now look at the entropy change for some non reversible processes.
Entropy change for mixing
We can consider the entropy change when two objects transfer heat from one to another, for example when we mix water at two different temperatures. When the temperature difference is fairly small we can approximate the entropy change for each process as
$\Delta S=\frac{Q}{\overline{T}}$
$\overline{T}$ is an average temperature for the process.
In the example of mixing equal quantities of water at different temperatures $T_{H}$ and $T_{L}$, and amount of heat $Q$ will be transferred from the hot water until the temperature is a new temperature $T_{M}=\frac{T_{H}+T_{L}}{2}$. For the water which is cooling the entropy change will be negative
$\Delta S_{cooling}=-\frac{Q}{T_{cooling}}$
where $T_{H}>T_{cooling}>T_{M}$
For the water whose temperature is increasing the change in entropy is
$\Delta S_{heating}=\frac{Q}{T_{heating}}$
where $T_{L}<T_{heating}<T_{M}$
As $T_{cooling}>T_{heating}$ we can see that the total entropy change of the system
$\Delta S_{cooling}+\Delta S_{heating}>0$
Transfer of entropy to the environment
In many cases when considering the total entropy change in a system we need to consider the entropy change to the environment. For example, if we take an object which is cooling by heat lost to the environment through a quasistatic reversible process where $dQ=mc\,dT$
$\Delta S_{object}=\int \frac{dQ}{T}=mc\int_{T_{1}}^{T_{2}}\frac{dT}{T}=mc\ln\frac{T_{2}}{T_{1}}=-mc\ln\frac{T_{1}}{T_{2}}$
If we consider the environment as thermal reservoir at a fixed temperature $T_{2}$ the entropy change is
$\Delta S_{environment}=mc\frac{T_{1}-T_{2}}{T_{2}}=mc(\frac{T_{1}}{T_{2}}-1)$
$\Delta S_{total}=mc((\frac{T_{1}}{T_{2}}-1)+\ln\frac{T_{1}}{T_{2}}))$ which we can see is always greater than zero.
Second law of thermodynamics from an entropy viewpoint
The examples we have discussed fit with our expectations of how things work. If we mix hot and cold water they will equilibrate to the same temperature and they won't spontaneously separate again in to hot and cold water (which would decrease entropy). If we have a hot object in an cooler environment it will transfer the heat to the environment, the reverse, which would decrease entropy won't happen.
We can express the second law of thermodynamics in terms of entropy:
The entropy of an isolated system never decreases, it either stays constant (for a reversible process) or increases (for irreversible processes). As all real processes are irreversible the total entropy for a system and it's environment increases as a result of any natural process.
While we can decrease the entropy of part of the universe, some other parts entropy will be increased by a greater amount, leading towards an continual overall increase of the universe's entropy.
Order, disorder and availability of energy
Entropy can be seen as a measure of the order of a system.
For example when we have hot and cold fluids we have a form of order, which is lost when we mix them together. We also lose the capacity to use them for work, while they were separated we could have used them to drive a heat engine, which requires a temperature gradient to do work, once they are mixed we cannot get work from them, even though no energy has been lost.
We can view the continual change of order to disorder as a gradual heating of the universe to a uniform temperature (expansion of the universe would complicate this, as this would result in a lower final temperature, which might eventually tend to absolute zero). The long term consequence of the universe acquiring a uniform temperature and maximal entropy state would be it's eventual heat death in which all mechanical energy would be lost.
Entropy and Statistics
So far we have considered entropy along the lines it was first proposed by Clausius.
Boltzmann was responsible for giving entropy a statistical basis.
So far when we have considered the state of the system we are referring to it's macrostate, ie. what is the pressure, volume, temperature etc. For each of these macrostates there is some set of microstates which give rise to the macrostate.
If we consider a gas knowing the microstate of the gas would imply that we know the velocity and position of each and every molecule. This is impossible to know, but to determine the probability of a given macrostate we don't need to know the details of the microstate, simply how many microstates there are which correspond to that macrostate. Those macrostates which have the greatest number of microstates have the greatest probability of occurrence. Boltzmann expressed entropy in terms of the number of microstates; the entropy of a given macrostate is
$S=k\ln W$
where $W$ is the number of microstates corresponding to that state.
Second law as a consequence of statistics
As we have now made a link between entropy and probability
we can now see that the second law of thermodynamics is simply a statement that a change of a system will be towards one that is more probable. For example, it is extremely unlikely that all the molecules of gas in a room will arrange themselves neatly ordered on one side of the room, because this would be a single microstate, whereas there are a very large number of microstates which obey the Maxwell distribution in which the molecules are evenly distributed and move randomly.
Maxwell's Demon
However, the statistical viewpoint actually leads us to the conclusion that the second law may not be as rigid as we have so far presented it. Processes which decrease entropy are not strictly forbidden, they are just very unlikely to occur, a concept Maxwell considered through the thought experiment known as Maxwell's demon. Irrespective of this idea, over time the occurrence of a few statistically unlikely events will have little effect on the overall direction.
This has not stopped the harnessing of such events being proposed as a means of faster than light travel.
An application of entropy closer to home
The salt we added to the ice at the beginning of the class lowers its melting point. This effect is a colligative property which means it depends on the number of a molecules in a solvent. Colligative properties are a consequence of entropy. When we add salt to the ice, it makes a solution with liquid water (which is always present on the surface of the ice, even if the temperature is below zero Celsius). This solution has higher entropy than pure liquid water. The salt has no effect on the entropy of the ice itself which remains pure. As we have increased the entropy of the solution, or in other words, the number of available configurations available in the liquid salt-water phase, compared to the number of configurations available in the solid phase we have increased the probability of transition from ice to the liquid phase and shifted the equilibrium temperature where salt-water and ice can co-exist to a lower temperature.
Because the water in the bottles is fairly pure and has been cooled slowly and without agitation we will hopefully find that at least some of the bottles are supercooled.
Brinicles
A consequence of the lower melting point of salty water is that ice in the sea actually forms relatively pure ice with channels of liquid brine (very salty water). When the air is very cold compared to the sea temperature the brine will have a temperature significantly less than the sea and so will seep out of the channels at the bottom of the ice, rapidly freezing the fresher water it comes in to contact with, forming a brinicle or finger of death.
Third law of thermodynamics
The statistical definition of entropy leads logically to the third law of thermodynamics which defines the absolute value of entropy.
As $S=k\ln W$
we can see that the entropy is equal to zero when there is a single microstate for the system, or $W=1$, which corresponds to a perfectly ordered state occurring at $T=0\mathrm{K}$.
phy141/lectures/38.txt · Last modified: 2013/11/25 17:22 by mdawber | CommonCrawl |
There are 250 scraps. For each 11 removed, one extra can be obtained. How much maximum is possible, if one cut by one unit?
I have read this somewhere, and one guy gave the answer as 25, but I can't understand why. Can anyone please explain?
10 - 11 = -1 ??????
You can't continue after doing this 24 times. Or you will get -1. As said by @Novarg , you can't take 11 things out of 10 things.
Assuming I understand your puzzle correctly: You have 250 'things'. Each time you remove 11, you get back one. How many 11s can you remove?
But here is another way to look at it without the "...".
If you have 250 "scraps", then for every 11, you can make a new one. There are 22 groups of 11 in 250 with 8 left over. So, from 250, you can use $22\times11=242$, and get 22 new ones.
Now, from these 30, you can get 2 more by using 22.
So, in total, you've made $22+2=24$ new scraps, you've used $242+22=264$ scraps in total, and have 10 left over. | CommonCrawl |
Painlevé equations from Nakajima-Yoshioka blow-up relations
arxiv.org. math. Cornell University, 2018
Bershtein M., Shchechkin A.
Gamayun, Iorgov and Lisovyy in 2012 proposed that tau function of the Painlevé equation equals to the series of c=1 Virasoro conformal blocks. We study similar series of c=−2 conformal blocks and relate it to Painlevé theory. The arguments are based on Nakajima-Yoshioka blow-up relations on Nekrasov partition functions. We also study series of q-deformed c=−2 conformal blocks and relate it to q-Painlevé equation. Using this we prove formula for the tau function of q-Painlevé A_7^{(1)′} equation.
Research target: Mathematics
Priority areas: mathematics
Text on another site
Keywords: Painleve equationsdifference Painleve equations Nakajima-Yoshioka blow-up equationsInstanton moduli space
Publication based on the results of: Теория представлений и математическая физика(2018)
On Divergence of Puiseux Series Asymptotic Expansions of Solutions to the Third Painlevé Equation
Parusnikova A., Vasilyev A. V. math. arxive. Cornell University, 2017. No. 1702.05758.
In this paper we present a family of values of the parameters of the third Painlevé equation such that Puiseux series formally satisfying this equation -- considered as series of z^{2/3} -- are series of exact Gevrey order one. We prove the divergence of these series and provide analytic functions which are approximated by them in sectors with the vertices at infinity.
Geometry of second adjointness for p-adic groups
Bezrukavnikov R., Kazhdan D. Representation Theory. 2015.
Geometry of second adjointness for p-adic groups.
Сабейские этюды
Коротаев А. В. М.: Восточная литература, 1997.
Added: Mar 9, 2013
Dynamics of Information Systems: Mathematical Foundations
Iss. 20. NY: Springer, 2012.
This proceedings publication is a compilation of selected contributions from the "Third International Conference on the Dynamics of Information Systems" which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.
Noncommutative geometry and Painleve equations
Okounkov A., Rains E. Algebra and Number Theory. 2015.
Added: Sep 4, 2015
Instantons and 2d Superconformal field theory
Bershtein M., Белавин А. А., Белавин В. А. Journal of High Energy Physics. 2011. Vol. 9. No. 117.
A recently proposed correspondence between 4-dimensional N=2 SUSY SU(k) gauge theories on R^4/Z_m and SU(k) Toda-like theories with Z_m parafermionic symmetry is used to construct four-point N=1 super Liouville conformal block, which corresponds to the particular case k=m=2.
The construction is based on the conjectural relation between moduli spaces of SU(2) instantons on R^4/Z_2 and algebras like \hat{gl}(2)_2\times NSR. This conjecture is confirmed by checking the coincidence of number of fixed points on such instanton moduli space with given instanton number N and dimension of subspace degree N in the representation of such algebra.
Instanton moduli spaces and $\mathscr W$-algebras
Braverman A., Finkelberg M. V., Nakajima H. arxiv.org. math. Cornell University, 2014. No. 2381.
We describe the (equivariant) intersection cohomology of certain moduli spaces ("framed Uhlenbeck spaces") together with some structures on them (such as e.g.\ the Poincar\'e pairing) in terms of representation theory of some vertex operator algebras ("W-algebras").
Pure SU(2) gauge theory partition function and generalized Bessel kernel
Gavrylenko P., Lisovyy O. math-ph. arXiv.org. arXiv.org, 2017. No. 1705.01869.
We show that the dual partition function of the pure $\mathcal N=2$ $SU(2)$ gauge theory in the self-dual $\Omega$-background (a) is given by Fredholm determinant of a generalized Bessel kernel and (b) coincides with the tau function associated to the general solution of the Painlev\'e III equation of type $D_8$ (radial sine-Gordon equation). In particular, the principal minor expansion of the Fredholm determinant yields Nekrasov combinatorial sums over pairs of Young diagrams.
Added: May 5, 2017
Model for organizing cargo transportation with an initial station of departure and a final station of cargo distribution
Khachatryan N., Akopov A. S. Business Informatics. 2017. No. 1(39). P. 25-35.
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traffic is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the final node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a finite-dimensional system of differential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the "correct" extension of solutions of a system of differential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.
Nullstellensatz over quasi-fields
Trushin D. Russian Mathematical Surveys. 2010. Vol. 65. No. 1. P. 186-187.
Деловой климат в оптовой торговле во II квартале 2014 года и ожидания на III квартал
Лола И. С., Остапкович Г. В. Современная торговля. 2014. № 10.
Added: Jan 29, 2015
Прикладные аспекты статистики и эконометрики: труды 8-ой Всероссийской научной конференции молодых ученых, аспирантов и студентов
Вып. 8. МЭСИ, 2011.
Added: Sep 13, 2013
Laminations from the Main Cubioid
Timorin V., Blokh A., Oversteegen L. et al. arxiv.org. math. Cornell University, 2013. No. 1305.5788.
According to a recent paper \cite{bopt13}, polynomials from the closure $\ol{\phd}_3$ of the {\em Principal Hyperbolic Domain} ${\rm PHD}_3$ of the cubic connectedness locus have a few specific properties. The family $\cu$ of all polynomials with these properties is called the \emph{Main Cubioid}. In this paper we describe the set $\cu^c$ of laminations which can be associated to polynomials from $\cu$.
Entropy and the Shannon-McMillan-Breiman theorem for beta random matrix ensembles
Bufetov A. I., Mkrtchyan S., Scherbina M. et al. arxiv.org. math. Cornell University, 2013. No. 1301.0342.
Bounded limit cycles of polynomial foliations of ℂP²
Goncharuk N. B., Kudryashov Y. arxiv.org. math. Cornell University, 2015. No. 1504.03313.
In this article we prove in a new way that a generic polynomial vector field in ℂ² possesses countably many homologically independent limit cycles. The new proof needs no estimates on integrals, provides thinner exceptional set for quadratic vector fields, and provides limit cycles that stay in a bounded domain.
Метод параметрикса для диффузий и цепей Маркова
Конаков В. Д. STI. WP BRP. Издательство попечительского совета механико-математического факультета МГУ, 2012. № 2012.
Added: Dec 5, 2012
Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action?
Colliot-Thélène J., Kunyavskiĭ B., Vladimir L. Popov et al. Compositio Mathematica. 2011. Vol. 147. No. 2. P. 428-466.
Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.
Absolutely convergent Fourier series. An improvement of the Beurling-Helson theorem
Vladimir Lebedev. arxiv.org. math. Cornell University, 2011. No. 1112.4892v1.
We obtain a partial solution of the problem on the growth of the norms of exponential functions with a continuous phase in the Wiener algebra. The problem was posed by J.-P. Kahane at the International Congress of Mathematicians in Stockholm in 1962. He conjectured that (for a nonlinear phase) one can not achieve the growth slower than the logarithm of the frequency. Though the conjecture is still not confirmed, the author obtained first nontrivial results.
Обоснование адиабатического предела для гиперболических уравнений Гинзбурга-Ландау
Пальвелев Р., Сергеев А. Г. Труды Математического института им. В.А. Стеклова РАН. 2012. Т. 277. С. 199-214.
Hypercommutative operad as a homotopy quotient of BV
Khoroshkin A., Markaryan N. S., Shadrin S. arxiv.org. math. Cornell University, 2012. No. 1206.3749.
We give an explicit formula for a quasi-isomorphism between the operads Hycomm (the homology of the moduli space of stable genus 0 curves) and BV/Δ (the homotopy quotient of Batalin-Vilkovisky operad by the BV-operator). In other words we derive an equivalence of Hycomm-algebras and BV-algebras enhanced with a homotopy that trivializes the BV-operator. These formulas are given in terms of the Givental graphs, and are proved in two different ways. One proof uses the Givental group action, and the other proof goes through a chain of explicit formulas on resolutions of Hycomm and BV. The second approach gives, in particular, a homological explanation of the Givental group action on Hycomm-algebras.
Cross-sections, quotients, and representation rings of semisimple algebraic groups
V. L. Popov. Transformation Groups. 2011. Vol. 16. No. 3. P. 827-856.
Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.
Математическое моделирование социальных процессов
Edited by: А. Михайлов Вып. 14. М.: Социологический факультет МГУ, 2012. | CommonCrawl |
Eisenstein ideal
In mathematics, the Eisenstein ideal is an ideal in the endomorphism ring of the Jacobian variety of a modular curve, consisting roughly of elements of the Hecke algebra of Hecke operators that annihilate the Eisenstein series. It was introduced by Barry Mazur (1977), in studying the rational points of modular curves. An Eisenstein prime is a prime in the support of the Eisenstein ideal (this has nothing to do with primes in the Eisenstein integers).
Definition
Let N be a rational prime, and define
J0(N) = J
as the Jacobian variety of the modular curve
X0(N) = X.
There are endomorphisms Tl of J for each prime number l not dividing N. These come from the Hecke operator, considered first as an algebraic correspondence on X, and from there as acting on divisor classes, which gives the action on J. There is also a Fricke involution w (and Atkin–Lehner involutions if N is composite). The Eisenstein ideal, in the (unital) subring of End(J) generated as a ring by the Tl, is generated as an ideal by the elements
Tl − l - 1
for all l not dividing N, and by
w + 1.
Geometric definition
Suppose that T* is the ring generated by the Hecke operators acting on all modular forms for Γ0(N) (not just the cusp forms). The ring T of Hecke operators on the cusp forms is a quotient of T*, so Spec(T) can be viewed as a subscheme of Spec(T*). Similarly Spec(T*) contains a line (called the Eisenstein line) isomorphic to Spec(Z) coming from the action of Hecke operators on the Eisenstein series. The Eisenstein ideal is the ideal defining the intersection of the Eisenstein line with Spec(T) in Spec(T*).
Example
• The Eisenstein ideal can also be defined for higher weight modular forms. Suppose that T is the full Hecke algebra generated by Hecke operators Tn acting on the 2-dimensional space of modular forms of level 1 and weight 12.This space is 2 dimensional, spanned by the Eigenforms given by the Eisenstein series E12 and the modular discriminant Δ. The map taking a Hecke operator Tn to its eigenvalues (σ11(n),τ(n)) gives a homomorphism from T into the ring Z×Z (where τ is the Ramanujan tau function and σ11(n) is the sum of the 11th powers of the divisors of n). The image is the set of pairs (c,d) with c and d congruent mod 691 because of Ramanujan's congruence σ11(n) ≡ τ(n) mod 691. The Hecke algebra of Hecke operators acting on the cusp form Δ is just isomorphic to Z. If we identify it with Z then the Eisenstein ideal is (691).
References
• Mazur, Barry (1977), "Modular curves and the Eisenstein ideal", Publications Mathématiques de l'IHÉS (47): 33–186, ISSN 1618-1913, MR 0488287
• Mazur, Barry; Serre, Jean-Pierre (1976), "Points rationnels des courbes modulaires X0(N) (d'après A. Ogg)", Séminaire Bourbaki (1974/1975), Exp. No. 469, Lecture Notes in Math., vol. 514, Berlin, New York: Springer-Verlag, pp. 238–255, MR 0485882
| Wikipedia |
A paper for G4G9
by Colin Wright
It's surprising, sometimes, how a large collection of apparently unconnected facts can sometimes be woven into whole cloth. Some months ago as I was drifting off to sleep I thought I saw how a bunch of stuff I knew all tied up together into a neat bundle.
Then I woke up, but surprisingly, it looked like it all still worked! Even more surprisingly, when I investigated further, it did, and here are the results.
As implausible as it may seem, we're going to compute the distance to the Moon using a few well-known facts, a few simple observations, a pendulum, and a stopwatch. Pretty much everything here was known to Isaac Newton in the late 1600's, and it's even been suggested that he performed pretty much exactly these calculations.
Maybe, maybe not. Let's just see what we can do with some really elementary reasoning.
We'll warm up with a well-known (in some circles!) question: How far is the horizon?
We'll pretend things are simple. We'll pretend the Earth is a sphere, and suppose we're at the top of a tall mountain, say, 5 metres high. (Yes - I know that's not very tall really, but bear with me ...)
We can create a right-angled triangle with one corner at the centre of the Earth, one corner at our position, and one where our line-of-sight tangents the Earth's surface.
Our good friend Pythagoras now steps up and says that we have that $R^2+H^2=(R+5)^2,$ which can be expanded and simplified and we get $H^2\approx{}10\times{}R.$
By the way, I'll keep the equations in the main text fairly simple and expand on them in the boxes on the right side of the page. Feel free to ignore them if you just want the main ideas, or if you want the challenge of working out the details yourself.
$R^2+H^2=(R+5)^2$
$R^2+H^2=R^2+2\times{}5\times{}R+5^2$
$R^2+H^2=R^2+10\times{}R+25$
$H^2=10\times{}R+25$
$H^2\approx{}10\times{}R$
Now, the original definition of the metre was "One ten-millionth of the distance from the North Pole to the Equator through Paris," which means the circumference of the Earth is 40 million metres, so the radius is roughly 6.4 million metres. Substituting this we get
$H^2\approx{}64\times{}10^6,$ and so $H\approx\sqrt{64\times{}10^6}=8000m$
So from a height of 5 metres, the distance to the horizon is about 8000 metres, or 8km.
Now let's turn it around. Suppose we're at sea level and 8000m from the top of a 5m high mountain. If we fire a projectile line-of-sight at the peak, ignore air resistance, and it gets there in 1 second (unikely, I know).
Since we know that projectiles move in parabolic arcs, this also serves to show that, to a certain level of accuracy, a circle is a parabola. This is because when $x$ is small, $sin(x)=x$ and $cos(x)=1-x^2.$
In one second it will fall about 5m, because acceleration due to gravity is about $10m/s^2,4 so by the time it gets there, it will still be at sea-level. In other words, it will be grazing the Earth's (perfectly spherical) surface.
It's in orbit.
So we've just shown that subject to all our approximations, orbital velocity at grazing altitude is 8 km/s. Quite astonishing how our good friend Pythagoras is, in some sense, "Rocket Science."
(Need to insert diagram of
cannon firing at 8km/s)
So now let's be a little more general. Instead of being exactly 5m high, let's pick an acceleration a and an amount of time t and suppose we are $at^2/2$ high.
Our Pythagorean triangle equation is now
$H^2+R^2=(R+at^2/2)^2$
We simplify, divide through by $t^2,$ throw away the irrelevant small part, and then remember that distance over time is velocity. That means we get
$v^2=aR$
That seems arbitrary, I know, but distance fallen in time t
under acceleration a is given by $d=at^2/2$ so we're at a height such that something will fall that distance in time t.
$H^2+R^2=R^2+2\times{}R\times{}at^2/2+(at^2/2)^2)$
$H^2=2\times{}R\times{}at^2/2$ (ignoring the last term)
$H^2=R\times{}a\times{}t^2$
$(H/t)^2=R\times{}a$
$v^2=R\times{}a$
$a=v^2/R$
Suddenly we have the formula for acceleration in a circle.
Which is nice.
What does this have to do with the Moon?
You may ask why something moving in a circle is accelerating? Well, its speed may not be changing, but its direction is. If left alone its direction wouldn't change, so something must be pushing on it, changing its direction. That comes out in the wash as meaning it's accelerating.
I've also been a little cavalier about ignoring small quantities and so forth. In truth there are some details about limits and such like, and that's where the serious calculus should be done.
If we suppose the Moon to be moving in a circular orbit, and supposing the radius of that orbit to be M, we can now say that its acceleration in orbit is $v^2/M.$ So if only we knew how far away it was, and its velocity, we would know its acceleration.
But if we know its distance then we do know its velocity, because we know it takes 29.53 days from full moon to full moon. Correcting for sidereal time, that means it takes 27.32 days to make a complete circuit of the Earth. Call that time P. Therefore the Moon's velocity in orbit is $(2\pi{}M)/P.$
When the Moon goes around the Earth, the Earth is also going around the Sun. In the 365.25 days it takes for a year the Moon goes around the Earth - apparently - some 365.25/29.53 times. However ...
The orbit of the Earth itself adds another complete rotation. From the point of view of the stars the Moon hasn't gone around 12.37 times, it's gone around 13.37 times, and that means that each orbit, from the point of view of the stars, takes 365.25/13.37 days, or 27.32 days.
Thus the sidereal orbital period of the Moon is 27.32 days.
So we have the formula for acceleration in a circle that needs the distance and velocity, but we now know both of those, so we can say that the Moon's acceleration is this:
$a=(\frac{2\pi}{P})^2M$
Is that of any use?
Well, we know that acceleration due to gravity is what holds the Moon in orbit, so if only we knew how hard the Earth is pulling the Moon, then we would know that.
But we do.
$a=v^2/M$
$\quad=[(2\pi{}M)/P]^2/M$
$\quad=(2\pi)^2M/P^2$
$\quad=(2\pi/P)^2M$
We know that acceleration at the Earth's surface is g, and that it falls off as an inverse square. Hence the acceleration due to gravity at any distance, say M, is given by:
$a=g.(R/M)^2$
where R is the Earth's radius.
But the Earth's radius is $40\times{}10^6/(2\pi),$ so putting it all together we get:
$a=(2\pi/P)^2M$ : acceleration in a circle
$a=g.(R/M)^2$ : acceleration due to gravity
$(2\pi/P)^2M=g.(R/M)^2$
$M^3=g(\frac{RP}{2\pi})^2$
$M^3=g\left(\frac{RP}{2\pi}\right)^2$
And we know everything on the right hand side, except g.
But we can find g with a pendulum and a stopwatch. (I knew you'd be wondering where they came in.)
We know that the time taken for a complete swing of a pendulum is given by the formula:
$T=2\pi\sqrt{L/g}$
Rearranging this we get:
$g=L.(2\pi/T)^2$
We can substitute that into our earlier formula and we get
$M^3=L.(RP/T)^2$
And now we know everything !!
Of course we have to go away and construct a pendulum, and then we have to measure how long it takes to swing. Typically we measure 10 swings, both back and forth, and then divide the total time by 10. We should also do that several times to make sure we get error bars on the result, because each one will vary slightly. There's lots to do here.
So what do we get?
$L=2.0m$ (Length of the pendulum)
$T=2.8s$ (Time for a swing of the pendulum)
$P=27.32\times{}86400s$ (Moon's orbital period)
$R=40\times{}10^6/(2\pi)$ (Radius of the Earth)
$M=383\times{}10^6m$
Which is the right answer.
Of course, the Moon's orbit isn't circular, the Earth isn't of constant radius, nor is it a sphere, and we've assumed that the metre is one ten millionth of the distance from the North Pole to the Equator. But even so, we're not just in the right ball park, we're smack in the middle of the true range.
Not bad for a few sums. | CommonCrawl |
In the previous chapters, we discussed the Bode plots. There, we have two separate plots for both magnitude and phase as the function of frequency. Let us now discuss about polar plots. Polar plot is a plot which can be drawn between magnitude and phase. Here, the magnitudes are represented by normal values only.
The polar form of $G(j\omega)H(j\omega)$ is
$$G(j\omega)H(j\omega)=|G(j\omega)H(j\omega)| \angle G(j\omega)H(j\omega)$$
The Polar plot is a plot, which can be drawn between the magnitude and the phase angle of $G(j\omega)H(j\omega)$ by varying $\omega$ from zero to ∞. The polar graph sheet is shown in the following figure.
This graph sheet consists of concentric circles and radial lines. The concentric circles and the radial lines represent the magnitudes and phase angles respectively. These angles are represented by positive values in anti-clock wise direction. Similarly, we can represent angles with negative values in clockwise direction. For example, the angle 2700 in anti-clock wise direction is equal to the angle −900 in clockwise direction.
Rules for Drawing Polar Plots
Follow these rules for plotting the polar plots.
Substitute, $s = j\omega$ in the open loop transfer function.
Write the expressions for magnitude and the phase of $G(j\omega)H(j\omega)$.
Find the starting magnitude and the phase of $G(j\omega)H(j\omega)$ by substituting $\omega = 0$. So, the polar plot starts with this magnitude and the phase angle.
Find the ending magnitude and the phase of $G(j\omega)H(j\omega)$ by substituting $\omega = \infty$. So, the polar plot ends with this magnitude and the phase angle.
Check whether the polar plot intersects the real axis, by making the imaginary term of $G(j\omega)H(j\omega)$ equal to zero and find the value(s) of $\omega$.
Check whether the polar plot intersects the imaginary axis, by making real term of $G(j\omega)H(j\omega)$ equal to zero and find the value(s) of $\omega$.
For drawing polar plot more clearly, find the magnitude and phase of $G(j\omega)H(j\omega)$ by considering the other value(s) of $\omega$.
Consider the open loop transfer function of a closed loop control system.
$$G(s)H(s)=\frac{5}{s(s+1)(s+2)}$$
Let us draw the polar plot for this control system using the above rules.
Step 1 − Substitute, $s = j\omega$ in the open loop transfer function.
$$G(j\omega)H(j\omega)=\frac{5}{j\omega(j\omega+1)(j\omega+2)}$$
The magnitude of the open loop transfer function is
$$M=\frac{5}{\omega(\sqrt{\omega^2+1})(\sqrt{\omega^2+4})}$$
The phase angle of the open loop transfer function is
$$\phi=-90^0-\tan^{-1}\omega-\tan^{-1}\frac{\omega}{2}$$
Step 2 − The following table shows the magnitude and the phase angle of the open loop transfer function at $\omega = 0$ rad/sec and $\omega = \infty$ rad/sec.
Frequency (rad/sec)
Phase angle(degrees)
0 ∞ -90 or 270
∞ 0 -270 or 90
So, the polar plot starts at (∞,−900) and ends at (0,−2700). The first and the second terms within the brackets indicate the magnitude and phase angle respectively.
Step 3 − Based on the starting and the ending polar co-ordinates, this polar plot will intersect the negative real axis. The phase angle corresponding to the negative real axis is −1800 or 1800. So, by equating the phase angle of the open loop transfer function to either −1800 or 1800, we will get the $\omega$ value as $\sqrt{2}$.
By substituting $\omega = \sqrt{2}$ in the magnitude of the open loop transfer function, we will get $M = 0.83$. Therefore, the polar plot intersects the negative real axis when $\omega = \sqrt{2}$ and the polar coordinate is (0.83,−1800).
So, we can draw the polar plot with the above information on the polar graph sheet. | CommonCrawl |
\begin{definition}[Definition:Integral Element of Algebra/Definition 2]
Let $A$ be a commutative ring with unity.
Let $f : A \to B$ be a commutative $A$-algebra.
Let $b\in B$.
The element $b$ is '''integral''' over $A$ {{iff}} the generated subalgebra $A \sqbrk b$ is a finitely generated module over $A$.
\end{definition} | ProofWiki |
Georgy Egorychev
Georgy Petrovich Egorychev (or Yegorychev) (Георгий Петрович Егорычев, born 1938) is a Russian mathematician, known for the Egorychev method.[1][2]
Biography
He graduated in mathematics from Ural State University and in 1960 became a teacher of mathematics in secondary school.[1]
In 1982 G. P. Egorychev and D. I. Falikman shared the Fulkerson Prize for (independently) proving van der Waerden's conjecture that the matrix with all entries equal has the smallest permanent of any doubly stochastic matrix.[3][4] Egorychev is now a professor in the Department of Mathematical Support of Discrete Devices and Systems, Institute of Mathematics and Fundamental Informatics at Siberian Federal University (Russian abbreviation is SFU, SibFU, or СФУ), founded in 2006.[1]
He was an Invited Speaker of the ICM in 1986 in Berkeley, California. He was awarded a Scholarship of the President of Russia in 1994–1996 and again in 1997–2000.[1]
Research
His research deals with combinatorial analysis, multidimensional complex analysis, and algorithms of integral representation and calculation of combinatorial sums and their applications in various fields of mathematics and science. In particular, his research has applied the Egorychev method to the basis of tensor calculus and to the theory of matrix functions, including permanents and determinants over various algebraic systems. He has published over 80 articles.[1]
Selected publications
• Егорычев Г.П. (2013). Новое семейство полиномиальных тождеств для вычисления детерминантов. Доклады Академии Наук, т. 452, No.1, с. 1–3. (A new family of polynomial identities for the calculation of determinants. Reports of the Academy of Sciences, vol. 452, No. 1, p. 1–3.)
• Egorychev G.P. (2009). Method of coefficients: an algebraic characterization and recent applications. Springer, Adv. in Combin. Math.; Math. Proc. of the Waterloo Workshop in Computer Algebra 2008, devoted to the 70th birthday of G. Egorychev, pp. 1–30.
• Egorychev G.P. and Zima E.V. (2008). Integral representation and Algorithms for closed form summation. Handbook of Algebra, vol. 5, ed. M. Hazewinkel, Elsevier, pp. 459–529.
• Егорычев Г.П. (2008). Дискретная математика. Перманенты. Учебное пособие. Красноярский государственный университет, Красноярск. 272 стр. (Discrete Math. Permanent. Tutorial. Krasnoyarsk State University, Krasnoyarsk. 272 pp.)
References
1. "Егорычев Георгий Петрович". Институт математики и фундаментальной информатики СФУ, sfu-kras.ru.
2. Egorychev, G. P. (1984). Integral representation and the Computation of Combinatorial sums. American Mathematical Society.
3. G. P. Egorychev, "The solution of van der Waerden's problem for permanents," Akademiia Nauk SSSR. Doklady 258: 1041–1044, 1981.
4. D. I. Falikman, "A proof of the van der Waerden conjecture on the permanent of a doubly stochastic matrix," Matematicheskie Zametki 29: 931–938, 1981.
Authority control: Academics
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| Wikipedia |
Filtering procedures for untargeted LC-MS metabolomics data
Courtney Schiffman ORCID: orcid.org/0000-0003-4126-90331,
Lauren Petrick2,6,
Kelsi Perttula4,
Yukiko Yano4,
Henrik Carlsson4,
Todd Whitehead5,6,
Catherine Metayer5,6,
Josie Hayes4,
Stephen Rappaport4,6 &
Sandrine Dudoit1,3
Untargeted metabolomics datasets contain large proportions of uninformative features that can impede subsequent statistical analysis such as biomarker discovery and metabolic pathway analysis. Thus, there is a need for versatile and data-adaptive methods for filtering data prior to investigating the underlying biological phenomena. Here, we propose a data-adaptive pipeline for filtering metabolomics data that are generated by liquid chromatography-mass spectrometry (LC-MS) platforms. Our data-adaptive pipeline includes novel methods for filtering features based on blank samples, proportions of missing values, and estimated intra-class correlation coefficients.
Using metabolomics datasets that were generated in our laboratory from samples of human blood, as well as two public LC-MS datasets, we compared our data-adaptive filtering method with traditional methods that rely on non-method specific thresholds. The data-adaptive approach outperformed traditional approaches in terms of removing noisy features and retaining high quality, biologically informative ones. The R code for running the data-adaptive filtering method is provided at https://github.com/courtneyschiffman/Metabolomics-Filtering.
Our proposed data-adaptive filtering pipeline is intuitive and effectively removes uninformative features from untargeted metabolomics datasets. It is particularly relevant for interrogation of biological phenomena in data derived from complex matrices associated with biospecimens.
Metabolomics represents the small-molecule phenotype that can be objectively and quantitatively measured in biofluids such as blood serum/plasma, urine, saliva, or tissue/cellular extracts [1–4]. Untargeted metabolomics studies allow researchers to characterize the totality of small molecules in a set of biospecimens and thereby discover metabolites that discriminate across phenotypes [1, 3, 5]. Among the techniques employed for untargeted metabolomics, liquid chromatography-high-resolution mass spectrometry (LC-HRMS) has become the analytical tool of choice due to its high sensitivity, simple sample preparation, and broad coverage of small molecules [2, 6]. However, many of the thousands of features detected by untargeted metabolomics are not biologically interesting because they represent background signals from sample processing or multiple signals arising from the same analyte (adducts, isotopes, in-source fragmentation) [7]. Furthermore, feature detection and integration with software such as XCMS [8] is imperfect, in that noise can erroneously be identified as a peak group, the domain of integration can be incorrect, etc. Thus, large metabolomics datasets can contain thousands of falsely identified features or features with imperfect integration (e.g., incorrect integration regions and missing values).
Inadequate feature filtering can affect subsequent statistical analysis. For example, if high quality features are erroneously filtered, they will not be considered as candidate biomarkers in univariate tests of significance for association with biological factors of interest or in metabolic pathway analysis. Furthermore, if one performs univariate tests of significance and ranks features based on p-values, biologically meaningful features could be lost in an abundance of noise without adequate feature filtering. Failure to filter noise could also result in false positives when assessing the significance of metabolic pathways with software such as Mummichog, which relies on sampling features from the entire dataset to create null distributions of pathway statistics [9].
Therefore, untargeted metabolomic data require a set of filtering methods to remove noise prior to investigating the biological phenomena of interest. Data normalization has received a lot of recent attention in untargeted metabolomics [10–14]. Feature filtering, however, remains a fairly automated, indelicate, and brief step in the preprocessing of untargeted metabolomic data. Many studies rely on valuable preprocessing pipelines offered from programs like Metaboanalyst and Workflow4Metabolomics to process their raw data. Such programs have greatly advanced the field of untargeted metabolomics and have improved data pre-processing and analysis and replication of results. However, many users of these programs rely on the provided, default cutoffs for feature filtering, and do not attempt to identify more appropriate, data-specific filtering cutoffs.
For example, MetaboAnalyst allows users to filter features based on mean/median value across samples, as wells as variability across biological samples and quality control (QC) samples. While these are indeed useful filtering metrics, most users do not determine the filtering thresholds appropriate for their specific data. Metaboanalyst suggests removing the lowest k percent of features based on the size of the dataset (e.g., lowest 40% of features for a dataset with more than one thousand features based on mean/median abundance across samples), and a relative standard deviation (RSD, the same as a coefficient of variation or CV) cutoff of 25% for LC-MS data [12]. While these are helpful guidelines for selecting cutoffs, users often fail to investigate whether they are appropriate for their data. Similarly, Workflow4Metabolomics, for good reasons, allows users to filter features based on variability across replicates and sample mean vs. blank mean ratios, but many users continue to rely on default or commonly used cutoffs. Here we offer researchers alternatives to default filtering cutoffs that may be more appropriate for their datasets.
We argue that filtering methods should be data-adaptive. A data-adaptive pipeline is one which tailors filtering to the specific characteristics of a given dataset, rather than using predefined methods. In what follows, we present a series of steps (Fig. 1) representing a data-adaptive pipeline for filtering untargeted metabolomics data prior to discovering metabolites and metabolic pathways of interest. Our data-adaptive filtering approach contains novel methods for removing features based on blank sample abundances, proportions of missing values, and estimated intra-class correlation coefficients (ICC). To create data-dependent thresholds for the above three feature characteristics, we propose visualizing the differences in the characteristics between known high and low quality features. By examining such differences for each dataset, one can minimize noise without compromising the underlying biological signal. Once this is done for several datasets generated from a given laboratory, the determined filtering cutoffs may be appropriate to other similar datasets. Properly filtered untargeted metabolomic data can then be used as input into valuable processing pipelines such as MetaboAnalyst and Workflow4Metabolomics for further preprocessing and data normalization. We compare our data-adaptive filtering method to common filtering methods using two untargeted LC-HRMS datasets that were generated in our laboratory (see also Additional file 1) and two public LC-MS datasets obtained on a different analytical platform. To compare the methods, we identified hundreds of high and low quality peaks in each dataset. We then showed how our data-adaptive pipeline surpasses workflows that use default cutoffs to remove low quality features while retaining high quality features.
Filtering flowchart. Flowchart of a data-adaptive filtering pipeline for untargeted metabolomics data. The pipeline is data-adaptive because the filtering cutoffs used are specific to the filtering needs of the data at hand
Visualizing high and low quality features
When working with untargeted LC-MS data, visualization of extracted ion chromatograms (EIC) of features can be used to optimize peak detection, peak quantification, and biomarker discovery [8, 15, 16]. We propose randomly sampling several hundred EICs after peak detection and quantification to visualize peak morphology and integration. The EICs can then be classified by the user as "high" or "low" quality (see Fig. 2). A high quality peak has good morphology (e.g., is bell-shaped, although this is not a necessary condition), the correct region of integration across all samples, and proper retention time alignment. Such visualization is made easy with plotting functions from peak detection software such as the 'highlightChromPeaks' function within XCMS [8]. In almost all cases, we find the distinction between high and low quality peaks to be clear, but when peaks are ambiguous we make the conservative choice to classify them as low quality. Once features are classified as high or low quality, their characteristics across samples such as average blank and biological sample abundance, percent missing, and ICC can be compared and used to perform feature filtering. While classification of high and low quality peaks is a time intensive step, we have found that visualization and inspection of hundreds of features takes between 1–2 h and greatly improves the ability to uncover biological variability in the data. Moreover, after feature visualization, executing the remaining steps of the filtering pipeline requires no more than 1 h.
Example of a high and low quality peak group. The peak groups depicted here are examples of features with (a) high and (b) low quality peak morphology and integration. We followed the XCMS R package vignette to process the example LC-MS dataset provided in the package [8]
Data-adaptive feature filtering
Example datasets
To help present and visualize our data-adaptive feature filtering methods, we introduce an untargeted LC-HRMS dataset generated in our laboratory on a platform consisting of an Agilent 1100 series LC coupled to an Agilent 6550 QToF mass spectrometer. The dataset contains the metabolomes of 36 serum samples from incident colorectal cancer (CRC) case-control pairs as described in [16, 17]. Over 21,000 features were detected in the 36 serum samples that were analyzed in one batch [16, 17]. We randomly sampled over 900 features from the dataset and classified these as "high" or "low" quality according to their peak morphology and integration quality. To demonstrate the performance of our data-adaptive pipeline, we split the known high and low quality features into a training set (60%) and a test set (40%). Features in the training set were used to visualize appropriate, data-dependent cutoffs, whereas features in the test set were used to evaluate the effectiveness of the selected cutoffs. At each stage of the data-adaptive filtering, we compared our method to more traditional filtering methods by examining the proportions of high and low quality features in the test set that were removed. An application of the data-adaptive pipeline to another untargeted LC-HRMS metabolomics dataset generated in our laboratory can be found in Additional file 1. This additional dataset represents the metabolomes of 4.7-mm punches from archived neonatal blood spots (NBS) of 309 incident case subjects that were obtained for the California Childhood Leukemia Study [15, 18]. For the sake of clarity, we do not include results for this second dataset in the main text, and the results can be found instead in Additional file 1.
We also visualized and classified over 200 features in each of two public LC-MS datasets. One of the public datasets was generated on a platform consisting of an Accela liquid chromatographic system (Thermo Fisher Scientific, Villebon-sur-Yvette, France) coupled to an LTQ-Orbitrap Discovery (Thermo Fisher Scientific, Villebon-sur-Yvette, France). This dataset contains the metabolomes of 189 human urine samples analyzed in negative mode. We took a subset of 45 of the urine samples in the first batch, along with 14 pooled QC samples and 5 blank samples. We processed this dataset using the original xcms functions and parameters used by the authors (W4M00002_Sacurine-comprehensive) [10, 19]. The second public dataset was generated on a platform consisting of an Accela II HPLC system (Thermo Fisher Scientific, Bremen, Germany) coupled to an Exactive Orbitrap mass spectrometer (Thermo Fisher Scientific) [20]. This dataset contains the metabolomes of epithelial cell lines treated with low and high concentrations of chloroacetaldehyde. We used all 27 cell line samples in negative mode treated with low concentrations, as well as 6 pooled QC and 11 blank samples. The original work did not use xcms to process the raw data, so we used the R package IPO to determine the xcms parameters [21].
Filtering features based on blank samples
Blank control samples, which are obtained from the solvents and media used to prepare biological samples, can help to pinpoint background features that contribute to technical variation [2, 3, 10, 22, 23]. A common filtering method is to use a fold-change (biological signal/blank signal) cutoff to remove features that are not sufficiently abundant in biological samples [3, 10, 12]. Rarely does the user examine the data to determine a suitable cutoff. We employ a data-adaptive procedure that takes into account the mean abundance of features in blank and biological samples, the difference between mean abundances in blank and biological samples, and the number of blank samples in which each feature is detected. Our method then assigns cutoffs according to the background noise and average level of abundance. If the dataset contains several batches, filtering is performed batch-wise.
We use a mean-difference plot (MD-plot) to visualize the relationship between feature abundances in the blank and biological samples and assess background noise (Fig. 3). Abundances are log transformed prior to all data pre-processing and visualization. The mean log abundances of each feature across biological and blank samples are then calculated and the average of and difference between these two means are then plotted on the x- and y-axes, respectively. The horizontal zero-difference line (blue lines in Fig. 3) represents the cutoff between features having higher mean abundances in the blank samples and those having higher mean abundances in the biological samples. If there are n blank samples in a batch, then n+1 clusters of features will typically be visually identifiable in the MD-plot, where cluster i=0,…,n is composed of features that are detected in i blank samples. For example, because three blank samples per batch were used in the example dataset, four clusters are identifiable in Fig. 3a. Similar clusters can be identified in all datasets generated from our laboratory (See Additional file 1: Figure S1) and in the public datasets. Filtering is then performed separately for each cluster. If a cluster contains no high quality features, as is often the case with clusters that contain lower abundance features, that cluster can be removed entirely.
MD-plot for the CRC dataset. a Four clusters of features can be identified in the MD-plot, corresponding to features detected in zero, one, two and all of the three blank samples. For this dataset, all high quality features (in red) are in the cluster of features with the highest average abundances that are detected in all three blank samples. b Because all of the high quality features in the training set are detected in all three blank samples, we remove any features detected in less than three blank samples. We filter features detected in all blank samples (shown here) by using the distribution of the known noise below the zero difference line (in blue) to estimate the noise above the zero difference line. We use the absolute value (green lines) of the lower quartile of the negative differences (purple lines) within each partition (20th, 40th, 60th, and 80th percentiles) as filtering cutoffs. Any features above the green lines are retained
The cluster corresponding to features detected in all n blank samples tends to have the highest number of features (around 95% of the total number of features), features with higher average abundances, and the highest number of high quality features. Therefore, careful, data-dependent filtering of this cluster is crucial for the success of subsequent analyses. This cluster also has a non-uniform distribution of mean feature abundances (Fig. 3b). This cluster is thus partitioned based on quantiles (20th, 40th, 60th, and 80th percentiles) of the empirical distribution of mean abundances (x-axis). This ensures that each partition has the same number of features and that the features are uniformly distributed throughout the dynamic range. Within each partition, the empirical distribution of abundances below the zero-difference line is used to estimate the technical variation above that line. The absolute value (green lines in Fig. 3b) of an appropriately identified percentile of the negative mean differences (purple lines in Fig. 3b) is used as a cutoff to remove uninformative features. Users may identify appropriate percentiles of the negative mean differences (purple lines) based on how many high quality features would be removed if the absolute values of those percentiles (green lines) were used as cutoffs. We find percentiles between the lower quartile and median to be appropriate for this cluster of features, because they remove as many low quality features as possible without removing high quality ones. Feature filtering in the remaining clusters can be performed in a similar manner, but without the need to partition features based on average abundance.
Using MD-plots to filter features allows for the simultaneous filtering of features by both the difference in abundance in blank and biological samples (y-axis) and average abundance (x-axis). Average abundance of features across biological samples is a commonly used filtering characteristic, but the filtering is often done using pre-specified cutoffs (e.g., lowest forty percent for datasets with more than one thousand features) (Fig. 4a) [10, 12]. Although we advocate for the filtering approach described previously, if users prefer to filter by just average abundance, the MD-plot allows for easy visualization of a data-dependent cutoff that removes as many low quality features as possible without removing high quality ones. The same can be said for identifying a data-adaptive fold-change (biological signal/blank signal) cutoff, rather than using default cutoffs provided in preprocessing workflows (Fig. 4b) [10]. While we recognize that the background signal can modify the biological signal (e.g., via ion suppression), we do not consider this source of variability.
Two traditional filtering cutoffs. a For datasets with more than one thousand features, MetaboAnalyst recommends removing the lowest 40% of features according to average abundance [12]. The vertical line shown here is an example of such a cutoff, with all features to the left of the vertical line being removed. With this cutoff, many features with higher average abundance in blank samples (below the blue line) are still retained. b Some filtering methods use a pre-specified cutoff for the ratio or difference between average abundance in blank versus biological samples [10]. The black horizontal line shown here represents such a cutoff. All features with a difference in average abundance between biological and blank samples less than two (below the black line) would be removed. If such a traditional filtering approach is to be used, the MD-plot can, at the very least, help users to identify a more appropriate, less arbitrary cutoff for their dataset that strikes a better balance between removing low quality features and retaining high quality ones
Filtering features by percent missing
As mentioned above, low-abundance metabolomic features tend to have a high proportion of undetected values across samples. In addition, when using software such as XCMS for peak detection and quantification, peaks can be missed by the first round of peak detection and integration. Functions such as 'fillChromPeaks' in XCMS are often used to integrate signals for samples for which no chromatographic peak was initially detected [8, 12]. Low quality peaks also tend to have higher proportions of missing values after initial peak identification and integration (Fig. 5 and Additional file 1: Figure S2).
Distributions of percent missing for high and how quality peaks in the training set. Using (a) box plots and (b) density plots of the percent missing values for high and low quality features in the training set, we chose a filtering cutoff of 68% missing, the median value of percent missing for the low quality features in the training set. The plots help to visualize and compare the modes and percentiles of the two distributions (for low and high quality features). The median of the distribution for low quality features is greater than the mode and even 90th percentile of the distribution for high quality features in the training set, making it an appropriate cutoff
To determine the appropriate filtering cutoff for percent missing, we create side-by-side box plots of percent missing values for the high and low quality features classified by visualization of EICs (Fig. 5a). The box plots help to compare the percentiles of the distributions of percent missing values for the high and low quality features, and to select an appropriate cutoff based on these percentiles. Density plots of percent missing values can also be used to visualize the modes and percentiles of the distributions for high and low quality features (Fig. 5b), and cutoffs can be determined based on these distributional properties. For example, appropriate cutoffs would be those that discriminate between the modes of the two distributions, that remove long tails of distributions of low quality features, that correspond to extreme percentiles of one distribution but intermediate percentiles of another, etc. To ensure that we do not remove features that are differentially missing between biological groups of interest (e.g., mostly missing in cases but not controls), we perform a Fisher exact test for each feature, comparing the number of missing and non-missing values against the biological groups of interest. A small p-value for a given feature would indicate that there is a significant dependence between the phenotype of interest and missing values. Features with a percent missing below the identified threshold or with a Fisher exact p-value less than some threshold (we recommend a small value such as the one hundredth percentile of the p-value distribution) are retained. This test of association between the phenotype of interest and missing values can easily be extended to studies where the biological factor of interest is a multilevel categorical variable or a continuous variable by using, for example, a Chi-Square test or a Wilcoxon rank-sum test, respectively.
Filtering features by ICC
High quality and informative features have relatively high variability across subjects (biological samples) and low variability across replicate samples [10, 12] (Fig. 6 and Additional file 1: Figure S3). Typically, the coefficient of variation (CV) is calculated across pooled QC samples for each feature and those with a CV above a predetermined cutoff (e.g., 20–30%) are removed [1, 2, 10, 12, 22]. However, we find that the CV is often a poor predictor of feature quality (Fig. 7 and Additional file 1: Figure S4) because it only assesses variability across technical replicates, without considering biologically meaningful variability across subjects. Instead, we propose examining the proportion of between-subject variation to total variation, otherwise known as the intra-class correlation coefficient (ICC) [24], as a characteristic for filtering. Since the ICC simultaneously considers both technical and biological variability, a large ICC for a given feature indicates that much of the total variation is due to biological variability regardless of the magnitude of the CV.
Distributions of estimated ICC values for high and low quality peaks in the training set for the CRC data. We use (a) box plots and (b) density plots to visualize the modes and percentiles of the distributions of the ICC values for high and low quality features in the training set, and choose a filtering cutoff of 0.43. The high quality features have a mode close to one, and the distribution of the low quality features has a larger left tail, suggesting that a cutoff to the left of the mode of the high quality features would be appropriate
Box plot of CV values in the CRC dataset. A typical CV filtering cutoff is 30% (horizontal black line) [10]. For the CRC dataset, this cutoff does not remove any of the low quality features
Our method for estimation of the ICC employs the following random effects model:
$$\begin{array}{@{}rcl@{}} Y_{i,j} = \mu_{j} + b_{i,j} + \epsilon_{i,j,k}, \end{array} $$
where Yi,j is the abundance of feature j in subject i, μj is the overall mean abundance of feature j, bi,j is a random effect for feature j in subject i, and εi,j,k is a random error for replicate measurement k for feature j in subject i. The ICC is estimated by taking the ratio of the estimated variance of bi,j (between-subject variance) to the estimated variance of bi,j+εi,j,k (total variance). If replicate specimens or LC-MS injections are analyzed for each subject, then application of Eq. 1 is straightforward. However, since metabolomics data are often collected with single measurements of each biospecimen and employ repeated measurements of pooled QC samples to estimate precision, then Eq. 1 can be fit by treating the pooled QC samples as repeated measures from a 'pseudo-subject'. As with percent missing, density plots and box plots of the estimated ICC values for high and low quality features can be compared to determine a data-specific filtering cutoff (Fig. 6). Again, we look to the modes and percentiles of the distributions of the high and low quality features to select an appropriate cutoff that strikes a balance between removing low quality peaks and retaining high quality ones. If multiple batches are involved, the final feature list represents the intersection of features from all batches.
The MD-plot for the CRC dataset shows that all high quality features in the training set are in the same cluster corresponding to features detected in all three blank samples (Fig. 3). Because features in this cluster have higher average abundances and lower percent missing than those in the other three clusters, it is not surprising that this cluster is comprised of many high quality peaks. We therefore remove features in the other three clusters for this dataset, and focus on the data-adaptive filtering of the cluster containing the high quality features (Fig. 3b). We use the lower-quartile of noisy features below the zero difference line to estimate the noise above the zero difference line because this cutoff removes a considerable number of low quality features without removing many of the high quality features (Fig. 3b). This threshold in the training set was then applied to the test set. In fact, this filtering step removed 68% of the 21,000 features, and 41% of the identified low quality features in the test set (Fig. 8). Almost all (95%) of the high quality features in the test set were retained (Fig. 8). A common approach to filtering would be to remove features based on their mean abundance, such as removing the lowest 40% [12]. If this threshold were used to filter the CRC dataset, only 31% of the identified low quality features in the test set would be removed, and many remaining features would have higher average abundances in the blank samples (Fig. 4). Another traditional approach is to arbitrarily select a cutoff (2–5) for the ratio between average biological and blank sample abundances. A similar cutoff applied to the CRC dataset (a cutoff of two for the difference between average log abundances in biological and blank samples) would remove only 36% of the low quality features and 10% of the high quality features in the test set, and would fail to remove many of the low quality features in the clusters that are removed by our data-adaptive filtering. Utilizing blank samples in filtering certainly helps to reduce the number of low quality features. Furthermore, utilizing data visualization helps to ensure that filtering is done appropriately, i.e. that an appropriate balance is struck between removing low quality features and retaining high quality ones.
Percent of high and low quality features in the test set remaining after each filtering step. Each step of the proposed data-adaptive filtering pipeline considerably reduces the number of remaining low quality features in the test set. The desired trade-off between removing low and high quality features can be obtained by adjusting the stringency of the cutoffs at each step. For the CRC dataset, 76% of the low quality features and 28% of the high quality features in the test set were removed. For the urine dataset, 74% of the low quality features and 17% of the high quality features in the test set were removed. For the cell line dataset, 76% of the low quality features and 21% of the high quality features in the test set were removed
The next step in the data-adaptive filtering is to visualize differences in percent missing among the remaining high and low quality features (Fig. 5). Using the information on distribution modes and percentiles provided by box plots and density plots of the data in the training set, we chose to remove features with more than 68% missing values (median of percent missing for low quality features). This threshold was then applied to the test set. When a Fisher exact test was used for each feature to detect significant associations between missing values and the biological factor of interest (CRC), 68 features had p-values less than 0.027 (the one hundredth percentile of the p-values) and were retained regardless of their percent missing values. Combining these two filtering criteria removed 47% of the remaining low quality features and only 11% of the remaining high quality features in the test set (Fig. 8).
We used the 12 QC samples from the CRC dataset to calculate ICC values for each of the remaining features. Using the information provided by the density and box plots, we chose to remove features with ICC values less than 0.43 (the lower hinge of the box plot for low quality features in the training set) (Fig. 6). This threshold was then applied to the test and removed 23% of the remaining low quality features and only 15% of the remaining high quality ones (Fig. 8). Compare this to using CV values to perform filtering, where a typical CV cutoff of 30% or even 20% (Fig. 7) [10] results in no further filtering of the remaining low quality features in the test set. With all steps of the data-adaptive pipeline, the CRC dataset was reduced to just 3,009 features. The data-adaptive filtering removed 76% of features identified as low quality and retained 72% of those identified as high quality in the test set (Fig. 8).
When the data-adaptive pipeline was applied to the publicly available urine dataset [19], 83% of the high quality features in the test set were retained and 74% of the low quality features in the test set were removed. We used a percent missing cutoff of 69% (median of percent missing in the low quality feature training set) and an ICC cutoff of 0.35 (lower whisker of the box plot of ICC values for low quality features in the training set). When the data-adaptive pipeline was applied to the public cell line dataset [20], 79% of the high quality features in the test set were retained and 76% of the low quality features in the test set were removed. We used a percent missing cutoff of 27% (median of percent missing values in the low quality feature training set) and an ICC cutoff of 3.8×10−9 (median of ICC values for low quality features in the training set).
We recognize that our data-adaptive pipeline involves several steps of manual work, such as the visual identification of high and low quality features and the selection of filtering cutoffs. Such methods do present the opportunity for user error, but we argue that such error will not effect the end results of a study. To our knowledge, xcms does not provide peak quality scores for an automated identification of high and low quality peaks. Furthermore, as stated previously, in the vast majority of cases the contrast between images of high and low quality features is striking. Occasional miss-classification of features as high or low quality will not considerably affect the distributions of the feature characteristics used to select the cutoffs, and therefore will not have a large impact on final filtering results. We see the manual selection of filtering cutoffs based on thorough data visualization as an advantage of our proposed pipeline. Researchers may likely have specific requirements for the balance between removing low quality and retaining high quality features depending on their scientific question of interest, their analysis plan or the size of their data. Manual selection of filtering cutoffs, as opposed to using pre-determined cutoffs, allows researchers to adjust the stringency of their feature filtering to fit the needs of their study.
Pipelines such as Workflow4Metabolomics and MetaboAnalyst have been crucial for advancing LC-MS based untargeted metabolomics. The aim of our work is to assist users in applying appropriate filtering methods for their specific data instead of relying on default, non-specific filtering parameters. Given the inherent heterogeneity of metabolomic studies, we argue that feature filtering should be data-adaptive. Here, we provide filtering criteria for each step in a metabolomic pipeline and discuss how to choose cutoffs based on data visualization and distributional properties of high and low quality features. Because of the random noise present in untargeted LC-MS data, we also encourage investigators to visually inspect features of interest for peak morphology and integration prior to including them in analyses of biological variability. We appreciate that our data-adaptive filtering method requires more effort than selecting default or common cutoffs, but argue that the improved data quality will greatly improve statistical analyses performed in applications involving biomarker discovery and pathway characterization leading to more robust and reproducible findings.
CRC:
Coefficient of variation
EIC:
Extracted ion chromatogram
ICC:
Intra-class correlation coefficient
LC-MS:
Liquid chromatography mass spectrometry
MD-plot:
Mean-difference plot
QC:
RSD:
Relative standard deviation
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We gratefully acknowledge the assistance of Agilent Technologies (Santa Clara, CA, USA) for the loan of the high-resolution mass spectrometer that was used to generate the sample datasets.
Research reported in this publication was supported by the National Institute Of Environmental Health Sciences of the National Institutes of Health under Award Numbers P01ES018172, P50ES018172, R01ES009137 and P42ES004705, and the CHEAR Resources Development and Untargeted Cores U2CES026561. This work was also supported by the U.S. Environmental Protection Agency through grants RD83451101 and RD83615901. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health or the USEPA. The funding bodies were not involved in the design of the study or in data collection, interpretation or analysis.
One of the datasets generated and analyzed during the current study, as well as R code for running the filtering pipeline are available at https://github.com/courtneyschiffman/Metabolomics-Filtering.
Division of Biostatistics, UC Berkeley, Berkeley, 94720, USA
Courtney Schiffman
& Sandrine Dudoit
The Senator Frank R. Lautenberg Environmental Health Sciences Laboratory, Department of Environmental Medicine and Public Health, Icahn School of Medicine at Mount Sinai, New York, USA
Lauren Petrick
Department of Statistics, UC Berkeley, Berkeley, 94720, USA
Sandrine Dudoit
Division of Environmental Health Sciences, UC Berkeley, Berkeley, 94720, USA
Kelsi Perttula
, Yukiko Yano
, Henrik Carlsson
, Josie Hayes
& Stephen Rappaport
Division of Epidemiology, UC Berkeley, Berkeley, 94720, USA
Todd Whitehead
& Catherine Metayer
Center for Integrative Research on Childhood Leukemia and the Environment, UC Berkeley, Berkeley, 94720, USA
, Todd Whitehead
, Catherine Metayer
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CS developed the pipeline, performed all filtering, and wrote the manuscript. LP helped to develop the pipeline, collected the NBS dataset, and assisted with writing the manuscript. SD and SR supervised data collection and pipeline development and contributed to writing the manuscript. KP collected the serum sample dataset. TW and CM contributed the NBS samples. JH processed the raw data for the CRC dataset. YY and HC helped with data collection, writing of the manuscript and pipeline development. All authors read and approved the final manuscript.
Correspondence to Courtney Schiffman.
The datasets used for benchmarking were obtained from two previously published studies. The study was approved by the University of California Committee for the Protection of Human Subjects, the California Health and Human Services Agency Committee for the Protection of Human Subjects, and the institutional review boards of all participating hospitals. The biospecimens (neonatal blood specimens) and corresponding data used in this study were obtained from the California Biobank Program (SIS request number(s) 26, Section 6555(b)), 17 CCR. The California Department of Public Health is not responsible for the results or conclusions drawn by the authors of this publication.
Both investigations obtained biospecimens from human subjects with informed written consent under protocols that had been approved by institutional review boards from all participating institutions. Written informed consent to participate in childhood leukemia research was obtained from the parents of all study subjects from the California Childhood Leukemia Study.
Filtering procedures for untargeted LC-MS metabolomics data. This file illustrates the application of the data-adaptive filtering pipeline to an additional dataset generated in our laboratory. (PDF 379 kb)
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TMF:
TMF, 2000, Volume 123, Number 2, Pages 299–307 (Mi tmf604)
This article is cited in 3 scientific papers (total in 3 papers)
The duality of quantum Liouville field theory
L. O'Raifeartaigh, J. M. Pawlowski, V. V. Sreedhar
Dublin Institute for Advanced Studies
Abstract: It has been found empirically that the Virasoro center and three-point functions of quantum Liouville field theory with the potential $\exp(2b\phi(x))$ and the external primary fields $\exp(\alpha\phi(x))$ are invariant with respect to the duality transformations $\hbar\alpha\rightarrow q-\alpha$, where $q=b^{-1}+b$. The steps leading to this result (via the Virasoro algebra and three-point functions) are reviewed in the path-integral formalism. The duality occurs because the quantum relationship between the $\alpha$ and the conformal weights $\Delta_\alpha$ is two-to-one. As a result, the quantum Liouville potential can actually contain two exponentials (with related parameters). In the two-exponential theory, the duality appears naturally, and an important previously conjectured extrapolation can be proved.
DOI: https://doi.org/10.4213/tmf604
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Theoretical and Mathematical Physics, 2000, 123:2, 663–670
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Citation: L. O'Raifeartaigh, J. M. Pawlowski, V. V. Sreedhar, "The duality of quantum Liouville field theory", TMF, 123:2 (2000), 299–307; Theoret. and Math. Phys., 123:2 (2000), 663–670
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\Bibitem{OraPawSre00}
\by L.~O'Raifeartaigh, J.~M.~Pawlowski, V.~V.~Sreedhar
\paper The duality of quantum Liouville field theory
\jour TMF
\vol 123
\issue 2
\mathnet{http://mi.mathnet.ru/tmf604}
\crossref{https://doi.org/10.4213/tmf604}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1794162}
\zmath{https://zbmath.org/?q=an:1031.81632}
\jour Theoret. and Math. Phys.
\crossref{https://doi.org/10.1007/BF02551399}
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This publication is cited in the following articles:
Blaszak M., "From bi-Hamiltonian geometry to separation of variables: Stationary Harry-Dym and the KdV dressing chain", Journal of Nonlinear Mathematical Physics, 9 (2002), 1–13, Suppl. 1
Blaszak, M, "Separability preserving Dirac reductions of Poisson pencils on Riemannian manifolds", Journal of Physics A-Mathematical and General, 36:5 (2003), 1337
Giribet G.E., Lopez-Fogliani D.E., "Remarks on free field realization of SL(2, R)(k)/U(1) x U(1) WZNW model", Journal of High Energy Physics, 2004, no. 6, 026
Full text: 95 | CommonCrawl |
\begin{document}
\begin{abstract} We introduce a new method for computing bounds on the independence number and fractional chromatic number of classes of graphs with local constraints, and apply this method in various scenarios. We establish a formula that generates a general upper bound for the fractional chromatic number of triangle-free graphs of maximum degree~$\Delta \ge 3$. This upper bound matches that deduced from the fractional version of Reed's bound for small values of~$\Delta$, and improves it when~$\Delta\ge 17$, transitioning smoothly to the best possible asymptotic regime, barring a breakthrough in Ramsey theory.
Focusing on smaller values of~$\Delta$, we also demonstrate that every
graph of girth at least~$7$ and maximum degree~$\Delta$ has fractional
chromatic number at most~$1+ \min_{k \in \mathbb{N}} \frac{2\Delta +
2^{k-3}}{k}$. In particular, the fractional chromatic number of a graph
of girth~$7$ and maximum degree~$\Delta$ is
at most~$\frac{2\Delta+9}{5}$ when~$\Delta \in [3,8]$,
at most~$\frac{\Delta+7}{3}$ when~$\Delta \in [8,20]$,
at most~$\frac{2\Delta+23}{7}$ when~$\Delta \in [20,48]$, and
at most~$\frac{\Delta}{4}+5$ when~$\Delta \in [48,112]$. In addition, we also obtain new lower bounds on the independence ratio of graphs of maximum
degree~$\Delta \in \{3,4,5\}$ and girth~$g\in \{6,\dotsc,12\}$,
notably~$1/3$ when~$(\Delta,g)=(4,10)$ and~$2/7$ when~$(\Delta,g)=(5,8)$. \end{abstract}
\title{Fractional chromatic number, maximum degree and girth}
\section{Introduction}
\subsection{A motivation coming from Ramsey theory} Since the seminal result of Ramsey~\cite{Ram30}, the branch of combinatorics now called ``Ramsey Theory'' has known an ever increasing range of interest from the community.
\begin{thm}[Ramsey, 1930] For every integers~$s,t\ge 2$, there exists a minimal integer~$R(s,t)$ such that every graph on~$n\ge R(s,t)$ vertices contains either a clique of size~$s$, or an independent set of size~$t$. \end{thm}
Computing the exact value of the \emph{Ramsey number}~$R(s,t)$ for all possible pairs~$(s,t)$ is a notorious problem, and only little progress has been made since the first quantitative result due to Erd\H os and Szekeres~\cite{ErSz35} that~$R(s,t) \le \binom{s+t-2}{s-1}$ for every~$s,t\ge 2$. Two particular regimes of the Ramsey numbers have attracted a particular focus, namely the \emph{diagonal Ramsey numbers}~$R(s,s)$, and the \emph{off-diagonal Ramsey numbers}~$R(s,t)$ where~$s$ is a fixed constant (typically~$s=3$) and~$t\to \infty$. The result of Erd\H os and Szekeres~\cite{ErSz35} implies that~$\frac{1}{s} \ln R(s,s) \le 4$, while Erd\H os~\cite{Erd47} showed in 1947 through an analysis of random graphs drawn from~$G(n,\frac{1}{2})$ that~$\frac{1}{s} \ln R(s,s) \ge \frac{1}{2}$. Reducing this gap is still an open problem from these days. Our work is mainly motivated by the off-diagonal regime. In this setting, it is relevant to introduce the \emph{maximum degree}~$\Delta(\cdot)$ as an additional parameter, since on one hand~$\Delta(G)$ is an direct lower bound on the independence number of a triangle-free graph~$G$, and on the other hand graphs with smaller maximum degree are easier to properly colour, and hence contain larger independent sets. The best general upper bound on~$R(3,t)$ to this date is due to Shearer~\cite{She83} and can be deduced from the following result.
\begin{thm}[Shearer, 1983] \label{thm:shearer} Every triangle-free graph~$G$ on~$n$ vertices and of average degree~$d$ contains an independent set of size at least~$\frac{d\ln d - d + 1}{(d-1)^2}n$. In particular, this implies that~$R(3,t)\lesssim \frac{t^2}{\ln t}$. \end{thm}
In this paper, we are interested in finding refinements of \Cref{thm:shearer}. We will study the \emph{fractional chromatic number} and \emph{Hall ratio} of graphs of given \emph{girth}. Before going further, we introduce the relevant notions.
\subsection{Definitions and observations}
The fractional chromatic number~$\chi_f(G)$ of a graph~$G$ is a refinement of the chromatic number. It is the fractional solution to a linear program, the integer solution of which is the chromatic number. Let~$G$ be a given graph; we define~$\mathscr{I}(G)$ to be the set of all independent sets of~$G$. We will often restrict to the set~$\mathscr{I}_{\max}(G)$ of all maximal independent sets of~$G$, or to the set~$\mathscr{I}_\alpha(G)$ of all maximum independent sets of~$G$. Then the \emph{fractional chromatic number}~$\chi_f(G)$ of~$G$ is the solution of the following linear program. \begin{align*}
& \quad \min \sum_{I \in \mathscr{I}(G)} w_I\\ \text{such that} & \begin{cases}
w_I \in [0,1] &\quad \text{for each~$I\in\mathscr{I}(G)$}\\
\displaystyle\sum_{\substack{I\in\mathscr{I}(G)\\v\in I}} w_I \ge 1&\quad\text{for each~$v\in V(G)$}. \end{cases} \end{align*}
A \emph{fractional colouring of weight~$w$ of~$G$} is any instance within the domain of the above linear program such that~$\sum w_I = w$. Observe that a fractional colouring of weight~$w$ remains valid after adding vertices to its non-maximal independent sets, and that this does not affect its weight. Therefore, the optimal value of the above linear program remains unchanged if one additionally requires that only maximal independent sets, i.e. those in~$\mathscr{I}_{\max}(G)$, are given a positive weight. A~$k$-colouring of~$G$ is a special case of a fractional colouring of weight~$k$ of~$G$, where~$w_I = 1$ if~$I$ is a monochromatic class of the $k$-colouring, and~$w_I = 0$ otherwise. Note also that if~$W$ is a clique in~$G$, then any fractional colouring of~$G$ is of weight at least~$\abs{W}$, and hence~$\chi_f(G) \ge \omega(G)$. More generally, if~$H$ is a subgraph of~$G$ then~$\chi_f(G)\ge\chi_f(H)$. Another elementary lower bound on the fractional chromatic number comes from the independence number~$\alpha(G)$. Indeed, the total weight induced by an independent set~$I$ of~$G$ on its vertex set is at most~$w_I \alpha(G)$, and so the weight of a fractional colouring of~$G$ is at least~$\abs{V(G)}/\alpha(G)$. These two lower bounds can be combined by the \emph{Hall ratio~$\rho(G)$} of~$G$, which is defined as~$\rho(G) \coloneqq \max\sst{\abs{V(H)}/\alpha(H)}{H \subseteq G}$. The above observations allow us to write the following inequalities: \[ \max\left\{\omega(G),\frac{\abs{V(G)}}{\alpha(G)}\right\} \le \rho(G) \le \chi_f(G) \le \chi(G) \le \Delta(G)+1, \] where~$\Delta(G)$ is the maximum degree of~$G$. If~$G$ is a perfect graph then equality holds between~$\omega(G)$ and~$\chi(G)$, and so in particular between~$\omega(G)$ and~$\chi_f(G)$. Perfect graphs are those graphs that contain no odd hole nor odd antihole, as was conjectured by Berge~\cite{Ber61} in~1961, and proved by Chudnovsky \emph{et al.}~\cite{CRST06} in~2006. On the other side, the characterisation of the graphs~$G$ for which equality holds between~$\chi(G)$ and~$\Delta(G)+1$ was established by Brooks~\cite{Bro41} in~1941, and those graphs are cliques and odd cycles. Since~$\chi_f(C_{2k+1}) = \frac{k}{2k+1}$, the only graphs~$G$ such that~$\chi_f(G)=\Delta(G)+1$ are cliques. Moreover, equality holds between the Hall ratio of~$G$ and its fractional chromatic number for example when~$G$ is vertex transitive.
\subsection{Previous results on the Hall ratio}
Since the Hall ratio is the hereditary version of the inverse of the independence ratio (defined as the independence number divided by the number of vertices), any result on the independence ratio in a hereditary class of graphs can be extended to the Hall ratio. The Hall ratio of a graph has often been studied in relation with the \emph{girth}, which is the length of a smallest cycle in the graph. A first result in this direction is the celebrated introduction of the so-called ``deletion method'' in graph theory by Erd\H{o}s, who used it to demonstrate the existence of graphs with arbitrarily large girth and chromatic number. The latter is actually established by proving that the Hall ratio of the graph is arbitrarily large. As a large girth is not strong enough a requirement to imply a constant upper bound on the chromatic number, a way to pursue this line of research is to express the upper bound in terms of the maximum degree~$\Delta(G)$ of the graph~$G$ considered. This also applies to the Hall ratio.
Letting~$\girth(G)$ stand for the girth of the graph~$G$, that is, the length of a shortest cycle in~$G$ if~$G$ is not a forest and~$+\infty$ otherwise, we define~$\rho(d,g)$ to be the supremum of the Hall ratios over all graphs of maximum degree at most~$d$ and girth at least~$g$. We also let~$\rho(d,\infty)$ be the limit as~$g\to \infty$ of~$\rho(d,g)$ --- note that if we fix~$d$ then~$\rho(d,g)$ is a non-increasing function of~$g$. In symbols,~$\rho(d,g) \coloneqq \sup\sst{\abs{V(G)}/\alpha(G)}{\text{$G$ graph with~$\Delta(G)\le d$ and~$\girth(G)\ge g$}}$, and~$ \rho(d,\infty) \coloneqq \lim_{g \to \infty}\limits \rho(d,g)$.
In~1979, Staton~\cite{Sta79} established that~$\rho(d,4)\le\frac{5d-1}{5}$, in particular implying that~$\rho(3,4)\le\frac{14}{5}$. The two graphs depicted in Figure~\ref{graph1}, called the graphs of Fajtlowicz and of Locke, have fourteen vertices each, girth~$5$, and no independent set of order~$6$. It follows that~$\rho(3,4)=\frac{14}{5}=\rho(3,5)$. It is known that the graphs of Fajtlowicz and of Locke are the only two cubic triangle-free and connected graphs with Hall ratio~$\frac{14}{5}$. This follows from a result of Fraughnaugh and Locke~\cite{FrLo95} for graphs with more than~$14$ vertices completed by an exhaustive computer check on graphs with at most~$14$ vertices performed by Bajnok and Brinkmann~\cite{BaBr98}.
\begin{figure}
\caption{The two cubic triangle-free connected graphs with Hall ratio~$\frac{14}{5}$.}
\label{graph1}
\end{figure}
In~1983, Jones~\cite{Jon84} reached the next step by establishing that~$\rho(4,4) = \frac{13}{4}$. Only one connected graph is known to attain this value: it has~$13$ vertices and is presented in Figure~\ref{graph2}. The value of~$\rho(d,4)$ when~$d \ge 5$ is still unknown; the best general upper bound is due to Shearer~\cite{She91}, and improves his bound stated in \Cref{thm:shearer}. He also provided an upper bound for~$\rho(d,6)$ as a consequence of a stronger result on graphs with no cycle of length~$3$ or~$5$.
\begin{figure}
\caption{The only known 4-regular triangle-free connected graph of Hall ratio~$\frac{13}{4}$.}
\label{graph2}
\end{figure}
\begin{thm}[Shearer, 1991]\label{thm:shearer1} Set~$f(0)\coloneqq 1$, and~$f(d) \coloneqq \frac{1+(d^2-d)f(d-1)}{d^2+1}$ for every integer~$d\ge 2$. If~$G$ is a triangle-free graph on~$n$ vertices with degree sequence~$d_1,\dotsc,d_n$, then~$G$ contains an independent of size~$\sum_{i=1}^n f(d_i)$. \end{thm}
\begin{thm}[Shearer, 1991]\label{thm:shearer2} Set~$f(0)\coloneqq 0$,~$f(1)\coloneqq \frac{4}{7}$, and~$f(d) \coloneqq \frac{1+(d^2-d)f(d-1)}{d^2+1}$ for every integer~$d\ge 2$. If~$G$ is a graph on~$n$ vertices with degree sequence~$d_1,\dotsc,d_n$ and with no~$3$-cycle and no~$5$-cycle, then~$G$ contains an independent of size~$\sum_{i=1}^n f(d_i) - \frac{n_{11}}{7}$, where~$n_{11}$ is the number of pairs of adjacent vertices of degree~$1$ in~$G$. \end{thm}
Theorems~\ref{thm:shearer1} and~\ref{thm:shearer2} allow us to compute upper bounds on~$\rho(d,4)$ and on~$\rho(d,6)$ for small values of~$d$, as indicated in Table~\ref{tab:1}. When~$d \ge 5$, these bounds are the best known ones.
\renewcommand{1}{1.2} \begin{table}[!ht]\centering
\begin{tabular}{@{}crlrl@{}}\toprule ~$d$ & \multicolumn{2}{c}{upper bound on~$\rho(d,4)$} & \multicolumn{2}{c}{upper bound on~$\rho(d,6)$} \\ \midrule $2$ &~$\frac{5}{2}$ &~$=2.5$ &~$\frac{7}{3}$ &$\approx 2.33333$ \\ $3$ &~$\frac{50}{17}$ &$\approx 2.94118$ &~$\frac{14}{5}$ &$= 2.8$ \\ $4$ &~$\frac{425}{127}$ &$\approx 3.34646$ &~$\frac{119}{37}$ &$\approx 3.21622$ \\ $5$ &~$\frac{2210}{593}$ &$\approx 3.72681$ &~$\frac{3094}{859}$ &$\approx 3.60186$ \\ $6$ &~$\frac{8177}{2000}$ &$\approx 4.0885$ &~$\frac{57239}{14432}$ &$\approx 3.96612$ \\ $7$ &~$\frac{408850}{92177}$ &$\approx 4.43549$ &~$\frac{408850}{94769}$ &$\approx 4.31417$ \\ $8$ &~$\frac{13287625}{2785381}$ &$\approx 4.77049$ &~$\frac{13287625}{2857957}$ &$\approx 4.64934$ \\ $9$ &~$\frac{1089585250}{213835057}$ &$\approx 5.09545$ &~$\frac{1089585250}{219060529}$ &$\approx 4.9739$ \\ $10$ &~$\frac{11004811025}{2033474038}$ &$\approx 5.41183$ &~$\frac{11004811025}{2080503286}$ &$\approx 5.28949$\\ \bottomrule \end{tabular}
\caption{Upper bounds on~$\rho(d,4)$ and~$\rho(d,6)$ for~$d\le 10$ derived from Theorems~\ref{thm:shearer1} and~\ref{thm:shearer2}.}\label{tab:1} \end{table} \renewcommand{1}{1}
We are not aware of any non-trivial lower bounds on~$\rho(5,4)$ and~$\rho(6,4)$. Figure~\ref{graph34} show graphs illustrating that~$\rho(5,4) \ge \frac{10}{3} \approx 3.33333$ and~$\rho(6,4) \ge \frac{29}{8}=3.625$. These two graphs are circulant graphs, which are Cayley graphs over~$\mathbb{Z}_n$.
\begin{figure}
\caption{Two possibly extremal regular triangle-free graphs for the Hall ratio.}
\label{graph34}
\end{figure}
The value of~$\rho(3,g)$ has also been studied when~$g$ goes to infinity. Kardoš, Král' and Volec~\cite{KKV11} proved the existence of an integer~$g_0$ such that~$\rho(3,g_0) \le 2.2978$. This result has been improved by Hoppen and Wormald~\cite{HoWo18} to~$\rho(3,g_0) \le 2.2854$. More strongly, these upper bounds hold for the fractional chromatic number of every (sub)cubic graph of girth at least~$g_0$. In the other direction, Bollobás~\cite{Bol81} proved a general lower bound on~$\rho(d,g)$.
\begin{thm}[Bollobás, 1981]\label{thm-bol}
Let~$d\ge3$. Let~$\alpha$ be a real number in~$(0,1)$ such that
\[\alpha(d\ln 2-\ln(\alpha))+(2-\alpha)(d-1)\ln(2-\alpha)+(\alpha-1)d\ln(1-\alpha)<2(d-1)\ln 2.\]
For every integer~$g$, there exists a~$d$-regular graph with
girth at least~$g$ and Hall ratio more than~$2/\alpha$. \end{thm} Theorem~\ref{thm-bol} allows us to compute lower bounds on~$\rho(d, \infty)$ for any value of~$d$, the smaller ones being presented in Table~\ref{tab:2}. All these values can be generalised into a looser but asymptotically equivalent general lower bound of~$d/(2\ln d)$~\cite[Corollary~3]{Bol81}.
\begin{table}[!ht]\centering \begin{tabular}{@{ }cc@{ }} \toprule $d$ & lower bound on~$\rho(d,\infty)$ \\ \midrule $2$ &~$2$ \\ $3$ &~$2.17835$ \\ $4$ &~$2.3775$ \\ $5$ &~$2.57278$ \\ $6$ &~$2.76222$\\ $7$ &~$2.94606$\\ $8$ &~$3.1249$\\ $9$ &~$3.29931$ \\ $10$ &~$3.46981$ \\ $d$ &~$d/(2\ln d)$\\ \bottomrule \end{tabular}
\caption{Lower bounds on~$\rho(d,\infty)$ implied by Theorem~\ref{thm-bol}.}\label{tab:2} \end{table}
\subsection{Previous results on the fractional chromatic number} Recently, Molloy~\cite{Mol19} proved the best known extremal upper bounds for the chromatic number of graphs of given clique number and maximum degree.
\vbox{ \begin{thm}[Molloy, 2019]\label{thm-molloy} Let~$G$ be a graph of maximum degree~$\Delta$. \begin{itemize} \item If~$G$ is triangle-free, then for every~$\varepsilon>0$, there exists~$\Delta_\varepsilon$ such that, assuming that~$\Delta \ge \Delta_\varepsilon$, \[ \chi(G) \le (1+\varepsilon)\frac{\Delta}{\ln \Delta}.\] \item If~$G$ has clique number~$\omega(G) > 2$, then \[ \chi(G) \le 200\omega(G) \frac{\Delta \ln \ln \Delta}{\ln \Delta}. \] \end{itemize} \end{thm} }
The first bound is sharp up to a multiplicative factor in a strong sense, since as shown by Bollobás~\cite[Corollaries~3 and~4]{Bol81} for all integers~$g$ and~$\Delta\ge3$ there exists a graph with maximum degree~$\Delta$, girth at least~$g$ and chromatic number at least~$\frac{\Delta}{2\ln \Delta}$.
\raggedbottom There remains however a substantial range of degrees not concerned by the bound for triangle-free graphs given by~Theorem~\ref{thm-molloy}, namely when~$\Delta$ is smaller than~$\Delta_\varepsilon$, which is larger than~$20^{2/\varepsilon}$. Determining the maximum value of~$\chi_f$ among triangle-free graphs of maximum degree~$3$ has been a long standing open problem, before it was settled~\cite{DSV14}. The authors showed it to be equal to~$\rho(3,4)$, namely~$14/5$. The same question for larger values of the maximum degree is still open; for graphs of maximum degree~$4$ that value lies between~$3.25$ and~$3.5$. To this date, the best known upper bound in terms of clique number and maximum degree (when those two parameters are not too far apart) for the \emph{fractional} chromatic number\footnote{\vtop{For the chromatic number, the reader is referred to a nice theorem of Kostochka~\cite{Kos76}, which for instance implies that every graph with maximum degree at most~$5$ and girth at least~$35$ has chromatic number at most~$4$ (Corollary~2 in \emph{loc.\ cit.}). The general upper bound on the chromatic number guaranteed by Kostochka's theorem is never less than the floor of half the maximum degree plus two.}} is due to Molloy and Reed~\cite[Theorem~21.7, p.~244]{MoRe02}.
\begin{thm}[Molloy and Reed, 2002]\label{thm:Reedsbound} For every graph~$G$, \[ \chi_f(G) \le \frac{\omega(G)+\Delta(G)+1}{2}. \] \end{thm}
If one considers a convex combination of the clique number and the maximum degree plus one for an upper bound on the (fractional) chromatic number of a graph, then because the chromatic number of a graph never exceeds its maximum degree plus one, the aim is to maximise the coefficient in front of the clique number. The convex combination provided by Theorem~\ref{thm:Reedsbound} (which is conjectured to hold, after taking the ceiling, also for the chromatic number), is best possible. Indeed, for every positive integer~$k$ the graph~$G_k \coloneqq C_5 \boxtimes K_k$ is such that~$\omega(G_k)=2k, \Delta(G_k)=3k-1, \chi_f(G_k)=\frac{5k}{2} = \frac{\omega(G_k)+\Delta(G_k)+1}{2}$.
A local form of \Cref{thm:Reedsbound} exists: it was first devised by McDiarmid (unpublished) and appears as an exercise in Molloy and Reed's book~\cite{MoRe02}. A published version is found in the Ph.\@D.\@ thesis of Andrew King~\cite[Theorem~2.10, p.~12]{Kin09}.
\begin{thm}[McDiarmid, unpublished]\label{thm-McDiarmid} Let~$G$ be a graph, and set~$f_G(v) \coloneqq
\frac{\omega_G(v)+\deg_G(v)+1}{2}$ for every~$v\in V(G)$,
where~$\omega_G(v)$ is the order of a largest clique in~$G$ containing~$v$.
Then
\[ \chi_f(G) \le \max\sst{f_G(v)}{v\in V(G)}. \] \end{thm} \noindent In Subsection~\ref{sub-reed}, we slightly strengthen the local property of \Cref{thm-McDiarmid} as a way to illustrate the arguments used later on.
\subsection{Our results}
Our first contribution is to establish a (non-explicit) formula for an upper bound on the fractional chromatic number of triangle-free graphs depending on their maximum degree~$\Delta$. The upper bound which can be effectively computed from this formula improves on the one which can be derived from \Cref{thm-McDiarmid} as soon as~$\Delta \ge 17$.
\begin{table}[!ht]\centering \begin{tabular}{@{ }ccccccc@{ }} \toprule ~$\Delta(G)$&\phantom{abc}&$k$&\phantom{abc}&$\lambda$&\phantom{abc}& upper bound on~$\chi_f(G)$\\ \midrule ~$1 \dotso 16$&\phantom{abc} &~$2$&\phantom{abc} &~$\infty$ &\phantom{abc}&$\frac{\Delta(G)+3}{2}$ \\ ~$17$ &\phantom{abc}&~$3$ &\phantom{abc}&~$3.41613$ &\phantom{abc} &~$9.91552$ \\ ~$18$ &\phantom{abc} &~$3$ &\phantom{abc} &~$3.50195$ &\phantom{abc} &~$10.3075$ \\ ~$19$ &\phantom{abc} &~$3$ &\phantom{abc} &~$3.58603$ &\phantom{abc} &~$10.6981$ \\ ~$20$ &\phantom{abc} &~$3$ &\phantom{abc} &~$3.66847$ &\phantom{abc} &~$11.0875$ \\ ~$50$ &\phantom{abc} &~$4$ &\phantom{abc} &~$2.04455$ &\phantom{abc} &~$ 22.1644$ \\ ~$100$ &\phantom{abc} &~$5$ &\phantom{abc} &~$1.48418~$ &\phantom{abc} &~$ 38.0697$ \\ ~$200$ &\phantom{abc} &~$6$ &\phantom{abc} &~$1.24061~$ &\phantom{abc} &~$66.151~$ \\ ~$500$ &\phantom{abc} &~$8$ &\phantom{abc} &~$0.915598~$ &\phantom{abc} &~$ 139.842$ \\ ~$1000$ &\phantom{abc} &~$10$ &\phantom{abc} &~$0.734978~$ &\phantom{abc} &~$ 249.058$ \\ \bottomrule \end{tabular}
\caption{Upper bounds on~$\chi_f(G)$ when~$G$ is triangle-free.}\label{tab:3} \end{table}
\begin{thm}\label{thm:triangle-free} For every triangle-free graph~$G$ of maximum degree~$\Delta$,
\[ \chi_f(G) \le 1 + \min_{k\in \mathbb{N}} \inf_{\lambda>0} \frac{(1+\lambda)^k+\lambda(1+\lambda)\Delta}{\lambda(1+k\lambda)}.\] \end{thm}
\Cref{thm:triangle-free} lets us derive the upper bounds for the fractional chromatic number of triangle-free graphs presented in Table~\ref{tab:3}. We note that considering the couple~$(k,\lambda) = (2,\infty)$, Theorem~\ref{thm:triangle-free} implies the fractional Reed bound of Theorem~\ref{thm:Reedsbound}. We also obtain the following upper bound as a corollary.
\begin{cor}\label{cor-general} For every triangle-free graph~$G$ of maximum degree~$\Delta\ge 2$, \[ \chi_f(G) \le 1+\pth{1+\frac{2}{\ln \Delta}} \frac{\Delta}{\ln \Delta - 2\ln \ln \Delta}.\] \end{cor} \begin{proof} The function~$f\colon \Delta \mapsto 1+\pth{1+\frac{2}{\ln \Delta}} \frac{\Delta}{\ln \Delta - 2\ln \ln \Delta}$ is bounded from below by~$4$ when~$\Delta >0$, so we may assume that~$\Delta \ge 4$, using the naive upper bound~$\Delta+1$ for smaller values of~$\Delta$.
It remains to apply \Cref{thm:triangle-free} and consider the
couple~$(k,\lambda) = (\lfloor \ln \Delta(\ln \Delta - 2\ln \ln
\Delta)\rfloor, \frac{1}{\ln \Delta})$, by noting that we then have~$k \lambda \le \ln \Delta - 2\ln \ln \Delta \le 1+k\lambda$, and~$(1+\lambda)^k \le e^{k\lambda} \le \Delta/(\ln \Delta)^2$. \end{proof}
We note that \Cref{cor-general} in particular implies the fractional version of the triangle-free bound of Theorem~\ref{thm-molloy}. Therefore Theorem~\ref{thm:triangle-free} yields a smooth transition for the fractional chromatic number of triangle-free graphs from Reed's bound to Molloy's bound, as~$\Delta$ increases.
In order to obtain upper bounds smaller than that of \Cref{thm-McDiarmid} for smaller values of the maximum degree, we need to consider graphs of higher girth. Our second contribution is to establish good upper bounds for the fractional chromatic number of graphs of girth~$7$. Moreover, these bounds have the same local property as those of \Cref{thm-McDiarmid}.
\begin{thm}\label{thm:girth7}
Let~$f(x) \coloneqq 1 + \min_{k \in \mathbb{N}} \frac{2x + 2^{k-3}}{k}$. If~$G$ is a graph of girth at least~$7$, then~$G$ admits a fractional colouring~$c$ such that for every induced subgraph~$H$ of~$G$, the restriction of~$c$
to~$H$ has weight at most~$f\left(\max\sst{\deg_G(v)}{v\in V(H)}\right)$. In particular, \[ \chi_f(G) \le f(\Delta(G)). \] \end{thm}
\begin{rk}
In Theorem~\ref{thm:girth7}, if~$x\ge3$ then
the minimum of the function~$k\to\tfrac{2x+2^{k-3}}{k}$ (over~$\mathbb{N}$)
is attained when~$k$ is the integer closest to~$4 + \log_2 x -
\log_2\log_2 x$. So if~$x\ge3$, then~$f(x) = (2\ln 2 + o(1))
x/ \ln x$, which is off by a multiplicative factor~$2\ln 2$ from the asymptotic value for triangle-free graphs which can be derived from \Cref{thm:triangle-free}. The turning point happens when the maximum degree is approximately~$3\cdot 10^6$.
Theorem~\ref{thm:girth7} lets us derive the upper bounds for the fractional chromatic number of graphs of girth at least~$7$ presented in Table~\ref{tab:4}. \end{rk}
\renewcommand{1}{1.4} \begin{table}[!ht]\centering \begin{tabular}{@{ }ccccc@{ }} \toprule ~$\Delta(G)$&\phantom{abc}&optimal~$k$&\phantom{abc}&upper bound on~$\chi_f(G)$\\ \midrule ~$3 \dotso 8$&\phantom{abc} &~$5$&\phantom{abc} &$\frac{2\Delta(G)+9}{5}$ \\ ~$8 \dotso 20$&\phantom{abc} &~$6$&\phantom{abc} &$\frac{\Delta(G)+7}{3}$ \\ ~$20 \dotso 48$&\phantom{abc} &~$7$&\phantom{abc} &$\frac{2\Delta(G)+23}{7}$ \\ ~$48 \dotso 112$&\phantom{abc} &~$8$&\phantom{abc} &$\frac{\Delta(G)}{4}+5$ \\ ~$112 \dotso 256$&\phantom{abc} &~$9$&\phantom{abc} &$\frac{2\Delta(G)+73}{9}$ \\ \bottomrule \end{tabular}
\caption{Upper bounds on~$\chi_f(G)$ when~$G$ has girth at least~$7$.}\label{tab:4} \end{table} \renewcommand{1}{1}
One could wonder to what extent our results extend to the chromatic number. This is the motivation of a follow-up work involving the authors~\cite{DKPS20+}, where the main theorem is a version of \Cref{algo} holding for DP-colourings, which is used in order to derive bounds on~$\chi_{\rm DP}$ for various classes of sparse graphs. However, the bound of that main theorem is looser than that of \Cref{algo} in several ways, which makes it irrelevant for small degree graphs. Finding a generic method allowing to compute relevant upper bounds for the chromatic number of classes of sparse graphs of small maximum degree is an enticing open problem.
Finally, we provide improved upper bounds on the Hall ratio of graphs of maximum degree in~$\{3,4,5\}$ and girth in~$\{6,\dotsc,12\}$. In particular, these are upper bounds on the fractional chromatic number of vertex-transitive graphs in these classes. These upper bounds are obtained \emph{via} a systematic computer-assisted method.
\begin{thm}\label{thm:ratio}
The values presented in \Cref{table-thm7} are upper bounds
on~$\rho(d,g)$ for~$d\in\{3,4,5\}$ and~$g\in\{6,\dotsc,12\}$. \end{thm}
\begin{table}[!ht] \begin{center}
\begin{tabular}{@{}cccccccc@{}}
\toprule \diagbox{$d$}{$g$} &~$6$ &~$7$ &~$8$ &~$9$ &~$10$ &~$11$ &~$12$ \\ \midrule $3$ &~$30/11 \approx 2.727272$ & \color{gray}~$30/11$ &~$2.625224$ &~$2.604167~$ &~$2.557176$ &~$2.539132$ &~$2.510378$ \\ $4$ &~$41/13 \approx 3.153846$ & \color{gray}~$41/13$ &~$3.038497$ &~$3.017382$ &~$3$ \\ $5$ & \color{gray}~$69/19 \approx 3.631579$ &~$3.6$ &~$ 3.5$ \\ \bottomrule \end{tabular} \end{center} \caption{Upper bounds on~$\rho(d,g)$ for~$d\in\{3,4,5\}$
and~$g\in\{6,\dotsc,12\}$.}\label{table-thm7} \end{table}
The bounds provided by Theorem~\ref{thm:ratio} when~$d\in\{3,4\}$ and~$g=7$ are the same as those for~$g=6$. It seems that this could be a general phenomenon. We therefore offer the following conjecture, implicitly revealing that we expect our method to produce an upper bound of~$2.5$ on~$\rho(3,13)$.
\begin{conj}
The values presented in \Cref{table-conj} are upper bounds on~$\rho(d,g)$ for~$d\in\{3,4,5\}$ and~$g\in\{6,8,10,12\}$. \end{conj}
\begin{table}[!th]
\begin{center}
\begin{tabular}{@{}ccccc@{}} \toprule \diagbox{$d$}{$g$} &~$6$ &~$8$ &~$10$ &~$12$ \\ \midrule $3$ & &~$2.604167~$ &~$2.539132$ &~$2.5$ \\ $4$ & &~$3.017382$ &~$3$ \\ $5$ &~$3.6$ &~$ 3.5$ \\ \bottomrule \end{tabular} \end{center} \caption{Conjectured upper bounds on~$\rho(d,g)$
for~$d\in\{3,4,5\}$ and~$g\in\{6,8,10,12\}$.}\label{table-conj} \end{table}
\subsection{Notations} We introduce some notations before establishing a few technical lemmas, from which we will prove Theorems~\ref{thm:girth7} and~\ref{thm:ratio}. If~$v$ is a vertex of a graph~$G$ and~$r$ a non-negative integer, then~$N^r_G(v)$ is the set of all vertices of~$G$ at distance exactly~$r$ from~$v$ in~$G$, while~$N^r_G[v]$ is~$\bigcup_{j=0}^{r}N^j_G(v)$. If~$u$ is also a vertex of~$G$, we write~$\di_G(u,v)$ for the distance in~$G$ between~$u$ and~$v$. Further, if~$J$ is a subset of vertices of~$G$, then we write~$N_G(J)$ for the set of vertices that are not in~$J$ and have a neighbour in~$J$, while~$N_G[J]$ is~$N_G(J)\cup J$. We will omit the graph subscript when there is no ambiguity, and sometimes write~$N_X(v)$ instead of~$N(v)\cap X$, for any subset of vertices~$X\subseteq V(G)$. The set of all independent sets of~$G$ is~$\mathscr{I}(G)$, while~$\cI_{\max}(G)$ is the set of all maximal independent sets of~$G$ and~$\cI_{\alpha}(G)$ is the set of all maximum independent sets of~$G$. If~$w$ is a mapping from~$\mathscr{I}(G)$ to~$\mathbb{R}$ then for every vertex~$v\in V(G)$ we set
\[
w[v]\coloneqq\sum_{\substack{I\in\mathscr{I}(G)\\v\in I}}w(I).
\] Further, if~$\mathscr{I}$ is a collection of independent sets of~$G$, then $w(\mathscr{I})\coloneqq\sum_{I\in \mathscr{I}}w(I)$. If~$I$ is an independent set of a graph~$G$, a vertex~$v$ is \emph{covered} by~$I$ if~$v$ belongs to~$I$ or has a neighbour in~$I$. A vertex that is not covered by~$I$ is \emph{uncovered} (by~$I$). If~$G$ is a graph rooted at a vertex~$v$, then for every positive integer~$d$, the set of all vertices at distance~$d$ from~$v$ in~$G$ is a \emph{layer} of~$G$.
\section{Technical lemmas} In this section we present the tools needed for the proofs of the main theorems.
\subsection{Greedy fractional colouring algorithm} Our results on fractional colouring are obtained using a greedy algorithm analysed in a recent work involving the first author~\cite{DJKP20}. This algorithm is a generalisation of an algorithm first described in the book of Molloy and Reed~\cite[p.~245]{MoRe02} for the uniform distribution over maximum independent sets. The setting here is, for each induced subgraph~$H$ of the graph we wish to fractionally colour, a probability distribution over the independent sets of~$H$.
\begin{lemma}[Davies \emph{et al.}, 2018]\label{algo} Let~$G$ be a graph given with parameters~$\alpha_v, \beta_v$ for every vertex~$v\in V(G)$. For every induced subgraph~$H$ of~$G$, let~$\mathbf{I}_H$ be a random independent set of~$H$ drawn according to a given probability distribution, and assume that \[ \alpha_v \pr{v\in \mathbf{I}_H} + \beta_v \esp{\abs{N(v)\cap \mathbf{I}_H}} \geq 1,\] for every vertex~$v\in V(H)$. Then the \emph{greedy fractional algorithm} defined by \Cref{alg-greedy} produces a fractional colouring~$w$ of~$G$ such that the restriction of~$w$ to any subgraph~$H$ of~$G$ is a fractional colouring of~$H$ of weight at most~$\max_{v\in V(H)} \limits \alpha_v + \beta_v \deg_G(v)$. In particular, \[\chi_f(G)\le \max_{v\in V(G)} \limits \alpha_v + \beta_v \deg_G(v).\] \end{lemma}
\begin{algorithm}\caption{The greedy fractional algorithm}\label{alg-greedy}
\begin{algorithmic}[0]
\For{$I\in \mathscr{I}(G)$}
\State~$w(I)\gets0$
\EndFor
\State~$H\gets G$
\While{$\abs{V(H)}>0$}
\State~$\displaystyle \iota \gets \min\left\{ \min_{v\in V(H)} \frac{1-w[v]}{\pr{v\in \mathbf{I}_H}}, \min_{v\in V(H)} \Big(\alpha_v+\beta_v\deg_G(v)\Big)-w\big(\mathscr{I}(G)\big)\right\}$
\For{$I\in\mathscr{I}(H)$}
\State~$w(I)\gets w(I)+\pr{\mathbf{I}_{H}=I} \iota$
\EndFor
\State~$H \gets H - \sst{v\in V(H)}{w[v]=1}$
\EndWhile
\end{algorithmic} \end{algorithm}
We note that in Lemma~\ref{algo}, although there is one probability distribution on each induced subgraph, the reals~$\alpha_v$ and~$\beta_v$ associated with each vertex are fixed once and for all, which somewhat ties together the different probability distributions involved.
\subsection{Hard-core model} In the setting of \Cref{algo}, we need a probability distribution over the independent sets of a given graph~$H$. For instance, Molloy and Reed used the uniform distribution over the maximum independent sets of~$H$, and obtained the fractional Reed bound as a result (see Theorem~\ref{thm:Reedsbound}). As we will show in \Cref{sub-reed}, this bound is best possible when restricting to the maximum independent sets, even for trees. Therefore, we need to include non-maximum independent sets with non-zero probability in order to hope for improved bounds. Moreover, in order to perform a local analysis of the possible random outcomes, we need our probability distribution to have good relative independence between the random outcomes in a local part of the graph, and the ones outside this part.
The probability distribution that we are going to use as a setting of \Cref{algo} is the hard-core distribution over the independent sets of a graph, which has the Spatial Markov Property. Given a family~$\mathscr{I}$ of independent sets of a graph~$H$, and a positive real~$\lambda$, a random independent set~$\mathbf{I}$ drawn according to the hard-core distribution at fugacity~$\lambda$ over~$\mathscr{I}$ is such that \[ \pr{\mathbf{I}=I} = \frac{\lambda^{\abs{I}}}{Z_{\mathscr{I}}(\lambda)},\] for every~$I\in \mathscr{I}$, where~$Z_\mathscr{I}(\lambda)=\sum_{J\in \mathscr{I}}\limits \lambda^{\abs{J}}$ is the \emph{partition function} associated with~$\mathbf{I}$.
Along this work, we consider two possible families~$\mathscr{I}$ of independent sets of~$H$, the first one being the whole set~$\mathscr{I}(H)$ of independent sets of~$H$. Note that when~$\mathscr{I}=\mathscr{I}(H)$, and~$\lambda\to \infty$, the hard-core distribution converges towards the uniform distribution over the maximum independent sets of~$H$.
\begin{lemma}[Spatial Markov Property]\label{lem:markov} Given a graph~$H$, and a real~$\lambda>0$, let~$\mathbf{I}$ be drawn according to the hard-core distribution at fugacity~$\lambda$ over the independent sets~$\mathscr{I}(H)$ of~$H$. Let~$X\subseteq V(H)$ be any given subset of vertices, and~$J$ any possible outcome of~$\mathbf{I}\setminus X$. Then, conditioned on the fact that~$\mathbf{I} \setminus X = J$, the random independent set~$\mathbf{I}\cap X$ follows the hard-core distribution at fugacity~$\lambda$ over the independent sets of~$H[X\setminus N(J)]$. \end{lemma}
The proof of this result is standard and follows from a simple consideration of the marginal probabilities. It remains valid when we fix~$\lambda=\infty$, i.e. the uniform distribution over the maximum independent sets of any graph~$H$ has the Spatial Markov Property. Things are more complicated with our second choice for~$\mathscr{I}$, that is the set~$\cI_{\max}(H)$ of maximal independent sets of~$H$. Indeed, in this setting, one has to make sure that the local outcome of the independent set is compatible with the fact that the global outcome of the independent set is maximal, i.e. there remains no uncovered vertices in~$H$. This adds a new level of dependency, and we need the extra assumption~\eqref{eq:markov} introduced below in Lemma~\ref{lem:markov-max}, to be able to handle it. For two disjoint subsets of vertices~$X$ and~$U$ of a graph~$G$, we define~$P^2_X(U)$ to be the set of vertices~$x\in X$ such that there exists a path~$ux'x$ of length~$2$ with~$u\in U$ and~$x'\in X$. In symbols, $P^2_X(U)=N\big(N(U)\cap X\big) \cap X$.
\begin{lemma}[Spatial Markov Property for maximal independent sets]\label{lem:markov-max} Given a graph~$H$, and a real~$\lambda>0$, let~$\mathbf{I}$ be drawn according to the
hard-core distribution at fugacity~$\lambda$ over the maximal independent
sets~$\cI_{\max}(H)$ of~$H$. Let~$X\subseteq V(H)$ be any given subset of
vertices,~$J$ any possible outcome of~$\mathbf{I}\setminus X$, and~$U\coloneqq
(V(H)\setminus X) \setminus N[J]$ the set of vertices outside of~$X$
that are uncovered (by~$J$). Moreover, we assume that \begin{equation}\label{eq:markov}
\abs{N(v) \cap X} \le 1\quad\text{for any vertex~$v\in V(H)\setminus X$.} \tag{$\star$} \end{equation} Then, conditioned on the fact that~$\mathbf{I} \setminus X
= J$, the random independent set~$\mathbf{I}\cap X$ follows the hard-core distribution
at fugacity~$\lambda$ over the maximal independent sets of
\[H[X\setminus (N(J)\cup P^2_X(U)].\] \end{lemma}
\begin{proof}
Set~$W \coloneqq X \setminus (N(J)\cup P^2_X(U))$. First let~$I_X$ be any possible realisation of~$\mathbf{I} \cap X$, conditioned on the fact
that~$\mathbf{I}\setminus X = J$.
We prove that~$I_X \in \cI_{\max}(H[W])$. To this end, we begin by showing that~$N_X(U) \subseteq I_X$. By the definitions of~$J$ and~$U$,
every vertex in~$U$ must be adjacent to a vertex in~$X$, and hence by~\eqref{eq:markov} for each~$u\in U$
there exists a unique vertex~$v_u$ in~$X$ that is adjacent to~$u$. It follows that~$N_X(U)$
is contained in~$I_X$. This in particular implies that no vertex in~$P^2_X(U)$
can belong to~$I_X$, and hence~$I_X\subseteq W$. We note for later that we just
established that~$N_X(U)$ is a set of isolated vertices of~$H[W]$ (that is, these
vertices belong to~$W$ and have no neighbour in~$H[W]$).
Next we observe that~$I_X$ is maximal in~$H[W]$. Indeed, let~$w\in W\setminus I_X$. Because~$I_X\cup J$
is a maximal independent set of~$H$, there exists~$v\in I_X\cup J$ that is adjacent to~$w$ in~$H$.
Since~$W\subseteq X\setminus N(J)$ by definition, we deduce that~$v\in I_X$ and hence~$I_X$ is maximal
in~$H[W]$.
Second, given any set~$I_X \in \cI_{\max}(H[W])$, the set~$I_X\cup J$ is a valid realisation of~$\mathbf{I}$. Indeed,~$I_X$ and~$J$ are independent sets, and so is their union
as~$I_X \cap N(J) = \varnothing$. To prove that~$I_X\cup J$ is maximal in~$H$, it
suffices to show that every vertex~$x$ in~$U\cup (X\setminus W)$ has a neighbour
in~$I_X$. As reported earlier,~$N_X(U)$ is contained in~$W$ and forms a set
of isolated vertices in~$H[W]$. Therefore,~$N_X(U)$ is contained in every maximal independent
of~$H[W]$, and hence in~$I_X$. Since every vertex in~$U$ has a neighbour in~$X$,
it therefore only remains to deal with the case where~$x\in X\setminus W$.
Then~$x\in N(J)\cup P^2_X(U)$, and hence~$x$ has a neighbour in~$J\cup N_X(U)$, which
is contained in~$I_X\cup J$.
In conclusion, the set of realisations of~$\mathbf{I} \cap X$ is exactly~$\cI_{\max}(H[W])$,
and each such realisation~$I_X$ has a probability proportional to ~$\lambda^{\abs{I_X}+\abs{J}}$, and hence proportional to ~$\lambda^{\abs{I_X}}$ since~$J$ is fixed. This finishes the proof. \end{proof}
\subsection{Independence ratio} We state two lemmas which can be proved in similar ways. We only present the proof of the second one, the argument for the first one being very close but a little simpler.
\begin{lemma}\label{ratio-vertex}
Let~$r$ be a positive integer and~$G$ be a~$d$-regular graph. Let~$\alpha_0,\dotsc,\alpha_r$ be real
numbers such that~$\sum_{i=1}^{r}\alpha_i(d-1)^{i-1}\ge0$. Assume that there exists a probability
distribution~$p$ on~$\mathscr{I}_{\max}(G)$ such that \begin{equation} \forall v\in V(G),\quad \sum_{i=0}^r \alpha_i\esp{\mathbf{X}_i(v)} \geq 1, \end{equation} where~$\mathbf{X}_i(v)$ is the random variable counting the number of paths of
length~$i$ between~$v$ and a vertex belonging to a random independent set~$\mathbf{I}$
chosen following~$p$. Then \begin{equation}
\frac{\abs{V(G)}}{\alpha(G)} \le \alpha_0 + \sum_{i=1}^{r} \alpha_i d(d-1)^{i-1}. \end{equation} \end{lemma}
\begin{lemma}\label{ratio-edge}
Let~$r$ be a positive integer and~$G$ be a~$d$-regular graph. Let~$\alpha_0,\dotsc,\alpha_r$ be real
numbers such that~$\sum_{i=0}^{r}\alpha_i(d-1)^{i}\ge0$. Assume that there exists a probability
distribution~$p$ on~$\mathscr{I}_{\max}(G)$ such that \begin{equation} \label{edge-inequality} \forall e\in E(G),\quad \sum_{i=0}^r \alpha_i\esp{\mathbf{X}_i(e)} \geq 1, \end{equation} where~$\mathbf{X}_i(e)$ is the random variable counting the number of paths of length~$i+1$ starting with~$e$ and ending at a vertex belonging to a random independent set~$\mathbf{I}$ chosen following~$p$. Then \begin{equation}
\frac{\abs{V(G)}}{\alpha(G)} \le \sum_{i=0}^{r} 2\alpha_i (d-1)^i. \end{equation} \end{lemma}
\begin{proof}
Given an integer~$i\in\{0,\dotsc,r\}$ and an edge~$e$ of~$G$, the contribution of an arbitrary vertex~$v\in\mathbf{I}$
to~$\mathbf{X}_i(e)$ is the number of paths of length~$i+1$ starting at~$v$
and ending with~$e$. It follows that the total contribution of any
vertex~$v\in\mathbf{I}$ to~$\sum_{e\in E(G)}\mathbf{X}_i(e)$ is the number of
paths of~$G$ with length~$i+1$ that start at~$v$, which is at most~$d(d-1)^i$
since~$G$ is a~$d$-regular graph. Consequently,
\[ \esp{\sum_{e\in E(G)} \mathbf{X}_i(e)} \le \sum_{v\in V(G)} \pr{v\in \mathbf{I}} d(d-1)^i. \] We now sum~\eqref{edge-inequality} over all edges of~$G$.
\begin{align*}
\sum_{e\in E(G)} \sum_{i=0}^r \alpha_i \esp{\mathbf{X}_i(e)} &\ge \abs{E(G)} = \frac{d\cdot\abs{V(G)}}{2} \\
\sum_{i=0}^r \alpha_i \sum_{e \in E(G)} \esp{\mathbf{X}_i(e)} &\ge \frac{d\cdot\abs{V(G)}}{2} \\
\sum_{i=0}^r \alpha_i \sum_{v\in V(G)} \pr{v\in \mathbf{I}} d(d-1)^i &\ge \frac{d\cdot\abs{V(G)}}{2} \\
\sum_{i=0}^r 2 \alpha_i \esp{\abs{\mathbf{I}}} (d-1)^i &\ge \abs{V(G)} \\
\sum_{i=0}^{r} 2\alpha_i (d-1)^i &\ge \frac{\abs{V(G)}}{\alpha(G)}. \qedhere \end{align*} \end{proof}
The next lemma allows us to generalise Lemmas~\ref{ratio-vertex} and~\ref{ratio-edge} to non-regular graphs. To this end, we use a standard argument coupled with the existence of specific vertex-transitive type-$1$ regular graphs with any given degree and girth. These are provided by a construction of Exoo and Jajcay~\cite{ExJa08} in the proof of their Theorem~19, which is a direct generalisation of a construction for cubic graphs designed by Biggs~\cite[Theorem~6.2]{Big98}. We slightly reformulate their theorem, the mentioned edge-colouring and transitivity property following simply from the fact that the graph constructed is a Cayley graph obtained from a generating set consisting only of involutions. Given a graph~$G$ endowed with an edge-colouring~$c$, an automorphism~$f$ of~$G$ is \emph{$c$-preserving} if~$c(\{f(u),f(v)\}) = c(u,v)$ for each edge~$\{u,v\}$ of~$G$. The graph~$G$ is \emph{$c$-transitive} if for every pair~$(u,v)$ of vertices of~$G$ there exists a~$c$-preserving automorphism~$f$ of~$G$ such that~$f(u)=v$.
\begin{thm}[Exoo \& Jajcay, 2013]\label{thm-exja} For every integers~$d$ and~$g$ both at least~$3$, there exists a ~$d$-regular graph~$H$ with girth at least~$g$ along with a proper edge-colouring~$c$
using~$d$ colours such that~$H$ is~$c$-transitive. \end{thm}
\begin{lemma}\label{regular} From any graph~$G$ of maximum degree~$d$ and girth~$g$, we can construct a $d$-regular graph~$\varphi(G)$ of girth~$g$ whose vertex set can be partitioned into induced copies of~$G$, and such that any vertex~$v\in G$ can be sent to any of its copies through an automorphism. \end{lemma}
\begin{proof} Set~$k \coloneqq \sum_{v\in G} (d-\deg(v))$. Let~$G'$ be the supergraph of~$G$ obtained by
adding~$k$ vertices~$v'_1,\dotsc,v'_k$ each of degree~$1$, such that all
other vertices have degree~$d$. We let~$e'_i$ be the edge of~$G'$ incident to~$v'_i$,
for each~$i\in\{1,\dotsc,k\}$. By Theorem~\ref{thm-exja}, there exists a ~$k$-regular graph~$H$ of girth at least~$g$ together with a proper edge-colouring~$c$
using~$k$ colours, such that~$H$ is~$c$-transitive. Let~$n(H)$ be the number of
vertices of~$H$ and write~$V(H)=\{1,\dotsc,n(H)\}$.
We construct~$\varphi(G)$ by starting from the disjoint union of~$n(H)$
copies~$G_1,\dotsc,G_{n(H)}$ of~$G$. For each edge~$e=\{i,j\}\in E(H)$, letting~$u_e$
be the vertex of~$G$ incident to the edge~$e'_{c(e)}$ in~$G'$, we add an edge between
the copy of~$u_e$ in~$G_i$ and that in~$G_j$.
Any cycle in~$\varphi(G)$ either is a cycle in~$G$, and hence has length at least~$g$, or
contains all the edges of a cycle in~$H$, and hence has length at least~$g$. It follows
that~$\varphi(G)$ has girth~$g$.
The last statement follows directly from the fact that~$H$ is~$c$-transitive. \end{proof}
\begin{cor}
Let~$d$ and~$g$ be integers greater than two. If there exists a constant~$B=B(d,g)$
such that every~$d$-regular graph~$H$ with girth~$g$ has independence ratio
at least~$B$, then every graph~$G$ with maximum degree~$d$ and
girth~$g$ also has independence ratio at least~$B$. In particular, if \Cref{ratio-vertex} or \Cref{ratio-edge} can be applied to
the class of~$d$-regular graphs of girth~$g$, then the conclusion also holds
for the class of graphs with maximum degree~$d$ and
girth~$g$, that is, for~$\rho(d,g)$. \end{cor}
\begin{proof}
Let~$G$ be a graph with maximum degree~$d$ and girth~$g$ on~$n$ vertices. Let~$\varphi(G)$
be the graph provided by \Cref{regular}.
In particular,~$\abs{V(\varphi(G))}=kn$ where~$k$ is the number of induced copies of~$G$ partitioning~$V(\varphi(G))$.
By assumptions,~$\varphi(G)$ contains an independent set~$I$ of order at least~$B\cdot kn$. Letting~$I_i$ be the
set of vertices of the~$i$-th copy of~$G$ contained in~$I$, by the pigeon-hole principle there exists~$i\in\{1,\dotsc,k\}$
such that~$\abs{I_i}\ge B\cdot n$, and hence~$G$ has independence ratio at least~$B$. \end{proof}
\section{Fractional colourings}
\subsection{A local version of Reed's bound}\label{sub-reed} For the sake of illustration, we begin by showing how \Cref{algo} can be used to prove \Cref{thm:Reedsbound}. We actually establish a slight strengthening of \Cref{thm-McDiarmid}, the local form of \Cref{thm:Reedsbound}. The argument relies on the relation~\eqref{reedinequality} below~\cite[Lemma~2.11]{Kin09}, which is a local version of the relation~(21.10) appearing in Molloy and Reed's book~\cite{MoRe02}. The short argument, however, stays the same and we provide it here only for explanatory purposes, since it is the inspiration for the argument used in the proof of \Cref{thm:girth7}.
\begin{prop}\label{prop-McDiarmidlike} Let~$G$ be a graph, and set~$f_G(v) \coloneqq
\frac{\omega_G(v)+\deg_G(v)+1}{2}$ for every~$v\in V(G)$,
where~$\omega_G(v)$ is the order of a largest clique in~$G$ containing~$v$.
Then~$G$ admits a fractional colouring~$c$ such that the restriction of~$c$
to any induced subgraph~$H$ of~$G$ has weight at most~$\max_{v\in V(H)}\limits f_G(v)$.
In particular,
\[ \chi_f(G) \le \max\sst{f_G(v)}{v\in V(G)}. \] \end{prop}
\begin{proof}
We demonstrate the statement by applying \Cref{algo}. To this end, we use
the uniform distribution on maximum independent sets, which corresponds to the
hard-core distribution at fugacity~$\lambda=\infty$. \begin{as}\label{reedinequality} For every induced subgraph~$H$ of~$G$, let~$\mathbf{I}_H$ be a maximum independent set of~$H$,
drawn uniformly at random. Then for every vertex~$v\in V(H)$, \[\frac{\omega(v)+1}{2} \pr{v\in \mathbf{I}_H} + \frac{1}{2} \esp{\abs{N(v)\cap \mathbf{I}_H}} \ge 1.\] \end{as} \noindent The conclusion follows by applying \Cref{algo} with ~$\alpha_v=\frac{\omega(v)+1}{2}$ and~$\beta_v=\frac12$ for every~$v\in V(G)$.
It remains to establish~\eqref{reedinequality}. We let~$J$ be any possible outcome
of~$\mathbf{I}_H \setminus N[v]$, and~$W = N[v]\setminus N(J)$. We condition on the random
event~$E_J$ that~$\mathbf{I}_H \setminus N[v]=J$, and the Spatial Markov Property of the
uniform distribution over the maximum independent sets of~$H$ ensures that~$\mathbf{I}_H \cap W$ is
a uniform random maximum independent set of~$H[W]$. There are two cases.
\begin{enumerate}[label=(\roman*)]
\item If~$W$ is a clique of size~$k \le \omega(v)$, then exactly one vertex from~$W$ belongs to~$\mathbf{I}_H$, and every vertex in~$W$ has equal probability~$1/k$ to be in~$\mathbf{I}_H$. So, in this case,
\[ \frac{\omega(v)+1}{2} \pr{v\in \mathbf{I}_H \mid E_J} + \frac{1}{2}
\esp{\abs{N(v)\cap \mathbf{I}_H} \mid E_J} = \frac{\omega(v)+1}{2k} +
\frac{k-1}{2k} \ge 1. \] \item If~$W$ is not a clique, then~$\abs{W\setminus \{v\} \cap \mathbf{I}_H} \ge 2$ and~$v\notin \mathbf{I}_H$, since~$\mathbf{I}_H$ is a maximum independent set.
So, in this case,
\[\frac{\omega(v)+1}{2} \pr{v\in \mathbf{I}_H \mid E_J} + \frac{1}{2} \esp{\abs{N(v)\cap \mathbf{I}_H} \mid E_J} \ge \frac{1}{2} \times 2 = 1.
\] \end{enumerate}
The validity of~\eqref{reedinequality} follows by summing over all possible realisations~$J$ of~$\mathbf{I}_H \setminus N[v]$. \end{proof}
We finish by noting that the bound provided by \Cref{thm-McDiarmid} is best possible over the class of unicyclic triangle-free graphs if one uses the fractional greedy colouring of \Cref{algo} together with any probability distribution on the \emph{maximum} independent sets of the graph.
\begin{lemma}\label{lem-best}
If the probability distribution used in \Cref{algo} gives positive
probability only to maximum independent sets, then the greedy fractional
colouring algorithm can return a fractional colouring of weight up
to~$\frac{d+3}{2}$ in general for graphs of degree~$d$, should they be
acyclic when~$d$ is odd, or have a unique cycle (of length~$5$) when~$d$ is
even. \end{lemma}
\begin{proof} We prove the statement by induction on the positive integer~$d$. \begin{itemize} \item If~$d=1$, then let~$G_1$ consist only of an edge. The algorithm returns a fractional colouring of~$G_1$ of weight~$2$. \item If~$d=2$, then let~$G_2$ be the cycle of length~$5$. The algorithm returns a fractional colouring of~$G_2$ of weight~$\frac{5}{2}$. \item If~$d>2$, then let~$G_d$ be obtained from~$G_{d-2}$ by adding two neighbours of
degree~$1$ to every vertex. This creates no new cycles, so~$G_d$ is acyclic
when~$d$ is odd, and contains a unique cycle, which is of length~$5$,
when~$d$ is even.
For every~$d\ge3$, the graph~$G_d$ contains a unique maximum independent set,
namely~$I_0 \coloneqq V(G_d)\setminus V(G_{d-2})$. After the first
step of the algorithm applied to~$G_d$, all the vertices in~$I_0$ have
weight~$1$, and we are left with the graph~$G_{d-2}$ where every vertex
has weight~$0$. By the induction hypothesis, the total weight of the
fractional colouring returned by the algorithm is therefore~$1 +
\frac{(d-2)+3}{2} = \frac{d+3}{2}$.
\qedhere \end{itemize} \end{proof}
\subsection{Triangle-free graphs} Using a similar approach, it is possible to obtain improved bounds for the fractional chromatic number of a given triangle-free graph~$G$, as stated in Theorem~\ref{thm:triangle-free}.
\begin{proof}[Proof of Theorem~\ref{thm:triangle-free}]
Let~$\lambda>0$ be any positive real. For every induced subgraph~$H$ of~$G$, we let~$\mathbf{I}_H$ be a random independent set drawn according to the hard-core distribution at fugacity~$\lambda$ over the set~$\mathscr{I}(H)$ of all independent sets of~$H$. We first assert the following.
\begin{as}\label{triangle-free-occupancy} For every vertex~$v\in V(H)$, and every integer~$k \ge 1$, \[\pth{1+\frac{(1+\lambda)^k}{\lambda(1+k\lambda)}} \pr{v\in \mathbf{I}_H} + \frac{1+\lambda}{1+k\lambda}\esp{\abs{N(v)\cap \mathbf{I}_H}} \ge 1.\] \end{as}
\noindent Note that when~$\omega=2$, we deduce~\eqref{reedinequality} from~\eqref{triangle-free-occupancy} by taking~$k=2$ and letting~$\lambda$ go to infinity.
The result follows from \eqref{triangle-free-occupancy} by applying \Cref{algo} with~$\alpha_v = 1+\frac{(1+\lambda)^k}{\lambda(1+k\lambda)}$ and~$\beta_v = \frac{1+\lambda}{1+k\lambda}$ for every~$v\in V(G)$. There remains to prove \eqref{triangle-free-occupancy}.
We let~$J$ be any possible realisation of~$\mathbf{I}_H \setminus N[v]$. By the
Spatial Markov Property of the hard-core distribution, if we condition on
the event~$E_J$ that~$\mathbf{I}_H \setminus N[v] = J$ and write~$W \coloneqq
N[v]\setminus N(J)$, then~$\mathbf{I}_H \cap N[v]$ follows the hard-core
distribution at fugacity~$\lambda$ over~$\mathscr{I}(H[W])$. Since~$G$ (and
therefore also~$H$) is triangle-free,~$H[W]$ is a star~$K_{1,d}$, for
some integer~$d\in\{0,\dotsc,\Delta(G)\}$. An analysis of the hard-core
distribution over the independent sets of a star yields that
\begin{enumerate}[label=(\roman*)] \item~$\pr{v\in \mathbf{I}_H \mid E_J} = \dfrac{\lambda}{\lambda+(1+\lambda)^d}$, and \item~$\esp{\abs{N(v)\cap \mathbf{I}_H}\mid E_J} = \dfrac{d\lambda(1+\lambda)^{d-1}}{\lambda+(1+\lambda)^d}$,\label{ite-itr}
\end{enumerate}
where~\ref{ite-itr} uses that~$\sum_{i=0}^d i\binom{d}{i}\lambda^i = d\lambda{(1+\lambda)}^{d-1}$.
For some positive real numbers~$\alpha$ and~$\beta$, we let \[g(x) \coloneqq \alpha \dfrac{\lambda}{\lambda+(1+\lambda)^x} + \beta \dfrac{x\lambda(1+\lambda)^{x-1}}{\lambda+(1+\lambda)^x}.\] We observe that~$g$ is a convex function, and therefore its minimum over the
(non-negative) reals is reached at its unique critical point~$x^*$ such
that~$g'(x^*)=0$ (if it exists). Moreover, if there are numbers~$y$ and~$z>y$ such that ~$g(y)=g(z)$, then Rolle's theorem ensures that~$x^* \in (y,z)$, and~$g(x) \ge g(y)$ for every~$x\notin(y,z)$.
Let~$k$ be a positive integer. We now fix \[ \alpha \coloneqq 1+\frac{(1+\lambda)^k}{\lambda(1+k\lambda)} \quad \et \quad \beta = \frac{1+\lambda}{1+k\lambda}.\] One can easily check that with these values for~$\alpha$ and~$\beta$, it holds that \[ g(k-1)=1=g(k).\] We conclude that~$g(d) \ge 1$ for every non-negative integer~$d$, which means that \begin{equation}\label{eq:triangle-free} \pth{1+\frac{(1+\lambda)^k}{\lambda(1+k\lambda)}} \pr{v\in \mathbf{I}_H \mid E_J} + \frac{1+\lambda}{1+k\lambda}\esp{\abs{N(v)\cap \mathbf{I}_H} \mid E_J} \ge 1, \end{equation} for any possible realisation~$J$ of~$\mathbf{I}_H\setminus N[v]$. The conclusion follows again by taking the convex combination of~\eqref{eq:triangle-free} over all possible values of~$J$. \end{proof}
\subsection{A stronger bound for graphs of girth 7}
\begin{figure}
\caption{A schematic view of the situation for the proof of~(C). No vertex in~$J$ has a neighbour in~$U$. Each vertex in~$U$ has a unique neighbour in~$N^2(v)$, which belongs to~$\mathbf{I}_H$ with probability~$1$ since~$U$ must be covered. All vertices that belong to~$\mathbf{I}_H$ with probability~$1$ are coloured black, and every neighbour of a black vertex belongs to~$\mathbf{I}_H$ with probability~$0$; such vertices are coloured white. The white vertices in~$N^2[v]$ are either contained in~$N(J)$, or in~$P^2_{N^2[v]}(U)=\{b,c\}$. The grey vertices are those whose presence in~$\mathbf{I}_H$ is not determined; together they form the set~$W=\{v\}\cup W_1\cup W_2$, with~$W_1=\{a,d,e,f\}$ and~$W_2=\{d_1,d_2,e_1\}$.
We have~$W_{1,0}=\{a,f\}$,~$W_{1,1}=\{e\}$,~$W_{1,2}=\{d\}$ and~$W_{1,j}=\varnothing$ if~$j\ge3$. In particular~$x_{1,0}=2$ and~$x_{1,j}=1$ if~$j\in\{1,2\}$. Note that~$X_0=\{v,a,f\}$ while~$X_1=\cup_{j\ge1}W_{1,j}=\{d,e\}$. Note also that~$H[X_0]$ is a star centered at~$v$, and all connected components of~$H[X_1\cup W_2]$ are stars centered in~$X_1$.}
\label{fig-C}
\end{figure}
Let~$G$ be a graph of girth (at least)~$7$ and~$H$ an induced subgraph of~$G$. We wish to apply \Cref{algo} with an independent set~$\mathbf{I}_H$ drawn according to the hard-core distribution at fugacity~$\lambda$ over the set~$\cI_{\max}(H)$ of all \emph{maximal} independent sets of~$H$, for the specific value~$\lambda=4$. We begin by establishing the following assertion.
\begin{as}\label{as-A} For every given vertex~$v\in V(H)$ and every integer~$k\ge 4$, \[\frac{2^{k-3}+k}{k}\pr{v\in \mathbf{I}_H} + \frac{2}{k}\esp{\abs{N(v)\cap \mathbf{I}_H}} \geq 1.\] \end{as}
\begin{proof}
Figure~\ref{fig-C} provides a schematic illustration of the following. Let~$J$ be any possible realisation of~$\mathbf{I}_H \setminus N^2[v]$. We are going to condition on the random
event~$E_J$ that~$\mathbf{I}_H \setminus N^2[v]=J$. Let~$U \coloneqq (V(H)\setminus N^2[v])\setminus N[J]$ be
the set of vertices at distance more than~$2$ from~$v$ that are uncovered (by~$J$), and set~$W'\coloneqq
N^2[v]\setminus(N(J)\cup P^2_{N^2[v]}(U))$ (in that scenario we have $P^2_{N^2[v]}(U)=N^2(U)\cap N(v)$). It follows from the definitions that~$v$ belongs to~$W'$.
Because the girth of~$G$ is greater than~$6$, no vertex outside of~$N^2[v]$ has more than one neighbour
in~$N^2[v]$. So \Cref{lem:markov-max} ensures that~$\mathbf{I}_H \cap W'$ follows the hard-core distribution at
fugacity~$\lambda$ over the maximal independent sets of~$H[W']$. As the hard-core distribution behaves
independently on each connected component, it suffices to work with the connected component of~$H[W']$
that contains~$v$; letting~$W$ be the vertices in this connected component, we simply ignore the vertices in~$W'\setminus W$.
(Actually, the vertices in~$W'\setminus W$ are isolated in~$H[W']$ and hence belongs to~$\mathbf{I}_H$ with probability~$1$.)
We let~$W_i$ be the set of vertices in~$W$ at distance~$i$ from~$v$ in~$H[W]$, for~$i\in \{0,1,2\}$,
and~$W_{1,j}$ be the subset of vertices of~$W_1$ with~$j$ neighbours in~$W_2$. We set~$x_j \coloneqq
\abs{W_{1,j}}$. Finally, we let~$X_0 \coloneqq \{v\}\cup W_{1,0}$ and~$X_1 \coloneqq W_1 \setminus W_{1,0}$,
and for every~$i\in \{0,1\}$ we let~$\mathbf{I}_i$ be the random outcome of~$\mathbf{I}_H \cap X_i$ under the
condition~$E_J$.
We first assume that~$x_0\ge 1$. If~$v\notin \mathbf{I}_0$, then the maximality of~$\mathbf{I}_H$ ensures that~$\mathbf{I}_H
\setminus (X_1\cup W_2)$ equals~$J\cup W_{1,0}$, which we call~$J_0$. Since~$v$ is covered by~$J_0$, \Cref{lem:markov-max} implies
that, under the additional condition that $v\notin \mathbf{I}_0$, the random set~$\mathbf{I}_1$ follows the hard-core
distribution at fugacity~${\lambda=4}$ over the maximal independent sets of~$H[X_1\cup W_2]$. Moreover,~$\mathbf{I}_1$
behaves independently on each connected component of~$H[X_1\cup W_2]$, which are all stars with center in~$X_1$. So for every
integer~$j\in\{1,\dotsc,d-1\}$ and every vertex~$u\in W_{1,j}$, we have \[ \pr{u \in \mathbf{I}_1 \mid v\notin \mathbf{I}_0} = \frac{1}{1+\lambda^{j-1}}. \] It follows that
\[ \pr{\mathbf{I}_1 =\varnothing \mid v\notin \mathbf{I}_0} =\prod_{j=1}^{d-1} \pth{\frac{\lambda^{j-1}}{1+\lambda^{j-1}}}^{x_j},
\] and we define~$\zeta$ to be this value. Let~$p\coloneqq \pr{v\in \mathbf{I}_0}$; we have \begin{align*} \pr{\mathbf{I}_1 = \varnothing} &= \pr{\mathbf{I}_1 = \varnothing \mid v\in \mathbf{I}_0} \pr{v\in \mathbf{I}_0} + \pr{\mathbf{I}_1 = \varnothing \mid v\notin \mathbf{I}_0} \pr{v\notin \mathbf{I}_0} = p +(1-p)\zeta. \end{align*} If $\mathbf{I}_1=\varnothing$, then the maximality of~$\mathbf{I}_H$ ensures that~$\mathbf{I}_H \setminus X_0$ equals~$J \cup W_2$, which we call~$J_1$.
Since every vertex of~$X_1$ is covered by~$J_1$, \Cref{lem:markov-max} implies that, under the
additional condition that~$\mathbf{I}_1=\varnothing$, the random set~$\mathbf{I}_0$ follows the hard-core distribution
at fugacity~$\lambda$ over the maximal independent sets of~$H[X_0]$. Hence, since~$v$ cannot belong to~$\mathbf{I}_H$
as soon as~$\mathbf{I}_1\neq\varnothing$, \begin{align*} p = \pr{v\in \mathbf{I}_0} &= \pr{v\in \mathbf{I}_0 \mid \mathbf{I}_1=\varnothing}\pr{\mathbf{I}_1=\varnothing} = \frac{1}{1+\lambda^{x_0-1}}\Big(p + (1-p)\zeta\Big),\\
\intertext{so} p \pth{1+\lambda^{x_0-1}} &= \zeta+p(1-\zeta), \\
\intertext{and hence} p &= \frac{\zeta}{\zeta+\lambda^{x_0-1}}. \end{align*}
There remains to evaluate the expectancy of~$\abs{\mathbf{I}_H \cap N(v)}$, which is \begin{align*} \esp{\abs{N(v)\cap \mathbf{I}_H} \mid E_J} &= \pr{v\notin \mathbf{I}_0}\Big(x_0 + \esp{\abs{\mathbf{I}_1} \mid v\notin \mathbf{I}_0}\Big) = \frac{\lambda^{x_0-1}}{\zeta+\lambda^{x_0-1}} \pth{x_0 + \sum_{j=1}^{d-1} \frac{x_j}{1+\lambda^{j-1}}}. \end{align*}
We now assume that~$x_0=0$.
The probability distribution of~$\mathbf{I}_H \cap W$ is obtained from that above by forbidding the outcomes that correspond to a non-maximal independent set when~$x_0=0$. There is only one such outcome, which corresponds to the event~$v\notin \mathbf{I}_0$
and~$\mathbf{I}_1 = \varnothing$ (because~$v$ is no longer covered by~$W_{1,0}$ in that case). This means that $\mathbf{I}_H \cap N^2[v]=W_2$, hence that outcome has zero contribution to both $\pr{v\in \mathbf{I}_H \mid E_J}$ and $\esp{|N(v)\cap \mathbf{I}_H| \mid E_J}$; forbidding that outcome therefore increases those quantities. It follows that the previously computed values are lower bounds in the case $x_0=0$.
We conclude that regardless of the value of $x_0$, we have
\begin{align} \pr{v\in \mathbf{I}_H \mid E_J} &\ge \dfrac{\zeta}{\zeta + \lambda^{x_0-1}}, \quad \text{and}\label{eq:occupancy1}\\ \esp{\abs{N(v)\cap \mathbf{I}_H} \mid E_J} &\ge \dfrac{\lambda^{x_0-1}}{\zeta + \lambda^{x_0-1}}\left(x_0 + \sum_{j=1}^{d-1} \frac{x_j}{1+\lambda^{j-1}} \right).\label{eq:occupancy2} \end{align}
\noindent There remains to check that, when~$\lambda=4$, it holds that \begin{equation}\label{eq:occupancy} \frac{2^{k-3}+k}{k} \pr{v\in \mathbf{I}_H \mid E_J} + \frac{2}{k}\esp{\abs{N(v)\cap \mathbf{I}_H} \mid E_J} \geq 1. \end{equation}
By combining~\eqref{eq:occupancy1},~\eqref{eq:occupancy2}, and~\eqref{eq:occupancy}, it is enough to prove that \[\pth{2^{k-3}+k}\zeta + 2\lambda^{x_0-1}\pth{x_0 + \sum_{j=1}^{d-1} \frac{x_j}{1+\lambda^{j-1}}} \geq k(\zeta + \lambda^{x_0-1}), \] that is, multiplying both sides by the positive real~$\lambda^{1-x_0}/\zeta$, \begin{equation}\label{eq-calculus}
\pth{2^{k-3}+k}\lambda^{1-x_0} + \frac2\zeta\pth{x_0 + \sum_{j=1}^{d-1} \frac{x_j}{1+\lambda^{j-1}}} \geq k\lambda^{1-x_0} + \frac{k}{\zeta}. \end{equation} Let us set
\[ R\coloneqq {\zeta}^{-1}\cdot\left(k - 2x_0 - 2\sum_{j=1}^{d-1} \frac{x_j}{1+\lambda^{j-1}} \right) = \prod_{j=1}^{d-1} \left( 1 + \frac{1}{\lambda^{j-1}}\right)^{x_j}\left(k - 2x_0 - 2\sum_{j=1}^{d-1} \frac{x_j}{1+\lambda^{j-1}} \right), \] so that~\eqref{eq-calculus} is equivalent to \begin{equation} \label{eq:R} 2^{k-3}\lambda^{1-x_0} \geq R. \end{equation}
\noindent Observe that the left side of~\eqref{eq:R} is positive, and recall that, by definition, each value~$x_j$ is a non-negative integer. We may therefore assume that~$x_0<\frac{k}{2}$, since otherwise~$R\le 0$, which directly implies~\eqref{eq:R}. Likewise,~$R\le 0$ if~$x_1 \ge k-2x_0$, hence we may moreover assume that~$0\le x_1 \le k-2x_0-1$.
Let us now fix~$\lambda=4$ and prove $\eqref{eq:R}$ in that case, that is~$R\le 2^{k-2x_0-1}$ for every non-negative integer~$x_0$. Since the inequality is verified if~$R\le 0$, we assume from now on that~$R$ is positive. We define~$y_j \coloneqq \dfrac{x_j}{1+\lambda^{j-1}}$ for every~$j\in\{1,\dotsc,d-1\}$, and~$y \coloneqq \sum_{j=2}^{d-1}\limits y_j$. If~$y>0$ then we set $j_0 \coloneqq \min \{j\ge 2, x_j\neq 0\}$, and otherwise we set~$j_0\coloneqq\infty$. In particular,~$y\ge \frac{1}{1+\lambda^{j_0}}$. We have
\begin{align*} R &= 2^{x_1} \cdot \prod_{j=2}^{d-1} \left( 1 + \frac{1}{\lambda^{j-1}}\right)^{x_j}\left(k-2x_0-x_1 - 2\sum_{j=2}^{d-1} \frac{x_j}{1+\lambda^{j-1}} \right) \\ &= 2^{x_1} \cdot \prod_{j=j_0}^{d-1} \left( 1 + \frac{1}{\lambda^{j-1}}\right)^{(1+\lambda^{j-1})y_j}\left(k-2x_0-x_1 - 2\sum_{j=j_0}^{d-1} y_j\right)\\ &\le 2^{x_1} \cdot \prod_{j=j_0}^{d-1} \left( 1 + \frac{1}{\lambda^{j_0-1}}\right)^{(1+\lambda^{j_0-1})y_j}\left(k-2x_0-x_1 - 2\sum_{j=j_0}^{d-1} y_j\right)\\ &= 2^{x_1}\left( 1 + \frac{1}{\lambda^{j_0-1}}\right)^{(1+\lambda^{j_0-1})y} (k-2x_0-x_1 - 2y), \end{align*} where the inequality uses that~$\displaystyle x\mapsto{\left(1+\frac1{\lambda^{x}}\right)}^{(1+\lambda^x)}$ is decreasing (to~$1$) as~$x$ increases in~$\mathbb{R}_{>0}$.
\noindent
We let~$A\coloneqq \left( 1 + \frac{1}{\lambda^{j_0-1}}\right)^{1+\lambda^{j_0-1}}$,~$B \coloneqq k-2x_0-x_1$, and~$f\colon x \mapsto A^x(B-2x)$, so that~$R \le 2^{x_1} f(y)$. Since $j_0 \ge 2$, we have $A \le \frac{3125}{1024}$, and by the assumptions~$B$ is a positive integer.
We are going to use the following fact in order to systematically obtain an upper bound on~$f(y)$. \begin{description}
\item[Fact 1] For all real numbers~$y_0$,~$A$ and~$B$ with~$A>1$ and~$B>0$,
the maximum of the function~$f\colon x \mapsto A^x(B-2x)$ on~$\mathbb{R}$
is~$\frac{2A^{B/2}}{e\ln A}$, and if~$B/2-1/\ln A \le y_0$ then
the maximum on the domain~$[y_0,+\infty)$ is~$f(y_0)$. \end{description}
\noindent There are now three cases. \begin{enumerate}[label=(\roman*)]
\item If~$B=1$, then~$\frac{B}{2}-\frac{1}{\ln A} < 0$, hence~$f(y)\le f(0) = 1$. So~$R\le 2^{x_1} = 2^{k-2x_0-B}= 2^{k-2x_0-1}$. \item If~$B=2$, we have \begin{align*} \frac{B}{2} - \frac{1}{\ln A} &= 1 - \frac{1}{(1+\lambda^{j_0-1})\ln(1+1/\lambda^{j_0-1})} \\ & \le 1 - \frac{1}{(1+\lambda^{j_0-1})\cdot 1/\lambda^{j_0-1}} = \frac{1}{1+\lambda^{j_0-1}} \le y. \end{align*} So $f(y) \le f\hspace{-3pt}\pth{\frac{1}{1+\lambda^{j_0-1}}} = 2$, and $R\le 2^{x_1+1} = 2^{k-2x_0-B+1} = 2^{k-2x_0-1}$. \item Finally, if~$B \ge 3$, then since~$f(y) \le \frac{2A^{B/2}}{e \ln A}$ we deduce that~$R\le 2^{x_1}f(y) \le 2^{k-2x_0} \cdot \underbrace{\tfrac{2(A/4)^{B/2}}{e\ln A}}_{< 1/2} < 2^{k-2x_0-1}$.
\end{enumerate}
This finishes to prove that~$R\le2^{k-2x_0-1}$. This establishes~\eqref{eq-calculus}, and therefore~\eqref{eq:occupancy}.
The conclusion follows by taking the convex combination of~\eqref{eq:occupancy} over all possible values of~$J$. \end{proof}
We are now ready to prove \Cref{thm:girth7}.
\begin{proof}[Proof of \Cref{thm:girth7}] We set~$\lambda\coloneqq4$, and apply \Cref{algo} with \begin{align*} \alpha_v = 1 + \frac{2^{k(v)-3}}{k(v)} \quad \et \quad \beta_v = \frac{2}{k(v)} \end{align*} for every vertex~$v\in V(G)$, where~$k(v)$ is chosen such that $\frac{2\deg(v)+2^{k-3}}{k}$ is minimised when~$k=k(v)$.
The results follows from \eqref{as-A}. \end{proof}
\section{Bounds on the Hall ratio}\label{sec-hall} We focus on establishing upper bounds on the Hall ratios of graphs with bounded maximum degree and girth. These bounds are obtained by using the uniform distribution on~$\mathscr{I}_{\alpha}(G)$, for~$G$ in the considered class of graphs, into Lemma~\ref{ratio-vertex} or Lemma~\ref{ratio-edge}.
\subsection{Structural analysis of a neighbourhood} We start by introducing some terminology. \begin{defi}\label{pattern}\mbox{} \begin{enumerate} \item A \emph{pattern of depth~$r$} is any graph~$P$ given with a root vertex~$v$ such that \[\forall u\in V(G),\quad \di_G(u,v) \leq r.\] The \emph{layer at depth~$i$} of~$P$ is the set of vertices at distance~$i$ from its root vertex~$v$. \item A pattern~$P$ of depth~$r$ and root~$v$ is \emph{$d$-regular} if all its vertices have degree exactly~$d$, except maybe in the two deepest layers where the vertices are only required to have degree at most~$d$.
\end{enumerate} \end{defi} \begin{defi}
Let~$P$ be a pattern with depth~$r$ and root~$v$. Let~$\mathbf{I}$ be a uniform random maximum
independent set of~$P$. We define~$e_i(P) \coloneqq \esp{\abs{\mathbf{I} \cap N^i_P(v)}}$ for each~$i
\in\{0,\dotsc,r\}$. \begin{enumerate} \item The \emph{constraint} associated to~$P$ is the pair~$c(P)=(\mathbf{e}(P),n(P))$, where ~$\mathbf{e}(P)=(e_i(P))_{i=0}^r \in{\left(\mathbb{Q}^+\right)}^{r+1}$, and~$n(P)\ \in \mathbb{N}$ is the
\emph{cardinality} of the constraint, which is the number of maximum independent sets of~$P$.
Most of the time, we only need to know the value of~$\mathbf{e}(P)$, in which case we characterise the constraint~$c(P)=(\mathbf{e}(P),n(P))$ only by~$\mathbf{e}(P)$. The value of~$n(P)$ is only needed for a technical reason, in order to be able to compute constraints inductively. \item Given two constraints~$\mathbf{e}, \mathbf{e}' \in
\left(\mathbb{Q}^+\right)^{r+1}$, we say that~$\mathbf{e}$ is
\emph{weaker} than~$\mathbf{e}'$ if, for any vector~$\bm{\alpha} \in
\left(\mathbb{Q}^+\right)^{r+1}$ it holds that \[ \bm{\alpha}^{\!\top} \mathbf{e}' \ge 1 \quad \implies \quad \bm{\alpha}^{\!\top} \mathbf{e} \ge 1.\]
If the above condition holds only for all
vectors~$\bm{\alpha}\in{\left(\mathbb{Q}^+\right)}^{r+1}$ with
non-increasing coordinates, then we say that~$\mathbf{e}$ is
\emph{relatively weaker} than~$\mathbf{e}'$. \end{enumerate} \end{defi}
\noindent Note that~$\mathbf{e}$ is weaker than~$\mathbf{e}'$ if and only if $e_i \ge e'_i$ for every $i \in \{0,\dotsc,r\}$, and~$\mathbf{e}$ is relatively weaker than~$\mathbf{e}'$ if and only if $\sum_{j=0}^i \limits e_j \ge \sum_{j=0}^i \limits e'_j$ for every $i \in \{0,\dotsc,r\}$.
\begin{rk}\label{deg2} Let~$P$ be a pattern such that one of its vertices~$u$ is adjacent with some
vertices~$u_1,\dotsc, u_k$ of degree~$1$ in the next layer, where~$k\ge 2$. Then every
maximum independent set of~$P$ contains~$\{u_1,\dotsc u_k\}$ and not~$u$.
Consequently,~$\mathbf{e}(P)$ is weaker than~$\mathbf{e}(P\setminus \{u_3, \dotsc, u_k\})$ since,
letting~$i$ be the distance between~$u_1$ and the root of~$P$, one has \[ e_j(P) = \begin{cases}
e_j(P\setminus \{u_3, \dotsc, u_k\}) & \quad \text{if~$j\neq i$, and} \\
e_i(P\setminus \{u_3, \dotsc, u_k\}) + (k-2) & \quad \mbox{if j=i.} \end{cases} \] \end{rk}
\subsection{Tree-like patterns} \subsubsection{Rooting at a vertex}\label{root-vertex} Fix an integer~$r \ge 2$. Let~$G$ be a~$d$-regular graph of girth at least~$2r+2$, and let~$\mathbf{I}$ be a uniform random maximum independent set of G. For any fixed vertex~$v$, we let~$J$ be any possible realisation of~$\mathbf{I}\setminus N^r[v]$, and~$W\coloneqq N^r[v]\setminus N(J)$. By the Spatial Markov Property of the uniform distribution over the maximum independent sets of~$G$, the random independent set~$\mathbf{I}\cap N^r[v]$ follows the uniform distribution over the maximum independent sets of~$G[W]$. Now, observe that~$G[W]$ is a~$d$-regular pattern of depth~$r$ with root vertex~$v$, and since~$G$ has girth at least~$2r+2$, this pattern is moreover a tree. Let~$\mathcal{T}_r(d)$ be the set of acyclic~$d$-regular patterns of depth~$r$.
Let us define~$\mathbf{X}_i(v) \coloneqq \mathbf{I} \cap N^i(v)$, for every~$i\in\{0,\dotsc,r\}$. We seek parameters~$(\alpha_i)_{i=0}^{r}$ such that the inequality~$\sum_{i=0}^r \alpha_i \esp{\abs{\mathbf{X}_i(v)}} \ge 1$ is satisfied regardless of the choice of~$v$. To this end, it is enough to pick the rational numbers~$\alpha_i$ in such a way that the inequality is satisfied in any tree~$T\in \mathcal{T}_r(d)$, when~$v$ is the root vertex. In a more formal way, given any~$T\in \mathcal{T}_r(d)$, the vector~$\bm{\alpha} = (\alpha_0, \dotsc, \alpha_r)$ must be \emph{compatible} with the constraint~$\mathbf{e}(T)$, that is,~$\bm{\alpha}^{\!\top}\mathbf{e}(T) \ge 1$ for each~$T\in \mathcal{T}_r(d)$.
An application of Lemma~\ref{ratio-vertex} then lets us conclude that $\abs{V(G)}/\alpha(G)$ is bounded from above by the solution to the following linear program.
\begin{align}\label{lp} \text{Minimise} &\displaystyle \quad\quad \alpha_0 + \sum_{i=1}^r \alpha_i d(d-1)^{i-1}\\ \text{such that} & \begin{cases}
\forall T\in \mathcal{T}_r(d),& \displaystyle\sum_{i=0}^r \alpha_i e_i(T) \ge 1 \\
\forall i \le r, & \alpha_i \ge 0. \end{cases}\notag \end{align}
The end of the proof is made by computer generation of~$\mathcal{T}_r(d)$, in order to generate the desired linear program, which is then solved again by computer computation. For the sake of illustration, we give a complete human proof of the case where~$r=2$ and~$d=3$. There are~$10$ trees in~$\mathcal{T}_2(3)$. One can easily compute the constraint~$(e_0(T),e_1(T),e_2(T))$ for each~$T\in \mathcal{T}_2(3)$; they are depicted in Figure~\ref{trees}. Note that constraints~$\mathbf{e}_8$,~$\mathbf{e}_9$ and~$\mathbf{e}_{10}$ are weaker than constraint~$\mathbf{e}_7$, so we may disregard these constraints in the linear program to solve. Note also that constraint~$\mathbf{e}_0$ is relatively weaker than constraint~$\mathbf{e}_1$, and so may be disregarded as well, provided that the solution of the linear program is attained by a vector~$\bm{\alpha}$ with non-increasing coordinates, which will have to be checked. The linear program to solve is therefore the following.
\begin{align*} &\text{Minimise } \quad\quad\alpha_0 + 3\alpha_1 + 6\alpha_2 \\ & \text{such that } \left\{\begin{aligned}
& 5/2 \cdot \alpha_1 + 1/2 \cdot \alpha_2 &{}\ge{}& &1\\
& 2 \alpha_1 + 2\alpha_2 &{}\ge{}& &1 \\
& 1/5 \cdot \alpha_0 + 8/5 \cdot \alpha_1 + 6/5 \cdot \alpha_2 &{}\ge{}& &1 \\
& 1/3 \cdot \alpha_0 + \alpha_1 + 8/3\cdot \alpha_2 &{}\ge{}& &1 \\
& 1/2\cdot \alpha_0 + 1/2\cdot \alpha_1 + 4\alpha_2 &{}\ge{}& &1 \\
& \alpha_0 + 3\alpha_2 &{}\ge{}& &1 \\
& \alpha_0, \alpha_1, \alpha_2 \ge 0. & & \end{aligned}\right. \end{align*}
The solution of this linear program is~$\frac{85}{31}\approx 2.741935$, attained by~$\bm{\alpha} = \left(\frac{19}{31},\frac{14}{31},\frac{4}{31} \right)$, which indeed has non-increasing coordinates. This is an upper bound on~$\rho(3,6)$, though we prove a stronger one through a more involved computation in Section~\ref{sec:edge-root}.
\begin{figure}
\caption{An enumeration of~$\mathbf{e}(T)$ for all trees~$T\in \mathcal{T}_2(3)$.}
\label{trees}
\end{figure}
To compute~$\mathbf{e}(T)$ for each~$T\in \mathcal{T}_r(d)$, one can enumerate all the maximum independent sets of~$T$ and average the size of their intersection with each layer of~$T$. For general graphs, there might be no better way of doing so, however the case of~$\mathcal{T}_r(d)$ can be treated inductively by a standard approach: we distinguish between the maximum independent sets that contain the root and those that do not. To this end, we slightly extend our notion of ``constraint'' to any pair~$(\mathbf{e},n)$ where~$\mathbf{e}\in{\left(\mathbb{Q}^+\right)}^{r+1}$ and~$n$ is a non-negative integer; the constraints we shall use have a combinatorial interpretation with respect to some pattern.
\begin{defi}\label{def-cjoin} Let~$c=(\mathbf{e},n)$ and~$c'=(\mathbf{e}',n')$ be two constraints. \begin{enumerate} \item The operation~$\vee$ on~$c$ and~$c'$ returns the constraint \[ c \vee c' \coloneqq \begin{cases}
\pth{\dfrac{n}{n+n'} \mathbf{e} + \dfrac{n'}{n+n'}\mathbf{e}', n+n'}
& \quad\text{if }\lVert\mathbf{e}\rVert_1=\lVert\mathbf{e}'\rVert_1,\\
c & \quad\text{if }\lVert\mathbf{e}\rVert_1>\lVert\mathbf{e}'\rVert_1,\\
c' & \quad\text{if }\lVert\mathbf{e}\rVert_1<\lVert\mathbf{e}'\rVert_1. \end{cases}\] \item The operation~$\oplus$ on~$c$ and~$c'$ returns the constraint~$c \oplus c' \coloneqq \big(\mathbf{e} + \mathbf{e}', n\cdot n' \big)$. \end{enumerate} \end{defi}
For a given tree~$T\in \mathcal{T}_r(d)$ with root~$v$, let~$c_1(T)$ be the constraint associated to~$T$ where~$v$ is forced (that is, we restrict to the maximum independent sets that contain~$v$ when computing the constraint~$c_1(T)$), and let~$c_0(T)$ be the constraint associated to~$T$ where~$v$ is forbidden. It readily follows from Definition~\ref{def-cjoin} that \[c(T) = c_0(T) \vee c_1(T).\]
If~$(T_i)_{i=1}^{d}$ are the (possibly empty) subtrees of~$T$ rooted at the children of the root~$v$, then
\begin{align*}
c_0(T) &= \big( (0,\mathbf{e}), n \big) && \mbox{ where } (\mathbf{e},n) = \bigoplus_{i=1}^{d} c(T_i), \quad\text{and}\\
c_1(T) &= \big( (1,\mathbf{e}), n \big) && \mbox{ where } (\mathbf{e},n) = \bigoplus_{i=1}^{d} c_0(T_i). \end{align*} We thus obtain an inductive way of computing~$\mathbf{e}(T)$ by using the following initial values.
\begin{align*} c_0(\varnothing) &\coloneqq \big((0),1\big) & c_1(\varnothing) &\coloneqq \big((0),0\big) \\ c_0(\{v\}) &\coloneqq \big((0),1\big) & c_1(\{v\}) &\coloneqq \big((1),1\big).\\ \end{align*}
Using this inductive way to enumerate the vectors~$\mathbf{e}(T)$ for~$T\in \mathcal{T}_r(d)$, the following statement is obtained by computer calculus.
\begin{lemma}\label{vertex-bounds} The solution to the linear program~\eqref{lp} is \begingroup \allowdisplaybreaks \begin{align*} \mathcal{T}_3(3) \colon \quad& \frac{5849}{2228} \approx 2.625224 && \text{with } \bm{\alpha} = \left(\frac{953}{2228}, \frac{162}{557}, \frac{81}{557}, \frac{21}{557} \right),\\ \mathcal{T}_4(3) \colon \quad& \frac{2098873192}{820777797} \approx 2.557176 &&\text{with } \bm{\alpha} = \left(\frac{225822361}{820777797},\frac{18575757}{91197533},\frac{10597368}{91197533},\right.&\\* &&&\multicolumn{1}{r}{$\left.\dfrac{5054976}{91197533},\dfrac{1172732}{91197533}\right),$}\\ \mathcal{T}_5(3) \colon\quad& \frac{29727802051155412}{11841961450578397} \approx 2.510378 && \multicolumn{2}{l}{with~$\bm{\alpha} = \left(\dfrac{3027359065168972}{11841961450578397},\dfrac{2216425114872980}{11841961450578397},\right.$}\\* & \multicolumn{4}{r}{$\left.\dfrac{2224040336719575}{23683922901156794},\dfrac{2026654050681425}{47367845802313588},\dfrac{403660478424775}{23683922901156794},\dfrac{51149140376400}{11841961450578397}\right),$} \\ \mathcal{T}_3(4) \colon\quad& \frac{7083927}{2331392} \approx 3.038497 &&\multicolumn{2}{l}{with~$\bm{\alpha} = \left(\dfrac{123345}{333056},\dfrac{68295}{291424},\dfrac{12283}{145712},\dfrac{2911}{145712}\right),$} \\ \mathcal{T}_4(4) \colon\quad& 3 && \text{with } \bm{\alpha} = \left(\frac{7}{43},\frac{6}{43}, \frac{19}{258}, \frac{7}{258}, \frac{1}{258} \right),\\ \mathcal{T}_2(5) \colon\quad& \frac{69}{19} \approx 3.631579 &&\text{with } \bm{\alpha} = \left(\frac{37}{57}, \frac{6}{19}, \frac{4}{57}\right), \\ \mathcal{T}_3(5) \colon\quad& \frac{7}{2} = 3.5 && \text{with } \bm{\alpha} = \left(\frac{77}{282},\frac{25}{141},\frac{17}{282},\frac{2}{141}\right). \end{align*} \endgroup \end{lemma}
\subsubsection{Rooting in an edge}\label{sub-edge-rooted} Definition~\ref{pattern} can be extended to a pattern with a root-edge instead of a root-vertex. The distance in a pattern~$P$ between a vertex~$w$ and an edge~$uv$ is defined to be~$\min\{\di_P(w,u),\di_P(w,v)\}$. The depth of a pattern~$P$ rooted in an edge~$e$ is then the largest distance between~$e$ and a vertex in~$P$. It is possible to follow the same analysis as in Section~\ref{root-vertex} with edge-rooted patterns: in order for the edge-rooted pattern of depth~$r$ to always be a tree, the graph~$G$ must have girth at least~$2r+3$. Let~$\mathcal{T}'_r(d)$ be the set of acyclic edge-rooted~$d$-regular patterns of depth~$r$. By \Cref{ratio-edge}, the linear program to solve is now the following.
\begin{align}\label{lp-edge}
\frac{\abs{V(G)}}{\alpha(G)} &\le \displaystyle\min \quad 2 \sum_{i=0}^r \alpha_i (d-1)^i\\ \text{such that} & \begin{cases}
\forall T\in \mathcal{T}'_r(d),& \displaystyle\sum_{i=0}^r \alpha_i e_i(T) \ge 1 \\
\forall i \le r, & \alpha_i \ge 0. \end{cases}\notag \end{align}
For a given tree~$T \in \mathcal{T}'_r(d)$ rooted in~$e=uv$, it is possible to compute~$\mathbf{e}(T)$ using the constraints associated to vertex-rooted trees. If~$T_u$ and~$T_v$ are the subtrees of~$T$ respectively rooted at~$u$ and at~$v$, then it readily follows from Definition~\ref{def-cjoin} that \begin{equation}\label{eq-prod} c(T) = \Big(c_0(T_u)\oplus c_0(T_v)\Big) \vee \Big(c_0(T_u)\oplus c_1(T_v)\Big) \vee \Big(c_1(T_u)\oplus c_0(T_v)\Big). \end{equation}
Following the enumeration of the vectors~$\mathbf{e}(T)$ for~$T\in \mathcal{T}'_r(d)$ described earlier, the next statement is obtained by computer calculus.
\begin{lemma}\label{edge-bounds} The solution to the linear program~\eqref{lp-edge} is \begin{align*} \mathcal{T}'_2(3)\colon&\quad \frac{30}{11} \approx 2.72727 && \text{with } \bm{\alpha} {}={} \pth{\frac{1}{2},\frac{13}{44},\frac{3}{44}},\\ \mathcal{T}'_3(3)\colon&\quad \frac{125}{48} \approx 2.604167 && \text{with } \bm{\alpha} {}={} \left(\frac{11}{32},\frac{5}{24},\frac{3}{32}, \frac{1}{48}\right),\\ \mathcal{T}'_4(3)\colon&\quad \frac{14147193}{5571665} \approx 2.539132 &&\text{with } \bm{\alpha} {}={} \left(\frac{98057}{506515},\frac{159348}{1114333},\frac{3688469}{44573320}, \frac{1752117}{44573320}, \frac{402569}{44573320}\right),\\ \mathcal{T}'_2(4)\colon&\quad \frac{41}{13} \approx 3.153846 && \text{with } \bm{\alpha} {}={} \left(\frac{11}{26},\frac{3}{13},\frac{2}{39}\right),\\ \mathcal{T}'_3(4)\colon&\quad \frac{127937}{42400} \approx 3.017382 && \text{with } \bm{\alpha} {}={} \left(\frac{5539}{16960}, \frac{1737}{10600}, \frac{257}{5300}, \frac{399}{42400}\right),\\ \mathcal{T}'_2(5)\colon&\quad \frac{18}{5} = 3.6 && \text{with } \bm{\alpha} {}={} \left(\frac{17}{45},\frac{8}{45},\frac{2}{45}\right). \end{align*} \end{lemma}
The bounds obtained in Lemma~\ref{edge-bounds} are valid for graphs of girth at least~$2r+3$. It turns out that the same bounds, with the same~$\bm{\alpha}$, remain valid for graphs of girth~$2r+2=6$, when~$r=2$ and~$d \in \{3,4\}$. We were not able to check this for higher values of~$r$ or~$d$, but we propose the following conjecture which would explain and generalise this phenomenon.
\begin{conj}\label{conj} Let~$P$ be a~$d$-regular edge-rooted pattern of depth~$r$ and of girth~$2r+2$. Then the constraint~$\mathbf{e}(P)$ is weaker than some convex combination of constraints~$\mathbf{e}(T)$ with~$T\in \mathcal{T}'_r(d)$. More formally, there exist~$T_1, \dotsc, T_m \in \mathcal{T}'_r(d)$ and $\lambda_1, \dotsc, \lambda_m \in [0,1]$ with~$\sum_{i=1}^m \lambda_i = 1$ such that for any~$\bm{\alpha}\in{\left(\mathbb{Q}^+\right)}^{r+1}$, \[ \bm{\alpha}^{\!\top}\left(\sum_{i=1}^m \lambda_i \mathbf{e}(T_i) \right) \ge 1 \quad \implies \quad \bm{\alpha}^{\!\top} \mathbf{e}(P) \ge 1.\] \end{conj}
\subsection{More complicated patterns} \subsubsection{Rooting at a vertex} Let us fix a depth~$r \ge 2$. Let~$G$ be a~$d$-regular graph of girth~$g \le 2r+1$. We repeat the same analysis as in Section~\ref{root-vertex}: we end up having to find a vector~$\bm{\alpha} \in \mathbb{Q}^{r+1}$ compatible with all the constraints generated by vertex-rooted~$d$-regular patterns of depth~$r$ and girth~$g$. Letting~$\mathcal{P}_r(d,g)$ be the set of such patterns, we thus want that \[ \forall P \in \mathcal{P}_r(d,g), \quad \bm{\alpha}^{\!\!\top} \mathbf{e}(P) \ge 1. \]
In this setting, we could do no better than performing an exhaustive enumeration of every possible pattern~$P \in \mathcal{P}_r(d,g)$, and computing the associated constraint~$\mathbf{e}(P)$ through an exhaustive enumeration of $\mathscr{I}_{\alpha}(P)$. The complexity of such a process grows fast, and we considered only depth~$r\le2$ and degree~$d\le 4$. Since the largest value of the Hall ratio over the class of~$3$-regular graphs of girth~$4$ or~$5$ is known to be~$\frac{14}{5}=2.8$, and the one of~$4$-regular graphs of girth~$4$ is known to be~$\frac{13}{4}=3.25$, the only open value in these settings is for the class of~$4$-regular graphs of girth~$5$. Unfortunately, this method is not powerful enough to prove an upper bound lower than~$\frac{13}{4}$, the obtained bound for~$\mathcal{P}_2(4,5)$ being~$\frac{82}{25} = 3.28$. It is more interesting to root the patterns in an edge.
\subsubsection{Rooting in an edge}\label{sec:edge-root} Similarly, we define~$\mathcal{P}'_r(d,g)$ to be the set of edge-rooted $d$-regular patterns of girth~$g$. For fixed~$r$ and~$g$, we seek for the solution of the following linear program.
\begin{align}\label{lp-edge-6}
\frac{\abs{V(G)}}{\alpha(G)} &\le \min \quad 2 \sum_{i=0}^r \alpha_i (d-1)^i\\ \text{such that} & \begin{cases}
\forall P\in \mathcal{P}'_r(d,g),& \displaystyle\sum_{i=0}^r \alpha_i e_i(P) \ge 1 \\
\forall i \le r, & \alpha_i \ge 0. \end{cases}\notag \end{align}
Again, our computations were limited to the cases where~$r\le2$ and~$d\le 4$. However, we managed to prove improved bounds for girth~$6$ when~$d\in \{3,4\}$, which seems to support Conjecture~\ref{conj}.
\begin{lemma}\label{lem:program-girth6} The solution to the linear program~\eqref{lp-edge-6} is \begin{equation*}
\begin{aligned}[c] \mathcal{P}'_2(3,6)\colon\qquad & \frac{30}{11} \approx 2.72727\\ \mathcal{P}'_2(4,6)\colon\qquad & \frac{41}{13} \approx 3.153846\\ \end{aligned} \qquad
\begin{aligned}[c]
\text{with } \bm{\alpha} &{}={} \left(\frac{1}{2},\frac{13}{44},\frac{3}{44}\right),\\
\text{with } \bm{\alpha} &{}={} \left(\frac{11}{26},\frac{3}{13},\frac{2}{39}\right). \end{aligned} \end{equation*} \end{lemma}
\subsection{Proof of \texorpdfstring{\Cref{thm:ratio}}{Theorem 11}}
We are now ready to give a proof of the values presented in \Cref{tab:4}.
\begin{proof}[Proof of \Cref{thm:ratio}] We want to establish the upper bounds on~$\rho(d,g)$ presented in \Cref{tab:4}. They all follow from an application of \Cref{ratio-vertex} or \Cref{ratio-edge} using the solution of the appropriate linear program. \begin{itemize}
\item The upper bounds for~$\rho(3,6)$ and~$\rho(4,6)$ follow from \Cref{lem:program-girth6} by applying \Cref{ratio-edge}.
\item The upper bounds for~$\rho(5,6)$, for~$\rho(d,8)$ with~$d\in\{3,4,5\}$, for~$\rho(d,10)$ with~$d\in\{3,4\}$
and for~$\rho(3,12)$ follow from \Cref{vertex-bounds} by applying \Cref{ratio-vertex}.
\item The upper bounds for~$\rho(d,7)$ with~$d\in\{3,4,5\}$, for~$\rho(d,9)$ with~$d\in\{3,4\}$ and for~$\rho(3,11)$
follow from \Cref{edge-bounds} by applying \Cref{ratio-edge}.
\qedhere \end{itemize} \end{proof}
\section{Conclusion} We finish with a discussion about our method, its possible future applications, and its limitations.
\subsection{Going further} We have decided to restrict ourselves in this work to graphs with given girth. Our general framework reduces the remaining work to an analysis of the behaviour of the hard-core distribution in trees of a given depth, which minimises the need to use massive computations through a computer in order to derive our results. In the future, it will be interesting to apply our framework to other classes of graphs. We can imagine any class of graphs which is locally constrained, that is graphs of maximum degree~$d$ whose neighbourhoods up to a fixed depth~$r$ is a strict subset of graphs of maximum degree~$d$ and radius~$r$. For instance, one could consider the Hall ratio and fractional chromatic number of~$C_\ell$-free graphs for a fixed value of~$\ell$, of~$K_4$-free graphs, or of squares of graphs of girth at least some constant~$g$.
It would be interesting to be able to feed our framework with other probability distributions on the independent sets. A promising one would be the following: let~$\mathbf{I}$ be a random maximal independent set obtained greedily by fixing a uniform random priority ordering on the vertices. Shearer~\cite{She83} observed that~$\esp{\mathbf{I}}$ matches the bound given in \Cref{thm:shearer}. However, such a distribution has a strong global dependency; it might be really difficult --- if possible at all --- but probably fruitful, to find a spatial Markov property satisfied by this distribution.
\subsection{On the hard-core model} In the present work, we have fed our method with the hard-core distribution at different regimes. We have considered all the independent sets with a fugacity~$\lambda$ tending to~$0$, restricted ourselves to the maximal independent sets with a fugacity~$\lambda$ equal to~$4$, and then to the maximum independent sets, which forces the uniform distribution with~$\lambda=\infty$.
We observe that when we let the fugacity~$\lambda$ grow, and hence favour larger independent sets, we obtain better bounds for really small values of the degree. On the other hand, we are more constrained in our choices, which translates into worse asymptotic bounds.
It could be unsettling that restricting to maximal independent sets does not yield a strict improvement in the bound obtained with our method; indeed it is always possible to find an optimal fractional colouring using only maximal independent sets as colour classes, up to covering some vertices with weight more that~$1$. However, because we fix the realisation of the random independent set on a large part of the graph, we suffer from a deeper propagation of the constraints due to the maximality of the independent set. When working with a partial realisation of a maximal independent set, vertices at distance~$2$ from that realisation can be forbidden, hence the need to work with deeper patterns (we must always have the freedom to pick the root of the pattern in the independent set).
Another unsettling observation comes from the fact that the optimised fugacity when working with the whole set of independent sets is~$\lambda$ tending to~$0$ as the degree~$d$ tends to infinity. This means that the independent set that receives the largest weight in our fractional colouring is the empty independent set, which seems rather wasteful. This is however needed in order to assign a non negligible weight to the independent set that contains only the root (when we work with patterns of depth 1). Indeed, there is only one independent set that contains the root, while there are exponentially many independent sets that do not contain it. This is no longer the case when we restrict to maximal independent sets, which explains why the optimal value of the fugacity is a constant in this case.
\subsection{Limitations of the method} While our framework gives a lot of freedom in its possible applications, it still suffers from some limitations which we discuss hereafter.
We cannot improve the bounds for the fractional chromatic number by trying to increase the girth. Indeed, since the class of patterns that we need to consider in our framework must be hereditary in that case, increasing the depth would only strictly increase the number of patterns (the ones of smaller depth are still present). Therefore, the girth of the class of graphs for which we can derive our bound is determined by the choice of the probability distribution, and the depth of its dependencies.
Working with maximum independent sets, it appears that our method cannot prove a better bound for~$\rho(d,\infty)$ than~$d/2 + 1$, which is reached at some girth~$g_0$. We could not find an explanation for this boundary, although such an explanation would have a strong theoretical interest. So, in order to obtain new bounds with our method for~$\rho(d,g)$, one would have to use the hard-core model in different regimes than the one where~$\lambda$ is infinite.
The bounds obtained appear not to be tight, if we consider the pairs~$(d,g)$ for which we know the value of~$\rho$ and~$\chi_f$ and compare it with the bound obtained using our method. However, obtaining these tight bounds systematically requires a deep structural analysis already when~$d=3$, and an even deeper one when~$d=4$. It is unlikely that these structural analyses could be generalised in order to cover larger values of~$d$, while our method provides a smooth transition between the small values of~$d$ and the asymptotic regime.
\subsection{Note added} After this work first appeared in a public preprint repository, a work from Cames van Batenburg, Goedgebeur, Joret~\cite{CGJ20} improved our bound for~$\rho(3,6)$, by establishing that it is at most~$8/3$. This work relies again on a deep structural analysis of cubic graphs avoiding some finite family of graphs (all of which contain a~$C_5$) as a subgraph.
\end{document} | arXiv |
\begin{document}
\allowdisplaybreaks \title{Inexact-Proximal Accelerated Gradient Method for Stochastic Nonconvex Constrained Optimization Problems
}
\author{Morteza Boroun, Afrooz Jalilzadeh\footnote{ Department of Systems and Industrial Engineering, The University of Arizona, 1127 E James Rogers Way, Tucson, AZ. Email: [email protected], [email protected].}}
\date{}
\maketitle \begin{abstract} Stochastic nonconvex optimization problems with nonlinear constraints have a broad range of applications in intelligent transportation, cyber-security, and smart grids. In this paper, first, we propose an inexact-proximal accelerated gradient method to solve a nonconvex stochastic composite optimization problem where the objective is the sum of smooth and nonsmooth functions, the constraint functions are assumed to be deterministic and the solution to the proximal map of the nonsmooth part is calculated inexactly at each iteration. We demonstrate an asymptotic sublinear rate of convergence for stochastic settings using increasing sample-size considering the error in the proximal operator diminishes at an appropriate rate. Then we customize the proposed method for solving stochastic nonconvex optimization problems with nonlinear constraints and demonstrate a convergence rate guarantee. Numerical results show the effectiveness of the proposed algorithm. \end{abstract} \section{INTRODUCTION} \label{sec:intro}
There is a rapid growth in the global urban population and the concept of smart cities is proposed to manage the impact of this surge in urbanization. Intelligent transportation, cyber-security, and smart grids are playing vital roles in smart city projects which are highly influenced by big data analytic and effective use of machine learning techniques \cite{ullah2020applications}. As data gets more complex and applications of machine learning algorithms for decision-making broaden and diversify, recent research has been shifted to constrained optimization problems with nonconvex objectives \cite{ma2017demand} to improve efficiency and scalability in smart city projects.
Consider the following constrained optimization problem with a stochastic and nonconvex objective: \begin{align}\label{p0} \nonumber \min_{x\in X} \quad &f(x)\triangleq \mathbb E[F(x,\zeta(\omega))]\\
\mbox{s.t.} \quad &\phi_i(x)\leq 0,\quad i=1,\hdots,m, \end{align} where $ {\zeta}: \Omega \rightarrow \mathbb{R}^o$, ${F}: \mathbb{R}^n \times \mathbb{R}^o \rightarrow \mathbb{R}$, {and} $(\Omega,\mathcal{F},\mathbb{P})$ denotes the associated probability space. We consider function $f(x): \mathbb R^n\to \mathbb R$ is smooth and possibly nonconvex, $\phi_i(x):\mathbb R^n\to \mathbb R$ are \af{deterministic,} convex, and smooth for all $i$, and set $X$ is convex and compact. To solve this problem, first we propose an algorithm for solving the following composite optimization problem \begin{align}\label{p1} &\min_{x\in \mathbb R^n} g(x)\triangleq f(x)+h(x), \end{align} where $h(x): \mathbb R^n\to \mathbb R$ is a convex function and possibly nonsmooth. Using the indicator function $\mathbb{I}_\Theta(\cdot)$, where $\mathbb{I}_\Theta(x)=0$ if $x\in \Theta$ and $\mathbb{I}_\Theta(x)=+\infty$ if $x\notin \Theta$, one can write problem \eqref{p0} in the form of problem \eqref{p1} by choosing
$h(x)= \mathbb{I}_\Theta(x)$ and $\Theta=\{x\mid x\in X, \ \phi_i(x)\leq 0, \ \forall i=1,\hdots,m\}$. Moreover, we show that how to customize the proposed method to solve problem \eqref{p0}. Indeed, proximal-gradient methods are an appealing approach for solving \eqref{p1} due to their computational efficiency and fast theoretical convergence guarantee. In deterministic and convex regime, subgradient methods have been shown to have a convergence rate of $\mathcal O(1/\sqrt{T})$, however, proximal-gradient methods can achieve a faster rate of \af{$\mathcal O(1/ T)$, \af{where $T$ is the total number of iterations}}. Each iteration of a proximal-gradient method requires solving the following: \begin{align}\label{prox}
\mbox{prox}_{\gamma,h}(y)=\underset{u\in \mathbb R^n}{\mbox{argmin}}\{h(u)+{1\over 2\gamma}\|u-y\|^2\}. \end{align} In many scenarios, computing the exact solution of the proximal operator may be expensive or may not have an analytic solution. In this work, we propose a gradient-based scheme to solve the nonconvex optimization problem \eqref{p1} by computing the proximal operator inexactly at each iteration.
Next, we introduce important notations that we use throughout the paper and then briefly summarize the related research. \subsection{Notations}
We denote the optimal objective value (or solution) of \eqref{p1} by $g^*$ (or $x^*$) and the set of the optimal solutions by $X^*$, which is assumed to be nonempty. For any $a\in \mathbb R$, we define $[a]_+=\max\{0,a\}$. $\mathbb{E}[\bullet]$ denotes the expectation with respect to the probability measure $\mathbb{P}$ and $\mathcal B(s)=\{x\in \mathbb R^n \mid \|x\|\leq s\}$. $\Pi_\Theta(\cdot)$ denotes the projection onto convex set $\Theta$ and $\mbox{\bf relint}(X)$ denotes the relative interior of the set $X$. Throughout the paper, $\mathcal{\tilde O}$ is used to suppress all the logarithmic terms. \subsection{Related Works} There has been a lot of studies on first-order methods for convex optimization with convex constraints, see \cite{tran2014primal,xu2019iteration} for deterministic constraints and \cite{basu2019optimal,lan2016algorithms} for stochastic constraints. Nonconvex optimization problems without constraints or with easy-to-compute projection on the constraint set have been studied by \cite{ghadimi2013stochastic,zhang2018convergence,lan2019accelerated}. When the function $f$ in problem \eqref{p1} is convex and $h$ is a nonsmooth function, \cite{schmidt2011convergence} showed that even with errors in the computation of the gradient and the proximal operator, the inexact proximal-gradient method achieves the same convergence rates as the exact counterpart, if the magnitude of the errors is controlled in an appropriate rate. In nonconvex setting, assuming the proximal operator has an exact solution, \cite{ghadimi2016accelerated} obtained a convergence rate of $\mathcal O(1/T)$, using accelerated gradient scheme for deterministic problems and in stochastic regime using increasing sample-size they obtained the same convergence rate. Inspired by these two works, we present accelerated inexact proximal-gradient framework that can solve problems \eqref{p0} and \eqref{p1}. {In deterministic regime,} \cite{kong2019complexity} analyzed the iteration-complexity of a quadratic penalty accelerated inexact proximal point method for solving linearly constrained nonconvex composite programs with iteration complexity of $ \mathcal {\tilde O}(\epsilon^{-3})$. Inexact proximal-point penalty method introduced by \cite{lin2019inexact} and \cite{li2021rate} can solve nonlinear constraints with complexity of $\mathcal{\tilde O}(\epsilon^{-2.5})$ and $ \mathcal {\tilde O}(\epsilon^{-3})$ {for affine equality constraints} and nonconvex constraints, respectively. {Recently, \cite{li2020augmented} showed complexity result of $\mathcal{\tilde O}(\epsilon^{-2.5})$ for deterministic problems with nonconvex objective and convex constraints with nonlinear functions to achieve $\epsilon$-KKT point.}
{In stochastic regime,} \cite{boob2019stochastic} has studied functional constrained optimization problems and obtained a non-asymptotic convergence rate of $\mathcal{ O}(\epsilon^{-2})$ for stochastic problems with convex constraints to achieve $\epsilon^2$-KKT point. {In this paper, we obtain the same convergence rate under weaker assumptions. In particular, in contrast to \cite{boob2019stochastic}, our analysis does not require the objective function to be Lipschitz and we prove an asymptotic convergence rate result.} Next, we outline the contributions of our paper.
\subsection{Contributions} In this paper, we consider a stochastic nonconvex optimization problem with convex nonlinear constraints. We propose an inexact proximal accelerated gradient (IPAG) method where at each iteration the projection onto the nonlinear constraints is solved inexactly. By improving the accuracy of the approximate solution of the proximal subproblem (projection step) at an appropriate rate and ensuring feasibility at each iteration combined with a variance reduction technique, we demonstrate a convergence rate of $\mathcal O(1/T)$, where $T$ is the total number of iterations, and the oracle complexity (number of sample gradients) of $\mathcal O(1/\epsilon^2)$ to achieve an $\epsilon$-first-order optimality of problem \eqref{p0}. \af{To accomplish this task, first we} analyze the proposed method for the composite optimization problem \eqref{p1} which can be specialized to \eqref{p0} using an indicator function. Moreover, our proposed method requires weaker assumptions compare to \cite{boob2019stochastic}.
Next, we state the main definitions and assumptions that we need for the convergence analysis. In Section \ref{conv}, we introduce the IPAG algorithm to solve the composite optimization problem and then in Section \ref{const opt} we show that IPAG method can be customized to solve a nonconvex stochastic optimization problem with nonlinear constraints \eqref{p0}. Finally, in section \ref{numer} we present some empirical experiments to show the benefit of our proposed scheme in comparison with a competitive scheme.
\subsection{Assumptions and Definitions}\label{asd}
Let $\rho$ be the error in the calculation of the proximal objective function achieved by $\tilde x$, i.e.,
\begin{align}\label{prox error}
{1\over 2\gamma}\|\tilde x-y\|^2+h(\tilde x)\leq \rho+\min_{x\in \mathbb R^n}\left\{{1\over 2\gamma}\|x-y\|^2+h(x)\right\}, \end{align} and we call $\tilde x$ a $\rho$-approximate solution to the proximal problem. Next, we define $\rho$-subdifferential and then we state a lemma to characterize the elements of the $\rho$-subdifferential of $h$ at $x$. \begin{definition}[$\rho$-subdifferential]\label{subgrad} Given a convex function $h(x):\mathbb R^n\to \mathbb R$ and a positive scalar $\rho$, the $\rho$-approximate subdifferential of $h(x)$ at a point $x\in \mathbb R^n$, denoted as $\partial_\rho h(x)$, is $$\partial_\rho h(x)=\{d\in \mathbb R^n: h(y)\geq h(x)+\langle d,y-x\rangle-\rho\}.$$ Therefore, when $d\in\partial_\rho h(x)$, we say that $d$ is a $\rho$-subgradient of $h(x)$ at point $x$. \end{definition} \begin{lemma}\label{error subdef}
If $\tilde x$ is a \af{$\rho$-approximate} solution to the proximal problem \eqref{prox} in the sense of \eqref{prox error}, then there exists $v$ such that $\|v\|\leq \sqrt{2\gamma\rho}$ and $$\tfrac{1}{\gamma}\left(y-\tilde x-v\right)\in \partial_{\rho} h(\tilde x).$$ \end{lemma} Proof of Lemma \ref{error subdef} can be found in \cite{schmidt2011convergence}. Throughout the paper, we exploit the following basic lemma. \begin{lemma}\label{3norms} Given a symmetric positive definite matrix $Q$, we have the following for any $\nu_1,\nu_2,\nu_3$:
\begin{align*}(\nu_2-\nu_1)^TQ(\nu_3-\nu_1)={1\over 2}(\|\nu_2-\nu_1\|^2_Q+\|\nu_3-\nu_1\|^2_Q-\|\nu_2-\nu_3\|^2_Q), \mbox{ where } \|\nu\|_Q \triangleq \sqrt{\nu^TQ\nu}.\end{align*} \end{lemma} In our analysis we use the following lemma \cite{ghadimi2016accelerated}. \begin{lemma}\label{bound gamma} Given a positive sequence $\alpha_k$, define $\Gamma_k=1$ for $k=1$ and $\Gamma_k= (1-\alpha_k)\Gamma_{k-1}$ for $k>1$. Suppose a sequence $\{\chi_k\}_k$ satisfies $\chi_k\leq (1-\alpha_k)\chi_{k-1}+\lambda_k$, where $\lambda_k>0$. Then for any $k\geq1$, we have that $\chi_k\leq \Gamma_k\sum_{j=1}^k \gamma_j/\Gamma_j$. \end{lemma} The following assumptions are made throughout the paper. \begin{assumption}\label{assump1} The following statements hold: \begin{itemize} \item[(i)] A slater point of problem \eqref{p0} is available, i.e., there exists $x^{\circ}\in \mathbb R^n$ such that $\phi_i(x^{\circ})<0$ for all $i=1,\hdots,m$ and $x^{\circ}\in \mbox{\bf relint}(X)$. \item[(ii)] Function $f$ is smooth and weakly-convex with Lipschitz continuous gradient, i.e. there exists $L,\ell\geq0$ such that
$-\tfrac{\ell}{2}\|y-x\|^2\leq f(x)-f(y)-\langle \nabla f(x),y-x\rangle \leq \tfrac{L}{2} \|y-x\|^2$.
\item[(iii)] There exists $C>0$ such that $\|\mbox{prox}_{\gamma,h}(y)\|\leq C$ for any $\gamma>0$ and $y\in \mathbb R^n$.
\item[(iv)] $\mathbb{E}[ \xi_k \mid \mathcal{F}_k] = 0$ holds a.s., where $ \xi_k \triangleq \nabla f(z_k) - \nabla F(z_k,\omega_k)$. Also, there exists $\tau>0$ such that {$\mathbb{E}[\|\bar \xi_k\|^2\mid \mathcal{F}_k] \leq {\tau^2\over N_k}$} holds a.s. for all $k$ and $\mathcal{F}_k \triangleq \sigma\left(\{z_0,\bar \xi_0, z_1,\bar \xi_1 \hdots, z_{k-1},\bar\xi_{k-1}\}\right)$, where $ \bar \xi_k\triangleq \frac{\sum_{j=1}^{N_k} \nabla f(z_k)-\nabla F(z_k,\omega_{j,k})}{N_k}$. \end{itemize} \end{assumption} \af{Note that Assumption 1 is a common assumption in nonconvex and stochastic optimization problems and it holds for many real-world problems such as problem of non-negative principal component analysis and classification problem with nonconvex loss functions \cite{pham2020proxsarah}.} \section{CONVERGENCE ANALYSIS}\label{conv}
In this section, we propose an inexact-proximal accelerated gradient scheme for solving problem \eqref{p1} assuming that an inexact solution to the proximal subproblem exists through an inner algorithm $\mathcal M$. Later in section \ref{const opt}, we show that how the inexact solution can be calculated at each iteration for problem \eqref{p0}. Since problem \eqref{p1} is nonconvex, we demonstrate the rate result in terms of \af{$\|{z-\mbox{prox}_{\lambda h}(z-\lambda\nabla f(z))}\|$} which is a standard termination criterion for solving constrained or composite nonconvex problems \cite{nemirovski1983problem,ghadimi2014mini,ghadimi2016accelerated}. \af{For problem \eqref{p0}, the first-order optimality condition is equivalent to find $z^*$ such that $z^*=\Pi_\Theta(z^*-\lambda\nabla f(z^*))$ for some $\lambda>0$. Hence, we show the convergence result in terms of $\epsilon$-first-order optimality condition for a vector $z$, i.e., $\|z-\Pi_\Theta(z-\lambda\nabla f(z))\|^2\leq \epsilon$.}
\begin{algorithm}[htbp] \caption{Inexact-proximal Accelerated Gradient Algorithm (IPAG)} \label{alg1}
{\bf input:} \af{$x_0,y_0 \in \mathbb R^n$, positive sequences $\{\alpha_k,\gamma_k,\lambda_k\}_k$ and Algorithm $\mathcal M$ satisfying Assumption \ref{assump2}}; \\
{\bf for} $k=1\hdots T$ {\bf do} \\ \mbox{(1)}\quad $z_k =(1-\alpha_k)y_{k-1} +\alpha_kx_{k-1}$; \\ \mbox{(2)}\quad $x_k\approx \mbox{prox}_{\gamma_{k}h}\left(x_{k-1}-\gamma_k (\nabla f(z_k )+\bar \xi_k)\right)$ (solved inexactly by algorithm $\mathcal M$ with $q_k$ iterations); \\ \mbox{(3)}\quad $y_k \approx \mbox{prox}_{\lambda_{k}h}\left(z_k -\lambda_k (\nabla f(z_k )+\bar \xi_k)\right)$ (solved inexactly by algorithm $\mathcal M$ with $p_k$ iterations);\\ {\bf end for}\\ {\bf Output:} \af{$z_N$} where $N$ is randomly selected from $\{T/2,\hdots,T\}$ with $\mbox{Prob}\{N=k\}=\frac{1}{\sum_{k=\lfloor T/2\rfloor}^T \tfrac{1-L \lambda_k}{16\lambda_k \Gamma_k}} \left(\tfrac{1-L \lambda_k}{16\lambda_k \Gamma_k}\right)$. \end{algorithm}
\af{\begin{assumption}\label{assump2} For a given $c\in\mathbb R^n$ and $\gamma>0$, consider the problem $\tilde u\triangleq\mbox{prox}_{\gamma h}\left(c\right)$.
An algorithm $\mathcal M$ with an initial point $ u_0$, output $u$ and convergence rate of $\mathcal O(1/t^2)$ within $t$ steps exists, such that $\|u-\tilde u\|^2\leq (a_1\| u_0-\tilde u\|^2+a_2)/t^2$ for some $a_1,a_2>0$.
\end{assumption} Suppose the solutions of proximal operators $\tilde x_k\triangleq\mbox{prox}_{\gamma_{k}h}\left(x_{k-1}-\gamma_k (\nabla f(z_k )+\bar \xi_k)\right)$ and $\tilde y_k \triangleq \mbox{prox}_{\lambda_{k}h}(z_k \break -\lambda_k (\nabla f(z_k )+\bar \xi_k))$ are not available exactly, instead an $e_k$-subdifferential solution $x_k$ and $\rho_k$-subdifferential solution $y_k$ are available, respectively. In particular, given $\bar \xi_k$ for the proximal subproblem in step (2) and (3) of Algorithm \ref{alg1} at iteration $k$,
Assumption \ref{assump2} immediately implies that after $q_k$ and $p_k$ steps of Algorithm $\mathcal M$ with initial point $x_{k-1}$ and $y_{k-1}$, we have $e_k=\gamma_k(c_1\|x_{k-1}-\tilde x_k\|^2+c_2)/q_k^2$ and $\rho_k=\lambda_k(b_1\|y_{k-1} -\tilde y_k \|^2+b_2)/p_k^2$, for some $c_1,c_2,b_1,b_2>0$ where $\gamma_k, \lambda_k$ represents strong convexity of the subproblems, respectively.
Later, in Section \ref{const opt}, we show the existence of Algorithm $\mathcal{M}$ such that it satisfies Assumption \ref{assump2}. \begin{remark} Note that from Assumption \ref{assump1}(iii) and \ref{assump2}, we can show the following for all $k>0$: \begin{align}\label{bound x}
\nonumber&\|x_k-\tilde x_k\|^2\leq \tfrac{1}{q_k^2}\big[2c_1(\|x_{k-1}-\tilde x_{k-1}\|^2+\|\tilde x_{k-1}-\tilde x_k\|^2)+c_2\big]\leq \|x_{k-1}-\tilde x_{k-1}\|^2+\tfrac{8C^2+c_2}{q_k^2}\\
&\implies \|x_k-\tilde x_k\|^2\leq\|x_0-\tilde x_0\|^2+\sum_{j=1}^k \tfrac{8C^2+c_2}{q_j^2} \implies \|x_k\|\leq C+\sqrt{\|x_0-\tilde x_0\|^2+\tilde C} \triangleq B_1, \end{align}
where $\tilde C\triangleq \sum_{j=1}^k \tfrac{8C^2+c_2}{q_j^2}$ and we used the fact that $ \|\tilde x_k\|\leq C$. Similarly for step (3) of Algorithm \ref{alg1}, there exist $B_2,B_3>0$ such that the followings hold for all $k>0$, \begin{align}\label{bound xx}
\|y_k \|\leq B_2, \qquad \|z_k \|\leq B_3. \end{align} \end{remark}} Next, we state our main lemma that provides a bridge towards driving rate statements. \begin{lemma}\label{main lemma} Consider Algorithm \ref{alg1} and suppose \af{Assumption \ref{assump1} and \ref{assump2} hold} and choose stepsizes $\alpha_k$, $\gamma_k$ and $\lambda_k$ such that $\alpha_k\gamma_k\leq\lambda_k$. Let $\hat y_k\approx \mbox{prox}_{\lambda_{k}h}\left(z_k -\lambda_k \nabla f(z_k)\right)$ \aj{in the sense of \eqref{prox error} and $\hat y_k^r\triangleq \mbox{prox}_{\lambda_{k}h}\left(z_k -\lambda_k \nabla f(z_k)\right)$ for any $k\geq 1$}, then the following holds for all $T>0$. \begin{align}\label{lem result}
&\nonumber\mathbb E[\| \hat y_N -z_N \|^2+\| \hat y_N^r -z_N \|^2] \\
&\nonumber\quad \leq \left(\sum_{k=\lfloor T/2\rfloor}^T\tfrac{1-L \lambda_k}{16\lambda_k \Gamma_k} \right)^{-1}\Big[\tfrac{\alpha_1}{2\gamma_1 \Gamma_1}\|x_0-x^*\|^2+\tfrac{\ell}{2}\sum_{k=1}^T \tfrac{\alpha_k}{\Gamma_k}\big[2B_3^2+C^2+\alpha_k(1-\alpha_k)(2B_2^2+B_1^2)\big]\\ &\quad +\sum_{k=1}^T \left(\tfrac{\lambda_k\tau^2}{\Gamma_k N_k(1-L\lambda_k)}+\tfrac{2e_k}{\Gamma_k}+\tfrac{B_1^2+C^2}{\gamma_k\Gamma_k}+\tfrac{\rho_k(1+k)}{\Gamma_k}+\tfrac{B_1^2+B_2^2}{k\lambda_k\Gamma_k}+\tfrac{\aj{5\lambda_k}\tau^2(1-L \lambda_k)}{8\Gamma_kN_k}+\tfrac{\rho_k(1-L \lambda_k)}{\lambda_k^2 \Gamma_k}\right)\Big]. \end{align} \end{lemma}
\begin{proof} First of all from the fact that $\nabla f(x)$ is Lipschitz, for any $k\geq1$, the following holds: \begin{align}\label{lip}
f(y_k )\leq f({z_k })+\langle \nabla f(z_k ),y_k -z_k \rangle +\tfrac{L}{2}\|y_k -z_k \|^2.\end{align} Using Assumption \ref{assump1}(ii), for any $\alpha_k\in(0,1)$ one can obtain the following: \begin{align}\label{bound} \nonumber&f(z_k )-[(1-\alpha_k)f(y_{k-1} )+{\alpha_k} f(x)]\\ \nonumber&=\alpha_k[f(z_k {)}-f(x)]+(1-\alpha_k)[f({z_k )-f(y_{k-1} )}]\\ \nonumber&
\leq \alpha_k[\langle \nabla f(z_k ),z_k -x\rangle+\tfrac{{\ell}}{2}\|z_k -x\|^2]+(1-\alpha_k)[\langle \nabla f(z_k ),{z_k }-y_{k-1} \rangle+\tfrac{{\ell}}{2}\|x_u -y_{k-1} \|^2]\\ \nonumber&
= \langle \nabla f(z_k ),z_k -\alpha_kx-(1-\alpha_k)y_{k-1} \rangle+\tfrac{\ell\alpha_k}{2}\|z_k -x\|^2+\tfrac{\ell(1-\alpha_k)}{2}\|z_k -y_{k-1} \|^2\\ &\leq
\langle \nabla f(z_k ),z_k -\alpha_kx-(1-\alpha_k)y_{k-1} \rangle+\tfrac{\ell\alpha_k}{2}\|z_k -x\|^2+\tfrac{\ell\alpha_k^2(1-\alpha_k)}{2}\|{y_{k-1} }-x_{k-1}\|^2, \end{align}
where in the last inequality we used the fact that $z_k -y_{k-1} =\alpha_k(x_{k-1}-y_{k-1} )$. From Lemma \ref{error subdef}, if $e_k$ be the error in the proximal map of update $x_k$ in Algorithm \ref{alg1} there exists $v_k$ such that $\|v_k\|\leq \sqrt{2\gamma_k e_k}$ and $\tfrac{1}{\gamma_k}\left(x_{k-1}-x_k-\gamma_k(\nabla f(z_k )+\bar \xi_k)-v_k\right)\in {\partial}_{e_k} h(x_k)$. Therefore, from Definition \ref{subgrad}, the following holds:
\begin{align*} h(x)&\geq h(x_k)+\langle \tfrac{1}{\gamma_k}(x_{k-1}-x_k)-\nabla f(z_k )-\bar \xi_k-\tfrac{1}{\gamma_k}v_k, x-x_k\rangle-e_k\\ &\Longrightarrow \langle \nabla f(z_k )+\bar \xi_k,x_k-x\rangle + h(x_k)\leq h(x)-\tfrac{1}{\gamma_k}\langle v_k,x_k-x \rangle+e_k+\tfrac{1}{\gamma_k}\langle x_{k-1}-x_k,x_k-x\rangle{.} \end{align*}
From Lemma \ref{3norms}, we have that $\tfrac{1}{\gamma_k}\langle x_{k-1}-x_k,x_k-x\rangle=\tfrac{1}{2\gamma_k}[\|x_{k-1}-x\|^2-\|x_k-x_{k-1}\|^2-\|{x_k}-x\|^2]$, therefore, \begin{align}\label{bound1}
\nonumber&\langle \nabla f(z_k )+\bar \xi_k,x_k-{x} \rangle+h(x_k)\\ &\quad\leq h(x)-\tfrac{1}{\gamma_k}\langle v_k,x_k-x\rangle+e_k+\tfrac{1}{2\gamma_k}[\|x_{k-1}-x\|^2-\|{x_k-x_{k-1}}\|^2-\|{x_k}-x\|^2]. \end{align}
Similarly if $\rho_k$ be the error of computing the proximal map of update $y_k $ in Algorithm \ref{alg1}, then there exists $w_k$ such that $\|w_k\|\leq \sqrt{2\lambda_k \rho_k}$ and one can obtain the following: \begin{align}\label{b3} \nonumber&\langle\nabla f(z_k )+\bar \xi_k,y_k -x\rangle+h(y_k )\\
&\quad\leq h(x)-\tfrac{1}{\lambda_k}\langle w_k,y_k -x\rangle+\rho_k +\tfrac{1}{2\lambda_k}[\|z_k -x\|^2-\|{y_k }-z_k \|^2-\|{y_k -x}\|^2]. \end{align} Letting $x=\alpha_k x_k + (1-\alpha_k)y_{k-1} $ in \eqref{b3} for any $\alpha_k\geq 0$, the following holds: \begin{align*} &\langle\nabla f(z_k )+ \bar \xi_k,y_k -\alpha_k x_k-(1-\alpha_k)y_{k-1} \rangle+h(y_k )\\
&\quad \leq h(\alpha_k x_k+(1-\alpha_k)y_{k-1} )-\tfrac{1}{\lambda_k}\langle w_k,y_k -\alpha_k x_k-(1-\alpha_k)y_{k-1} \rangle+\rho_k\\&\qquad+\tfrac{1}{2\lambda_k}[\|{z_k }-\alpha_kx_k-(1-\alpha_k)y_{k-1} \|^2-\|y_k -z_k \|^2]. \end{align*} From convexity of $h$ and step (1) of algorithm \ref{alg1} we obtain: \begin{align}\label{bound2} \nonumber &\langle\nabla f (z_k )+\bar \xi_k,{y_k }-\alpha_kx_k-(1-\alpha_k)y_{k-1} \rangle+h(y_k )\\ \nonumber &\quad \leq \alpha_k h(x_k)+(1-\alpha_k)h(y_{k-1} {)}-\tfrac{1}{\lambda_k}\langle w_k,y_k -\alpha_kx_k-(1-\alpha_k)y_{k-1} \rangle +\rho_k\\
&\qquad+\tfrac{1}{2\lambda_k}[\alpha_k^2\|x_k-x_{k-1}\|^2-\|y_k -z_k \|^2. \end{align} Multiplying \eqref{bound1} by $\alpha_k$ and then sum it up with \eqref{bound2} gives us the following \begin{align}\label{sum} \nonumber&\langle \nabla f(z_k )+\bar \xi_k,y_k -\alpha_kx-(1-\alpha_k){y_{k-1} }\rangle+h(y_k )\\
\nonumber&\quad \leq(1-\alpha_k)h(y_{k-1} )+\alpha_k h(x)-\tfrac{\alpha_k}{2\gamma_k}[\|x_{k-1}- x\|^2-\|{x_k}-x\|^2]-\tfrac{1}{\gamma_k}\langle v_k,x_k-x\rangle\\
&\qquad+e_k+\underbrace{\tfrac{\alpha_k(\gamma_k\alpha_k-\lambda_k)}{2\gamma_k\lambda_k}}_{\text{term (a)}}\|x_k-x_{k-1}\|^2-\tfrac{1}{2\lambda_k}\|y_k -z_k \|^2-\tfrac{1}{\lambda_k}\langle w_k,{y_k }-\alpha_kx_k-(1-\alpha_k)y_{k-1} \rangle+\rho_k. \end{align} By choosing $\gamma_k$ such that $\alpha_k\gamma_k\leq\lambda_k$, one can easily confirm that term (a)$\leq 0$. Now combining \eqref{lip}, \eqref{bound} and \eqref{sum} and using the facts that $g(x)=f(x)+h(x)$ and $z_k =y_{k-1} +\alpha_k(x_{k-1}-y_{k-1} )$, we get the following:
\begin{align}\label{bound g}
\nonumber g(y_k )&\leq(1-\alpha_k)g(y_{k-1} )+\alpha_kg(x)-\tfrac{1}{2}(\tfrac{1}{\lambda_k}-L)\|y_k -z_k \|^2+\overbrace{\langle\bar \xi_k,\alpha_k(x-x_{k-1})+z_k -y_k \rangle}^{\text{term (b)}}\\ \nonumber&\quad+\tfrac{\alpha_k}{2\gamma_k}[\|{x_{k-1}-x}\|^2-\|x_k-x\|^2]+\tfrac{{\ell\alpha_k}}{2}\|x_{md}-x\|^2+\tfrac{{\ell\alpha_k^2}(1-\alpha_k)}{2}\|y_{k-1} -x_{k-1}\|^2\\ &\quad-\tfrac{1}{\gamma_k}\langle v_k,x_k-x\rangle+e_k-\tfrac{1}{\lambda_k}\langle w_k,{y_k }-\alpha_kx_k-(1-\alpha_k)y_{k-1} \rangle+\rho_k. \end{align} Moreover one can bound term (b) as follows using the Young's inequality. \begin{align}\label{termb} \nonumber \langle\bar \xi_k,\alpha_k(x-x_{k-1})+z_k -y_k \rangle&=\langle\bar \xi_k,\alpha_k(x-x_{k-1})\rangle+\langle\bar \xi_k,z_k -y_k \rangle\\
&\leq \langle\bar \xi_k,\alpha_k(x-x_{k-1})\rangle+\tfrac{\lambda_k}{1-L\lambda_k}\|z_k -y_k \|^2+\tfrac{1-L\lambda_k}{4\lambda_k}\|\bar \xi_k\|^2. \end{align} Using \eqref{termb} in \eqref{bound g}, we get the following. \begin{align*}
g(y_k )&\leq(1-\alpha_k)g(y_{k-1} )+\alpha_kg(x)-\tfrac{1}{4}(\tfrac{1}{\lambda_k}-L)\|y_k -z_k \|^2+\langle\bar \xi_k,\alpha_k(x-x_{k-1})\rangle+\tfrac{\lambda_k}{1-L\lambda_k}\|\bar \xi_k\|^2\\ &\quad+\tfrac{\alpha_k}{2\gamma_k}[\|{x_{k-1}-x}\|^2-\|x_k-x\|^2]+\tfrac{{\ell\alpha_k}}{2}\|x_{md}-x\|^2+\tfrac{{\ell\alpha_k^2}(1-\alpha_k)}{2}\|y_{k-1} -x_{k-1}\|^2\\ &\quad-\tfrac{1}{\gamma_k}\langle v_k,x_k-x\rangle+e_k-\tfrac{1}{\lambda_k}\langle w_k,{y_k }-\alpha_kx_k-(1-\alpha_k)y_{k-1} \rangle+\rho_k. \end{align*} Subtract $g(x)$ from both sides, using lemma \ref{bound gamma}, assuming ${\tfrac{\alpha_k}{\lambda_k\Gamma_k}}$ is a non-decreasing sequence and summing over $k$ from $k=1$ to T, the following can be obtained.
\begin{align*}
&\tfrac {g(x_T )-g(x)}{\Gamma_T}+\sum_{k=1}^T \tfrac{1-L \lambda_k}{4\lambda_k \Gamma_k}\|y_k -z_k \|^2\\
&\quad \leq \tfrac{\alpha_1}{2\gamma_1 \Gamma_1}\|x_0-x\|^2-\tfrac{\alpha_{T+1}}{2\gamma_{T+1}\Gamma_{T+1}}\|x_T-x\|^2+\tfrac{\ell}{2}\sum_{k=1}^T \tfrac{\alpha_k}{\Gamma_k}\big[\|{z_k }-x\|^2+\alpha_k(1-\alpha_k)\|{y_{k-1} }-x_{k-1}\|^2\big]\\
&\qquad+\sum_{k=1}^T\tfrac{\alpha_k}{\Gamma_k} \langle\bar \xi_k,x-x_{k-1}\rangle+\sum_{k=1}^T\tfrac{\lambda_k}{\Gamma_k(1-L\lambda_k)}\|\bar \xi_k\|^2\\ &\qquad-\sum_{k=1}^T\big[\tfrac{1}{\gamma_k\Gamma_k}\langle v_k, x_k-x\rangle+\tfrac{e_k}{\Gamma_k}-\tfrac{1}{\lambda_k\Gamma_k}\langle w_k,{y_k }-\alpha_kx_k-(1-\alpha_k){y_{k-1} }\rangle+\tfrac{\rho_k}{\Gamma_k}\big]. \end{align*}
Letting $x=x^*$ and using Assumption \ref{assump1}(iii), inequalities \eqref{bound x} and \eqref{bound xx} and the fact that $\|v_k\|\leq \sqrt{2\gamma_k e_k}$ and $\|w_k\|\leq \sqrt{2\lambda_k \rho_k}$, we can simplify the above inequality as follows: \begin{align*}
\tfrac {g(x_T )-g(x^*)}{\Gamma_T}+\sum_{k=1}^T \tfrac{1-L \lambda_k}{4\lambda_k \Gamma_k}\|y_k -z_k \|^2 &\leq \tfrac{\alpha_1}{2\gamma_1 \Gamma_1}\|x_0-x^*\|^2+\tfrac{\ell}{2}\sum_{k=1}^T \tfrac{\alpha_k}{\Gamma_k}\big[2B_3^2+C^2+\alpha_k(1-\alpha_k)(2B_2^2+B_1^2)\big]\\
&+\sum_{k=1}^T\tfrac{\alpha_k}{\Gamma_k} \langle\bar \xi_k,x^*-x_{k-1}\rangle+\sum_{k=1}^T\tfrac{\lambda_k}{\Gamma_k(1-L\lambda_k)}\|\bar \xi_k\|^2\\ &+\sum_{k=1}^T \big(\tfrac{2e_k}{\Gamma_k}+\tfrac{B_1^2+C^2}{\gamma_k\Gamma_k}+\tfrac{\rho_k(1+k)}{\Gamma_k}+\tfrac{B_1^2+B_2^2}{k\lambda_k\Gamma_k}\big). \end{align*}
Using the fact that $g(x_T )-g(x^*)\geq 0$, taking conditional expectation from both sides and applying Assumption \ref{assump1}(iv) on the conditional first and second moments, we get the following. \begin{align}\label{exp}
\nonumber\sum_{k=1}^T \tfrac{1-L \lambda_k}{4\lambda_k \Gamma_k}\mathbb E[\| y_k -z_k \|^2\mid \mathcal F_k] &\leq \tfrac{\alpha_1}{2\gamma_1 \Gamma_1}\|x_0-x^*\|^2+\tfrac{\ell}{2}\sum_{k=1}^T \tfrac{\alpha_k}{\Gamma_k}\big[2B_3^2+C^2+\alpha_k(1-\alpha_k)(2B_2^2+B_1^2)\big]\\
&+\sum_{k=1}^T\tfrac{\lambda_k\tau^2}{\Gamma_k N_k(1-L\lambda_k)}+\sum_{k=1}^T \big(\tfrac{2e_k}{\Gamma_k}+\tfrac{B_1^2+C^2}{\gamma_k\Gamma_k}+\tfrac{\rho_k(1+k)}{\Gamma_k}+\tfrac{B_1^2+B_2^2}{k\lambda_k\Gamma_k}\big). \end{align}
\aj{To bound the left-hand side we use the following inequality by defining $y^r_k\triangleq \mbox{prox}_{\lambda_{k}h}\left(z_k -\lambda_k (\nabla f(z_k)+\bar \xi_k)\right)$ and $\hat y^r_k\triangleq \mbox{prox}_{\lambda_{k}h}\left(z_k -\lambda_k \nabla f(z_k)\right)$. \begin{align*}
\|y_k -z_k \|^2&={1\over 2}\|y_k -z_k \|^2+{1\over 2}\|y_k -z_k \|^2\\
&\geq {1\over 4}\|\hat y_k -z_k \|^2-{1\over 2}\|\hat y_k -y_k \|^2+{1\over 4}\|\hat y_k^r -z_k \|^2-{1\over 2}\|\hat y_k^r -y_k \|^2\\
&\geq {1\over 4}\|\hat y_k -z_k \|^2+{1\over 4}\|\hat y_k^r -z_k \|^2-{3\over 2}\|\hat y_k -\hat y_k^r \|^2-{5\over 2}\|\hat y_k^r -y_k^r \|^2-{5\over 2}\| y_k^r -y_k \|^2, \end{align*} where we used the fact that for any $a,b\in \mathbb R$, we have that $(a-b)^2\geq\tfrac{1}{2}a^2-b^2$ and for any $a_i\in \mathbb R$, $(\sum_{i=1}^ma_i)^2\leq m\sum_{i=1}^ma_i^2$.
From Assumption \ref{assump1}(iv), we know that $\|\hat y^r_k -y^r_k \|^2\leq \lambda_k^2\tau^2/N_k$, also we know that $\|\hat y_k -\hat y^r_k \|^2\leq \rho_k/\lambda_k$ and similarly $\| y_k - y^r_k \|^2\leq \rho_k/\lambda_k$. Therefore, one can conclude that $\|y_k -z_k \|^2\geq \tfrac{1}{4}\|\hat y_k -z_k \|^2+\tfrac{1}{4}\|\hat y_k^r -z_k \|^2-\tfrac{5}{2}\lambda_k^2\tau^2/N_k-4\rho_k/\lambda_k$}. Hence, by taking another expectation from \eqref{exp} and then using this bound, the following can be obtained. \begin{align*}
&\nonumber\sum_{k=1}^T \tfrac{1-L \lambda_k}{16\lambda_k \Gamma_k}\mathbb E[\|\hat y_k -z_k \|^2+\|\hat y_k^r -z_k \|^2] \\
&\quad \leq \tfrac{\alpha_1}{2\gamma_1 \Gamma_1}\|x_0-x^*\|^2+\tfrac{\ell}{2}\sum_{k=1}^T \tfrac{\alpha_k}{\Gamma_k}\big[2B_3^2+C^2+\alpha_k(1-\alpha_k)(2B_2^2+B_1^2)\big]\\%+\sum_{k=1}^T\tfrac{\lambda_k\tau^2}{\Gamma_k N_k(1-L\lambda_k)}\\ &\quad +\sum_{k=1}^T \left(\tfrac{\lambda_k\tau^2}{\Gamma_k N_k(1-L\lambda_k)}+\tfrac{2e_k}{\Gamma_k}+\tfrac{B_1^2+C^2}{\gamma_k\Gamma_k}+\tfrac{\rho_k(1+k)}{\Gamma_k}+\tfrac{B_1^2+B_2^2}{k\lambda_k\Gamma_k}+\tfrac{\aj{5\lambda_k}\tau^2(1-L \lambda_k)}{8\Gamma_kN_k}+\tfrac{\rho_k(1-L \lambda_k)}{\lambda_k^2 \Gamma_k}\right).
\end{align*}
Using the fact that $\sum_{k=\lfloor T/2\rfloor}^T A_t\leq \sum_{k=1}^T A_t$ where $A_t=\tfrac{1-L \lambda_k}{16\lambda_k \Gamma_k}\mathbb E[\|\hat y_k -z_k \|^2+\|\hat y_k^r -z_k \|^2]$, dividing both side by $\sum_{k=\lfloor T/2\rfloor}^T \tfrac{1-L \lambda_k}{16\lambda_k \Gamma_k}$ and using definition of $N$ in Algorithm \ref{alg1}, the desired result can be obtained. \end{proof} We are now ready to prove our main rate results. \begin{theorem}\label{th1} Let $\{y_k ,x_k,z_k \}$ generated by Algorithm \ref{alg1} such that at each iteration $k\geq 1$, \aj{$e_k$-approximate solution of step (2) and $\rho_k$-approximate solution of step (3) are available through an inner algorithm $\mathcal M$}. Suppose \af{Assumption \ref{assump1} and \ref{assump2} hold} and we select the parameters in Algorithm \ref{alg1} as $\alpha_k=\tfrac{2}{k+1}$, $\gamma_k=\tfrac{k}{4L}$, $\lambda_k=\tfrac{1}{2L}$, $\Gamma_k=\tfrac{2}{k(k+1)}$ and $N_k=k+1$. Then for $B=B_1^2+B_2^2+B_3^2+C^2$, the following holds for all $T>0$. \begin{align}\label{th result}
\mathbb E[\|\hat y_N -z_N \|^2+\|\hat y_N^r -z_N \|^2]\leq \tfrac{128}{LT^3}\left[ 2BT(T+1)\left(\tfrac{\ell}{4}+\tfrac{13\tau^2}{64LB}+4L\right)+\sum_{k=1}^T \left(\tfrac{2e_k}{\Gamma_k}+\tfrac{\rho_k(1+k)}{\Gamma_k}+\aj{\tfrac{4L^2\rho_k}{\Gamma_k}}\right)\right], \end{align} where $\hat y_k\approx \mbox{prox}_{\lambda_{k}h}\left(z_k -\lambda_k \nabla f(z_k)\right)$ \aj{in the sense of \eqref{prox error}, and $\hat y_k^r= \mbox{prox}_{\lambda_{k}h}\left(z_k -\lambda_k \nabla f(z_k)\right)$ for any $k\geq 1$}. \end{theorem}
\begin{proof} Using the definition of $\lambda_k$ and $\Gamma_k$, we get the following. \begin{align}\label{sum1} \sum_{k=\lfloor T/2\rfloor}^T \tfrac{1-L \lambda_k}{16\lambda_k \Gamma_k}=\sum_{k=\lfloor T/2\rfloor}^T \tfrac{Lk(k+1)}{32}=\tfrac{L}{32}\left[\tfrac{7T^3}{24}+T^2+\tfrac{5T}{6}\right]\geq \tfrac{LT^3}{128}. \end{align} \af{Next, using the definition of parameters specified in the statement of the theorem we have that} \begin{align}\label{sum2} \nonumber&\sum_{k=1}^T \tfrac{\alpha_k}{\Gamma_k}=\sum_{k=1}^T k=\tfrac{T(T+1)}{2}\af{,}\qquad\qquad\qquad\qquad \sum_{k=1}^T \tfrac{\tau^2}{\Gamma_kN_k}=\sum_{k=1}^T \tfrac{\tau^2k}{2}=\tfrac{\tau^2T(1+T)}{4}\af{,}\\ &\sum_{k=1}^T \tfrac{1}{\gamma_k\Gamma_k}=\sum_{k=1}^T2L(k+1)=2LT(T+3)\af{,}\qquad \sum_{k=1}^T\tfrac{1}{k\lambda_k\Gamma_k}=\sum_{k=1}^TL(k+1)=LT(T+3). \end{align} Using \eqref{sum1} and \eqref{sum2} in \eqref{lem result} \af{and} the fact that $\alpha_k(1-\alpha_k)\leq 1$, $\tfrac{T+3}{T+1}\leq 2$ and defining $B=B_1^2+B_2^2+B_3^2+C^2$ we get the desired result.
\end{proof}
\begin{corollary}\label{cor1}
Let $\{y_k ,x_k,z_k \}$ \aj{be} generated by Algorithm \ref{alg1} such that at each iteration $k\geq 1$, \aj{$e_k$-approximate solution of step (2) and $\rho_k$-approximate solution of step (3) are calculated} by an inner algorithm $\mathcal M$ \aj{where} $e_k=\gamma_k(c_1\|x_{k-1}-\tilde x_k\|^2+c_2)/q_k^2$ and $\rho_k=\lambda_k(b_1\|y_{k-1} -\tilde y_k \|^2+b_2)/p_k^2$. Suppose \af{Assumptions \ref{assump1} and \ref{assump2} hold} and $p_k=k+1$ and $q_k=k$. If we choose the stepsize parameters as in Theorem \ref{th1}, then the following holds for all \aj{$T\geq 1$}. \begin{align}\label{conv rate}
&\mathbb E[\|\hat y_N -z_N \|^2+\|\hat y_N^r -z_N \|^2]\leq\af{ \tfrac{D_1}{T}+\tfrac{D_2}{T^2}},\\ \nonumber & D_1\triangleq\tfrac{128}{L}\left[4B\left(\tfrac{\ell}{4}+\tfrac{13\tau^2}{64LB}+4L\right)+\left(\tfrac{c_1(2B_1^2+C^2)+c_2}{L}\right)+\left(\tfrac{b_1(2B_2^2+C^2)+b_2}{4L}\right)\right], \\& \nonumber D_2\triangleq{128}\left({b_1(2B_2^2+C^2)+b_2}\right), \end{align} where $\hat y_k\approx \mbox{prox}_{\lambda_{k}h}\left(z_k -\lambda_k \nabla f(z_k)\right)$ \aj{in the sense of \eqref{prox error} and $\hat y_k^r= \mbox{prox}_{\lambda_{k}h}\left(z_k -\lambda_k \nabla f(z_k)\right)$ for any $k\geq 1$}.
The oracle complexity (number of gradient samples) to achieve $\mathbb E[\|\hat y_N -z_N \|^2+\|\hat y_N^r -z_N \|^2]\leq \epsilon$ is $\mathcal O(1/\epsilon^2)$. \end{corollary} \begin{proof} Using the definition of the stepsizes, $p_k$, $e_k$, and $\rho_k$ one can obtain the following: \begin{align*} &\sum_{k=1}^T \tfrac{2e_k}{\Gamma_k}\leq \tfrac{c_1(2B_1^2+C^2)+c_2}{4L}\sum_{k=1}^T (k+1)=\left(\tfrac{c_1(2B_1^2+C^2)+c_2}{4L}\right)T(T+3).\\ &\sum_{k=1}^T \tfrac{\rho_k(1+k)}{\Gamma_k}\leq \tfrac{b_1(2B_2^2+C^2)+b_2}{4L}\sum_{k=1}^T k=\left(\tfrac{b_1(2B_2^2+C^2)+b_2}{8L}\right)T(T+1).\\ &\aj{\sum_{k=1}^T\tfrac{\rho_k}{\Gamma_k}=\left(\tfrac{b_1(2B_2^2+C^2)+b_2}{4L}\right)\sum_{k=1}^T \tfrac{k(k+1)}{(k+1)^2}\leq \left(\tfrac{b_1(2B_2^2+C^2)+b_2}{4L}\right)\sum_{k=1}^T 1=\left(\tfrac{b_1(2B_2^2+C^2)+b_2}{4L}\right)T.} \end{align*}
Using the above inequalities in \eqref{th result}, we get the desired convergence result. Additionally, \aj{the} total number of sample gradients of the objective is $\sum_{k=1}^T N_k=\sum_{k=1}^T (k+1)=T(T+3)$ and \aj{the} total number of gradients of the constraint is $\sum_{k=1}^T p_k+q_k=\sum_{k=1}^T 2k+1=T(T+2)$. From \eqref{conv rate}, we have that $\mathbb E[\|\tilde y_N -z_N \|^2]\leq \mathcal O(1/T)=\epsilon$, hence, $\sum_{k=1}^T N_k=\mathcal O(1/\epsilon^2)$ and similarly $\sum_{k=1}^T p_k+q_k=\mathcal O(1/\epsilon^2)$. \end{proof}
\af{In the next corollary, we justify our choice of measure. We show that if $\mathbb E[\|\hat y_N^r -z_N \|^2]\leq\epsilon$, then the first order optimality condition \aj{for problem \eqref{p1}} holds within a ball \aj{with} radius $\sqrt \epsilon$.
\begin{corollary} Under the premises of Corollary \ref{cor1}, after running Algorithm \ref{alg1} for $T\geq D/\epsilon$ iterations, where $D\triangleq D_1+D_2$, the following holds. $$0\in \mathbb E[\nabla f(\aj{\hat y_N^r})]+\mathbb E[\partial \aj{h(\hat y_N^r)}]+\mathcal{B}\left(3L\sqrt \epsilon\right).$$\end{corollary}
\begin{proof} Suppose $\hat y_N^r$ is a solution of $ \mbox{prox}_{\lambda_{N}h}\left(z_N -\lambda_N \nabla f(z_N)\right)$. Then $0\in \partial h(\hat y_N^r)+\nabla f(z_N)+(\hat y_N^r-z_N)/\lambda. $ Adding and subtracting $\nabla f(\hat y_N^r)$ form the right-hand side of the above inequality, gives the following: \begin{align}\label{subgrad h} 0\in \partial h(\hat y_N^r)+\nabla f(z_N)+{1/\lambda}(\hat y_N^r-z_N)\pm \nabla f(\hat y_N^r). \end{align}
Moreover, using the fact that $T\geq D/\epsilon$ and $\mathbb E[\|\hat y_N^r -z_N \|^2]\leq\tfrac{D}{T}=\epsilon$ one can show the following result. \begin{align*}
\mathbb E\left[\|\nabla f(z_N)-\nabla f(\hat y_N^r)+1/\lambda(\hat y_N^r-z_N)\|\right]&\leq \mathbb E\left[L\|\hat y_N^r-z_N\|+1/\lambda\|\hat y_N^r-z_N\|\right]\leq 3L\sqrt {\epsilon}, \end{align*} where we use the fact that $\lambda=1/(2L)$. Using the above inequality and taking expectation from \eqref{subgrad h} the desired result can be obtained. \end{proof}}
In the next section, we show how Algorithm \ref{alg1} can be customized to solve problem \eqref{p0}. \subsection{Constrained Optimization}\label{const opt} Recall that problem \eqref{p0} can be written in a composite form using an indicator function, i.e. problem \eqref{p0} is equivalent to $\min_x g(x)=f(x)+h(x)$, where $h(x)= \mathbb{I}_\Theta(x)$ and $\Theta=\{x\mid x\in X, \ \phi_i(x)\leq 0, \ \forall i=1,\hdots,m\}$. In step (2) and (3) of Algorithm \ref{alg1}, one needs to compute the proximal operators inexactly which are of the following form: \begin{align}\label{co}
&\min_{u\in X} \quad {1\over 2\gamma}\left\|u- y\right\|^2\quad \mbox{s.t.} \quad \phi_i(u)\leq 0,\quad i=1,\hdots,m, \end{align} for some $y\in\mathbb{R}^n$. Problem \eqref{co} has a strongly convex objective function with convex constraints, and there has been variety of methods developed to solve such problems. One of the efficient methods for solving large-scale convex constrained optimization problem with strongly convex objective \af{that satisfies Assumption \ref{assump2}} is first-order primal-dual scheme that guarantees a convergence rate of $\mathcal O(1/\sqrt \epsilon)$ in terms of suboptimality and infeasibility, e.g., \cite{he2015mirror,hamedani2018primal}. Next, we discuss some details of implementing such schemes as an inner algorithm for solving the subproblems in step (2) and (3) of Algorithm \ref{alg1}.
Based on Corollary \ref{cor1}, to obtain a convergence rate of $\mathcal O(1/T)$, one needs to find an $e_k$- and $\epsilon_k$-approximated solution in the sense of \eqref{prox error}. Note that since the nonsmooth part of the objective function, $h(x)$, in the proximal subproblem is an indicator function, \eqref{prox error} implies that the approximate solution of the subproblem has to be feasible, otherwise the indicator function on the left-hand side of \eqref{prox error} goes to infinity. However, the first-order primal-dual methods mentioned above find an approximate solution which might be infeasible. To remedy this issue, let $x^{\circ}$ be a slater feasible point of \eqref{co} (i.e., $\phi_i(x^{\circ})<0$ for all $i=1,\hdots,m$) and let $\hat x$ be the output of the inner algorithm $\mathcal M$ such that it is $\epsilon$-suboptimal and $\epsilon$-infeasible, then $\tilde x=\kappa x^{\circ}+(1-\kappa)\hat x$ is a feasible point of \eqref{co} for $\kappa\triangleq \max_i \tfrac{[\phi_i(\hat x)]_+}{[\phi_i(\hat x)]_+-\phi_i(x^{\circ})}$ which is $\mathcal O(\epsilon)$-suboptimal, see the next lemma for the proof. \begin{algorithm}[htbp] \caption{IPAG for constrained optimization} \label{alg2}
{\bf input:} \af{$x^{\circ},x_0,y_0 \in \mathbb R^n$ and positive sequences $\{\alpha_k,\gamma_k,\lambda_k\}_k$, and Algorithm $\mathcal M$ satisfying Assumption \ref{assump2}}; \\
{\bf for} $k=1\hdots T$ {\bf do} \\ \mbox{(1)}\quad $z_k =(1-\alpha_k)y_{k-1} +\alpha_kx_{k-1}$; \\ \mbox{(2)}\quad $x\approx \Pi_\Theta\left(x_{k-1}-\gamma_k (\nabla f(z_k )+\bar \xi_k)\right)$ (solved inexactly by algorithm $\mathcal M$ with $q_k$ iterations); \\ \mbox{(3)}\quad $y \approx\Pi_\Theta\left(z_k -\lambda_k (\nabla f(z_k )+\bar \xi_k)\right)$ (solved inexactly by algorithm $\mathcal M$ with $p_k$ iterations);\\ \mbox{(4)}\quad $\kappa=\max_i \tfrac{[\phi_i(x)]_+}{[\phi_i(x)]_+-\phi_i(x^{\circ})}$ and $\tilde \kappa=\max_i \tfrac{[\phi_i(y)]_+}{[\phi_i(y)]_+-\phi_i(x^{\circ})}$;\\ \mbox{(5)}\quad $ x_k=\kappa x^{\circ}+(1-\kappa) x$;\\ \mbox{(6)}\quad $ y_k=\tilde \kappa x^{\circ}+(1-\tilde \kappa) y$;\\ {\bf end for}\\ {\bf Output:} \af{$z_N$ where $N$ is randomly selected} from $\{T/2,\hdots,T\}$ with $\mbox{Prob}\{N=k\}=\frac{1}{\sum_{k=\lfloor T/2\rfloor}^T \tfrac{1-L \lambda_k}{16\lambda_k \Gamma_k}} \left(\tfrac{1-L \lambda_k}{16\lambda_k \Gamma_k}\right)$. \end{algorithm}
\begin{lemma}\label{convex comb} Let $x^{\circ}$ be a strictly feasible point of \eqref{co} and $\hat x$ be the output of an inner algorithm $\mathcal M$ such that it is $\epsilon$-suboptimal and $\epsilon$-infeasible solution of \eqref{co}. Then $\tilde x=\kappa x^{\circ}+(1-\kappa)\hat x$ is a feasible point of \eqref{co} and an $\mathcal O(\epsilon)$-approximate solution in the sense of \eqref{prox error} where $\kappa= \max_i \tfrac{[\phi_i(\hat x)]_+}{[\phi_i(\hat x)]_+-\phi_i(x^{\circ})}$. \end{lemma}
\begin{proof} Let $x^*$ be the optimal solution of \eqref{co}. Since $\hat x$ is $\epsilon$-suboptimal and $\epsilon$-infeasible solution, $\hat x\in X$ and the following holds: \begin{align*}
\big|\tfrac{1}{2\gamma}\|\hat x-y\|^2-\tfrac{1}{2\gamma}\|x^*-y\|\big|\leq \epsilon,\quad \mbox{and}\quad [\phi_i(\hat x)]_+\leq \epsilon, \ \forall i\in\{1,\hdots,m\}. \end{align*} Since $X$ is a convex set and $x^{\circ},\hat x\in X$, then clearly $\kappa x^{\circ}+(1-\kappa)\hat x\in X$ for any $\kappa\in[0,1]$. Moreover, $\phi_i(x^\circ)<0$ for all $i$, hence $\kappa= \max_i \tfrac{[\phi_i(\hat x)]_+}{[\phi_i(\hat x)]_+-\phi_i(x^{\circ})}\in [0,1]$ and $\kappa\leq \tfrac{\epsilon}{\min_i\{-\phi_i(x^{\circ})\}}$. From convexity of $\phi_i(\cdot)$, one can show the following for all $i=1,\hdots,m$. \begin{align*} \phi_i(\tilde x)\leq \kappa\phi_i(x^{\circ})+(1-\kappa)\phi_i(\hat x)\leq 0, \end{align*} where we used the definition of $\kappa$. Hence, $\tilde x$ is a feasible point of \eqref{co}. Next, we verify $\tilde x$ satisfies \eqref{prox error}. \begin{align*}
&\tfrac{1}{2\gamma}\|\tilde x-y\|^2+\mathbb I_\Theta(\tilde x)-\tfrac{1}{2\gamma}\|x^*-y\|^2-\mathbb I_\Theta(x^*)\\
&\quad =\tfrac{1}{2\gamma}\|\tilde x-y\pm x^{\circ}\|^2-\tfrac{1}{2\gamma}\|x^*-y\|^2\\
&\quad \leq \tfrac{\kappa^2}{2\gamma}\|x^\circ-y\|^2+\tfrac{(1-\kappa)^2}{2\gamma}\|\hat x-y\|^2+\tfrac{\kappa(1-\kappa)}{\gamma}\|x^\circ-y\|^2\|\hat x-y\|^2-\tfrac{1}{2\gamma}\|x^*-y\|^2\\
&\quad = \tfrac{\kappa^2}{2\gamma}\|x^\circ-y\|^2+\tfrac{\kappa(1-\kappa)}{\gamma}\|x^\circ-y\|^2\|\hat x-y\|^2+(1-\kappa^2)\left[\tfrac{1}{2\gamma}\|\hat x-y\|-\tfrac{1}{2\gamma}\|x^*-y\|\right]\\
&\qquad-\tfrac{1-(1-\kappa^2)}{2\gamma}\|x^*-y\|^2\\
&\quad \leq \tfrac{\kappa^2}{2\gamma}\|x^\circ-y\|^2+\tfrac{\kappa(1-\kappa)}{\gamma}\|x^\circ-y\|^2\|\hat x-y\|^2+\epsilon\leq \mathcal O(\epsilon), \end{align*}
where we used the fact that $\hat x, x^*$ are feasible, $\hat x$ is $\epsilon$-suboptimal and $\kappa\leq \tfrac{\epsilon}{\min_i\{-\phi_i(x^{\circ})\}}$.
\end{proof}
In the following corollary, we show that the output of Algorithm \ref{alg2} is feasible to problem \eqref{p0} and satisfies $\epsilon$-first-order optimality condition. \begin{corollary}\label{cor4}
Consider problem \eqref{p0}. Suppose \af{Assumption \ref{assump1} and \ref{assump2} hold} and let $\{y_k ,x_k,z_k \}$ \aj{be} generated by Algorithm \ref{alg2} such that the stepsizes and parameters are chosen as in Corollary \ref{cor1}. Then the iterates are feasible and $\mathbb E\left[\|z_N-\Pi_\Theta\left(z_N-\lambda_N\nabla f(z_N)\right)\|^2\right]\leq \mathcal O(\epsilon)$ holds with an oracle complexity $\mathcal O(1/\epsilon^2)$.
\end{corollary} \begin{proof}
From Lemma \ref{convex comb} we know that the iterates are feasible and from Corollary \ref{cor1}, we conclude that $\mathbb E[\hat \|y_N^r-z_N\|^2]\leq \epsilon$ with an oracle complexity $\mathcal O(1/\epsilon^2)$. Considering problem \eqref{p0}, definition of $\hat y_N^r$ is equivalent to $\hat y_N^r=\Pi_\Theta\left(z_N-\lambda_N\nabla f(z_N)\right)$ which implies the desired result. \end{proof}
\section{NUMERICAL EXPERIMENTS}\label{numer} The goal of this section is to present some computational results to compare the performance of the IPAG method with another competitive scheme. \aj{For Algorithm \ref{alg2}, we consider accelerated primal-dual algorithm with backtracking (APDB)} method introduced by \cite{hamedani2018primal} as the inner algorithm $\mathcal M$. \aj{In particular, APDB is a primal-dual scheme with a convergence guarantee of \af{$\mathcal O(1/T^2)$} in terms of suboptimality and infeasibility when implemented for solving \eqref{co} which satisfies the requirements of Corollary \ref{cor4}, i.e., produces approximate solutions for the proximal subproblems.}
\indent {\bf Example.} The IPAG method is benchmarked against the inexact constrained proximal point algorithm (ICPP) introduced by \cite{boob2019stochastic}. Consider the following \aj{stochastic} quadratic programming problem: \begin{align*}
\min_{-10\leq x\leq 10} \ &f(x)\triangleq-\tfrac{\epsilon}{2}\|DBx\|^2+ \tfrac{\tau}{2}\mathbb E[\|Ax-b(\xi)\|^2]\\ \text{s.t.} \quad &\ \tfrac{1}{2}x^T Q_{i} x+d_{i}^Tx-c_i \leq 0, \quad \forall i=1\hdots m, \end{align*} where $ \aj{A} \in \mathbb R^{p\times n}$, $p=n/2$, $B \in \mathbb R^{n\times n}, D \in \mathbb R^{n\times n}$ is a diagonal matrix, $b(\xi)= b+\omega \in \mathbb R^{p\times 1}$, where the elements of $\omega$ have an i.i.d. \aj{standard normal distribution}. \mb{ The entries of matrices $A$, $B$, and vector $b$ are generated by sampling from the uniform distribution ${U}$[0,1] and the diagonal entries of matrix $D$ are generated by sampling from the discrete uniform distribution ${U}$\{1,1000\}}. Moreover, $(\delta, \tau) \in \mathbb R_{++}^2$ , $ Q_i \in \mathbb R^{n\times n}$, $ d_i \in \mathbb R^{n\times 1}$ and $ c_i \in \mathbb R$ for all $i\in\{1,\hdots,m\}$. We chose scalers $\delta$ and $\tau$ such that $\lambda_{min}(\nabla^2f)<0$, i.e., minimum eigenvalue of the Hessian is negative. Note that Assumption \ref{assump1}(i) holds for $x^{\circ}=\mathbf 0$, where $\mathbf 0$ is the vector of zeros.
\begin{table}[htb]\scriptsize
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|} \hline
\multicolumn{2}{|c|}{} & \multicolumn{3}{c|}{IPAG} & \multicolumn{3}{c|}{ICPP} \\ \hline n & m & $f(x_T)$ & Infeas. & CPU(s) & $f(x_T)$ & Infeas. & CPU(s) \\ \hline 100 & 25 & -6.78e+5 & 0 & 12.10 & -4.85e+4 & 3.56e-1 & 32.99 \\ \hline 100 & 50 & -8.53e+5 & 0 & 31.76 & -2.42e+4 & 3.23e-1 & 65.79 \\ \hline 100 & 75 & -4.18e+5 & 0 & 52.43 & -2.16e+4 & 3.75e-1 & 110.53 \\ \hline 200 & 25 & -3.22e+6 & 0 & 65.56 & -1.81e+5 & 2.56e-1 & 132.18 \\ \hline 200 & 50 & -1.85e+6 & 0 & 90.49 & -8.45e+4 & 4.54e-1 & 208.84 \\ \hline 200 & 75 & -1.33e+6 & 0 & 138.75 & -7.78e+4 & 3.93e-1 & 287.20 \\ \hline \end{tabular}
\begin{tabular}{|c|c|c|c|c|} \hline
\multicolumn{3}{|c|}{} &{IPAG} &{ICPP} \\ \hline n & m &std. & $f(x_T)$ & $f(x_T)$ \\ \hline 100 & 25 &1&-6.7866e+5 & -4.8563e+4 \\ \hline 100 & 25 &5&-6.5288e+5&-4.8596e+4 \\ \hline 100 & 25 & 10&-6.2336e+5&-4.8528e+4 \\ \hline 200 & 50 & 1& -1.8552e+6 & -8.4550e+4\\ \hline 200 & 50 & 5&-1.8452e+6&-8.5264e+4 \\ \hline 200 & 50 &10&-1.8383e+6&-8.6096e+4 \\ \hline \end{tabular}\caption{Comparing IPAG and ICPP.} \label{tab1} \end{table}
In Table \ref{tab1} (left), we compared the objective value, CPU time, and infeasibility \af{(Infeas.)} of our proposed method with ICPP \cite{boob2019stochastic} and in Table \ref{tab1} (right) we compared \aj{the} methods for different choices of standard deviation (std.) \af{of $\omega$}. To have a fair comparison, we fixed the oracle complexity (i.e. the number of computed gradients is equal for both methods). As it can be seen in the table, for different choices of $m$ and $n$, IPAG scheme outperforms ICPP. For instance, when we have 25 constraints and $n=100$, the objective value for our scheme \aj{reaches} $f(x_T)=-6.78e+5 $ which is significantly smaller than $-4.85e+4$ for ICPP method. Note that our scheme, in contrast to ICPP, obtains a feasible solution \aj{at} each iteration. Similar behavior can be \aj{observed} for different choices of the standard deviation \af{in Table \ref{tab1} (right)}.
standard deviation.
\footnotesize
\end{document} | arXiv |
\begin{document}
\title{Warped product pointwise bi-slant submanifolds in metallic Riemannian manifolds}
\markboth{{\small\it {\hspace{2cm} Warped product pointwise bi-slant submanifolds in metallic Riemannian manifolds}}}{\small\it{Warped product pointwise bi-slant submanifolds in metallic Riemannian manifolds \hspace{2cm}}}
\textbf{Abstract:} In this paper, we study some properties of warped product pointwise bi-slant submanifolds in locally metallic Riemannian manifolds and we construct some examples in Euclidean spaces.\\
{ \textbf{2020 Mathematics Subject Classification:} 53B20, 53B25, 53C42, 53C15. }
{ \textbf{Keywords:} Metallic Riemannian structure; Warped product bi-slant submanifold; Pointwise slant submanifold. }
\section{Introduction}
Metallic Riemannian manifolds and their submanifolds were defined and investigated by C. E. Hretcanu, M. Crasmareanu and A. M. Blaga in (\cite{Hr5}, \cite{Hr4}), as a generalization of Golden Riemannian manifolds studied in (\cite{CrHr}, \cite{Hr3}, \cite{Hr2}). The authors of the present paper obtained some properties of invariant, anti-invariant and slant submanifolds (\cite{Blaga_Hr}), semi-slant submanifolds (\cite{Hr6}) and, respectively, hemi-slant submanifolds (\cite{Hr7}) in metallic and Golden Riemannian manifolds and they provided some integrability conditions for the distributions involved in these types of submanifolds. Moreover, properties of metallic and Golden warped product Riemannian manifolds were presented in some previous works of the authors (\cite{Blaga1}, \cite{Blaga2}, \cite{Hr10}). In the last years, the study of submanifolds in metallic Riemannian manifolds has been continued by many authors (\cite{bea1}, \cite{fee1}, \cite{fee2}), which introduced the notion of lightlike submanifold of a metallic semi-Riemannian manifold.
\section{Preliminaries}
The name of metallic number is given to the positive solution of the equation \linebreak $x^{2}-px-q=0$, which is $\sigma _{p,q}=\frac{p+\sqrt{p^{2}+4q}}{2}$ (\cite{Spinadel}), where $p$ and $q$ are positive integer values. The metallic structure is a particular case of polynomial structure on a manifold, which was generally defined in (\cite{Goldberg2}, \cite{Goldberg1}).
Let $\overline{M}$ be an $m$-dimensional manifold endowed with a tensor field $J$ of type $(1,1)$. Then $J$ is called a \textit{metallic structure} if it satisfies: \begin{equation}\label{e1} J^{2}= pJ+qI, \end{equation} for $p$, $q\in\mathbb{N}^*$, where $I$ is the identity operator on $\Gamma(T\overline{M})$. If a Riemannian metric $\overline{g}$ is $J$-compatible, i.e.: \begin{equation} \label{e2} \overline{g}(JX, Y)= \overline{g}(X, JY), \end{equation} for any $X$, $Y \in \Gamma(T\overline{M})$, then $(\overline{M},\overline{g},J)$ is called a {\it metallic Riemannian manifold} (\cite{Hr4}). In this case, $\overline{g}$ verifies: \begin{equation} \label{e3} \overline{g}(JX, JY)=\overline{g}(J^{2}X, Y) =p \overline{g}(JX,Y)+q \overline{g}(X,Y), \end{equation} for any $X$, $Y \in \Gamma(T\overline{M})$. \normalfont
For $p=q=1$ one obtain the \textit{Golden structure} $J$ which satisfies $J^{2}= J + I$. If ($\overline{M}, \overline{g})$ is a Riemannian manifold endowed with a Golden structure $J$ such that the Riemannian metric $\overline{g}$ is $J$-compatible, then $(\overline{M},\overline{g},J)$ is called a {\it Golden Riemannian manifold} (\cite{CrHr}).
Let $M$ be an isometrically immersed submanifold in the metallic Riemannian manifold ($\overline{M}, \overline{g},J)$. The tangent space $T_x\overline{M}$ of $\overline{M}$ in a point $x \in M$ can be decomposed into the direct sum $T_x\overline{M}=T_x M\oplus T_x^{\perp}M,$ for any $x\in M$, where $T_{x}^{\bot }M$ is the normal space of $M$ in $x$. Let $i_{*}$ be the differential of the immersion $i: M \rightarrow\overline{M}$. Then the induced Riemannian metric $g$ on $M$ is given by $g(X, Y)=\overline{g}(i_{*}X, i_{*}Y)$, for any $X$, $Y \in \Gamma(TM)$. In all the rest of the paper, we shall denote by $X$ the vector field $i_{*}X$, for any $X \in \Gamma(TM)$.
For any $X \in \Gamma(TM)$, let $TX:=(J X)^T$ and $NX:=(J X)^{\perp}$ be the tangential and normal components, respectively, of $JX$ and for any $V \in \Gamma(T^{\perp}M)$, let $tV:=(J V)^{T}$, $nV:=(J V)^{\perp}$ be the tangential and normal components, respectively, of $JV$. Then we have: \begin{equation}\label{e4} JX = TX + NX, \end{equation} \begin{equation}\label{e5} JV = tV + nV, \end{equation} for any $X \in \Gamma(TM)$ and $V \in \Gamma(T^{\perp}M)$.
The maps $T$ and $n$ are $\overline{g}$-symmetric (\cite{Blaga_Hr}): \begin{equation}\label{e6} \overline{g}(TX,Y)=\overline{g}(X,TY), \end{equation} \begin{equation}\label{e7} \overline{g}(nU,V)=\overline{g}(U,nV) \end{equation} and \begin{equation}\label{e8}
\overline{g}(NX,V)=\overline{g}(X,tV), \end{equation} for any $X$, $Y \in \Gamma(TM)$ and $U$, $V\in \Gamma(T^{\perp}M)$.
\pagebreak We also obtain (\cite{Hr7}): \begin{equation} \label{e9} T^{2}X = pTX+qX-tNX, \end{equation} \begin{equation} \label{e10} pNX= NTX+nNX, \end{equation} \begin{equation} \label{e11} n^{2}V = pnV+qV-NtV, \end{equation} \begin{equation} \label{e12} ptV= TtV+tnV, \end{equation} for any $X \in \Gamma(TM)$ and $V \in \Gamma(T^{\bot}M)$.
Let $\overline{\nabla}$ and $\nabla $ be the Levi-Civita connections on $(\overline{M},\overline{g})$ and on its submanifold $(M,g)$, respectively. The Gauss and Weingarten formulas are given by: \begin{equation}\label{e13} \overline{\nabla}_{X}Y=\nabla_{X}Y+h(X,Y), \end{equation} \begin{equation}\label{e14} \overline{\nabla}_{X}V=-A_{V}X+\nabla_{X}^{\bot}V, \end{equation} for any $X$, $Y \in \Gamma(TM)$ and $V \in \Gamma(T^{\bot}M)$, where $h$ is the second fundamental form and $A_{V}$ is the shape operator, which satisfy \begin{equation}\label{e15}
\overline{g}(h(X, Y),V)=\overline{g}(A_{V}X, Y).
\end{equation}
For any $X$, $Y \in \Gamma(TM)$, the covariant derivatives of $T$ and $N$ are given by: \begin{equation}\label{e16} (\nabla_{X}T)Y=\nabla_{X}TY - T(\nabla_{X}Y), \end{equation} \begin{equation}\label{e17} (\overline{\nabla}_{X}N)Y=\nabla_{X}^{\bot}NY - N(\nabla_{X}Y). \end{equation}
For any $X \in \Gamma(TM)$ and $V \in \Gamma(T^{\bot}M)$, the covariant derivatives of $t$ and $n$ are given by: \begin{equation}\label{e18} (\nabla_{X}t)V=\nabla_{X}tV - t(\nabla_{X}^{\bot}V), \end{equation} \begin{equation}\label{e19} (\overline{\nabla}_{X}n)V=\nabla_{X}^{\bot}nV - n(\nabla_{X}^{\bot}V). \end{equation}
From $(\ref{e1})$ we obtain: \begin{equation} \label{e20} \overline{g}((\overline{\nabla}_XJ)Y,Z)=\overline{g}(Y,(\overline{\nabla}_XJ)Z), \end{equation} for any $X$, $Y$, $Z\in \Gamma(T\overline{M})$, which implies (\cite{Blaga3}):
\pagebreak \begin{equation}\label{e21} \overline{g}((\nabla_X T)Y,Z)=\overline{g}(Y,(\nabla_X T)Z), \end{equation} \begin{equation}\label{e22} \overline{g}((\overline{\nabla}_{X}N)Y,V )= \overline{g}(Y,(\nabla_{X}t)V), \end{equation} for any $X$, $Y$, $Z\in \Gamma(TM)$ and $V \in \Gamma(T^{\bot}M)$.
The analogue concept of locally product manifold is considered in the context of metallic geometry, having the name of \textit{locally metallic manifold} (\cite{Blaga2}). Thus, we say that the metallic Riemannian manifold $(\overline{M},\overline{g}, J)$ is \textit{locally metallic} if $J$ is parallel with respect to the Levi-Civita connection $\overline{\nabla}$ on $\overline{M}$ (i.e. $\overline{\nabla}J=0$).
\begin{remark} In (\cite{Hr4}) we obtained that any almost product structure $F$ on $\overline{M}$ induces two metallic structures on $\overline{M}$: \begin{equation}\label{e105} J_{1}= \frac{2\sigma-p}{2}F+\frac{p}{2}I, \end{equation} \begin{equation}\label{e106} J_{2}=-\frac{2\sigma-p}{2}F+\frac{p}{2}I, \end{equation} where $\sigma=\sigma _{p, q}=\frac{p+\sqrt{p^{2}+4q}}{2}$, with $p, q$ positive integer numbers. \end{remark}
Also, for an almost product structure $F$ and for any $X \in \Gamma(TM)$ and $V \in \Gamma(T^{\perp}M)$, the decompositions into the tangential and normal components of $FX$ and $FV$ are given by: \begin{equation}\label{e107} FX = fX + \omega X, \end{equation} \begin{equation}\label{e108} FV = BV + CV, \end{equation} where $fX:=(F X)^T$, $\omega X:=(FX)^{\perp}$, $BV:=(F V)^T$ and $CV:=(F V)^{\perp}$.
Moreover, the maps $f$ and $C$ are $\overline{g}$-symmetric (\cite{Li&Liu}): \begin{equation}\label{e109} \overline{g}(fX,Y)=\overline{g}(X,fY), \end{equation} \begin{equation} \overline{g}(CU,V)=\overline{g}(U,CV), \end{equation} for any $X, Y\in \Gamma(TM)$ and $U, V\in \Gamma(T^{\perp}M)$.
\begin{remark}(\cite{Hr6}) If $M$ is a submanifold in the almost product Riemannian manifold $(\overline{M}, \overline{g}, F)$ and $J$ is a metallic structure induced by $F$ on $\overline{M}$, then: \begin{equation} \label{e110} TX = \frac{p}{2}X \pm \frac{2\sigma-p}{2}fX, \end{equation} \begin{equation} \label{e111} NX= \pm \frac{2\sigma-p}{2}\omega X, \end{equation} for any $X \in \Gamma(TM)$. \end{remark}
\section{Pointwise slant submanifolds in metallic Riemannian manifolds}
B.-Y. Chen studied CR-submanifolds of a K\"{a}hler manifold which are warped products of holomorphic and totally real submanifolds, respectively (\cite{Chen3}, \cite{Chen1}, \cite{Chen2}). Also, in his new book (\cite{ChenBook}), he presents a multitude of properties for warped product manifolds and submanifolds, such as: warped product of Riemannian and K\"{a}hler manifolds, warped product submanifolds of K\"{a}hler manifolds (with the particular cases: warped product CR-submanifolds, warped product semi-slant or hemi-slant submanifolds of K\"{a}\-hler manifolds), CR-warped products in complex space forms and so on.
We shall state the notion of pointwise slant submanifold in a metallic Riemannian manifold, following Chen's definition (\cite{Chen5}, \cite{Chen4}) of pointwise slant submanifold of an almost Hermitian manifold.
\begin{definition} A submanifold $M$ of a metallic Riemannian manifold $(\overline{M}, \overline{g}, J)$ is called \textit{pointwise slant} if the angle $\theta_x(X)$ between $JX$ and $T_xM$ (called the \textit{Wirtinger angle}) is independent of the choice of the tangent vector $X \in T_{x}M\setminus\{0\}$, but it depends on $x \in M$. The Wirtinger angle is a real-valued function $\theta$ (called the Wirtinger function), verifying \begin{equation}\label{e27}
\cos\theta_x =\frac{\| TX \|}{\| JX \|}, \end{equation} for any $x\in M$ and $X \in T_{x}M\setminus\{0\}$. \end{definition}
A pointwise slant submanifold of a metallic Riemannian manifold is called \textit{slant submanifold} if its Wirtinger function $\theta$ is globally constant.
In a similar manner as in (\cite{Chen5}) we obtain: \begin{proposition} If $M$ is an isometrically immersed submanifold in the metallic Riemannian manifold $(\overline{M}, \overline{g}, J)$, then $M$ is a pointwise slant submanifold if and only if \begin{equation}\label{e28} T^2=(\cos^2\theta)(pT+qI), \end{equation} for some real-valued function $\theta$. \end{proposition}
From (\ref{e9}) and (\ref{e28}) we have: \begin{proposition} Let $M$ be an isometrically immersed submanifold in the metallic Riemannian manifold $(\overline{M}, \overline{g}, J)$. If $M$ is a pointwise slant submanifold with the Wirtinger angle $\theta$, then: \begin{equation}\label{e29} \overline{g}(NX,NY)=(\sin^2\theta)[p\overline{g}(TX,Y)+q\overline{g}(X,Y)] \end{equation} and \begin{equation}\label{e30}
tNX=(\sin^2\theta)(pTX+qX), \end{equation} for any $X$, $Y\in \Gamma(TM)$. \end{proposition}
From (\ref{e28}), by a direct computation, we obtain: \begin{proposition} Let $M$ be an isometrically immersed submanifold in the metallic Riemannian manifold $(\overline{M}, \overline{g}, J)$. If $M$ is a pointwise slant submanifold with the Wirtinger angle $\theta$, then: \begin{equation}\label{e31} (\nabla_XT^2)Y=p(\cos^2\theta)(\nabla_XT)Y-\sin(2\theta)X(\theta)(pTY+qY), \end{equation} for any $X$, $Y\in \Gamma(TM)$. \end{proposition}
\section{Pointwise bi-slant submanifolds in metallic Riemannian manifolds}
In this section we introduce the notion of pointwise bi-slant submanifold in the metallic context.
\begin{definition} \label{d2} Let $M$ be an immersed submanifold in a metallic Riemannian manifold $(\overline{M},\overline{g},J)$. We say that $M$ is a {\it pointwise bi-slant submanifold} of $\overline{M}$ if there exists a pair of orthogonal distributions $D_{1}$ and $D_{2}$ on $M$ such that
(i) $TM = D_{1}\oplus D_{2}$;
(ii) $J(D_{1}) \bot D_{2}$ and $J(D_{2}) \bot D_{1}$;
(iii) the distributions $D_{1}$, $D_{2}$ are pointwise slant. \end{definition}
If ${\theta_{1}}$ and ${\theta_{2}}$ are the slant functions of $D_{1}$ and $D_{2}$, respectively, then the pair $\{ \theta_{1}, \theta_{2}\}$ is called the {\it bi-slant function}.
A pointwise slant submanifold $M$ is called {\it proper} if ${\theta_{1}}_x$, ${\theta_{2}}_x \neq 0; \frac{\pi}{2}$, for any $x\in M$ and both ${\theta_{1}}$, ${\theta_{2}}$ are not constant on $M$.
In particular, if ${\theta_{1}}=0$ and ${\theta_{2}} \neq 0; \frac{\pi}{2}$, then $M$ is called \textit{a pointwise semi-slant submanifold}; if ${\theta_{1}} = \frac{\pi}{2}$ and ${\theta_{2}} \neq 0; \frac{\pi}{2}$, then $M$ is called \textit{a pointwise hemi-slant submanifold}.
Remark that if $M$ is a pointwise bi-slant submanifold of $\overline{M}$, then the distributions $D_{1}$ and $D_{2}$ on $M$ verify $T(D_{1}) \subseteq D_{1}$ and $T(D_{2}) \subseteq D_{2}$.
\begin{example} Let $\mathbb{R}^{6}$ be the Euclidean space endowed with the usual Euclidean metric $\langle\cdot,\cdot\rangle$. Let $i: M \rightarrow \mathbb{R}^{6}$ be the immersion given by: $$i(u,v):=\left(\cos u \cos v, \cos u \sin v, \sin u \cos v, \sin u \sin v, \sin v, \cos v \right),$$ where $M :=\{(u, v) \mid u, v \in (0, \frac{\pi}{2})\}$.
\pagebreak A local orthogonal frame on $TM$ is given by:
$$Z_{1}= -\sin u \cos v \frac{\partial}{\partial x_{1}} -\sin u \sin v \frac{\partial}{\partial x_{2}}+ \cos u \cos v \frac{\partial}{\partial x_{3}} + \cos u \sin v \frac{\partial}{\partial x_{4}}$$
$$ Z_{2}= - \cos u \sin v \frac{\partial}{\partial x_{1}} +\cos u \cos v \frac{\partial}{\partial x_{2}} - \sin u \sin v \frac{\partial}{\partial x_{3}} +
\sin u \cos v \frac{\partial}{\partial x_{4}} +$$$$+ \cos v \frac{\partial}{\partial x_{5}} - \sin v \frac{\partial}{\partial x_{6}}.$$
We define the metallic structure $J : \mathbb{R}^{6} \rightarrow \mathbb{R}^{6} $ by: $$
J(X_{1},X_{2},X_{3},X_{4},X_{5},X_{6}):=(\sigma X_{1}, \overline{\sigma} X_{2},\sigma X_{3},\overline{\sigma} X_{4},\sigma X_{5}, \overline{\sigma} X_{6} ),
$$ where $\sigma:=\sigma_{p,q}=\frac{p+\sqrt{p^{2}+4q}}{2}$ is a metallic number ($p, q \in \mathbb{N}^{*}$) and $\overline{\sigma}=p-\sigma$.
We remark that $J$ verifies $J^{2}X=p J + q I$ and $\langle JX, Y\rangle = \langle X, JY\rangle$, for any $X$, $Y \in \mathbb{R}^{6}$. Also, we have
$$JZ_{1}=-\sigma\sin u \cos v \frac{\partial}{\partial x_{1}} -\overline{\sigma}\sin u \sin v \frac{\partial}{\partial x_{2}}+ \sigma\cos u \cos v \frac{\partial}{\partial x_{3}} + \overline{\sigma}\cos u \sin v \frac{\partial}{\partial x_{4}},$$
$$ JZ_{2}= -\sigma \cos u \sin v \frac{\partial}{\partial x_{1}} +\overline{\sigma}\cos u \cos v \frac{\partial}{\partial x_{2}} - \sigma\sin u \sin v \frac{\partial}{\partial x_{3}} +$$ $$+\overline{\sigma}
\sin u \cos v \frac{\partial}{\partial x_{4}} +\sigma \cos v \frac{\partial}{\partial x_{5}} -\overline{\sigma} \sin v \frac{\partial}{\partial x_{6}}.$$
We remark that $\langle JZ_{1}, Z_{2}\rangle =\langle JZ_{2}, Z_{1}\rangle =0 $, $\langle JZ_{1}, Z_{1}\rangle= \sigma \cos^{2}v + \overline{\sigma}\sin^{2}v$ and $\langle JZ_{2}, Z_{2}\rangle = p$.
On the other hand we get:
$$\|Z_{1}\|=1, \ \ \|Z_{2}\|= \sqrt{2},$$
$$\|J Z_{1}\|=\sqrt{\sigma^{2} \cos^{2}v + \overline{\sigma}^{2}\sin^{2}v}=\sqrt{p(\sigma \cos^{2}v + \overline{\sigma}\sin^{2}v)+q},$$ $$\|J Z_{2}\|=\sqrt{\sigma^{2}+\overline{\sigma}^{2}}=\sqrt{p^{2}+2q}.$$
We denote by $D_{1}:=span\{Z_{1}\}$ the pointwise slant distribution with the slant angle $\theta_{1}$, where $\cos \theta_{1} = \frac{f(u,v)}{\sqrt{pf(u,v)+q}}$, for $f(u,v):=\sigma \cos^{2}v + \overline{\sigma}\sin^{2}v$ a real function on $M$. Also, we denote by $D_{2}:=span\{Z_{2}\}$ the slant distribution with the slant angle $\theta_{2}$, where $\cos \theta_{2} =\frac{p}{\sqrt{2(p^{2}+2q})}$.
The distributions $D_{1}$ and $D_{2}$ satisfy the conditions from Definition \ref{d2}.
If $M_{1}$ and $M_{2}$ are the integral manifolds of the distributions $D_{1}$ and $D_{2}$, respectively, then $M := M_{1} \times_{\sqrt{2}} M_{2}$ with the Riemannian metric tensor $$g:= du^{2} + 2 d v^{2}$$ is a pointwise bi-slant submanifold in the metallic Riemannian manifold $(\mathbb{R}^{6}, \langle\cdot,\cdot\rangle, J)$. \end{example}
\begin{example} In particular, if $f$ is a metallic function (i.e. $f^{2}=pf+q$), then $\cos \theta_{1}=1$ and we remark that $M$ is a semi-slant submanifold in the metallic Riemannian manifold $(\mathbb{R}^{6}, \langle\cdot,\cdot\rangle, J)$, with the slant angle $\theta=\theta_{2}$. \end{example}
\begin{example} On the other hand, if $f=0$ (i.e. $\tan v=\sqrt{- \frac{\sigma}{\overline{\sigma}}}=\frac{\sqrt{p^{2}+4q}+p}{2\sqrt{q}}$), then $\cos \theta_{1}=0$ and we remark that $M$ is a hemi-slant submanifold in the metallic Riemannian manifold $(\mathbb{R}^{6}, \langle\cdot,\cdot\rangle, J)$, with the slant angle $\theta=\theta_{2}$. \end{example}
If we denote by $P_{i}$ the projections from $TM$ onto $D_{i}$, for $i \in \{1,2 \}$, then $X=P_{1}X+P_{2}X$, for any $X \in \Gamma(TM)$. In particular, if $X \in D_{i}$, then $X=P_{i}X$, for $i \in \{1,2 \}$.
If we denote by $T_{i}=P_{i}\circ T$, for $i \in \{1,2 \}$, then, from (\ref{e4}), we obtain: \begin{equation}\label{e32} JX=T_{1}X+T_{2}X+NX. \end{equation}
In a similar manner as in (\cite{Chen4}), we get: \begin{lemma} Let $M$ be a pointwise bi-slant submanifold of a locally metallic Riemannian manifold $(\overline{M},\overline{g},J)$ with pointwise slant distributions $D_{1}$ and $D_{2}$, having slant functions ${\theta_{1}}$ and ${\theta_{2}}$. Then
(i) for any $X$, $Y\in D_1$ and $Z\in D_2$, we have: \begin{equation}\label{e33} (\sin^{2}\theta_{1}-\sin^{2}\theta_{2})\overline{g}(\nabla_{X}Y,pT_{2}Z+qZ)= \end{equation} $$=p[\overline{g}(\nabla_{X}Y,T_{2}Z)+\overline{g}(\nabla_{X}Z,T_{1}Y)]+ p(\cos^{2}\theta_{1}+1)\overline{g}(A_{NZ}Y+A_{NY}Z,X)-$$ $$-\overline{g}(A_{NT_{1}Y}Z+A_{NT_{2}Z}Y,X)-\overline{g}(A_{NZ}T_{1}Y+A_{NY}T_{2}Z,X); $$
(ii) for any $X\in D_1$ and $Z$, $W\in D_2$, we have: \begin{equation}\label{e34} (\sin^{2}\theta_{2}-\sin^{2}\theta_{1})\overline{g}(\nabla_{Z}W,pT_{1}X+qX)= \end{equation} $$=p[\overline{g}(\nabla_{Z}W,T_{1}X)+\overline{g}(\nabla_{Z}X,T_{2}W)]+ p(\cos^{2}\theta_{2}+1)\overline{g}(A_{NX}W+A_{NW}X,Z)-$$ $$-\overline{g}(A_{NT_{2}W}X+A_{NT_{1}X}W,Z)-\overline{g}(A_{NW}T_{1}X+A_{NX}T_{2}W,Z). $$ \end{lemma}
\proof From (\ref{e1}) we have: \begin{equation}\label{e35} q\overline{g}(\nabla_{X}Y,Z)=q\overline{g}(\overline{\nabla}_{X}Y,Z)=\overline{g}(J^{2}\overline{\nabla}_{X}Y,Z)-p\overline{g}(J\overline{\nabla}_{X}Y,Z), \end{equation} for any $X$, $Y\in D_{1}$ and $Z \in D_{2}$.
By using (\ref{e2}) and $(\overline{\nabla}_{X}J)Y=0$, we obtain: \begin{equation}\label{e36} q\overline{g}(\nabla_{X}Y,Z)= \overline{g}(\overline{\nabla}_{X}J^2Y,Z)-p\overline{g}(\overline{\nabla}_{X}JY,Z). \end{equation}
From (\ref{e32}) we get $JX = T_{1}X+NX$, $JY = T_{1}Y+NY$ and $JZ = T_{2}Z+NZ$, for any $X$, $Y \in D_1$ and $Z\in D_2$ and from here we obtain: $$ q\overline{g}(\nabla_{X}Y,Z)=\overline{g}(\overline{\nabla}_{X}JT_{1}Y,Z)+\overline{g}(\overline{\nabla}_{X}JNY,Z) -p\overline{g}(\overline{\nabla}_{X}(T_{1}Y+NY),Z)= $$ $$ =\overline{g}(\overline{\nabla}_{X}T_{1}^{2}Y,Z)+\overline{g}(\overline{\nabla}_{X}NT_{1}Y,Z)+\overline{g}(\overline{\nabla}_{X}NY,JZ) -p\overline{g}(\overline{\nabla}_{X}T_{1}Y,Z)+$$$$+p\overline{g}(A_{NY}X,Z)= $$ $$ =\overline{g}(\overline{\nabla}_{X}(\cos^{2} \theta_{1}(pT_{1}Y+qY)),Z)-\overline{g}(A_{NT_{1}Y}X,Z)+\overline{g}(\overline{\nabla}_{X}NY,T_{2}Z+NZ)+ $$ $$ +p\overline{g}(T_{1}Y,\overline{\nabla}_{X}Z)+p\overline{g}(A_{NY}X,Z). $$
Thus, we get: $$ q\overline{g}(\nabla_{X}Y,Z)=\cos^{2} \theta_{1}\overline{g}(\overline{\nabla}_{X}(pT_{1}Y+qY),Z)-\sin 2 \theta_{1}X(\theta_{1})\overline{g}(pT_{1}Y+qY,Z)-$$$$-\overline{g}(A_{NT_{1}Y}X,Z)- \overline{g}(A_{NY}X,T_{2}Z)-\overline{g}(\overline{\nabla}_{X}NZ,JY) +\overline{g}(\overline{\nabla}_{X}NZ,T_{1}Y)+$$$$+p\overline{g}(T_{1}Y,\overline{\nabla}_{X}Z)+p\overline{g}(A_{NY}X,Z). $$
By using $\overline{g}(pT_{1}Y+qY,Z)=0$, we obtain: $$ q\sin^{2}\theta_{1}\overline{g}(\nabla_{X}Y,Z)= p\cos^{2}\theta_{1}\overline{g}(\overline{\nabla}_{X}T_{1}Y,Z)-\overline{g}(A_{NT_{1}Y}Z+A_{NY}T_{2}Z,X)+ $$ $$ +\overline{g}(JNZ,\overline{\nabla}_{X}Y)-\overline{g}(A_{NZ}X,T_{1}Y)+p\overline{g}(T_{1}Y,\overline{\nabla}_{X}Z)+p\overline{g}(A_{NY}Z,X). $$
Using (\ref{e10}) and (\ref{e30}), we find: $$\overline{g}(JNZ,\overline{\nabla}_{X}Y) = \overline{g}(tNZ+nNZ,\overline{\nabla}_{X}Y) = $$ $$= \sin^{2}\theta_{2}\overline{g}(\nabla_{X}Y,qZ+pT_{2}Z)+\overline{g}(pNZ-NT_ 2Z,\overline{\nabla}_{X}Y)=$$ $$= q\sin^{2}\theta_{2}\overline{g}(\nabla_{X}Y,Z)+p\sin^{2}\theta_{2}\overline{g}(\nabla_{X}Y,T_{2}Z)-$$$$-p\overline{g}(\overline{\nabla}_{X}NZ,Y)+ \overline{g}(\overline{\nabla}_{X}NT_ 2Z,Y)$$ and from $$\overline{g}(\overline{\nabla}_{X}T_{1}Y,Z)=-\overline{g}(JY-NY,\overline{\nabla}_{X}Z)=\overline{g}(\overline{\nabla}_{X}Y,JZ)-\overline{g}(\overline{\nabla}_{X}NY,Z)=$$ $$=\overline{g}(\overline{\nabla}_{X}Y,T_ 2Z)-\overline{g}(Y,\overline{\nabla}_{X}NZ)-\overline{g}(\overline{\nabla}_{X}NY,Z) =$$ $$=\overline{g}(\overline{\nabla}_{X}Y,T_ 2Z)+\overline{g}(Y,A_{NZ}X)+\overline{g}(A_{NY}X,Z) $$ we have: $$ q(\sin^{2}\theta_{1}-\sin^{2}\theta_{2})\overline{g}(\nabla_{X}Y,Z)= p(1-\sin^{2}\theta_{1})\overline{g}(\nabla_{X}Y,T_{2}Z)+$$$$+p\cos^{2}\theta_{1}\overline{g}(A_{NZ}Y+A_{NY}Z,X)+\sin^{2}\theta_{2}\overline{g}(\nabla_{X}Y,pT_{2}Z)- $$$$-\overline{g}(A_{NT_{1}Y}Z+A_{NY}T_{2}Z+A_{NZ}T_{1}Y+A_{NT_{2}Z}Y,X)- $$ $$-p\overline{g}(Y,\overline{\nabla}_{X}NZ)+p\overline{g}(T_{1}Y,\nabla_{X}Z)+p\overline{g}(A_{NY}Z,X)$$ and from here we get (\ref{e33}).
In the same manner we find (\ref{e34}).
\begin{proposition} Let $M$ be a pointwise semi-slant submanifold in a locally metallic Riemannian manifold $(\overline{M},\overline{g},J)$ with pointwise slant distributions $D_{1}$ and $D_{2}$, having slant functions $\theta_{1}$ and $\theta_{2}$.
(i) If $\theta_{1}=0$ and $\theta_{2}=\theta$, we obtain: \begin{equation}\label{e350} \sin^{2}\theta\overline{g}(\nabla_{X}Y,pT_{2}Z+qZ)=-p[\overline{g}(\nabla_{X}Y,T_{2}Z)+\overline{g}(\nabla_{X}Z,T_{1}Y)] - \end{equation} $$ -2p\overline{g}(A_{NZ}Y,X)+\overline{g}(A_{NZ}T_{1}Y+A_{NT_{2}Z}Y,X), $$ for any $X$, $Y\in D^T$ and $Z\in D^{\theta}$, and \begin{equation}\label{e37} \sin^{2}\theta\overline{g}(\nabla_{Z}W,pT_{1}X+qX)=p[\overline{g}(\nabla_{Z}W,T_{1}X)+\overline{g}(\nabla_{Z}X,T_{2}W)]+ \end{equation} $$ +p(\cos^{2}\theta+1)\overline{g}(A_{NW}X,Z)-\overline{g}(A_{NT_{2}W}X+A_{NW}T_{1}X,Z), $$ for any $X\in D^T$ and $Z$, $W \in D^{\theta}$.
(ii) If $\theta_{1}=\theta$ and $\theta_{2}=0$, we obtain: \begin{equation}\label{e360} \sin^{2}\theta\overline{g}(\nabla_{X}Y,T^{2}_{2}Z)=p\overline{g}(\nabla_{X}Y,T_{2}Z)-p\overline{g}(\nabla_{X}T_{1}Y,Z)+ \end{equation} $$ +p(\cos^{2}\theta+1)\overline{g}(A_{NY}X,Z)-\overline{g}(A_{NT_{1}Y}Z+A_{NT_{2}Z}Y,X)-\overline{g}(A_{NY}T_{2}Z,X), $$ for any $X, Y \in D^{\theta}$ and $Z\in D^{T}$, and \begin{equation}\label{e370} \sin^{2}\theta\overline{g}(\nabla_{Z}W,pT_{1}X+qX)=-p\overline{g}(\nabla_{Z}W,T_{1}X)+p\overline{g}(\nabla_{Z}T_{2}W,X)- \end{equation} $$ -2p\overline{g}(A_{NX}Z,W)+\overline{g}(A_{NT_{2}W}Z,X)+\overline{g}(A_{NT_{1}X}Z,W)+\overline{g}(A_{NX}T_{2}W,Z), $$ for any $X\in D^{\theta}$ and $Z$, $W \in D^T$. \end{proposition}
\begin{proposition} Let $M$ be a pointwise hemi-slant submanifold in a locally metallic Riemannian manifold $(\overline{M},\overline{g},J)$ with pointwise slant distributions $D_{1}$ and $D_{2}$, having slant functions $\theta_{1}$ and $\theta_{2}$.
(i) If $\theta_{1}=\frac{\pi}{2}$ and $\theta_{2}=\theta$, we obtain: \begin{equation}\label{e38} \cos^{2}\theta\overline{g}(\nabla_{X}Y,pT_{2}Z+qZ)=p\overline{g}(\nabla_{X}Y,T_{2}Z)+ \end{equation} $$+p\overline{g}(A_{NZ}Y+A_{NY}Z,X)-\overline{g}(A_{NT_{2}Z}Y+A_{NY}T_{2}Z,X),$$ for any $X$, $Y\in D^{\perp}$ and $Z\in D^{\theta}$, and \begin{equation}\label{e39} q\cos^{2}\theta\overline{g}(\nabla_{Z}W,X)=-p\overline{g}(\nabla_{Z}X,T_{2}W)- \end{equation} $$- p(\cos^{2}\theta+1)\overline{g}(A_{NX}W+A_{NW}X,Z)+\overline{g}(A_{NT_{2}W}X+A_{NX}T_{2}W,Z), $$ for any $X\in D^{\perp}$ and $Z$, $W\in D^{\theta}$.
(ii) If $\theta_{1}=\theta$ and $\theta_{2}=\frac{\pi}{2}$, we obtain: \begin{equation}\label{e309} q\cos^{2}\theta\overline{g}(\nabla_{X}Y,Z)=-p\overline{g}(\nabla_{X}Z,T_{1}Y)- \end{equation} $$- p(\cos^{2}\theta+1)\overline{g}(A_{NZ}Y+A_{NY}Z,X)+\overline{g}(A_{NT_{1}Y}Z+A_{NZ}T_{1}Y,X), $$ for any $X, Y\in D^{\theta}$ and $Z\in D^{\perp}$, and \begin{equation}\label{e310} \cos^{2}\theta\overline{g}(\nabla_{Z}W,pT_{1}X+qX)=p\overline{g}(\nabla_{Z}W,T_{1}X)+ \end{equation} $$+p\overline{g}(A_{NX}W+A_{NW}X,Z)-\overline{g}(A_{NT_{1}X}W+A_{NW}T_{1}X,Z),$$ for any $X\in D^{\theta}$ and $Z$, $W\in D^{\perp}$. \end{proposition}
\section{Warped product pointwise bi-slant submanifolds in metallic Riemannian manifolds}
In (\cite{Blaga1}), the authors of this paper introduced the Golden warped product Riemannian manifold and provided a necessary and sufficient condition for the warped product of two locally Golden Riemannian manifolds to be locally Golden. Moreover, the subject was continued in the papers (\cite{Blaga2}, \cite{Hr8}), where the authors characterized the metallic structure on the product of two metallic manifolds in terms of metallic maps and provided a necessary and sufficient condition for the warped product of two locally metallic Riemannian manifolds to be locally metallic.
Let $(M_1,g_1)$ and $({M_2},g_2)$ be two Riemannian manifolds (of dimensions $n_{1}>0$ and $n_{2}>0$, respectively) and let $\pi_1$, $\pi_2$ be the projection maps from the product manifold ${M_1}\times {M_2}$ onto ${M_1}$ and ${M_2}$, respectively. We denote by $\widetilde{\varphi}:=\varphi \circ \pi_1$ the lift to ${M_1}\times {M_2}$ of a smooth function $\varphi$ on ${M_1}$. Then ${M_1}$ is called \textit{the base} and ${M_2}$ is called \textit{the fiber} of ${M_1}\times {M_2}$. The unique element $\widetilde{X}$ of $\Gamma(T({M_1}\times {M_2}))$ that is $\pi_1$-related to $X\in \Gamma(T{M_1})$ and to the zero vector field on ${M_2}$ will be called the \textit{horizontal lift of $X$} and the unique element $\widetilde{V}$ of $\Gamma(T({M_1}\times {M_2}))$ that is $\pi_2$-related to $V\in \Gamma(T{M_2})$ and to the zero vector field on ${M_1}$ will be called the \textit{vertical lift of $V$}. We denote by $\mathcal{L}({M_1})$ the set of all horizontal lifts of vector fields on ${M_1}$ and by $\mathcal{L}({M_2})$ the set of all vertical lifts of vector fields on ${M_2}$.
For $f: M_1 \rightarrow (0,\infty)$ a smooth function on ${M_1}$, we consider the Riemannian metric $g$ on $M:={M_1}\times {M_2}$: \begin{equation}\label{e40} g:=\pi_1^* g_1+(f \circ \pi_1)^2 \pi_2^*g_2. \end{equation}
\begin{definition} \label{d3} The product manifold of ${M_1}$ and ${M_2}$ together with the Riemannian metric $g$ is called \textit{the warped product} of ${M_1}$ and ${M_2}$ by the warping function $f$ (\cite{Bishop}).
A warped product manifold $M:={M_1}\times_f {M_2}$ is called \textit{trivial} if the warping function $f$ is constant. In this case, $M$ is the Riemannian product ${M_1}\times {M_2}_f$, where ${M_2}_f$ is the manifold $M_2$ equipped with the metric $f^2 g_2$ (which is homothetic to $g_2$) (\cite{ChenBook}). \end{definition}
In the next considerations, we shall denote by $(f \circ \pi_1)^2=: f^2$, $\pi_1^* g_1=: g_1$ and $\pi_2^*g_2=:g_2$, respectively.
\begin{lemma} (\cite{ChenBook})\label{1} If $\nabla$ denotes the Levi-Civita connection on $M:={M_1}\times_f {M_2}$, then: \begin{equation}\label{e41}
\nabla_{X}Z=\nabla_{Z}X = X (\ln f) Z, \end{equation} for any $X$, $Y \in \Gamma(T{M_1})$ and $Z$, $W \in \Gamma(T{M_2})$. \end{lemma}
The warped product $M_{1} \times_{f} M_{2}$ of two pointwise slant submanifolds $M_{1}$ and $M_{2}$ of a metallic Riemannian manifold $(\overline{M},\overline{g},J)$ is called a \textit{warped product pointwise bi-slant submanifold}. Moreover, it is called \textit{proper} if both $M_{1}$ and $M_{2}$ are proper pointwise slant submanifolds in $(\overline{M},\overline{g},J)$.
In a similar manner as in (\cite{Hr8}), we get: \begin{proposition} Let $M:={M_1}\times_f {M_2}$ be a warped product pointwise bi-slant submanifold in a locally metallic Riemannian manifold $(\overline{M},\overline{g},J)$ with slant functions ${\theta_{1}}$, ${\theta_{2}}$ and warped function $f$. Then, for any $X$, $Y \in \Gamma(TM_{1})$ and $Z$, $W \in \Gamma(TM_{2})$, we have: \begin{equation}\label{e42}
\overline{g}(h(X,Y),NZ)=-\overline{g}(h(X,Z),NY), \end{equation} \begin{equation}\label{e43}
\overline{g}(h(X,Z),NW)=0, \end{equation} \begin{equation}\label{e44}
\overline{g}(h(Z,W),NX)= T_{1}X(\ln f)\overline{g}(Z,W)- X(\ln f)\overline{g}(Z,T_{2}W). \end{equation} \end{proposition}
\proof For any $X$, $Y \in \Gamma(TM_{1})$ and $Z \in \Gamma(TM_{2})$, by using (\ref{e2}), (\ref{e4}), (\ref{e13}), (\ref{e41}) and $\overline{\nabla}J=0$ we obtain: $$ \overline{g}(h(X,Y),NZ)= \overline{g}(\overline{\nabla}_{X}Y,JZ)-\overline{g}(\overline{\nabla}_{X}Y,T_{2}Z)=$$ $$=\overline{g}(\overline{\nabla}_{X}T_{1}Y,Z)+\overline{g}(\overline{\nabla}_{X}NY,Z)+\overline{g}(\overline{\nabla}_{X}T_{2}Z,Y)=$$ $$=-\overline{g}(\nabla_{X}Z,T_{1}Y)-\overline{g}(A_{NY}X,Z)+\overline{g}(Y,\nabla_{X}T_{2}Z)=$$ $$=-X(\ln f)\overline{g}(T_{1}Y,Z)-\overline{g}(h(X,Z),NY) + X(\ln f)\overline{g}(Y,T_{2}Z).$$
On the other hand, $\overline{g}(T_{1}Y,Z)=\overline{g}(JY,Z)=\overline{g}(Y,JZ)=\overline{g}(Y,T_{2}Z)$ and we obtain (\ref{e42}).
For any $X \in \Gamma(TM_{1})$ and $Z$, $W \in \Gamma(TM_{2})$, by using (\ref{e2}), (\ref{e4}), (\ref{e13}), (\ref{e41}) and $\overline{\nabla}J=0$ we obtain: $$ \overline{g}(h(X,Z),NW)= \overline{g}(\overline{\nabla}_{X}Z,JW)-\overline{g}(\overline{\nabla}_{X}Z,T_{2}W)=$$ $$= \overline{g}(\nabla_{X}T_{2}Z,W)-\overline{g}(A_{NZ}X,W)-\overline{g}(\nabla_{X}Z,T_{2}W)=$$ $$= X (\ln f)[\overline{g}(T_{2}Z,W)-\overline{g}(Z,T_{2}W)]-\overline{g}(h(X,W),NZ)$$ and using $$\overline{g}(T_{2}Z,W)-\overline{g}(Z,T_{2}W)=\overline{g}(JZ,W)-\overline{g}(Z,JW)=0,$$ we obtain \begin{equation}\label{e500} \overline{g}(h(X,Z),NW)= -\overline{g}(h(X,W),NZ). \end{equation}
On the other hand, after interchanging $Z$ by $X$, we have:
$$ \overline{g}(h(Z,X),NW)=\overline{g}(\nabla_{Z}T_{1}X,W)-\overline{g}(A_{NX}Z,W)-\overline{g}(\nabla_{Z}X,T_{2}W)=$$
$$ =T_{1}X (\ln f)\overline{g}(Z,W)-X (\ln f)\overline{g}(Z,T_{2}W)-\overline{g}(h(Z,W),NX)=\overline{g}(h(X,W),NZ)$$
and using (\ref{e500}) we get (\ref{e43}).
For any $X \in \Gamma(TM_{1})$ and $Z$, $W \in \Gamma(TM_{2})$, by using (\ref{e2}), (\ref{e4}), (\ref{e13}), (\ref{e41}) and $\overline{\nabla}J=0$ we obtain: $$ \overline{g}(h(Z,W),NX)= \overline{g}(\overline{\nabla}_{Z}W,JX)-\overline{g}(\overline{\nabla}_{Z}W,T_{1}X)=$$ $$= \overline{g}(\nabla_{Z}T_{2}W,X)+\overline{g}(\overline{\nabla}_{Z}NW,X)-\overline{g}(\nabla_{Z}W,T_{1}X)=$$ $$= -\overline{g}(T_{2}W,\nabla_{Z}X)-\overline{g}(A_{NW}Z,X)+\overline{g}(W,\nabla_{Z} T_{1}X)=$$ $$ = - X(\ln f)\overline{g}(Z,T_{2}W)+T_{1}X(\ln f)\overline{g}(Z,W)$$ and we get (\ref{e44}).
\begin{proposition} Let $M:={M_1}\times_f {M_2}$ be a warped product pointwise bi-slant submanifold in a locally metallic Riemannian manifold $(\overline{M},\overline{g},J)$ with slant functions ${\theta_{1}}$, ${\theta_{2}}$ and warped function $f$. Then, for any $X \in \Gamma(TM_{1})$ and $Z \in \Gamma(TM_{2})$, we have: \begin{equation}\label{e45} (\nabla_XT^2)Z=p(\cos^2\theta)(\nabla_XT)Z. \end{equation} \end{proposition}
\proof From (\ref{e1}) we have: \begin{equation}\label{e46} q\overline{g}(\nabla_{X}Z,W)=q\overline{g}(\overline{\nabla}_{X}Z,W)=\overline{g}(J^{2}\overline{\nabla}_{X}Z,W)-p\overline{g}(J\overline{\nabla}_{X}Z,W), \end{equation} for any $X \in\Gamma(TM_{1})$ and $Z,W \in \Gamma(TM_{2})$.
By using (\ref{e2}), (\ref{e41}) and $(\overline{\nabla}_{X}J)Z=0$, we obtain: \begin{equation}\label{e47} q X(\ln f)\overline{g}(Z,W)= \overline{g}(\overline{\nabla}_{X}JZ,JW)-p\overline{g}(\overline{\nabla}_{X}JZ,W) \end{equation} and using $JZ=T_ 2 Z+ NZ$, for any $Z \in \Gamma(TM_{2})$, we have: $$ q X(\ln f)\overline{g}(Z,W)= \overline{g}(\overline{\nabla}_{X}T_ 2 Z,T_ 2 W)+\overline{g}(\overline{\nabla}_{X}T_ 2Z,NW)+\overline{g}(\overline{\nabla}_{X}JNZ,W)- $$ $$ -p\overline{g}(\overline{\nabla}_{X}T_ 2 Z,W)-p\overline{g}(\overline{\nabla}_{X}NZ,W). $$
Thus, from (\ref{e10}) and (\ref{e41}) we get: $$ q X(\ln f)\overline{g}(Z,W)=X(\ln f)\overline{g}(T_ 2Z,T_ 2W)+\overline{g}(h(X,T_ 2Z),NW)+$$$$+\overline{g}(\overline{\nabla}_{X}tNZ,W)+ \overline{g}(\overline{\nabla}_{X}nNZ,W)-pX(\ln f)\overline{g}(T_ 2Z,W)+p\overline{g}(A_{NZ}X,W). $$
From (\ref{e43}) we obtain $\overline{g}(h(X,T_ 2Z), NW)=0$ and $\overline{g}(A_{NZ}X,W)=0$.
Thus, by using (\ref{e6}), (\ref{e28}) and (\ref{e30}), we have: $$ q X(\ln f)\overline{g}(Z,W)=X(\ln f)\overline{g}(\cos^{2}\theta_ 2(pT_ 2Z+qZ),W )+$$ $$+\overline{g}(\overline{\nabla}_X(\sin^{2}\theta_ 2(pT_ 2Z+qZ)),W )+
\overline{g}(\overline{\nabla}_X(pNZ-NT_ 2Z),W)-$$$$-pX(\ln f)\overline{g}(T_ 2Z,W) $$ which implies $$ \sin(2\theta_ 2)X(\theta_ 2)\overline{g}(pT_ 2Z+qZ,W )=p\overline{g}(h(X,W),NZ)-\overline{g}(h(X,W),T_2Z)=0. $$ Thus, from (\ref{e43}) and (\ref{e31}) we get (\ref{e45}).
\begin{example} Let $\mathbb{R}^{6}$ be the Euclidean space endowed with the usual Euclidean metric $\langle\cdot,\cdot\rangle$. Let $i: M \rightarrow \mathbb{R}^{6}$ be the immersion given by: $$i(u,v):=\left( u \sin v, u \cos v, u, u \cos v, u \sin v, v \right),$$ where $M :=\{(u, v) \mid u>0, v \in (0, \frac{\pi}{2})\}$.
A local orthogonal frame on $TM$ is given by:
$$Z_{1}= \sin v \frac{\partial}{\partial x_{1}} + \cos v \frac{\partial}{\partial x_{2}}+ \frac{\partial}{\partial x_{3}} + \cos v \frac{\partial}{\partial x_{4}}+\sin v \frac{\partial}{\partial x_{5}}$$
$$ Z_{2}= u \cos v \frac{\partial}{\partial x_{1}} - u \sin v \frac{\partial}{\partial x_{2}} - u \sin v \frac{\partial}{\partial x_{4}} + u \cos v \frac{\partial}{\partial x_{5}}+\frac{\partial}{\partial x_{6}}.$$
We define the metallic structure $J : \mathbb{R}^{6} \rightarrow \mathbb{R}^{6} $ by: $$
J(X_{1},X_{2},X_{3},X_{4},X_{5},X_{6}):=(\sigma X_{1}, \sigma X_{2},\sigma X_{3},\overline{\sigma} X_{4}, \overline{\sigma}X_{5}, \overline{\sigma} X_{6} ), $$ where $\sigma:=\sigma_{p,q}=\frac{p+\sqrt{p^{2}+4q}}{2}$ is a metallic number ($p, q \in \mathbb{N}^{*}$) and $\overline{\sigma}=p-\sigma$.
We remark that $J$ verifies $J^{2}X=p J + q I$ and $\langle JX, Y\rangle = \langle X, JY\rangle$, for any $X$, $Y \in \mathbb{R}^{6}$. Also, we have:
$$J Z_{1}= \sigma \sin v \frac{\partial}{\partial x_{1}} +\sigma \cos v \frac{\partial}{\partial x_{2}}+ \sigma \frac{\partial}{\partial x_{3}} + \overline{\sigma} \cos v \frac{\partial}{\partial x_{4}}+
\overline{\sigma} \sin v \frac{\partial}{\partial x_{5}}$$
$$ J Z_{2}= \sigma u\cos v \frac{\partial}{\partial x_{1}} - \sigma u\sin v \frac{\partial}{\partial x_{2}} - \overline{\sigma} u\sin v \frac{\partial}{\partial x_{4}} + \overline{\sigma} u \cos v \frac{\partial}{\partial x_{5}} + \overline{\sigma} \frac{\partial}{\partial x_{6}}.$$
We remark that $\langle JZ_{1}, Z_{2}\rangle =\langle JZ_{2}, Z_{1}\rangle =0 $, $\langle JZ_{1}, Z_{1}\rangle= 2\sigma +\overline{\sigma}$ and
$\langle JZ_{2}, Z_{2}\rangle = u^{2}(\sigma+\overline{\sigma})+\overline{\sigma}$.
On the other hand we get:
$$\|Z_{1}\|=\sqrt{3}, \ \ \|Z_{2}\|= \sqrt{2u^{2}+1},$$
$$\|J Z_{1}\|=\sqrt{2 \sigma^{2} + \overline{\sigma}^{2}}, \ \ \|J Z_{2}\|=\sqrt{u^{2}(\sigma^{2}+\overline{\sigma}^{2})+\overline{\sigma}^{2}}.$$
We denote by $D_{1}:=span\{Z_{1}\}$ the slant distribution with the slant angle $\theta_{1}$, where $\cos \theta_{1} = \frac{2\sigma+\overline{\sigma}}{\sqrt{3(2 \sigma^{2} + \overline{\sigma}^{2})}}$. Also, we denote by $D_{2}:=span\{Z_{2}\}$ the pointwise slant distribution with the slant angle $\theta_{2}$, where $\cos \theta_{2} = \frac{u^{2}(\sigma+\overline{\sigma})+\overline{\sigma}}{\sqrt{(2u^{2}+1)(u^{2}(\sigma^{2}+\overline{\sigma}^{2})+\overline{\sigma}^{2})}}$.
The distributions $D_{1}$ and $D_{2}$ satisfy the conditions from Definition \ref{d3}.
If $M_{1}$ and $M_{2}$ are the integral manifolds of the distributions $D_{1}$ and $D_{2}$, respectively, then $M := M_{1} \times_{\sqrt{2u^{2}+1}} M_{2}$ with the Riemannian metric tensor $$g:= 3 du^{2} + (2u^{2}+1) d v^{2}$$ is a warped product pointwise bi-slant submanifold in the metallic Riemannian manifold $(\mathbb{R}^{6}, \langle\cdot,\cdot\rangle, J)$. \end{example}
\section{Warped product pointwise semi-slant or hemi-slant submanifolds in metallic Riemannian manifolds}
In this section we get some properties of pointwise semi-slant and pointwise hemi-slant submanifolds in locally metallic Riemannian manifolds.
\begin{definition} \label{d30} Let $M:={M_1}\times_f {M_2}$ be a warped product bi-slant submanifold in a metallic Riemannian manifold $(\overline{M},\overline{g},J)$ such that one of the components $M_{i}$ ($i \in \{1,2\}$) is an invariant submanifold (respectively, anti-invariant submanifold) in $\overline{M}$ and the other one is a pointwise slant submanifold in $\overline{M}$, with the Wirtinger angle $\theta_x \in [0, \frac{\pi}{2}]$. Then we call the submanifold $M$ \textit{warped product pointwise semi-slant submanifold} (respectively, \textit{warped product pointwise hemi-slant submanifold}) in the metallic Riemannian manifold $(\overline{M},\overline{g},J)$. \end{definition}
In a similar manner as in Theorem 2 from (\cite{Hr8}), we obtain:
\begin{theorem} If $M:={M_{T}}\times_f {M_{\theta}}$ is a warped product pointwise semi-slant submanifold in a locally metallic Riemannian manifold $(\overline{M},\overline{g},J)$ with the pointwise slant angle $\theta_ x \in (0, \frac{\pi}{2})$, for $x \in M_{\theta}$, then the warping function $f$ is constant on the connected components of $M_{T}$. \end{theorem}
\proof For any $X \in \Gamma(TM_{T})$, $Z \in \Gamma(TM_{\theta})\setminus\{0\}$, by using (\ref{e13}) in $\overline{\nabla}_{Z}JX = J\overline{\nabla}_{Z}X$ and (\ref{e41}), we obtain: $$ TX(\ln f)Z+h(TX,Z)=T\nabla_{Z}X+N\nabla_{Z}X+t h(X,Z)+n h(X,Z). $$
From the equality of the normal components of the last equation, it follows \begin{equation}\label{e320}
h(TX,Z) = X(\ln f)NZ + n h(X,Z) \end{equation} and replacing $X$ with $TX=JX$ (for $X \in \Gamma(TM_{T})$) in (\ref{e320}), we obtain:
$$
h(J^{2}X,Z)=TX(\ln f)NZ+n h(TX,Z).
$$
Thus, we get:
$$ TX(\ln f)\overline{g}(NZ,NZ)= \overline{g}(h(J^{2}X,Z),NZ)-\overline{g}(n h(TX,Z),NZ)=
$$
$$ =p \overline{g}(h(TX,Z),NZ) +q \overline{g} (h(X,Z),NZ) -\overline{g}(nh(TX,Z),NZ),
$$
for any $X \in \Gamma(TM_{T})$ and $Z \in \Gamma(TM_{\theta})$.
From (\ref{e43}) we have $\overline{g}(h(TX,Z),NZ)= \overline{g} (h(X,Z),NZ) =0$, for any $X \in \Gamma(TM_{T})$ and $Z \in \Gamma(TM_{\theta})$ and by using (\ref{e29}), we get:
\begin{equation}\label{e303} TX(\ln f)\sin^{2}\theta [p \overline{g}(TZ,Z)+q \overline{g}(Z,Z)] = -\overline{g}(nh(TX,Z),NZ).
\end{equation}
On the other hand, for any $X \in \Gamma(TM_{T})$ and $Z \in \Gamma(TM_{\theta})$, we have $TX \in \Gamma(TM_{T})$ and $TZ \in \Gamma(TM_{\theta})$ and from (\ref{e43}), we obtain: $$\overline{g}(h(TX,Z),NZ)=\overline{g}(h(TX,Z),NTZ)=0.$$
Thus, by using (\ref{e1}) and (\ref{e7}), we have:
$$\overline{g}(nh(TX,Z),NZ)=\overline{g}(h(TX,Z),nNZ)=\overline{g}(h(TX,Z),J^{2}Z-JTZ)=$$
$$=p\overline{g}(h(TX,Z),NZ)+q\overline{g}(h(TX,Z),Z)-\overline{g}(h(TX,Z),NTZ)=0$$
and using (\ref{e303}), we obtain:
$$TX(\ln f)\tan^{2}\theta_x \overline{g}(TZ,TZ)=0,$$ for any $Z \in \Gamma(TM_{\theta})$ and $x \in M_{\theta}$.
Since $\theta_x \in (0, \frac{\pi}{2})$ and $TZ \neq 0$, we get $TX(\ln f) = 0$, for any $X \in \Gamma(TM_{T})$, which implies that the warping function $f$ is constant on the connected components of $M_{T}$.
\begin{theorem} If $M:={M_\theta}\times_f {M_T}$ is a warped product pointwise semi-slant submanifold in a locally metallic Riemannian manifold $(\overline{M},\overline{g},J)$ with the pointwise slant angle $\theta_x \in (0, \frac{\pi}{2})$, for $x \in M_\theta$, then $$(A_{NT_{1}Y}X - A_{NT_{1}X}Y) \in \Gamma(TM_{\theta}),$$ for any $X,Y \in \Gamma(TM_{\theta})$. \end{theorem}
\proof For any $X, Y\in \Gamma(TM_{\theta})$ and $Z\in \Gamma(TM_{T})\setminus\{0\}$, from (\ref{e36}) and the symmetry of the shape operator, we have: $$ \sin^{2}\theta\overline{g}([X,Y],T^{2}_{2}Z)=p\overline{g}([X,Y],T_{2}Z)-p\overline{g}(\nabla_{X}T_{1}Y-\nabla_{Y}T_{1}X,Z) + $$ $$ +p(\cos^{2}\theta+1)[\overline{g}(h(X,Z),NY)-\overline{g}(h(Y,Z),NX)]-\overline{g}(h(X,Z),NT_{1}Y)+ $$ $$+\overline{g}(h(Y,Z),NT_{1}X)+\overline{g}(h(X,Y),NT_{2}Z)-\overline{g}(h(Y,X),NT_{2}Z)-$$ $$-\overline{g}(h(X,T_{2}Z),NY)+\overline{g}(h(Y,T_{2}Z),NX).$$
Using (\ref{e2}) and (\ref{e42}), we obtain: $$ \overline{g}(\nabla_{X}T_{1}Y-\nabla_{Y}T_{1}X,Z)=\overline{g}(\nabla_{X}JY-\nabla_{Y}NX-\nabla_{Y}JX+\nabla_{Y}NX,Z)= $$ $$ =\overline{g}(\nabla_{X}Y,JZ)-\overline{g}(\nabla_{Y}X,JZ)+\overline{g}(A_{NY}X,Z)-\overline{g}(A_{NX}Y,Z)= $$ $$ =\overline{g}([X,Y],JZ)+\overline{g}(h(X,Z),NY)-\overline{g}(h(Z,Y),NX)=\overline{g}([X,Y],T_{2}Z). $$
From (\ref{e42}) we get: $$\overline{g}(h(X,Z),NY)=\overline{g}(h(Y,Z),NX)=-\overline{g}(h(X,Y),NZ).$$
Thus, using the symmetry of the shape operator, we have: $$ \overline{g}(h(X,T_{2}Z),NY)-\overline{g}(h(Y,T_{2}Z),NX)=$$$$=-\overline{g}(h(X,Y)NT_{2}Z)+\overline{g}(h(Y,X),NT_{2}Z)=0 $$ and $$\overline{g}(h(X,Z),NT_{1}Y)-\overline{g}(h(Y,Z),NT_{1}X)=\overline{g}(A_{NT_{1}Y}X-A_{NT_{1}X}Y,Z).$$
Thus, we obtain:
$$ \sin^{2}\theta\overline{g}([X,Y],T^{2}_{2}Z)=\overline{g}(A_{NT_{1}Y}X-A_{NT_{1}X}Y,Z), $$ which implies the conclusion.
Following the same steps such in (\cite{Hr8}), we can prove that:
\begin{theorem} If $M:={M_{\perp}}\times_f {M_{\theta}}$ (or $M:={M_{\theta}}\times_f {M_{\perp}}$) is a warped product pointwise hemi-slant submanifold in a locally metallic Riemannian manifold $(\overline{M},\overline{g},J)$ with the pointwise slant angle $\theta_x \in (0, \frac{\pi}{2})$, for $x \in M_\theta$, then the warping function $f$ is constant on the connected components of $M_{\perp}$ if and only if \begin{equation}\label{e10008} A_{NZ}X = A_{NX}Z, \end{equation} for any $X \in \Gamma(TM_{\perp})$ and $Z \in \Gamma(TM_{\theta})$ (or $X \in \Gamma(TM_{\theta})$ and $Z \in \Gamma(TM_{\perp})$, respectively). \end{theorem}
\textit{Cristina E. Hretcanu}
\textit{Stefan cel Mare University of Suceava, Romania}
\textit{[email protected]}
\textit{Adara M. Blaga}
\textit{West University of Timi\c{s}oara, Rom\^{a}nia}
\textit{[email protected]}
\end{document} | arXiv |
\begin{definition}[Definition:Neighborhood of Infinity (Complex Analysis)]
A '''neighborhood of $\infty$''' in $\C$ is a subset of the set of complex numbers $\C$ wich contains a set of the form $\{z \in \C : |z| > r\}$ for some $r\in\R$.
That is, a subset which contains all complex numbers whose modulus is sufficiently large.
\end{definition} | ProofWiki |
Khintchine inequality
In mathematics, the Khintchine inequality, named after Aleksandr Khinchin and spelled in multiple ways in the Latin alphabet, is a theorem from probability, and is also frequently used in analysis. Heuristically, it says that if we pick $N$ complex numbers $x_{1},\dots ,x_{N}\in \mathbb {C} $, and add them together each multiplied by a random sign $\pm 1$, then the expected value of the sum's modulus, or the modulus it will be closest to on average, will be not too far off from ${\sqrt {|x_{1}|^{2}+\cdots +|x_{N}|^{2}}}$.
Statement
Let $\{\varepsilon _{n}\}_{n=1}^{N}$ be i.i.d. random variables with $P(\varepsilon _{n}=\pm 1)={\frac {1}{2}}$ for $n=1,\ldots ,N$, i.e., a sequence with Rademacher distribution. Let $0<p<\infty $ and let $x_{1},\ldots ,x_{N}\in \mathbb {C} $. Then
$A_{p}\left(\sum _{n=1}^{N}|x_{n}|^{2}\right)^{1/2}\leq \left(\operatorname {E} \left|\sum _{n=1}^{N}\varepsilon _{n}x_{n}\right|^{p}\right)^{1/p}\leq B_{p}\left(\sum _{n=1}^{N}|x_{n}|^{2}\right)^{1/2}$
for some constants $A_{p},B_{p}>0$ depending only on $p$ (see Expected value for notation). The sharp values of the constants $A_{p},B_{p}$ were found by Haagerup (Ref. 2; see Ref. 3 for a simpler proof). It is a simple matter to see that $A_{p}=1$ when $p\geq 2$, and $B_{p}=1$ when $0<p\leq 2$.
Haagerup found that
${\begin{aligned}A_{p}&={\begin{cases}2^{1/2-1/p}&0<p\leq p_{0},\\2^{1/2}(\Gamma ((p+1)/2)/{\sqrt {\pi }})^{1/p}&p_{0}<p<2\\1&2\leq p<\infty \end{cases}}\\&{\text{and}}\\B_{p}&={\begin{cases}1&0<p\leq 2\\2^{1/2}(\Gamma ((p+1)/2)/{\sqrt {\pi }})^{1/p}&2<p<\infty \end{cases}},\end{aligned}}$
where $p_{0}\approx 1.847$ and $\Gamma $ is the Gamma function. One may note in particular that $B_{p}$ matches exactly the moments of a normal distribution.
Uses in analysis
The uses of this inequality are not limited to applications in probability theory. One example of its use in analysis is the following: if we let $T$ be a linear operator between two Lp spaces $L^{p}(X,\mu )$ and $L^{p}(Y,\nu )$, $1<p<\infty $, with bounded norm $\|T\|<\infty $, then one can use Khintchine's inequality to show that
$\left\|\left(\sum _{n=1}^{N}|Tf_{n}|^{2}\right)^{1/2}\right\|_{L^{p}(Y,\nu )}\leq C_{p}\left\|\left(\sum _{n=1}^{N}|f_{n}|^{2}\right)^{1/2}\right\|_{L^{p}(X,\mu )}$
for some constant $C_{p}>0$ depending only on $p$ and $\|T\|$.
Generalizations
For the case of Rademacher random variables, Pawel Hitczenko showed[1] that the sharpest version is:
$A\left({\sqrt {p}}\left(\sum _{n=b+1}^{N}x_{n}^{2}\right)^{1/2}+\sum _{n=1}^{b}x_{n}\right)\leq \left(\operatorname {E} \left|\sum _{n=1}^{N}\varepsilon _{n}x_{n}\right|^{p}\right)^{1/p}\leq B\left({\sqrt {p}}\left(\sum _{n=b+1}^{N}x_{n}^{2}\right)^{1/2}+\sum _{n=1}^{b}x_{n}\right)$
where $b=\lfloor p\rfloor $, and $A$ and $B$ are universal constants independent of $p$.
Here we assume that the $x_{i}$ are non-negative and non-increasing.
See also
• Marcinkiewicz–Zygmund inequality
• Burkholder-Davis-Gundy inequality
References
1. Pawel Hitczenko, "On the Rademacher Series". Probability in Banach Spaces, 9 pp 31-36. ISBN 978-1-4612-0253-0
1. Thomas H. Wolff, "Lectures on Harmonic Analysis". American Mathematical Society, University Lecture Series vol. 29, 2003. ISBN 0-8218-3449-5
2. Uffe Haagerup, "The best constants in the Khintchine inequality", Studia Math. 70 (1981), no. 3, 231–283 (1982).
3. Fedor Nazarov and Anatoliy Podkorytov, "Ball, Haagerup, and distribution functions", Complex analysis, operators, and related topics, 247–267, Oper. Theory Adv. Appl., 113, Birkhäuser, Basel, 2000.
| Wikipedia |
how to produce zooplankton
159.203.86.213. This method is called "biomanipulation" and is usually done by reducing predation on zooplankton by planktivorous fish either by directly removing these fish or adding a fish predator such as pike. Eventually, the whole zooplankton community becomes the bottom of a food chain for an entire food web stretching from the smallest fish to the largest whale. These keywords were added by machine and not by the authors. In:J.A. 2004). This procedure gives results which are comparable with those from other upwelling areas. Frontier (1963) has made some observations in the Guinea Current off Abidjan with a Hensen net (0.33 m²; 0.41 mm mesh). Requirements 1. The model then can be used to estimate the production of a sample Zooplankton species over a given time interval. This relationship implies that, when a stage lasts two days, one-half of the individuals will pass from that stage on one day and the other half on the following day, i.e., the number of individuals in a given stage will be inversely proportional to the duration of that stage. Limnol. For many areas for which radiocarbon observations are available, there are not adequate observations in zooplankton. Cruises - sample collection. If possible the estimates of the stock of zooplankton as ml/1000 m³ should be converted to measures of production. From 2004 - 2009, CMarZ carried out more than 80 cruises and collected samples from every ocean basin. Therefore the Pacific and Indian Ocean nets appear to sample the zooplankton populations adequately, excluding larger forms like euphausiids which were not caught by Hentschel's waterbottles. 1972. Can someone put One Tree Hill season 6 episode 1 online on 2nd sept please? 1. Counting chamber 4. The estimate of one third of the generation time was taken from the seasonal picture of intermittent upwellings off Southern California (California, Department of Fish and Game, 1953). Cite as. Where the upwelling is resumed, the algae continue to produce, but the grazing restraint is no longer there. So a new outburst is possible until the zooplankton has been regenerated. Oceanogr. The mesh size of the metre nets was usually about 0.25 to 0.31 mm and the net was hauled at about 1 m/sec. Zooplankton: Definition: Phytoplankton is a group of free-floating microalgae that drifts with the water current and forms an important part of the ocean, sea, and freshwater ecosystems. Ltd., Australia). 2nd Ed. © 2020 Springer Nature Switzerland AG. Observations in the Peru Current were confirmed by the large quantities of information on Hensen net hauls from 50 m to the surface (Flores, 1967; Flores and Elias, 1967 and Guillen and Flores, 1967). A number of models of Zooplankton production have been developed, but they fall into two general classes: (1) direct models based on time-dependent parameters of the zooplankton species [e.g., Edmondson and Winberg (1971) and Rigler and Downing (1984)], and (2) indirect models based on inferred rates of Zooplankton filtering, assimilation, and consumption by fish and other predators [e.g., Winberg (1971)]. Further, it is possible that the displacement volume included an unspecified volume of algae. Ver. Generally phytoplankton (plankton that use photosynthesis like plants) need … Tiny insects called zooplankton eat phytoplankton. Only in a few cases can the observations be averaged adequately through a season, but they are well enough established to estimate generation time. Each tank was inoculated with 2 million organisms obtained from pure stock of zooplankton maintained in the laboratory. Unable to display preview. Methods for the Estimation of Production of Aquatic Animals. Phytoplankton produce their own food by lassoing the energy of the sun in a process called photosynthesis. Zooplankton may have different types of epibionts, like diatoms, green algae, ciliates, bacteria, etc. The holoplanktonic species spend their entire lives in the pelagic environment; meroplanktonic forms are temporary members of the plankton, and include the eggs and larval stages of many benthic invertebrates and fish. You have a great fishing hole. Zooplankton can reproduce rapidly, and populations can increase by about 30 percent a day under favorable conditions. Zooplankton range from zooflagellates a few micrometres long, to large jellyfish. In Table I, where the data are presented in detail, column L gives the generation time with one third added. The bluegill are eaten by bass and BAM! Main Difference. Prepas, E. 1978. Place on the microscope andtry counting the number of organisms present When viewed under the microscope, the sample mayreveal such zooplankton as microcrustace… Few, if any, of the individuals present at the peak of an exponentially developing population were alive at the beginning of the exponential phase. Using a volumetric pipette(wide-mouth pipette), obtain 1 mL of the sample and add it into aSedgwick–Rafter chamber 3. The survival of animals across the upwelling area depends on the algal production. How can zooplankton production be made easier and more reliable? The freshwater zooplankton species was mass produced in lab until transfer to fiberglass tank 1-tonne (t) filled with filtered tap water. The marine zooplankton community includes many different species of animals, ranging in size from microscopic protozoans to animals of several metres in dimension. Some cyanobacteria can fix nitrogen and produce toxins. Blackwell, Oxford. This is a preview of subscription content. Perhaps the assumptions that the loss of nauplii is balanced by gain of algae, and that the proportions escaping are small, are roughly justified. and J. Many hauls for zooplankton have been made in the Pacific and Indian Oceans. Plankton come in two varieties: zooplankton and phytoplankton. Phytoplankton are the microscopic plants that absorb sunlight to produce sugars that form the base of the entire food web. There are unfortunately no observations of zooplankton made in the Canary Current or in the Benguela Current except those made on the Meteor Expedition. Prey are pulled into the polyps' mouths and digested in their stomachs. This process is experimental and the keywords may be updated as the learning algorithm improves. Zooplankton reach maturity quickly and live short, but productive lives. 1984. How would rhe zooplankton be affected if the fish population were to increase? The more food you produce from fertilization, the more fish you can grow. Then it is ready to produce more planktonic larvae. Although there is a large quantity of information on secondary production in the California Current, observations in other upwelling areas are sufficient only to establish an average, but not a seasonal trend, as for the radiocarbon measurements. DISCOVERY). To produce a global assessment of marine zooplankton biodiversity, including accurate and complete information on species diversity, biomass, biogeographical distribution and genetic diversity. Fish larvae are part of the zooplankton that eat smaller plankton, while fish eggs carry their own food supply. So must zooplankton, which feed on the phytoplankton. Downing, and F.H. Rigler, F.H. Academic Press, New York. An upwelling area might be 800 km in length by 200 km in width, with a current moving towards the equator at 20 km/d (Wooster and Reid, 1963). The estimated generation time has been arbitrarily lengthened by one third, to take account of the intermittent character of upwelling, due to variation in the wind stress (California, Department of Fish and Game, 1953). During the daylight hours, zooplankton generally drift in deeper waters to avoid predators. There is not enough information to draw any conclusion for the Canary Current in secondary production and that for the Guinea Current is based on very few observations. The Russians working in the Indian Ocean have used a large Juday net (0.5 m², with a mesh of 0.26 mm) and they express their results as mg/1000 m³ wet weight, essentially the same form of expression as used in the American and Australian work. Mitt. Download preview PDF. They help in regulating algal and microbial productivity through grazing and in the transfer of primary productivity to fish and other consumers (Dejen et al . Microscope Procedure 1. The zooplankton distributions off California (Thrailkill, 1956, 1957, 1959, 1961, 1963) reflect to some extent the transient structures of an upwelling region, possibly because the animals are vulnerable to food lack in the periods between upwellings. As a population changes by addition and growth over a given time interval, a demographic turnover occurs [see reviews of Edmondson (1974) and Rigler and Downing (1984)]. Zooplankton is found in an aquatic environment which serve as food to fishes in lakes, Zooplankton organisms are identified as important components of aquatic ecosystems. Where the upwelling is resumed, the algae continue to produce, but the grazing restraint is no longer there. It is possible that the intermittent halts in the upwelling process actually sustain greater levels of production. Summarizing, the estimates of secondary production are based on the use of one standard net, or its analogues. Edmondson, W.T. This procedure will be demonstrated using Daphnia. Most feed on smaller particles, including phytoplankton (microscopic plants), using sievelike devices which may function like flypaper rather than sieves because viscous … 358 pp. Crustaceans such as water fleas ( Daphnia), cyclops, and copepods are representatives of the consumers or zooplankton found in samples. A Manual on Methods for the Assessment of Secondary Productivity in Fresh Waters. It is hoped that the loss of zooplankton through the meshes is balanced by gain of algae by clogging. Blackwell, Oxford. Indeed the very meticulous methods used by Hentschel on the Meteor Expedition tend to confirm the results from the CALCOFI/POFI/IIOE nets. Fish can produce high numbers of eggs which are often released into the open water column. Zooplankton (pictured below) are a type of heterotrophic plankton that range from microscopic organisms to large species, such as jellyfish. Production, in the context of a population, then, is growth and is only one factor in the material or energy budget for the whole population (Edmondson, 1974): $$\frac{{\Delta N}}{{\Delta t}} = birth + growth - mortality$$, $${N_t} = {N_0} + birth + growth - mortality$$, Since it is necessary to measure recruitment or birth rate, and with continuous birth and death one cannot identify distinct cohorts, production (, $$P = \frac{{{N_1}\Delta {w_1}}}{{{T_1}}} + \frac{{{N_2}\Delta {w_2}}}{{{T_2}}} + \frac{{{N_3}\Delta {w_3}}}{{{T_3}}} + ...\frac{{{N_n}\Delta {w_n}}}{{{T_n}}}$$. Zooplankton are the animal component of the planktonic community ("zoo" comes from the Greek for animal). IBP Handbook 17. Asked By Wiki User. Kamshilov's (1951) formula converting length (0.6-1.0 mm) to weights for copepodites (0.6-1.0 mm) and nauplii (0.08-0.1 mm) has been used. This figure, raised to the area of the upwelling area (as defined above) and raised by the number of generations (as defined above), gives the quantity produced, in tons C/yr. Most Zooplankton, and some benthic animals, reproduce continuously. 19–58. However, the southwest Arabian upwelling, the Domes off Costa Rica and Java and the Indonesian area appear to be areas of medium zooplankton production. We want to capture children's imaginations through great storytelling, bringing the beauty, awe and fascination of the ocean and its inhabitants alive. An individual also undergoes a biochemical turnover during its lifetime so that, upon completing a mean lifespan, it will have assimilated several times its final mass. While many zooplankton exhibit DVM, Chaoborus is uniquely adapted to migrate vertically even when oxygen is not present. Zooplankton feed efficiently on algae, so culture systems are not fouled and contaminating organisms do not proliferate. Limnol. While many zooplaknton require high levels of oxygen to produce ATP, Chaoborus uses an anaerobic malate cycle to derive ATP when oxygen is … The Baleen zooplankton harvesting system (Frish Pty. The available data in different areas are not good enough to support or deny this speculation. Secondary production. To count the animals and estimate their volumes from an array of nets, each sampling a band of size properly, would take so much time that the results could not have been obtained as quickly for such extensive areas. There is no information about the mesh selection of these nets with respect to the composition of the displacement volume. Krill may be the most well-known type of zooplankton; they are a major component of the diet of humpback, right, and blue whales. The most important upwelling areas in terms of secondary production are those in the Peru Current and the Benguela Current, where production is of the order of 3-5.106 tons C/yr (using column N1). Rigler, Editors. pp 251-256 | Foggy White Water: This mainly comprises of zooplankton, clay particles and detritus. (10–50 x … Loss by escape has not been measured, although shown for some larger animals (Fleminger and Clutter, 1965; McGowan and Fraundorf, 1966), so it is possible that euphausiids everywhere are improperly sampled. So a new outburst is possible until the zooplankton has been regenerated. The first step is to determine the depth from which the nets were hauled, so that ml/1000 m³ are converted to ml/m² in a specified layer; fortunately the authors quoted above give the depths of sampling in the upwelling areas specified. On the basis of data given by Marshall and Orr (1955) on maturation and hatching, the duration of a full generation could be worked out at different temperatures. Plankton have evolved many different ways to keep afloat. In all the upwellings examined, temperature observations at the surface are available from the sources given in section 6.6. The same procedure should really be applied to the algae, but a short half of a week or so only reduced the rate of increase of algal production, whereas a week's halt in upwelling may cause the failure of a local brood of nauplii, and when upwelling returns it would take half a generation for a new brood to get underway. Likens, G.E. This process protects the zooplankton from being eaten by the predators especially diurnal and also support the phytoplankton to produce their food in the presence of sunlight. 6. Figure 5.3. During these episodes, the lakes turn green as if green paint had been spilled. and J.J. Gilbert. Limnol. Formalin solution (40percent formaldehyde) 2. Many members of the zooplankton community feed on other members of the population, and in turn become the meals of other larger predators. Columns N1, and N2 give the total secondary production as tons C.106/yr, with the lengthened generation time (N1) and with an uncorrected generation time (N2), so the quantity in column N2 is one quarter greater than that in N1. ... binders and stabilizers have the potential to produce a sustainable, totally aquatic, totally organic, quality fish feed. Like phytoplankton, zooplankton are usually weak swimmers and usually just drift along with the currents. Both discrete time-interval and instantaneous models are used to estimate production. There are times when Spring Lake and Muskegon Lake experience algal blooms. Plankton is composed of the phytoplankton (the plants of the sea) and zooplankton (zoh-plankton) which are typically the tiny animals found near the surface in aquatic environments. Winberg, G.G. 1970. The rate of increase of algal production is reduced and the zooplankton production is perhaps destroyed. Volumetric pipette 3. Marshall and Orr (1955) give the duration of copepodite stages as a proportion of the duration of copepodite stage I. McLaren (1965) gives the duration of some copepodite stages at different temperatures, so a relation between stage duration and temperature was constructed. However, with the present state of zooplankton sampling, the nets used probably represent the best compromise for estimates of zooplankton displacement volume over extensive areas. So we may consider the generation of zooplankton to be autonomous within the upwelling area. The quantity ml/m² (or g/m², wet weight) is converted to carbon by the factor 1/17.85 (Cushing, 1958). The model then can be used to estimate the production of a sample Zooplankton species over a given time interval. Notes on quantitative sampling of natural populations of planktonic rotifers. Sugar-coated Daphnia: A preservation technique for Cladocera. They are heterotrophic (other-feeding), meaning they cannot produce their own food and must consume instead other plants or animals as food. It is assumed that the animals have a uniform age distribution, which is not always the case. In fact a number of different designs have been used (see Reid's table 2) but the differences are not great enough to generate differences in quantities caught. Zooplankton is a group of small and floating organisms that form most of the heterotrophic animals in oceanic environments. The production of zooplankton is given by the average standing crop as ml/m², multiplied by the number of generations during the upwelling season. Int. Haney, J.F. Zooplankton are found within large bodies of water, including oceans and freshwater systems. Asexual reproduction is more common for holoplankton and can be accomplished through cell division, in which one cell divides in half to produce two cells, and so on. Aquafarmer is currently seeking Venture Capital and Partners to assist in product and protocol development. So a proportion of the zooplankton population escaped through the meshes and a further proportion evaded capture. The boat can be operated by one person and is powered by an outboard motor and auxiliary petrol engine to drive the pumps and hydraulic rams. and D.J. This migration is based on the season, size, age, and sex. In many ways, plankton rule the oceans. Winberg (eds). (ed.) and G.G. On that expedition numbers of "metazoa" were counted from 4 litres of water taken at the surface, at 50 m and 100 m (Hentschel, 1933). Zooplankton are drifting ecologically important organisms that are an integral component of the food chain. 175 pp. Oceanogr. So there is an input of zooplankton from the poleward end and it is generated from the coast by upwelling. At night, coral polyps come out of their skeletons to feed, stretching their long, stinging tentacles to capture critters that are floating by. A similar net was used by Tranter (1962) off northwest Australia and off Java. Fish eggs typically have a diameter of about 1 millimetre (0.039 in). It is an ideal environment for growth of fry or juvenile prawns since it provides natural feeds. Corals also eat by catching tiny floating animals called zooplankton. In the Indian Ocean, Wooster, Schaefer and Robinson (1967) gives samples from a 0.200 m water column from the Indian Ocean standard net, which is like that used in the Pacific, with a mesh size of 0.33 mm and a mouth opening of 1 m² (Currie, 1963). ), and tempora… Zooplankton Reproduction . . Assimilation (A) is the difference between ingestion and egestion (A = C - F). To analyze the production of populations with continuous reproduction, it is necessary to use methods that do not require complete evaluation of cohort differences. Hall. The graded and concentrated zooplankton is stored in wells in the floaters of the vessel and can be unloaded by pumping. In particular, this means they eat phytoplankton. Hatcheries often find it difficult to reliably produce enough algae for their zooplankton cultures. With continuous reproduction, the cohorts of the population overlap, so that it is either difficult or impossible to observe changes in abundance over time. By this means, the future population size of the species can be predicted on the basis of its present population structure and size, observed size-specific production of eggs, and certain assumed or known information on biomass and survival. The calculation of secondary productivity, pp. So for sunlight to reach them, they need to be near the top layer of the ocean. Very roughly it takes one zooplankton generation for water and zooplankton to be drifted across the length of an upwelling area. Not affiliated The main difference between Phytoplanktons and Zooplankton is that Phytoplanktons are a photosynthetic, microscopic organisms live in rivers, lakes, freshwater, and streams whereas Zooplanktons are small aquatic animals that also live in water bodies, but they cannot make their own food, and they are dependent on phytoplankton. Oceanogr. Sugar-frosted Daphnia: An improved fixation technique for Cladocera. Plankton are comprised of two main groups, permanent members of the plankton, called holoplankton (such as diatoms, radiolarians, dinoflagellates, foraminifera, amphipods, krill, copepods, salps, etc. 1971. These plankton "blooms" are common throughout the world's oceans and can be composed of phytoplankton, zooplankton, or gelatinous zooplankton, depending on the environmental conditions.
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Earth, Planets and Space
The LMF model and active tectonics
Correlation between seismicity and magnetic anomaly
A theoretical model-TESD
Conclusions and discussions
Full paper
Possible correlation between the vertical component of lithospheric magnetic field and continental seismicity
Yu Lei1,
Liguo Jiao1Email authorView ORCID ID profile and
Huaran Chen1
Earth, Planets and Space201870:179
Received: 8 April 2018
Accepted: 1 November 2018
Recent magnetic satellite missions facilitate new birth of large-scale geomagnetic field models and their applications to tectonics. Here, we directly compare the global geomagnetic field models NGDC-720 with the tectonics and seismicity in Mainland China and surroundings. It is found that the tectonics and seismicity in this area show remarkable correlation with the vertical component of lithospheric magnetic field (Bz) calculated at an altitude of 200 km. Previous thought was that earthquakes are more likely to occur in zero Bz belts or in obvious anomaly gradient belts. On the contrary, we find that more than half (53.2%) of the earthquakes occurred in areas with Bz of − 5 to − 3 nT or in areas with a relatively small horizontal gradient of Bz in the same time interval with the satellite data. The percentage seismic energy in these areas (− 5 nT < Bz < − 3 nT) is even as high as 94.6%. We explain this unexpected result with a two equivalent source dipole model, arguing that the viscosity difference caused by the temperature gradient within the lithosphere likely accounts for the correlation between magnetic anomalies and seismicity.
Lithosphere magnetic field
Continental seismicity
From 1999 to 2009, three magnetic satellites, Ørsted, SAC-C, and CHAMP, were launched along with gravitational satellites (including CHAMP), causing Christensen et al. (2009) to call this period the geopotential decade. New global lithospheric magnetic field (LMF) models were built by combining satellite observations with ground, marine and airborne data. These models include CM1-4 (Sabaka et al. 2004), MF1-7 (Maus et al. 2008), EMAG2-3 (Maus et al. 2009), and NGDC-720 (Maus 2010), Langel and Hinze (1998) provided a comprehensive overview of satellite LMF modeling, and Thébault et al. (2010) summarized these LMF models. After November 22, 2013, the launch of SWARM constellation, even more geomagnetic models were built or updated (Olsen et al. 2016). Because global and regional LMF models are valuable for plate tectonics and lithospheric dynamics, these new satellite observations and new models provide unprecedented opportunities to study the correlation between LMF and tectonics, furthermore seismicity on a large regional scale.
Earthquake occurrence is complex and nonlinear. Although timing is hard to predict, location does follow a few rules: Most middle and large earthquakes occur along boundaries of plates (Beroza and Kanamori 2007), or along boundaries of active tectonic blocks (Zhang et al. 2003, 2005). Seismicity also correlates with other geophysical parameters, such as lithospheric thickness (Chen et al. 2013), gravity field (Zhang et al. 2010; Mitsui and Yamada 2017), and aeromagnetic anomalies (Zhang et al. 2010). A common feature is that large earthquakes tend to occur along the remarkable gradient belts of these parameters.
The correlation between earthquakes and magnetic anomalies has been noticed for some time (Johnston 1997; Han et al. 2009; Huang 2011; Zhang et al. 2013; Marianna et al. 2014). Although authors have examined the temporal variation of magnetic anomalies near the epicenter before and after some large earthquakes, studies of the correlation between seismicity and magnetic anomalies are rare. An important hindrance to such studies has been that most geomagnetic data were obtained from ground-based observations. These observations face two difficulties: removing disturbances of near-surface magnetic sources and reducing all data to the same period. Either factor may introduce errors even stronger than the earthquake-related signals themselves. By avoiding the difficulties of near-surface data, satellite observations provide a better description of large-scale magnetic profile, making them more suitable for studying regional tectonics and seismicity. Taking advantage of the rapid development of satellite magnetic observations, several recent studies have applied these observations to earthquakes (Taylor et al. 2008; von Frese et al. 2008).
China lies at the junction of three tectonic plates: India, Pacific and Eurasia. Tremendous topographic contrasts combine with abnormally active tectonics to give China complex geography and serious seismic hazards. Previous studies (Deng et al. 2002; Zhang et al. 2003) have shown that plates in this area move as active, cohesive blocks. Different tectonic units (blocks, basins, orogens, faults, etc.) may exhibit diverse LMF characteristics. These factors make China and surroundings suitable and unique for studying the correlation between LMF and seismicity. Figure 1 presents the topography (ETOPO1), plate boundaries (UTIG), active blocks (Zhang et al. 2003), and GPS movements (Wang et al. 2001) in our research area (E72°–136°, N16°–56°).
The topography and active tectonic blocks of Mainland China and surroundings. White heavy solid lines mark plate boundaries. Blue lines mark the six level I blocks, which are divided by Level I faults. WKL West Kunlun, Altyn Altyn, HYQL Haiyuan-Qilian, MLM Mount Min-Mount Longmen, ANXJ Anning River-Xiaojiang, RR Red River, LR Lancang River fault, QLDB Qinling Dabie, HLS Mount Helan, YS Mount Yin, YBH Mount Yan-Bohai fault, TL Tanlu fault. Red lines mark Level II block boundaries. Other tectonic units mentioned in the text are also marked, such as JB Junggar Basin, TB Tarim Basin, SB Sichuan Basin, SLB Songliao Basin. Blue arrows mark the GPS velocity vectors (mm/year) with respect to the stable Eurasia
The distribution of satellite LMF in China has been studied for more than 20 years. As early as the MAGSAT era, An et al. (1992) studied the magnetic anomaly of China and its adjacent region with spherical cap harmonic analysis. Xu (1997) and Xu et al. (2000) calculated the apparent magnetization of Mainland China and compared it with geology and heat flux. Alsdorf and Nelson (1999) hypothesized that the negative anomaly of the Tibetan plateau results from a widespread melt crust. After CHAMP, Wang et al. (2008) and Kang et al. (2010) discussed the distributions of new global LMF models in China. Kang et al. (2010) found a positive correlation between LMF and the Curie depth. Using NGDC-720 model, they analyzed the crustal magnetic anomalies of the Tibetan plateau. Kang et al. (2013) used wavelet analysis to study the magnetic anomaly distribution and its attenuation with altitude around the eastern Himalayan syntaxis. Their research revealed that the magnetic anomaly boundaries agree well with the borders of the plateau regional tectonics.
These studies further verified the validity of satellite magnetic observations and the correspondence between LMF and tectonic features. Few studies, however, use the new magnetic satellite data or the new LMF models to examine the correlation between satellite LMF and seismicity. Taylor et al. (2008) compared the CHAMP scalar magnetic anomalies at an altitude of 350 km with the seismicity of the Korean Peninsula. They found that the zero contour of magnetic anomaly divides the seismicity of the peninsula into high and low parts. von Frese et al. (2008) compared the MAGSAT scalar magnetic anomalies at an altitude of 400 km with seismicity of the Transcontinental Magnetic Anomaly (TMA) of the USA. They found that earthquakes predominantly concentrate in both the western and the eastern margins of TMA. Wei and Yu (2012) compared the surface Bz calculated by the two CHAMP LMF models MF6 (Maus et al. 2008) and MF7 with the seismicity of Mainland China during 2004–2010. They found that most epicenters are located near areas with zero Bz.
These researchers (Taylor et al. 2008; von Frese et al. 2008; Wei and Yu 2012) reported correlation between seismicity and the satellite magnetic anomaly, but they did not perform a rigorous statistical inspection of this correlation. Furthermore, they attributed the seismic–magnetic correlation to the stress field. Wei and Yu (2012) even built a magneto-elastic model to explain such correlation. Stress variation is a key factor in triggering earthquakes, yet the piezomagnetic effect may not be large enough to generate magnetic anomaly which could be observed by satellite (Stuart et al. 1995). Other mechanisms for this correlation, however, need to be explored.
In this paper, we first reexamine the correlation between LMF and tectonics. We then use statistics to address how well LMF demonstrates seismicity. After this, we propose a two equal source dipole (TESD) model to explain the correlation between seismicity and LMF. Furthermore, we inspect how the TESD model changes with input parameters. Finally, we draw conclusions and make discussions.
The NGDC-720 LMF model
Maus (2010) built the NGDC-720 LMF model by combining data from satellites (for long wavelength) and from near the surface (for short wavelength). The broad spectrum of this model suits the study of tectonics at diverse lengths. It is represented via spherical harmonic analysis (SHA) and gives the components (Bx, By, Bz) directly on the surface. More details need calculation, such as magnetic anomalies at higher altitudes, and the anomaly gradients. In SHA, the magnetic potential ULMF is presented as:
$$U^{\text{LMF}} (r,\theta ,\phi ) = a\sum\limits_{n = 16}^{720} {\sum\limits_{m = 1}^{n} {\left( {\frac{a}{r}} \right)^{n + 1} \left( {g_{n}^{m} \cos m\varphi + h_{n}^{m} \sin m\varphi } \right)} } P_{n}^{m} (\cos \theta )$$
in which a is the Earth's reference radius (a = 6371.2 km), r, θ, and φ are the radius, geocentric co-latitude, and longitude, respectively, \(g_{n}^{m}\) and \(h_{n}^{m}\) are the spherical harmonic coefficients (also called Gauss coefficients), and \(P_{n}^{m}\) is the Schmidt semi-normalized associated Legendre function of integer degree n and order m. The degree n for LMF is conventionally taken from 16, because the rest part of LMF is thought to be masked by the main or core field (Olsen et al. 2007). The horizontal spatial wavelength λ (in km) is associated with each degree by Backus et al. (1996):
$$\lambda = \frac{2\pi a}{{\sqrt {n(n + 1)} }} \approx \frac{2\pi a}{n}$$
Thus, Formula (1) gives the magnetic potential for wavelengths between 56 km (n = 720) and 2500 km (n = 16). The Gauss coefficients of the NGDC-720 model were calculated from the EMAG2 grid (Maus et al. 2009), which is a combination of satellite, airborne, and marine data. Aeromagnetic data from Mainland China and surroundings used in EMAG2 were provided by GETECH (http://www.getech.com/) and CCOP (http://www.ccop.or.th/). The low degree (16 ≤ n ≤ 120) coefficients are replaced by the MF6 model (Maus et al. 2008), which was built with CHAMP low orbital (< 350 km) data from 2004 to 2007.
The intensity and all other components, including inclination and declination, can be calculated from Formula (1). Because of inclined magnetization, reduction to the pole (RTP) is routinely used to align scalar magnetic data to the source underground (Arkani 2007). While for the vector data, especially for the vertical (downward) component Bz, RTP is not a necessary process. Sometimes, Bz is used directly in tectonic applications (Purucker et al. 1998; Maule et al. 2005; Rajaram et al. 2009; Kang et al. 2012; Gao et al. 2013). Because of good correspondence between satellite Bz and tectonics, here we also use Bz:
$$B_{z} = - \sum\limits_{n = 16}^{720} {\sum\limits_{m = 0}^{n} {(n + 1)\left( {\frac{a}{r}} \right)^{n + 2} \left( {g_{n}^{m} \cos m\phi + h_{n}^{m} \sin m\phi } \right)} } P_{n}^{m} (\cos \theta )$$
To study the distribution of magnetic anomalies with different horizontal wavelengths, we may select different ranges of degree n, as did Kang et al. (2012). Since anomalies with different wavelengths have different attenuating characters, an alternative would be to select a different r. In a former study (Jiao et al. 2013), we visually compared seismicity to magnetic anomalies at altitudes (H = 0 km, 50 km, 100 km, 200 km and 400 km). We found that Bz at 200 km is mostly associated with seismicity. As a result, in this study, we focused on Bz at 200 km.
B z and active tectonics
The continental crust has stronger anisotropic structures than those of the oceanic crust. For Mainland China, the crustal tectonic heterogeneity is mainly characterized by horizontal blocks and vertical layers. Hinted by studies of active faults, folds, basins, volcanoes, earthquakes, and the GPS measurements of ground motion, the tectonics of Mainland China are divided by 6 level I and 22 level II active blocks (Deng et al. 2002; Zhang et al. 2003), as shown in Fig. 2. LMF is generated by magnetization of magnetic minerals residing in the crust and part of the upper mantle (Thébault et al. 2010). Because various minerals at different temperatures and pressures may exhibit diverse magnetization characteristics, the anisotropy of the continental crust could well be reflected in the distribution of LMF, as shown in direct comparisons (Frey 1982; Achache et al. 1987; Kang et al. 2012). To reexamine this reflection with recent LMF model, Bz at an altitude of 200 km is calculated (0.5° × 0.5° grids) by Formula (3), and plotted in colors in Fig. 2.
Comparisons of Bz with active tectonics and the 10-year (2000–2010) seismic catalog. White heavy solid lines mark plate boundaries. Blue lines mark the six level I blocks, red lines mark Level II block boundaries (see Fig. 1), and the names of level II blocks are given here. Focal mechanism with different colors and sizes indicate epicenters of focal depth and magnitude. Background colors are for Bz at 200 km altitude
Comparing LMF to tectonics (Fig. 2) reveals remarkable features. First, different types of tectonic units exhibit obviously different magnetic anomalies. Stable continents such as basins (Sichuan, Tarim, Junggar, Songliao) and cratons (North China) show obvious positive anomalies, whereas active orogens (Himalaya, Mount Tian and Mount Yin) and plateaus (Tibet) show obvious negative anomalies. Second, faults tend to lie along the boundaries of these distinct anomalies. For instance, the West Kunlun fault and the Altyn fault lie along the respective western and southern boundaries of the Tarim positive anomaly. The Mount Min-Mount Longmen fault and the Anning River-Xiaojiang fault follow the western boundary of the Sichuan basin positive anomaly, and the Red River fault lies on the northeastern boundary of the eastern Himalayan syntaxis (E92°–100°, N22°–28°) positive anomaly. Some Level II faults, though they have no obvious correlation with Bz at 200 km altitude, do correlate well with Bz at the surface (Kang et al. 2012).
To study the integral tectonics of the research area, the GPS movements and Bz are drawn together in Fig. 3, and the exponent of vertically averaged effective viscosity (calculated by the hard rheological model in Deng and Tesauro 2016) is drawn in Fig. 4. As a whole, the tectonics of Mainland China is controlled by the underthrusting of stiff (η in green and blue) India plate (where positive magnetic anomalies of > 16 nT dominate) from southwest and Pacific plate (η in green) from southeast (Zhao et al. 2011). The converging of plates generates heat via extrusion on the surface or friction near the boundary, thus uplifts the Curie isotherm, brings obvious negative anomaly, and weakens the lithospheric strength (η in red). Under these effects, negative anomalies (< −8 nT) dominate Tibet plateau. The extrusion (northward movement at a rate of 36–40 mm/year measured on the south of Himalaya, and eastward movement at a rate of 21–26 mm/year measured on the east boundary of the Tibet plateau, see Wang et al. (2001) and Zhang et al. (2004b) of Tibetan crust (weak, η in red) is resisted by the old and cold (strong) basins of Tarim (η in yellow and green, with GPS velocity decreasing to 4.7 mm/year NE in Tianshan) and Sichuan (η in yellow and green, with GPS velocity decreasing to 6–10 mm/year SE in South China), where obvious positive anomalies (> 24 nT) dominate. The northward extrusion of Tibet is also resisted by Qaidam basin (relative strong, η in orange), where the anomaly is only slightly positive (~ 4 nT). Extrusion thus finds its paths in the northeast and southeast direction, leading to an obvious negative anomaly (< −12 nT) in the northeast of Bayan Har block (14–17 mm/year NE) and a slightly negative anomaly (~ −4 nT) in Chuandian block (~ 20 mm/year SES). The northeast part develops continually northward, resulting in negative anomaly patches (− 8 nT) in Qilian (~ 7 mm/year NE) and even the west of Ordos block (~ 6 mm/year SE). Subjected to the combined effects of India and Pacific underthrusting, the ancient North China craton (~ 7 mm/year SE) is destroyed and only shows moderate positive anomaly (< 12 nT). The anomaly in North China craton is very different from the other two Precambrian cratons of Tarim (Tarim block) and Yangtze (in the northwest of South China Block), which may be a magnetic manifestation of craton destruction or lithospheric thinning (Zhu et al. 2012). The irregularity of magnetic distribution in North China also indicates the spatial complexity of craton destruction.
Comparisons of Bz at 200 km altitude with active tectonics and the megaseisms (> Ms7.0) catalog between 1976 and 2018. Background colors are for Bz at 200 km altitude. Orange arrows mark the GPS velocity vectors (mm/year). Lines with different colors mark boundaries of different tectonic units (see Figs. 1, 2)
Exponent of vertically averaged effective viscosity (η) variations of the lithosphere (viscosity data are provided by YF Deng, see Deng and Tesauro 2016). Lines with different colors mark boundaries of different tectonic units (see Figs. 1, 2)
In general, active blocks, faults, GPS movements and lithospheric strength in Mainland China all have close correlations with Bz. Boundaries of obvious magnetic anomalies often signify boundaries of active blocks or large faults (Fig. 3). Large GPS movements usually appear inside areas with obvious negative Bz (e.g. Tibet, Tianshan and Chuandian). Moreover, the GPS movements tend to decrease dramatically through negative Bz areas, which is most obvious in Tibet. Areas with obvious negative Bz often effect as absorbing the displacement, which result in crustal shortening and uplifting. By contrast, areas with obvious positive Bz (e.g. Tarim and Sichuan Basin) often effect as resisting, slowing and transmitting the crustal displacement, without obvious internal deformation. As for Bz and the lithospheric strength, strong lithosphere is commonly corresponding to obvious positive Bz (e.g. Tarim, Sichuan, Songliao and North China), and weak lithosphere is usually related to obvious negative Bz (e.g. Tibet and Chuandian, Fig. 4).
Seismicity and B z
The correlation between seismicity and the structure of fault zones or crustal deformation in Mainland China has been investigated (e.g. James et al. 1976; Wesnousky et al. 1984; Zhang et al. 2004a). In this study, we focus on the correlation between seismicity and magnetic anomaly.
The correlation between magnetic anomalies and tectonics lays the foundation for the correlation between magnetic anomalies and seismic activities. We use the International Seismological Centre (ISC) reviewed seismic catalogue to compare seismicity with LMF. Our research area is (E72°–136°, N16°–56°), where 595 earthquakes with magnitudes larger than Ms5.0 occurred in the continental area. The satellite magnetic data used by the NGDC-720 model range from 2004 to 2007. To compare seismicity in different periods and to check the validated period of the LMF model, we studied records of three periods: 2004–2007 (77 records), 2000–2010 (222 records) and 1978–2011 (595 records). Taking the 2000–2010 period as an example, in Fig. 2 we show the comparison between seismicity (marked by colored spheres) and Bz.
The spatial distribution of seismicity in Mainland China exhibits tremendous inhomogeneity. Most earthquakes occurred in west China, especially inside and near the boundary of Tibet plateau, and only one (Ms > 5.0) occurred in east China in this period. For the focal depth, shallow (< 30 km) earthquakes mostly occurred inside Tibet plateau and deep earthquakes (> 30 km) usually occurred near the north and east boundary of Tibet plateau. For the focal mechanisms, thrust earthquakes are mainly distributed on the Himalaya collision boundary, while extensional and strike-slip earthquakes are distributed inside Tibet plateau, and thrust and strike-slip earthquakes occurred along the north and east boundary of Tibet plateau. As for the correlation between Bz and earthquakes, Fig. 2 illustrates four phenomena: (1) More earthquakes have occurred in areas with negative anomalies than in areas with positive anomalies; (2) few earthquakes occurred inside predominant anomalies, either positive (e.g. basin anomalies) or negative (the two obvious anomalies located near the plate boundary of India and Eurasia); (3) numerous earthquakes occurred near zero Bz areas (around the Sichuan Basin anomaly), or near boundaries of obvious anomalies (such as the Tarim Basin anomaly); (4) for distributions of earthquakes that deviate from the above three rules, rules 1–3 could be reproduced by selecting earthquakes with magnitude larger than Ms7.0, or by selecting earthquakes with a focal depth deeper than 30 km, such as those near the boundary of the whole Tibetan negative anomaly (Jiao et al. 2013).
For a more directly illustration of above characters, only earthquakes with magnitude larger than Ms7.0 occurred in 1976–2018 were selected and drawn in Fig. 3. As can be seen, megaseisms are distributed mainly along boundaries of obvious magnetic anomalies. Earthquakes with all the three types of focal mechanisms could occur on the boundaries of positive or negative magnetic anomalies. No direct correlation between focal mechanism and magnetic anomaly is shown here.
To study the distribution signature of seismicity quantitatively and to compare earthquakes occurred in different periods, we performed a statistical analysis. The results are shown in Fig. 5. To specify the magnetic anomalies in the vicinity of the epicenters we noted the longitude and latitude of the epicenter, picked out the four nearby magnetic anomaly nodes, and calculated the average of these nodes. We uniformly divided the whole range of magnetic anomalies by an interval of 2 nT (depending on the amplitude of the anomaly), and we counted the number of earthquakes occurring in each interval. The statistical results for the three periods are shown in Fig. 5a–c, respectively.
Histograms and boxplots for the distribution of seismicity over Bz. a–c Different time intervals, respectively. d The statistical distribution of Bz grid nodes. See details in the text
Both of histograms and boxplots are shown in Fig. 5. First, let us check the histograms. Oblique numbers atop the corresponding histograms represent ratios of the maximum records within the magnetic interval to the total records, and oblique numbers inside the small box beside the histograms represent ratios of records within the negative anomalies to the total records. As is shown, most earthquakes concentrate in areas with small magnetic anomalies, while few earthquakes are located in areas with dominant anomalies. The records peak at Bz = −5 ~ −3 nT for all three periods. Increasing the time interval decreases the peak ratio from 53.2% (2004–2007) to 20.5% (1978–2011). Meanwhile, the ratio of records that occur inside negative anomalies also decreases from 81.8% (2004–2007) to 65.2% (1978–2011).
It is too early to conclude whether earthquakes tend to occur more frequently in areas with negative anomalies, or especially with Bz = −5 ~ −3 nT. What if the negative anomaly itself is dominant in the research area? Thus, the background distribution of Bz (Fig. 5d) needs to be checked. In other words, Fig. 5d shows the average statistical distribution of seismicity when there is no correlation between magnetic anomalies and seismicity. As can be seen, for the total 8932 grid nodes, the distribution peaks also at Bz = −5 ~ −3 nT, and the ratios are 52.7% (negative anomaly) and 11.3% (Bz = −5 ~ −3 nT), respectively.
Next, let us check the boxplots, which are on top of histograms in Fig. 5. Boxplot (Robert et al. 1978) is a method for graphically depicting groups of numerical data through their quartiles. In the boxplot, the median (red line), first quantile (blue vertical line on the left, Q1) and third quantile (blue vertical line on the right, Q3) of the data are first calculated. Then, the two limits are calculated customarily by extending 1.5 times of the box width (equals to Q3–Q1) to both sides. Data outside the two limits are outliers. Outliers are sometimes used to eliminate extreme or disturbed data. The coefficient 1.5 could be replaced by 3 in other cases, noting that the selection of the coefficient does not affect lines inside the box.
As is shown in Fig. 5, all the four boxplots exhibit concentrated distributions (a half samples are inside the narrow box nearby the median lines), and gradually increase in concentration appears when checking from Fig. 5d to a (both of the whole range and the half sample box become narrower). Especially, a quarter large earthquakes occurred inside a very narrow Bz box of 0.7 nT width (− 4.6 to − 3.9 nT) for the time period of 2004–2007. The medians shift left gradually from − 0.5 nT (grid distribution) to − 3.9 nT (2004–2007). Furthermore, there is good consistency between the left part of each box (limited by Q1 and the meridian) and the dominate histogram when comparing the boxplots and histogram plots for each time period.
All the four histogram plots in Fig. 5 failed to pass the rigorous normal distribution test; we thus seek for nonparametric statistical tools, e.g. the Kolmogorov–Smirnov test (Kolmogorov 1933; Smirnov 1948). The two-sample K–S test is one of the most useful and general nonparametric methods for comparing two samples, as it is sensitive to differences in both location and shape of the empirical cumulative distribution functions of the two samples. The null hypothesis for K–S test is that the two groups have the same overall distributions, against the alternative hypothesis that they have not. Using K–S test, we calculate the h values and p values between the background grid distribution and the seismic distributions for different time periods, which is shown in Table 1.
h and p value of K–S test
Statistic value/time period
2.19 × 10−10
5.48 × 10−7
The test result has two parameters: h and p. h is the hypothesis test result, returned as 1 or 0. If h = 1, it indicates the rejection of the null hypothesis at a certain significance level (defaulted as 5%). If h = 0, it indicates a failure to reject the null hypothesis at a certain significance level. The other parameter p is the probability, which denotes the degree of supporting the null hypothesis. If p < 0.05, it indicates that the null hypothesis is rejected at a significance of 5%.
As is shown in Table 1, h = 1 and p ≪ 0.05 hold for all the three time periods, which shows that in all the three time periods, the seismicity distributions are very different from the background grid distribution. Other statistical tools, e.g. Mann–Whitney U (Mann and Whitney 1947) and Kruskal–Wallis (Kruskal and Wallis 1952) test, have been also applied, and we got similar results. These indicate that the spatial distribution of earthquakes is influenced remarkably by Bz.
To study quantitatively the influence of Bz on the spatial distribution of continental seismicity, we propose an earthquake–magnetic ratio qEM to measure the magnetic seismic susceptibility as:
$$q_{\text{EM}} = {\raise0.7ex\hbox{${p_{\text{M}} }$} \!\mathord{\left/ {\vphantom {{p_{\text{M}} } {p_{\text{B}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${p_{\text{B}} }$}}$$
in which pM is the statistical percentage of seismic records in a certain magnetic interval and pB is the corresponding statistical percentage of the background grid nodes in the same magnetic interval. Where qEM > 1, earthquakes tend to occur in the given magnetic interval, so the risk of seismicity is higher. Where qEM < 1, the risk of seismicity is lower. During 2004–2007 (Fig. 5a), for example, the qEM for earthquakes that occur within a negative anomaly is qEM = 81.8%/52.7% = 1.6, while for Bz = −5 ~ −3 nT, the qEM = 53.2%/11.3% = 4.7. These calculations demonstrate the higher likelihood of seismicity for areas with negative Bz or Bz = −5 ~ −3 nT.
The qEM distribution for each time period is plotted in Fig. 6. As can be seen, qEM is always largest in the magnetic range of − 5 to – 3 nT for the three time periods, and it is most remarkable in 2004–2007. When the time intervals are extended, qEM decreases from 1.6 and 4.7 (2004–2007) to 1.2 and 1.8 (1978–2011), for negative Bz and Bz = −5 ~ −3 nT, respectively. Usually, larger qEM areas deserve more attention, while the large qEM areas in the both ends of Bz (Fig. 6b, c) should be noted as exceptions, because that few earthquakes occurred inside these areas.
qEM for each time period. a–c Different time intervals, respectively. d The statistical grid distribution of Bz
Seismicity and the gradient of B z
When comparing Bz to seismicity in Fig. 2, we realized numerous earthquakes were located near zero Bz belts (similar to Wei and Yu 2012), or near boundaries of strong anomalies (similar to von Frese et al. 2008). We have just studied the zero Bz belts, and now continue to study the boundaries, which resemble the gradient belts of anomalies. Former studies (e.g. Zhang et al. 2010) indicated that most earthquakes occurred along the horizontal gradient belts of the residual gravity anomaly. Since both magnetic field and gravity field are geopotential fields, we could also compare seismicity and magnetic anomaly gradient.
The gradient of Bz can be obtained by calculating the derivative of Formula (3) in spherical coordinates,
$$\begin{aligned} \nabla B_{z} & = \hat{e}_{r} \partial_{r} B_{z} + \hat{e}_{\theta } \frac{{\partial_{\theta } B_{z} }}{r} + \hat{e}_{\phi } \frac{{\partial_{\phi } B_{z} }}{r\sin \theta } \\ & = \hat{e}_{r} \sum\limits_{n = 16}^{720} {\sum\limits_{m = 0}^{n} {(n + 1)\left( {\frac{a}{r}} \right)^{n + 2} \frac{n + 2}{r}\left( {g_{n}^{m} \cos m\phi + h_{n}^{m} \sin m\phi } \right)} } P_{n}^{m} (\cos \theta ) \\ & \quad + \hat{e}_{\theta } \sum\limits_{n = 16}^{720} {\sum\limits_{m = 0}^{n} { - (n + 1)\left( {\frac{a}{r}} \right)^{n + 2} \left( {g_{n}^{m} \cos m\phi + h_{n}^{m} \sin m\phi } \right)} } \frac{{\partial P_{n}^{m} (\cos \theta )}}{r\partial \theta } \\ & \quad + \hat{e}_{\phi } \frac{1}{r\sin \theta }\sum\limits_{n = 16}^{720} {\sum\limits_{m = 0}^{n} {(n + 1)\left( {\frac{a}{r}} \right)^{n + 2} \left( {g_{n}^{m} \sin m\phi - h_{n}^{m} \cos m\phi } \right)} } mP_{n}^{m} (\cos \theta ) \\ \end{aligned}$$
and the horizontal gradient
$$|\nabla_{\text{H}} B_{z} | = \sqrt {(\partial_{x} B_{z} )^{2} + (\partial_{y} B_{z} )^{2} } = \sqrt {( - \partial_{\theta } B_{z} )^{2} + (\partial_{\varphi } B_{z} )^{2} }$$
We compared |∇HBz| with tectonics and seismicity in Fig. 7. As can been seen, some faults are along gradient belts, such as the West Kunlun fault, the Altyn fault, the Mount Yin fault and the Tanlu fault (a deep fracture first detected by a large NNE linear positive aeromagnetic anomaly in 1957). Some earthquakes were indeed located inside magnetic gradient belts, e.g. those near the northwest of Sichuan Basin, or in the west of Tarim Basin. In contrast to previous understandings, however, more earthquakes were located near boundaries of gradient belts, such as those to the north of the N40° line, or in areas without a large magnetic gradient, such as those inside the Tibetan block. Other gradient components, such as the vertical gradient of Bz, show a similar character (see Additional file 1: Figure S1).
Comparisons of Bz's horizontal gradient with active tectonics and the 10-year (2000–2010) seismic catalog. Background colors are for the horizontal gradient of Bz at 200 km altitude. Focal mechanism with different color and size are for earthquakes of different magnitudes and focal depths. Lines with different colors mark boundaries of different tectonic units (see Figs. 1, 2)
To obtain quantitative results, we also analyzed statistics from different time intervals. The results are shown in Fig. 8, and the gradient interval is 0.01 nT/km. As Fig. 8 shows, earthquakes are mostly located in areas with a relatively weak magnetic gradient. For the three different time periods, the intervals for the maximum number of seismic records are identical at |∇HBz| = 0.015–0.025 nT/km. The ratio of earthquakes occurring in this interval also decreases from 53.3 to 22.4% when the time interval is extended from 2004–2007 to 1978–2011. The background distribution of seismicity over |∇HBz| is shown in Fig. 8d. Here areas with |∇HBz| = 0.015–0.025 nT/km also dominate the whole research area. The earthquake–magnetic ratio qEM is 2.7, 1.6 and 1.1 for the time interval 2004–2007, 2000–2010, and 1978–2011, respectively.
The distribution of seismicity over the horizontal gradient of Bz (refer to Fig. 5 for legends)
For the total gradient |∇Bz|, we present the results directly. The peak value for the maximum number of earthquakes is located at |∇Bz| = 0.025–0.035 nT/km, and the qEM is 4.0, 2.6 and 1.7 for the time intervals of 2004–2007, 2000–2010, and 1978–2011, respectively. This magnetic gradient interval shifts to larger values than the horizontal gradient intervals because the vertical gradient is included. Also, in each period the qEM for the total gradient is larger than that for the horizontal gradient, which indicates that the former is more suitable for defining seismically dangerous areas. As an indicator, Bz (qEM is 4.7 for 2004–2007) is even better than |∇Bz|.
Seismicity for different altitudes and sub-regions
Table 2 shows similar statistical analyses of Bz for the same period 2004–2007 at different altitudes. The anomaly range for most earthquakes might not always be the same with the background grid distribution, such as the surface Bz. In this case, we selected the range for most earthquakes (5–15 nT). As Table 2 indicates, qEM for Bz at an altitude of 200 km has the largest value, thus most suitable to constrain seismicity. Furthermore, not only the earthquake number dominates in the anomaly range of − 5 to – 3 nT, but also the earthquake energy. The seismic energy in this range is about two orders larger than those in any other ranges, and it composed 94.6% of all the seismic energy in this period (see the graphical abstract). Bz at this altitude reflects well not only the seismicity, but also the topography (see Additional file 1: Figure S2). The statistical result is similar to our previous visual inspection (Jiao et al. 2013), while more accurate.
qEM at different altitudes
Altitudes (km)
Bz range for most earthquakes (nT)
− 10 to 0
− 5 to 5
− 5 to − 3
− 1.5 to − 0.5
qEM for Bz
Bz horizontal gradient range for most earthquakes (nT/km)
0.225–0.275
qEM for Bz horizontal gradient
To check the earthquake distributions in different regions, we divided the whole research area into 6 sub-regions: Northeast China, North China, Xinjiang, Tibet, Sichuan-Yunnan and South China. Results (Additional file 1: Figure S3) show that the anomaly range of − 5 to − 3 nT for most earthquakes is valid for Northeast China and Tibet, while invalid for the other four sub-regions. Furthermore, longer time interval (110 years) with more earthquakes (1147) is also inspected, which (Additional file 1: Figure S4) shows a consistent result with Fig. 5.
Our statistical analyses of seismicity showed earthquakes are more likely to occur in areas with certain magnetic anomalies. In order to explain the physical mechanism behind this phenomenon, Wei and Yu (2012) proposed a model of magnetic induction from line concentrated force in a magnetized half-plane. They based their explanations on the assumption that all the satellite magnetic anomalies are caused by concentrated stresses. This assumption, however, is far from the fact (Thébault et al. 2010). Furthermore, the small piezomagnetic effects (Stuart et al. 1995) make all the stress-related explanations futile. In our attempt to unfold the generating mechanism of the LFM, we put forward a new explanation.
Curie isotherm and the LMF
The LMF is generated by the magnetization of minerals in an ambient magnetic field, which is usually the main field. The two different magnetization mechanisms are: induced and remnant. The magnetic anomaly of the continental crust (especially tectonically stable basins and cratons) is primarily generated by induction in the current main field, while the anomaly of oceanic crust (especially the middle ocean ridges) is primarily generated by remnants in ancient main fields (Thébault et al. 2010). Since this research concerns continental seismicity, it examines magnetic anomalies generated by induction.
Studies about the distribution characteristics of satellite magnetic anomalies (e.g. Kang et al. 2010) argued that the magnitude of these anomalies correlated positively with the depth of the Curie isotherm. The Curie isotherm is a thermal boundary where ferromagnetism is converted into paramagnetism (Langel and Hinze 1998; Rajaram et al. 2009). During this process, magnetic susceptibility decreases dramatically. Therefore, the Curie isotherm is considered the lower magnetic boundary of the crust. The burial depth of the Curie isotherm is closely related to the surface heat flux (Maule et al. 2005), in that low heat flux areas are expected to have a relatively deep Curie isotherm, and vice versa. So to judge whether the Curie depth is critical for the LMF, we compared Bz with the surface heat flux in Fig. 9.
Comparison of Bz (colors) and surface heat flux (triangles). The heat flux data are provided by Hu et al. (2000, 2001). Lines with different colors mark boundaries of different tectonic units (see Figs. 1, 2)
As Fig. 9 reveals, the giant basins (Tarim, Junggar, Sichuan and Songliao) show a relatively low heat flux. Most are lower than 60 mW/m2 (blue filled triangles) with a few stand-out values not higher than 80 mW/m2 (green). These basins are all identified as strong positive anomalies. The heat flux in North China is usually lower than 80 mW/m2, where the magnetic anomaly is slightly positive. In contrast, high heat flux is observed in Tibet and at the southeastern coast. There, the heat flux is largely higher than 60 mW/m2 (green), and sometimes even higher than 140 mW/m2 (white). Most of these areas are identified as negative anomalies. These indicate that in most of the research area, the Curie depth is of key importance for LMF distribution.
Though intuitively magnetic low corresponds to high heat flow, some areas (such as South China) seem to be opposite. A magnetic low could come from a shallow Curie isotherm, as well as from a small magnetic susceptibility. In the latter case, it won't bring obvious heat flux. Furthermore, the surface heat flow relates to the Curie isotherm nonlinearly, and other parameters (such as the thermal conductivity, heat production rate, et al.) are also involved, while these won't change the dominate role of Curie depth for LMF in most continental regions.
TESD, a theoretical model
Enlightened by former studies (e.g. Maule et al. 2005; Kang et al. 2010) and our own comparisons (Fig. 9), we try to explain the correlation between seismicity and magnetic anomalies mainly via the effects of different Curie depths. To invert the Curie depth from satellite magnetic anomalies, equivalent source dipole (ESD) is often adopted for approximation (Mayhew et al. 1980; Purucker et al. 1998, 2002). A center dipole is used in ESD to describe the magnetic field generated by induction of a cuboid block. The susceptibility is assumed to be uniform, and the dipole moment mI is proportional to the cuboid volume. The formula is
$${\mathbf{m}}_{\text{I}} = {\mathbf{M}}_{\text{I}} V = \frac{{\kappa {\mathbf{B}}_{\text{M}} }}{{\mu_{0} }}V$$
where MI is the intensity of magnetization, V is the volume of the cuboid, к is the uniform susceptibility, BM is the main field, and μ0 is the permeability of the vacuum. The magnetic field at r generated by mI is
$${\mathbf{B}}_{I} = - \frac{{\mu_{0} }}{{4\pi r^{3} }}[{\mathbf{m}}_{\text{I}} - 3({\mathbf{m}}_{\text{I}} \cdot {\hat{\mathbf{r}}}){\hat{\mathbf{r}}}]$$
For simplicity, we set the length and width of the cuboid both at L and considered only the vertical component, ignoring the effects of horizontal magnetization and spherical geometry. The vertical induced field is
$$B_{Iz} = - \frac{{\kappa L^{2} dB_{Mz} }}{{2\pi r^{3} }}\left[ {1 - \frac{{3(d + H)^{2} }}{{r^{2} }}} \right]$$
where d is the burial depth of the dipole, which is half the height of the cuboid, BMz is the vertical main field, H is the altitude (set at 200 km), and r is the distance between the source point and the field point.
To explain the correlation between seismicity and the LMF, we put forward a model using two ESDs and labeled it TESD. In our TESD model, as is shown in Fig. 10c, two nearby cuboid blocks are combined, and the magnetic fields are approximated by two dipoles. We checked the distribution characteristics of the magnetic anomaly and compared them with our statistical results.
The TESD (Two Equivalent Source Dipole) model. a Bz (solid line) and its y-gradient (dashed line, absolute value); b the distribution of the 2D Bz. The two black dots are the centers of the two ESD; c the TESD sketch. The altitude is not in scale; d the distribution of 2D gradient
For simplicity, we assumed the two cuboids both have length and width L, but have respective Curie depths of 2d1 and 2d2. In Fig. 10c, point A is any point at height H in the same vertical plane with the two dipoles. The horizontal distance between point A and the dipole center is h. The distance r is calculated by
$$r = \sqrt {h^{2} + (d + H)^{2} }$$
Parameters are selected as
$$L = 300\;{\text{km}}, \, \quad d_{1} = 10\;{\text{km}}, \, \quad d_{2} = 25\;{\text{km}}, \, \quad B_{Mz} = 3 \times 10^{4} \;{\text{nT}},\quad \, \kappa = 0.035\;{\text{SI}}$$
The parameters (d1, d2, BMz) are selected roughly according to the Tibetan block and the Sichuan basin. The selection of the susceptibility is according to Purucker et al. (2002), Maule et al. (2005), and Rajaram et al. (2009).
In Eq. (3), only the short wavelengths (λ ≤ 2500 km) were included in Bz for the NGDC-720 model; thus when we compared the magnetic anomaly generated by the TESD model, the long wavelength should also be subtracted. A rigorous study requires that the total magnetic field generated by cuboid 1 and cuboid 2 should be expanded by SHA; then the long wavelength magnetic field of n = 1–15 is subtracted. The SHA should be done globally, and the Curie depths of areas outside cuboid 1 and cuboid 2 should also be known. These rigorous measures, however, make the computation too complex, so we used a much simpler filter. Instead of the SHA filter, we took the regional longitudinal average. The LMF then became the residual of the total field minus the average field, a process often adopted in analyses of local anomalies.
Figure 10a outlines Bz and its gradient (the absolute value) when point A moving across the TESD from west to east. Here the gradient is calculated by forward difference. To improve understanding, the 2D Bz and its gradient are also plotted in Fig. 10b, d. The centers of the two ESDs are located at x = 150 km and x = 450 km, respectively, while the boundary between the two cuboids is located at x = 300 km.
Of the research areas which point A passes through, we derived that 63.5% are related to negative anomalies (Fig. 10a). We noted that neither Bz nor its gradient is zero or maximal at the boundary between the two cuboids. Bz has a relatively low negative value − 7.2 nT over the whole research area, varying in magnitude from – 29 nT to 30 nT. The gradient is also relatively small (0.16 nT/km, with a maximum magnitude of 0.40 nT/km). According to our statistical analysis above, the negative anomalies constitute about 52.7% of the whole area. The magnetic anomaly interval for most earthquake records is − 5 to − 3 nT (maximum range for the whole area is − 16 to 28 nT). The magnetic gradient interval is 0.015–0.025 nT/km, with a maximum magnitude of 0.14 nT/km. Although some discrepancies exist between results derived from the TESD model and those from statistical analyses, the ratio (0.25) of the magnetic anomaly on the boundary (− 7.2 nT) to the total magnitude (− 29 nT) is comparable to our statistical result (− 5 ~ −3/−16 ≈ 0.19–0.31). Furthermore, the ratio (0.25) of anomaly gradient on the boundary (0.16) to the largest anomaly gradient (0.40) differs little from our statistical result (0.015–0.025/0.14 ≈ 0.11–0.18).
The selection of d1 = 10 km, d2 = 25 km corresponds to the Curie depths of 20 km and 50 km, respectively; thus the bottom temperatures of the two cuboids are both in the Curie isothermal, which is usually set to 580 °C (Frost and Shive 1986). The resulting temperature difference at the boundary of the two cuboids is remarkable. With a simple linear estimation, the temperature difference will be 348 °C at the bottom of cuboid 1. Then differences of parameters directly related to earthquakes or lithospheric dynamics will be evident, such as the effective viscosity (Shi and Cao 2008), defined by:
$$\eta_{\text{eff}} = \frac{{\tau_{\text{creep}} }}{{\dot{\varepsilon }}} = \frac{{\dot{\varepsilon }^{{\frac{1 - n}{n}}} }}{{2A^{{\frac{1}{n}}} }}\exp \left( {\frac{E}{nRT}} \right)$$
in which τcreep is the shearing stress for creep deformation, \(\dot{\varepsilon }\) is the stain rate, A, n, E are parameters which could be estimated by rock experiments, R is the universal gas constant, and T is the absolute temperature. The influence of the temperature on ηeff can be calculated by
$$\frac{{\partial \left( {\lg \eta_{\text{eff}} } \right)}}{\partial T} = - \frac{C}{{T^{2} }}$$
Usually we have
$$C = 0.4343E/nR \approx 4.3 \times 10^{3}$$
In our TESD model case,
$$T = 580 + 273 = 853\;{\text{K}}, \, \quad \Delta T = 348\;{\text{K}}$$
Finally, the computation results in ηeff2:ηeff1 ≈ 114, which means that at the boundary and at a depth of 20 km, the viscosity between the two blocks can be different by a factor of 100 or greater. Note that a remarkable difference of rheology will cause large faults and earthquakes to appear at the boundary (Burov 2007). A viscosity difference of up to two orders of magnitude at a depth of 25 km between the Tibetan plateau and the Sichuan basin is also supported by other studies (e.g. Shi and Cao 2008; Deng and Tesauro 2016). The possible correlation between Curie depth and earthquakes was formerly presented by other studies (e.g. Rajaram et al. 2009; Gao et al. 2015a, b).
For further study, we also present the magnetic anomaly and its gradient in two dimensions in Fig. 10b, d, where both the magnetic anomaly (− 7.2 nT) on line A and its gradient (0.16 nT/km) are the maximum magnitude for the whole boundary of the two cuboids. The latitude-averaged anomaly is close to − 5 nT, and the latitude-averaged anomaly gradient is about 0.1 nT/km for the whole boundary line. This brings the output of the TESD model even closer to our statistical results.
In addition, we compared the above results with the 2D anomaly Bz and its gradient (as shown in Figs. 2, 3, 7) regarding the Tibetan block and the Sichuan basin. The positive anomaly shows many similarities, while the negative anomaly shows differences. This distinction results from the side length of the cuboid not being in scale with the geographical dimensions of the Tibetan block. Moreover, unlike the Sichuan Basin, the Tibet block could not be approached as if it were composed of a single ESD.
Parameter dependence of the TESD model
During our TESD explanation, we tested only one group of input parameters defined by Formula (11). Various tectonic regions, however, have various parameters. For example, regions near magnetic poles have a larger ambient field BMz, and those near the magnetic equators have a smaller BMz. In this situation, the magnetic anomaly and its gradient will change in proportional to BMz. The anomaly changes in proportional to the magnetic susceptibility also. Therefore, we were more interested in changes in the Curie depth and in the side length.
In Fig. 11 we changed the ESD depths (d1 and d2) and plotted the latitude-averaged boundary magnetic anomaly (at 200 km altitude) by fixing other parameters as in Formula (11). As Fig. 11 shows, for the parameters selected, the average boundary anomaly is always negative, though not less than − 5 nT. For magnetic anomalies between − 5 and − 3 nT, as was shown in our statistical maximum earthquake span (Fig. 5), an ESD depth difference as large as 10 km is preferred but not necessary. At least one block with an ESD depth lower than about 15 km (equivalent to a Curie depth of 30 km) is needed to generate the corresponding anomaly.
The averaged boundary anomaly for two nearby blocks with various ESD depths
Next we evaluated the TESD dependence on side length (L) with susceptibilities ranging from к = 0.01 to к = 0.05. Similarly, we kept other parameters the same as formula (11). Figure 12 shows the latitude-averaged magnetic anomaly on the boundary, revealing an obvious critical point, before which the average boundary anomaly is always positive and after which that anomaly is always negative. For the parameter selected, the critical side length is 250 km. To get an average anomaly between − 5 and − 3 nT, the side length of the two blocks must be larger than 250 km, but smaller than a certain value which depends on susceptibility. For example, to make the anomaly larger than − 5 nT, the maximum side length for к = 0.04 is about 300 km. By the same token, however, a larger side length requires a smaller susceptibility.
The averaged boundary anomaly for two nearby blocks with various side lengths and various susceptibilities
In this study, we compared active tectonics and seismicity to the satellite magnetic anomalies in continental China and surroundings. Our surveys of the distribution of active blocks, faults and GPS movements confirm that the surficial tectonic structures are well reflected by magnetic anomalies measured at high altitudes (Figs. 1, 2, 3). In general, stable cratons and basins showed notable positive anomalies, whereas active orogens exhibited strong negative anomalies. Boundaries of active blocks (i.e. faults) and those of obvious magnetic anomalies often overlap. These phenomena establish a physical foundation for the correlation between seismicity and magnetic anomalies.
Before this study, we usually understood that earthquakes are more likely to occur in areas with zero Bz, or in high gradient belts. In contrast with these previous understandings, the statistics of this study yielded a consistent correlation between seismicity and Bz at 200 km altitude (Figs. 2, 3, 5, 6). For different time intervals, areas showing Bz between − 5 and − 3 nT (with a total regional magnitude of − 16 to 28 nT) or areas showing a horizontal gradient (Figs. 7, 8) between 0.015 and 0.025 nT/km (with a total regional magnitude of 0–0.16 nT/km) always revealed seismic susceptibility.
To identify the most suitable parameter among Bz and its gradients for constraining seismicity, we intentionally proposed a ratio qEM to measure the seismic susceptibility. A larger qEM also corresponds to a higher probability of seismicity. The qEM for each magnetic anomaly range was calculated (Fig. 6), while the range with most earthquakes is most indicative. By comparing Bz and its gradient at various altitudes, we found that seismicity is best constrained by Bz at 200 km (Table 2). The corresponding seismic susceptibility for the period 2004–2007 is 4.7, which means the probability for earthquakes with Ms > 5.0 occurring in areas with Bz = −5 ~ −3 nT will be 4.7 times that of the statistical average. If Bz and its gradient are applied together, they might provide an improved constraint on the spatial distribution of seismicity.
To explain the correlation between magnetic anomalies and seismicity, we presented a TESD model (Fig. 10). Based on the relation between the Curie isotherm and viscosity, along with our TESD model, we arrived at certain understandings. Most earthquakes tend to occur at the boundary of two distinct blocks with contrasting rheology. The LMF above this boundary does not have to be zero. Instead, because negative anomalies often correlate with active tectonics, the LMF can depart a little from zero toward the negative. Meanwhile, the magnetic anomaly gradient above this boundary can be relatively small or only depart a little from zero. Earthquakes barely occurred in areas with strong positive or strong negative anomalies, because these areas are either too "stiff" (cold) or too "soft" (warm). The parameters selected in the TESD model made an output of magnetic anomaly very close to the statistical result. This set of parameter selection, however, is only one of many possibilities. Other sets of parameters may yield the same or even better outputs. For instance, we checked the parameter dependence of the TESD model (Figs. 11, 12). We found that a Curie depth of at least one block larger than 30 km, and a side length between 250 and 300 km are needed for the selected ambient vertical main field and susceptibility. For seismically active regions with Bz between − 5 and − 3 nT, an obvious Curie depth difference (> 10 km) for the two nearby blocks is indicated. Yet for seismic inactive regions—also with Bz between − 5 and − 3 nT—the Curie depth difference may be small, but the Curie depth of at least one block must be deeper than some certain value.
Rajaram et al. (2009) found that large magnitude earthquakes are associated with high gradients in Curie depth. For the magnetic anomaly in our study, earthquakes especially those with large magnitudes are associated with the boundaries of obvious anomaly or the boundaries of obvious anomaly gradients. Whether boundaries of obvious anomaly or anomaly gradients coincide with high gradients in Curie depth, however, needs further research. Gao et al. (2015a, b) found that strong earthquakes primarily occurred in areas with the uplift of Curie surface or with shallow Curie surface. Generally, areas with shallow Curie surface exhibit negative anomalies. Our study shows that EQs tend to occur inside areas with negative anomalies (with a percentage of 81.8% for the year 2004–2007), which is consistent with Gao et al.'s results.
It should be noted that different types of activity can't be all simply explained by blocks with contrasting Curie depth. Contrasting stress is also a key factor. GPS measurements, strain rates, diverse sort of focal mechanisms and frequency of seismicity, are all controlled by the stress field and lithosphere strength. A recent study (Chen et al. 2017) showed that the lithosphere deformation and structures are primarily controlled by the strength heterogeneity of the crust. The lithosphere strength or rheology is affected remarkably by temperature. From this point of view, contrasting viscosity could affect seismic activities to a very large extent.
In the TESD model, the susceptibility κ is identical for both blocks. This may not always be true for all adjacent blocks, especially for areas with high heat flux but exhibit a relatively large positive anomaly or vice versa (Fig. 9). In this case (though only a small part of the research area), differences in Curie depth may be small. Different κ might also indicate different cuboid compositions, or different types of tectonic blocks, either of which could cause the boundary to be seismically active.
Although 53.2% of earthquakes have occurred in areas with Bz = −5 ~ −3 nT in the time of 2004–2007, where the probability of seismicity is 4.7 times the statistical average, 46.8% of moderate and large earthquakes have occurred outside these areas. Possible reasons may lie in different Curie depth distributions, different length scales, anisotropy of susceptibility, or effects of remanence, all of which require further research.
LMF:
TESD:
two equivalent source dipole
transcontinental magnetic anomaly
SHA:
spherical harmonic analysis
RTP:
reduction to the pole
ISC:
International Seismological Centre
LJ and YL analyzed the results and drafted the manuscript. LJ designed the methodology. HC proposed the research plan. All authors read and approved the final manuscript.
We thank Shi YL and Xu WY for their suggestions on theory, Yang T for many helpful discussions, and David Parks for reviewing the manuscript and editing the English grammar. We thank Hu SB for providing the heat flux data, Deng YF for the viscosity data and Wang H for the tectonic data. Comments by two anonymous reviewers and the handling editor, H Oda, substantially improved the manuscript.
The Gauss coefficients for calculating Bz are from the NGDC-720 model (http://geomag.org/models/ngdc720.html). For topography data, we used the ETOPO1 model (http://www.ngdc.noaa.gov/mgg/global/global.html). We downloaded seismic records are downloaded from the International Seismological Centre (ISC), On-line Bulletin, http://www.isc.ac.uk. Heat flux data were provided by S. B. Hu, which could also be downloaded from http://www.heatflow.und.edu/. We prepared Figs. 1, 2, 3, 4, 7 and 9 by using GMT (Wessel and Smith 1998).
This work is funded by National Key R&D Program of the Ministry of Science and Technology of China with the Project "Integration Platform Construction for Joint Inversion and Interpretation of Integrated Geophysics (Grant No. 2018YFC0603502)," and National Basic Research Program of China "973" (Grant No. 2014CB845906).
Open AccessThis 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.
40623_2018_949_MOESM1_ESM.docx Additional file 1. Supplementary figures for comparisons between Bz and tectonics.
Institute of Geophysics, China Earthquake Administration, Beijing, 100081, China
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1. Geomagnetism
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Recent Advances in Geo-, Paleo- and Rock- Magnetism | CommonCrawl |
\begin{definition}[Definition:Definitional Abbreviation]
When discussing a formal language, some particular WFFs may occur very often.
If such WFFs are very unwieldy to write and obscure what the author tries to express, it is convenient to introduce a shorthand for them.
Such a shorthand is called a '''definitional abbreviation'''.
It does ''not'' in any way alter the meaning or formal structure of a sentence, but is purely a method to keep expressions readable to human eyes.
\end{definition} | ProofWiki |
\begin{document}
\maketitle
\begin{abstract} We show that if $K$ is any knot whose Ozsv\'ath--Szab\'o concordance invariant $\tau(K)$ is positive, the all-positive Whitehead double of any iterated Bing double of $K$ is topologically but not smoothly slice. We also show that the all-positive Whitehead double of any iterated Bing double of the Hopf link (e.g., the all-positive Whitehead double of the Borromean rings) is not smoothly slice; it is not known whether these links are topologically slice. \end{abstract}
\section{Introduction}
A knot in the $3$-sphere is called \emph{topologically slice} if it bounds a locally flatly embedded disk in the $4$-ball, and \emph{smoothly slice} if the disk can be taken to be smoothly embedded. Two knots are called (topologically or smoothly) \emph{concordant} if they are the ends of an embedded annulus in $S^3 \times I$; thus, a knot is slice if and only if it is concordant to the unknot. More generally, a link is (topologically or smoothly) \emph{slice} if it bounds a disjoint union of appropriately embedded disks. The study of concordance --- especially regarding the relationship between the notions of topological and smooth sliceness --- is one of the major areas of active research in knot theory, and it is closely tied to the perplexing differences between topological and smooth $4$-manifold theory.
Given a knot $K \subset S^3$, the \emph{(untwisted) positive and negative Whitehead doubles} of $K$, $Wh_+(K)$ and $Wh_-(K)$, and the \emph{Bing double} of $K$, $B(K)$, are the satellites of $K$ illustrated in Figure \ref{fig:figure8} (for $K$ the figure-eight knot). The Whitehead doubles of a link $L$ are obtained by doubling the individual components of $L$, with a choice of sign for each component. In particular, we denote the all-positive and all-negative Whitehead doubles of $L$ by $Wh_+(L)$ and $Wh_-(L)$, respectively.
Whitehead and Bing doubling play a central role in the study of concordance. In the topological setting, Freedman \cite{FreedmanNewTechnique, FreedmanQuinn} proved that any Whitehead double of a knot or, more generally, a \emph{boundary link} (a link whose components bound disjoint Seifert surfaces) is topologically slice. Moreover, the surgery conjecture for $4$-manifolds with arbitrary fundamental group --- the central open problem in four-dimensional topology --- is equivalent to the conjecture that the Whitehead double of any link whose linking numbers are all zero is freely topologically slice.\footnote{A link $L$ is \emph{freely slice} if it bounds slice disks in $B^4$ whose complement has free fundamental group.} This conjecture is true for two-component links \cite{FreedmanWhitehead3} but open in general. To disprove the surgery conjecture for manifolds with free fundamental group, it would thus suffice to show that one such link --- e.g., a Whitehead double of the Borromean rings --- is not (freely) topologically slice. However, all such links have resisted all attempts to determine whether or not they are topologically slice.
\psfrag{Wh+(K)}{$Wh_+(K)$} \psfrag{Wh-(K)}{$Wh_-(K)$} \psfrag{BD(K)}{$BD(K)$} \psfrag{K}{$K$}
\begin{figure}
\caption{The positive and negative Whitehead doubles and the Bing double of the figure-eight knot.}
\label{fig:figure8}
\end{figure}
Around the same time, the advent of Donaldson's gauge theory made it possible to show that some of Freedman's examples of topologically slice knots are not smoothly slice. Akbulut [unpublished] first proved in 1983 that the positive, untwisted Whitehead double of the right-handed trefoil is not smoothly slice. Later, using results of Kronheimer and Mrowka on Seiberg--Witten theory, Rudolph \cite{RudolphObstruction} showed that any nontrivial knot that is \emph{strongly quasipositive} cannot be smoothly slice. In particular, the positive, untwisted Whitehead double of a strongly quasipositive knot is strongly quasipositive; thus, by induction, any iterated positive Whitehead double of a strongly quasipositive knot is topologically but not smoothly slice. Bi\v{z}aca \cite{Bizaca} used this result to give explicit constructions of exotic smooth structures on $\mathbb{R}^4$. Later, Hedden \cite{HeddenWhitehead} generalized this result to any knot $K$ whose Ozsv\'ath--Szab\'o invariant $\tau(K)$ (an integer-valued concordance invariant coming from the knot Floer homology of $K$ \cite{OSz4Genus, RasmussenThesis}) is positive. It is conjectured \cite[Problem 1.38]{KirbyList} that if $Wh_\pm(K)$ is smoothly slice, then $K$ itself must also be smoothly slice.
\begin{figure}
\caption{A binary tree $T$ and the corresponding iterated Bing double $B_T(K)$.}
\label{fig:tree}
\end{figure}
We may consider \emph{partially iterated Bing doubles} of any link: at each stage in the iteration, we replace some component by its Bing double. Specifically, given a knot $K$, a binary tree $T$ specifies such a link $B_T(K)$, as illustrated in Figure \ref{fig:tree}, with one component for each leaf of $T$. For a link $L=K_1 \cup \cdots \cup K_n$ and binary trees $T_1, \dots, T_n$, we may similarly obtain a link $B_{T_1, \dots, T_n}(L) = B_{T_1}(K_1) \cup \cdots \cup B_{T_n}(K_n)$. In particular, if $H$ is the Hopf link, the links obtained in this manner are known as \emph{generalized Borromean links}, since Bing doubling one component of $H$ yields the Borromean rings.
Using the author's work in \cite{LevineDoublingOperators} --- a lengthy computation of $\tau$ for a particular family of satellite knots --- we shall prove:
\begin{thm} \label{thm:notslice} \begin{enumerate} \item Let $K$ be a knot with $\tau(K)>0$, and let $T$ be any binary tree. Then the all-positive Whitehead double of $B_T(K)$, $Wh_+(B_T(K))$, is not smoothly slice. \item Let $H = K_1 \cup K_2$ denote the Hopf link, and let $T_1,T_2$ be binary trees. Then $Wh_+(B_{T_1,T_2}(H))$ is not smoothly slice. \end{enumerate} \end{thm}
Note that for any knot $K$, $B_T(K)$ is always a boundary link (see \cite{CimasoniSlicing} for a proof), so any Whitehead double of $B_T(K)$ (with clasps of either sign) is topologically slice. Thus, part (1) of Theorem \ref{thm:notslice} provides a large family of links that are topologically but not smoothly slice.\footnote{The question of when iterated Bing doubles of a knot are slice is also quite challenging, since the classical sliceness obstructions vanish for iterated Bing doubles. Recent papers by Cimasoni \cite{CimasoniSlicing} and Cha--Livingston--Ruberman \cite{ChaLivingstonRuberman}, Cha--Kim \cite{ChaKim}, and Van Cott \cite{VanCott} show that if an iterated Bing double of $K$ is topologically slice, then $K$ is algebraically slice; if it is smoothly slice, then $\tau(K)=0$. Also, Cochran, Harvey, and Leidy \cite{CochranHarveyLeidyDoubling} have used $L^2$ signatures to find algebraically slice knots with non-slice iterated Bing doubles.} On the other hand, it is unknown whether the links in part (2) --- the all-positive Whitehead doubles of the generalized Borromean links --- are topologically slice. Indeed, Freedman \cite{FreedmanQuinn} showed that the family of Whitehead doubles (with any signs) of generalized Borromean links is ``atomic'' for the surgery problem: whether or not they are freely topologically slice is equivalent to the surgery conjecture. Most experts nowadays conjecture that these links are \emph{not} topologically slice, but the problem remains unsolved after nearly twenty-five years.
The requirement that we consider all-positive Whitehead doubles is necessary for our proof of Theorem \ref{thm:notslice}. By taking mirrors, we also see that the all-negative Whitehead doubles of iterated Bing doubles of knots with $\tau(K)<0$ or of generalized Borromean links are not smoothly slice, but our method always fails when both positive and negative Whitehead doubling are used. Indeed, all of the gauge-theoretic invariants known to date suffer from the same asymmetry. It is still not known whether, for instance, the positive untwisted Whitehead double of the left-handed trefoil is smoothly slice.
\subsection*{Acknowledgments} This paper, along with \cite{LevineDoublingOperators}, made up a large portion of the author's thesis at Columbia University. The author is grateful to his advisor, Peter Ozsv\'ath, and the other members of his defense committee, Robert Lipshitz, Dylan Thurston, Paul Melvin, and Denis Auroux, for their suggestions; to Rumen Zarev, Ina Petkova, and Jen Hom for many helpful conversations about bordered Heegaard Floer homology; and to Rob Schneiderman, Charles Livingston, and Matthew Hedden for their suggestions regarding link concordance questions.
\section{Definitions}
We begin by giving more precise definitions of some of the terms used in the Introduction.
\subsection{Infection and doubling operators} We always work with \emph{oriented} knots and links. For any knot $K \subset S^3$, let $K^r$ denote $K$ with reversed orientation, let $\bar K$ denote the mirror of $K$ (the image of $K$ under a reflection of $S^3$), and let $-K = \bar K^r$. As ${K} \# {-K}$ is always smoothly slice, the concordance classes of $K$ and $-K$ are inverses in $\CC_1$, which justifies this choice of notation. Note that the invariants coming from Heegaard Floer homology ($\HFK(S^3,K)$, $\tau(K)$, etc.) are sensitive to mirroring but not to reversing the orientation of a knot.
Suppose $L$ is a link in $S^3$, and $\gamma$ is an oriented curve in $S^3 \smallsetminus} \def\co{\colon\thinspace L$ that is unknotted in $S^3$. For any knot $K \subset S^3$ and $t \in \mathbb{Z}$, we may form a new link $I_{\gamma,K,t}(L)$, the \emph{$t$-twisted infection of $L$ by $K$ along $\gamma$}, by deleting a neighborhood of $\gamma$ and gluing in a copy of the exterior of $K$ by a map that takes a Seifert-framed longitude of $K$ to a meridian of $\gamma$ and a meridian of $K$ to a $t$-framed longitude of $\gamma$. Since $S^3 \smallsetminus} \def\co{\colon\thinspace \gamma = S^1 \times D^2$, the resulting $3$-manifold is simply $\infty$ surgery on $K$, i.e. $S^3$; the new link $I_{\gamma,K,t}(L)$ is defined as the image of $L$. Alternately, let $\hat K \subset D^2 \times I$ be the $(1,1)$-tangle obtained by cutting $K$ at a point, oriented from $\hat K \cap D^2 \times \{0\}$ to $\hat K \cap D^2 \times \{1\}$. If $D$ is an oriented disk in $S^3$ with boundary $\gamma$, meeting $L$ transversely in $n$ points, we may obtain $I_{\gamma,K,t}(L)$ by cutting open $L$ along $D$ and inserting the tangle consisting of $n$ parallel copies of $\hat K$, following the $t$ framing. In a link diagram, a box labeled $K,t$ in a group of parallel strands indicates $t$-twisted infection by $K$ along the boundary of a disk perpendicular to those strands. To be precise, we adopt the following orientation convention: If the label $K,t$ is written horizontally and right-side-up, then $\hat K$ is oriented either from bottom to top or from left to right, depending on whether the strands meeting the box are positioned vertically or horizontally.\footnote{We allow both types of notation to avoid writing labels vertically.}
Given unlinked infection curves $\gamma_1, \gamma_2$, the image of $\gamma_2$ in $I_{\gamma_1, K_1, t_1}(L \cup \gamma_2)$ is again an unknot, so we may then infect by another pair $K_2,t_2$. We obtain the same result if we infect along $\gamma_2$ first and then $\gamma_1$. In general, given an unlink $\gamma_1, \dots, \gamma_n$, we may infect simultaneously along all the $\gamma_i$; the result may be denoted $I_{\gamma_1,K_1,t_1; \, \cdots; \, \gamma_n,K_n,t_n}(L)$, and the order of the tuples $(\gamma_i,K_i,t_i)$ does not matter.
If $P$ is a knot (or link) in the standardly embedded solid torus in $S^3$ and $K$ is any knot, the \emph{$t$-twisted satellite of $K$ with pattern $P$}, $P(K,t)$, is defined as $I_{\gamma,K,t}(P)$, where $\gamma$ is the core of the complementary solid torus. The knot $K$ is called the \emph{companion}. More generally, if we have a link $L$, we may replace a component of $L$ by its satellite with pattern $P$, working in a tubular neighborhood disjoint from the other components.
\begin{figure}
\caption{The Borromean rings.}
\label{fig:borromean}
\end{figure}
Let $B = B_1 \cup B_2 \cup B_3$ denote the Borromean rings in $S^3$, oriented as shown in Figure \ref{fig:borromean}. The $\pm$ $t$-twisted Whitehead double of $K$, is defined as \[ Wh_\pm(K,t) = I_{B_1,O,\mp 1; B_2,K,t}(B_3), \] where $O$ denotes the unknot. (Note the sign conventions: a left-handed twist in a pair of opposite strands is a positive clasp.) Moreover, we define the following generalization of Whitehead doubling: for knots $J$ and $K$ and integers $s$ and $t$, define $D_{J,s}(K,t)$ as the knot obtained from $B_3$ by performing $s$-twisted infection by $J$ along $B_1$ and $t$-twisted infection by $K$ along $B_2$: \[ D_{J,s}(K,t) = I_{B_1,J,s; \, B_2,K,t}(B_3). \] (See Figure \ref{fig:djskt}.) The symmetries of the Borromean rings imply: \begin{gather*} D_{J,s}(K,t)^r = D_{J^r,s}(K,t) = D_{J,s}(K^r,t) = D_{K,t}(J,s) \\ \overline{D_{J,s}(K,t)} = D_{\bar{J},-s}(\bar{K},-t) \end{gather*} We also introduce the convention that when the $t$ argument is omitted, it is taken to be zero: $D_{J,s}(K) = D_{J,s}(K,0)$.
\begin{figure}
\caption{The satellite knot $D_{J,s}(K,t)$.}
\label{fig:djskt}
\end{figure}
The Bing double of $K$ may be defined as $BD(K) = I_{B_1,K,0}(B_2 \cup B_3)$; we may also see this as a satellite operation where the pattern is a two-component link.
\subsection{Heegaard Floer homology and the $\tau$ invariant}
In the 2000s, Ozsv\'ath and Szab\'o \cite{OSz3Manifold, OSz4Manifold} introduced \emph{Heegaard Floer homology}, a package of invariants for $3$- and $4$-dimensional manifolds that are conjecturally equivalent to earlier gauge-theoretic invariants but whose construction is much more topological in flavor. In its simplest form, given a \emph{Heegaard diagram} $\HH$ for a $3$-manifold $Y$ (a certain combinatorial description of the manifold), the theory assigns a chain complex $\CF(\HH)$ whose chain homotopy type is independent of the choice of diagram; thus, the homology $\HF(Y) = H_*(\CF(\HH))$ is an invariant of the $3$-manifold. A $4$-dimensional cobordism between two $3$-manifolds induces a well-defined map between their Heegaard Floer homology groups. Ozsv\'ath and Szab\'o \cite{OSzKnot} and Rasmussen \cite{RasmussenThesis} also showed that a nulhomologous knot $K \subset Y$ induces a filtration on the chain complex of a suitably defined Heegaard diagram, yielding an knot invariant $\HFK(Y,K)$ that is the $E^1$ page of a spectral sequence converging to $\HF(Y)$. For knots in $S^3$, the invariant $\HFK(S^3,K)$ categorifies the Alexander polynomial $\Delta_K$, and it is powerful enough to detect the unknot \cite{OSzGenus} and whether or not $K$ is fibered \cite{Ghiggini, NiFibered}.
Furthermore, the spectral sequence from $\HFK(S^3,K)$ to $\HF(S^3) \cong \mathbb{Z}$ provides an integer-valued concordance invariant $\tau(K)$, which yields a lower bound on genus of smooth surfaces in the four-ball bounded by $K$: $\abs{\tau(K)} \le g_4(K)$ \cite{OSz4Genus}. In particular, any smoothly slice knot must have $\tau(K)=0$. Moreover, this genus bound applies not only for surfaces in $B^4$ but for surfaces in any rational homology $4$-ball.
The $\tau$ invariant can be used to extend many of the earlier results obstructing smooth sliceness. Hedden \cite{HeddenWhitehead} computed the value of $\tau$ for all twisted Whitehead doubles in terms of $\tau$ of the original knot: \begin{equation} \label{eq:tau-Wh} \tau(Wh_+(K,t)) = \begin{cases} 1 & t < 2 \tau(K) \\ 0 & t \ge 2 \tau(K). \end{cases} \end{equation} (An analogous formula for negative Whitehead doubles follows from the fact that $\tau(\bar K) = - \tau(K)$.) In particular, if $\tau(K)>0$, then $\tau(Wh_+(K,0))=1$, so $Wh_+(K,0)$ (the untwisted Whitehead double of $K$) is not smoothly slice. Since the $\tau$ invariant of a strongly quasipositive knot is equal to its genus \cite{LivingstonComputations}, Rudolph's earlier result follows from Hedden's.
The author's main theorem in \cite{LevineDoublingOperators} is a computation of $\tau$ for all knots of the form $D_{J,s}(K,t)$: \begin{thm} \label{thm:taudjskt} Let $J$ and $K$ be knots, and let $s,t \in \mathbb{Z}$. Then \[ \tau(D_{J,s}(K,t)) = \begin{cases} 1 & s<2\tau(J) \text{ and } t<2 \tau(K) \\ -1 & s>2\tau(J) \text{ and } t>2 \tau(K) \\ 0 & \text{otherwise}. \end{cases} \] \end{thm} In particular, note that if $\tau(K)>0$ and $s<2\tau(J)$, or if $\tau(K)<0$ and $s>2\tau(J)$, then $D_{J,s}(K,0)$ is topologically slice (as its Alexander polynomial is $1$) but not smoothly slice in any rational homology $4$-ball.
The proof of Theorem \ref{thm:taudjskt} is an involved computation using the theory of \emph{bordered Heegaard Floer homology}, developed recently by Lipshitz, Ozsv\'ath, and Thurston \cite{LOTBordered, LOTBimodules}. Briefly, the bordered theory associates to a $3$-manifold with boundary a module over an algebra associated to the boundary, so that if $Y$ is obtained by gluing together manifolds $Y_1$ and $Y_2$ along their common boundary, the chain complex $\CF(Y)$ may be computed as the derived tensor product of the invariants associated to $Y_1$ and $Y_2$. If a knot $K$ is contained in, say, $Y_1$, then we may obtain the filtration on $\CF(Y)$ corresponding to $K$ via a filtration on the algebraic invariant of $Y_1$. This technique is thus well-suited to the problem of computing Heegaard Floer invariants for knots obtained through infection operations. For more details, see \cite{LevineDoublingOperators}.
\subsection{Covering link calculus}
The proof of Theorem \ref{thm:notslice} makes use of \emph{covering link calculus}, first developed by Cochran and Orr \cite{CochranOrrBoundary} and used more recently by Cha and Kim \cite{ChaKim} and others \cite{CimasoniSlicing, ChaLivingstonRuberman, VanCott}. Let $R$ denote any of the rings $\mathbb{Z}$, $\mathbb{Q}$, or $\mathbb{Z}_{(p)}$ (for $p$ prime). A link $L$ in an $R$-homology $3$-sphere $Y$ is called \emph{topologically (resp.~smoothly) $R$-slice} if there exists a topological (resp.~smooth) $4$-manifold $X$ such that $\partial X = Y$, $H_*(X;R)=H_*(B^4;R)$, and $L$ bounds a locally flat (resp.~smoothly embedded), disjoint union of disks in $X$. A link that is $\mathbb{Z}$-slice (in either category) is $\mathbb{Z}_{(p)}$-slice for all $p$, and a link that is $\mathbb{Z}_{(p)}$-slice for some $p$ is $\mathbb{Q}$-slice. Also, a link in $S^3$ that is slice (in $B^4$) is clearly $\mathbb{Z}$-slice. The key result of Ozsv\'ath and Szab\'o \cite{OSz4Genus} is that the $\tau$ invariant of any knot that is smoothly $\mathbb{Q}$-slice is $0$.
Define two moves on links in $\mathbb{Z}_{(p)}$-homology spheres, called \emph{covering moves}: \begin{enumerate} \item Given a link $L\subset Y$, consider a sublink $L' \subset L$. \item Given a link $L \subset Y$, choose a component $K$ with trivial self-linking. For any $a \in \mathbb{N}$, the $p^a$-fold cyclic branched cover of $Y$ branched over $K$, denoted $\tilde Y$, is a $\mathbb{Z}_{(p)}$-homology sphere, and we consider the preimage $L'$ of $L$ in $\tilde Y$. \end{enumerate} We say that $L' \subset Y'$ is a \emph{$p$-covering link} of $L \subset Y$ if $L'$ can be obtained from $Y'$ using these moves.
The key fact is the following: \begin{prop} If $L$ is (topologically or smoothly) $\mathbb{Z}_{(p)}$-slice, then any $p$-covering link of $L$ is also (topologically or smoothly) $\mathbb{Z}_{(p)}$-slice. \end{prop} To prove that the second covering move preserves $\mathbb{Z}_{(p)}$-sliceness, we take the branched cover of the $X$ over the slice disk for $K$; the resulting $4$-manifold is a $\mathbb{Z}_{(p)}$-homology $4$-ball by a well-known argument (see, e.g., \cite[page 346]{KauffmanOnKnots}). Thus, a strategy for showing a link $L$ is not slice is to find a knot that is a covering link of $L$ and has a non-vanishing $\mathbb{Q}$-sliceness obstruction, such as $\tau$.
Note that if $L$ is a link in $S^3$ whose components are unknotted, then the branched cover branched over one component is again $S^3$. The putative $4$-manifold containing a slice disk, however, may change.
Henceforth, we restrict to the case where $p=2$ and omit further reference to $p$.
\section{Proof of Theorem \ref{thm:notslice}} \label{sec:covering}
The strategy for proving the first part of Theorem \ref{thm:notslice} is to obtain a knot $K'$ of the form \[ K' = D_{J_1,s_1} \circ \dots \circ D_{J_n,s_n}(K), \] where $s_i < 2\tau(J_i)$ for each $i$, as a covering link of $Wh_+(B_T(K))$. If $\tau(K)>0$, induction using Theorem \ref{thm:taudjskt} (which we prove below) shows that $\tau(K')=1$, so $K'$ cannot be rationally smoothly slice, so $Wh_+(D_T(K))$ cannot be smoothly slice. A similar argument works for the second part of the theorem.
The following lemmas are inspired by Van Cott's work on the sliceness of iterated Bing doubles \cite{VanCott}:
\begin{lemma} \label{lemma:solidtorus} Let $L$ be a link in $S^3$, and suppose there is an unknotted solid torus $U \subset S^3$ such that $L \cap U$ consists of two components $K_1,K_2$ embedded as follows: if $A_1, A_2$ are the components of the untwisted Bing double of the core $C$ of $U$, then $K_1 = D_{P_k,s_k} \circ \dots \circ D_{P_1,s_1}(A_1)$ and $K_2 = D_{Q_l,t_l} \circ \dots \circ D_{Q_1,t_1}(A_2)$, for some knots $P_1,\dots,P_k, Q_1,\dots,Q_l$ and integers $s_1, \dots, s_k, t_1, \dots, t_l$. Let $L'$ be the link obtained from $L$ by replacing $K_1$ and $K_2$ by the satellite knot \begin{equation} \label{eq:solidtorus1} C' = D_{P_k,s_k} \circ \dots \circ D_{P_1,s_1} \circ D_{R,u} (C) \end{equation} of $C$, where \begin{equation} \label{eq:solidtorus2} (R,u) = \begin{cases}
(Q_1 \# Q_1^r, \ 2t_1) & l=1 \\
(D_{Q_1,t_1} \circ \dots \circ D_{Q_{l-2},t_{l-2}} (D_{Q_{l-1},t_{l-1}}(Q_l \# Q_l^r, 2t_l)), \ 0) & l>1. \end{cases} \end{equation} Then $L'$ is a covering link of $L$. \end{lemma}
\psfrag{(a)}{(a)} \psfrag{(b)}{(b)} \psfrag{(c)}{(c)} \psfrag{(d)}{(d)} \psfrag{K1}{{\scriptsize $K_1$}} \psfrag{K2}{{\scriptsize \color{red} $K_2$}} \psfrag{T}{{\scriptsize $T$}} \psfrag{Tb}{{\scriptsize \color{blue} $T$}}
\psfrag{Q0t0}{{\scriptsize $Q_0,t_0$}} \psfrag{Q1t1}{{\scriptsize $Q_1,t_1$}} \psfrag{Q2t2}{{\scriptsize $Q_2,t_2$}} \psfrag{Q3t3}{{\scriptsize $Q_3,t_3$}} \psfrag{Qltl}{{\scriptsize $Q_l,t_l$}} \psfrag{QlQl2tl}{{\scriptsize $Q_l\#Q_l^r,2t_l$}} \psfrag{Q1Q12t1}{{\scriptsize $Q_1\#Q_1^r,2t_1$}}
\psfrag{Ql-1tl-1}[cc][cc]{{\scriptsize \shortstack[c]{$Q_{l-1},$ \\ $t_{l-1}$}}}
\psfrag{DPkskDP1s1}{{\scriptsize $[D_{P_k,s_k} \cdots D_{P_1,s_1}]$}} \psfrag{DPkskDP1s1b}{{\scriptsize \color{blue} $[D_{P_k,s_k} \cdots D_{P_1,s_1}]$}} \psfrag{DQltlDQ0t0}{{\scriptsize \color{red} $[D_{Q_l,t_l} \cdots D_{Q_0,t_0}]$}}
\psfrag{DQltlDQ1t1}[tl][tl]{{\scriptsize \color{red} \shortstack[l]{$[D_{Q_l,t_l} \cdots $ \\ \quad $D_{Q_1,t_1}]$ }}} \psfrag{DQltlDQ2t2}[tl][tl]{{\scriptsize \color{red} \shortstack[l]{$[D_{Q_l,t_l} \cdots $ \\ \quad $D_{Q_2,t_2}]$ }}} \psfrag{DQltlDQ3t3}[tl][tl]{{\scriptsize \color{red} \shortstack[l]{$[D_{Q_l,t_l} \cdots $ \\ \quad $D_{Q_3,t_3}]$ }}}
\psfrag{DQltlDQ3t3oneline}{{\scriptsize \color{red} $[D_{Q_l,t_l} \cdots D_{Q_3,t_3}]$}}
\psfrag{2l}{{\scriptsize $2^l$}} \psfrag{2l-1}{{\scriptsize $2^{l-1}$}} \psfrag{2l-2}{{\scriptsize $2^{l-2}$}} \psfrag{2l-1-2}{{\scriptsize $2^{l-1}-2$}}
\begin{figure}\label{fig:solidtorus1}
\end{figure}
\begin{figure}\label{fig:solidtorus2}
\end{figure}
\begin{figure}
\caption{The preimage of the link in Figure \ref{fig:solidtorus2} in the double-branched cover of $S^3$ over $K_2$ (shown without the upstairs branch set).}
\label{fig:solidtorus3}
\end{figure}
\begin{figure}\label{fig:solidtorus4}
\end{figure}
\psfrag{DQltlDQ1t1}{{\scriptsize \color{red} $[D_{Q_l,t_l} \cdots D_{Q_1,t_1}]$}}
\begin{figure}
\caption{The knot $R$ in the proof of Lemma \ref{lemma:solidtorus}.}
\label{fig:solidtorus5}
\end{figure}
\begin{proof} Let $T = S^3 \smallsetminus} \def\co{\colon\thinspace U$; then $L \smallsetminus} \def\co{\colon\thinspace (K_1 \cup K_2)$ is contained in $T$. Note that $K_1$ and $K_2$ are each unknotted, since $D_{J,s}(O,0) = O$ for any $J,s$. We may untangle $K_2$ as in Figures \ref{fig:solidtorus1}--\ref{fig:solidtorus2}. Specifically, $L$ is shown in Figures \ref{fig:solidtorus1}(a) and (b). To obtain Figure \ref{fig:solidtorus1}(c), we pull the two strands of the companion curve for $K_1$ through the infection region marked $Q_1,t_1$, and then untangle the companion curve for $K_2$. We then repeat this procedure to obtain Figure \ref{fig:solidtorus1}(d), and $l-2$ more times to obtain Figure \ref{fig:solidtorus2}.
The branched double cover of $S^3$ branched along $K_2$ is again $S^3$; consider the preimage of $K_1 \cup (L \cap T)$, shown in Figure \ref{fig:solidtorus3}. (The knot orientation conventions for infections are important here, since the knots $Q_i$ need not be reversible.) Since $T$ is contained in a ball disjoint from $K_1$, the sublink $L \cap T$ lifts to two identical copies, each contained in a solid torus. The preimage of $K_2$ also consists of two components, and each is the $D_{P_k,s_k} \circ \cdots \circ D_{P_1,s_1}$ satellite of the companion curve shown. A sublink consisting of one lift of each component (either the blue or the black part of Figure \ref{fig:solidtorus3}) is redrawn in Figure \ref{fig:solidtorus4}(a) in the case where $l=1$ and in Figure \ref{fig:solidtorus4}(b) in the case where $l>1$. In the former case, the companion curve shown is $D_{Q_l \# Q_l^r, 2t_l}(C)$, where $C$ is the core circle of the complement of $T$. In the latter case, it is $D_{R,0}(C)$, where we obtain $R$ by connecting the ends of one of the two parallel strands that pass through the red box in Figure \ref{fig:solidtorus4}(b). (A local computation shows that the linking number of these two strands is zero, so $D_{R,0}$ is the correct operator.) The knot $R$, shown in Figure \ref{fig:solidtorus5}, is then identified as \[ D_{Q_1,t_1} \circ \dots \circ D_{Q_{l-2},t_{l-2}} (D_{Q_{l-1},t_{l-1}}(Q_l \# Q_l^r, 2t_l)). \qedhere \] \end{proof}
\begin{lemma} \label{lemma:bing} Let $C$ be a knot, let $U$ be a regular neighborhood of $C$, and let $A_1, A_2 \subset U$ be the components of $BD(C)$. Let $K_1 = D_{P_k,s_k} \circ \dots \circ D_{P_1,s_1}(A_1)$ and $K_2 = D_{Q_l,t_l} \circ \dots \circ D_{Q_1,t_1}(A_2)$, for some knots $P_1,\dots,P_k, Q_1,\dots,Q_l$ and integers $s_1, \dots, s_k, t_1, \dots, t_l$. Let $C'$ be the knot defined by \eqref{eq:solidtorus1} and \eqref{eq:solidtorus2}. Then $C'$ is a covering link of $K_1 \cup K_2$. \end{lemma}
\begin{proof} The proof is almost identical to that of Lemma \ref{lemma:solidtorus}. The only difference is that $S^3 \smallsetminus} \def\co{\colon\thinspace U$ is now a knot complement rather than a solid torus containing some additional link components. The double branched cover over $K_2$ contains consists of the complement of the two solid tori shown in Figure \ref{fig:solidtorus3}, glued to two copies of $S^3 \smallsetminus} \def\co{\colon\thinspace U$, gluing Seifert-framed longitude to meridian and vice versa. The resulting manifold is again $S^3$, however. The rest of the proof proceeds \emph{mutatis mutandis}. (Alternately, we may simply replace each of the solid tori in Figures \ref{fig:solidtorus1}--\ref{fig:solidtorus5} by a box marked $C,0$, and proceed as before.) \end{proof}
A \emph{labeled binary tree} is a binary tree with each leaf labeled with a satellite operation. Given a knot $K$ and binary tree $\TT$ with underlying tree $T$, let $S_\TT(K)$ be the link obtained from $B_T(K)$ by replacing each component with the satellite specified by the label of the corresponding leaf. If $\TT$ has two adjacent leaves labeled $D_{P_k,s_k} \circ \dots \circ D_{P_1,s_1}$ and $D_{Q_l,t_l} \circ \dots \circ D_{Q_1,t_1}$, form a new labeled tree $\TT'$ by deleting these two leaves and labeling the new leaf either $D_{P_k,s_k} \circ \dots \circ D_{P_1,s_1} \circ D_{Q_1 \# Q_1^r, 2t_1} (C)$ or $D_{P_k,s_k} \circ \dots \circ D_{P_1,s_1} \circ D_{R,0}$, according to whether $l=1$ or $l>1$, respectively, where, $R = D_{Q_1,t_1} \circ \dots \circ D_{Q_{l-2},t_{l-2}} (D_{Q_{l-1},t_{l-1}}(Q_l \# Q_l^r, 2t_1))$ in the latter case. We call this move a \emph{collapse}. Lemmas \ref{lemma:solidtorus} and \ref{lemma:bing} then say that $S_{\TT'}(K)$ is a covering link of $S_\TT(K)$.
Theorem \ref{thm:taudjskt} and equations \eqref{eq:solidtorus1} and \eqref{eq:solidtorus2}, along with the additivity of $\tau$ under connect sum, imply:
\begin{prop} \label{prop:collapse} Suppose $\TT'$ is obtained from $\TT$ by collapsing leaves labeled $D_{P_k,s_k} \circ \dots \circ D_{P_1,s_1}$ and $D_{Q_l,t_l} \circ \dots \circ D_{Q_1,t_1}$, where $s_i < 2\tau(P_i)$ and $t_i < 2\tau(Q_i)$ for all $i$. Then the label of the new leaf of $\TT'$ has the form $D_{R_{k+1},u_{k+1}} \circ \cdots \circ D_{R_1,u_1}$, where $u_i < 2 \tau(R_i)$. \qed \end{prop}
\begin{proof}[Proof of Theorem \ref{thm:notslice}] For the first part of the theorem, note that in the new notation, $Wh_+(B_T(K)) = S_\TT(K)$, where every leaf of $\TT$ is labeled $D_{O,-1}$. Every label in $\TT$ satisfies the hypotheses of Proposition \ref{prop:collapse}. Using this proposition, we inductively collapse every pair of leaves of $\TT$ until we have a single vertex labeled $D_{P_k,s_k} \circ \cdots \circ D_{P_1,s_1}$, for knots $P_1, \dots, P_k$ and integers $s_1, \dots, s_k$ with $s_i < 2\tau(P_i)$. Thus, the knot $D_{P_k,s_k} \circ \cdots \circ D_{P_1,s_1} (K)$ is a covering link of $Wh_+(B_T(K))$. By Theorem \ref{thm:taudjskt}, $\tau(D_{P_k,s_k} \circ \cdots \circ D_{P_1,s_1} (K))=1$. Thus, $D_{P_k,s_k} \circ \cdots \circ D_{P_1,s_1} (K)$ cannot be smoothly slice in a rational homology $4$-ball, so $Wh_+(B_T(K))$ cannot be smoothly slice.
\begin{figure}
\caption{The proof of the second part of Theorem \ref{thm:notslice}.}
\label{fig:hopf}
\end{figure}
For the second part, the same argument as above shows that by using covering moves, we may replace $Wh_+(B_{T_1}(K_1) \cup B_{T_2}(K_2))$ with a two-component link of the form \[ D_{P_k,s_k} \circ \cdots \circ D_{P_1,s_1} (K_1) \cup D_{Q_l,t_l} \circ \cdots \circ D_{Q_0,t_0} (K_2), \] shown in Figure \ref{fig:hopf}(a), where $s_i < 2\tau(P_i)$ and $t_i < 2\tau(Q_i)$ for all $i$. (We start with $Q_0$ and $t_0$ for notational reasons.) After the isotopies in Figure \ref{fig:hopf}(a--c), note the similarity to Figure \ref{fig:solidtorus1}. We may thus proceed just as in the proof of Lemma \ref{lemma:solidtorus}, with suitable modifications to Figures \ref{fig:solidtorus2}--\ref{fig:solidtorus4}, to obtain the knot shown in Figure \ref{fig:hopf}(d) as a covering link of $Wh_+(B_{T_1}(K_1) \cup B_{T_2}(K_2))$. This knot is \[ D_{P_k,s_k} \circ \dots \circ D_{P_1,s_1} (D_{R,u} (Q_0,t_0)), \] where $(R,u)$ is as in \eqref{eq:solidtorus2}. This knot has $\tau=1$ by Theorem \ref{thm:taudjskt}, completing the proof. \end{proof}
\section{Strongly quasipositive knots and sliceness} \label{sec:quasipositive}
We conclude with a brief discussion of strongly quasipositive knots, which played a role in an earlier version of this paper.
A knot or link $L$ is called \emph{quasipositive} if it is the closure of a braid that is the product of conjugates of the standard positive braid generators $\sigma_i$ (but not their inverses). It is called \emph{strongly quasipositive} if it is the closure of a braid that is the product of words of the form $\sigma_i \dots \sigma_{j-1} \sigma_j \sigma_{j-1}^{-1} \dots \sigma_i^{-1}$ for $i<j$. A strongly quasipositive link naturally admits a particular type of Seifert surface determined by this braid form, and an embedded surface in $S^3$ is called \emph{quasipositive} if it is isotopic to such a surface. In other words, a link is strongly quasipositive if and only if it bounds a quasipositive Seifert surface.
A link $L$ is quasipositive if and only if it is a \emph{transverse $\mathbb{C}$-link}: the transverse intersection of $S^3 \subset \mathbb{C}^2$ with a complex curve $V$. If $L$ is strongly quasipositive, then the Seifert surface determined by the braid form is isotopic to $V \cap B^4$.
For a knot $K$ and $t \in \mathbb{Z}$, let $A(K,t)$ be an annulus in $S^3$ whose core circle is $K$ and whose two boundary components are $t$-framed longitudes of the core. Given two unlinked annuli $A$ and $A'$, let $A*A'$ denote the surface obtained by plumbing $A$ and $A'$ together. (To be precise, we must orient the core circles of $A$ and $A'$ and specify the sign of their intersection in $A * A'$.)
The following is a summary of some of Rudolph's results \cite{RudolphAnnuli, RudolphObstruction, RudolphPlumbing} on strongly quasipositive knots: \begin{thm} \label{thm:rudolph} $ \ $ \begin{enumerate} \item If $K$ is a strongly quasipositive knot other than the unknot, then $K$ is not smoothly slice. \item A knot $K$ is strongly quasipositive if and only if $A(K,0)$ is a quasipositive surface. \item If $K$ and $K'$ are strongly quasipositive, then $K \# K'$ is strongly quasipositive. \item The annulus $A(K,t)$ is quasipositive if and only if $t \le TB(K)$, where $TB(K)$ denotes the maximal Thurston--Bennequin number of $K$. \item If $A$ and $A'$ are annuli, then the surface $A*A'$ is quasipositive if and only if $A$ and $A'$ are both quasipositive. \end{enumerate} \end{thm}
Rudolph's original proof of (1) relies on the fact that complex curves are genus-minimizing, a major theorem proven by Kronheimer and Mrowka \cite{KronheimerMrowka} using gauge theory. Since a strongly quasipositive knot $K$ has a Seifert surface that is isomorphic to a complex curve, we thus see that $g_4(K) = g(K)$; in particular, if $K$ is nontrivial, then $g_4(K)>0$. Subsequently, Livingston \cite{LivingstonComputations} proved that both of these genera are equal to $\tau(K)$ when $K$ is strongly quasipositive. (For more on the relationship between $\tau$ and quasipositivity, see Hedden \cite{HeddenPositivity}.)
The untwisted $\pm$ Whitehead double of $K$, $Wh_{\pm}(K)$, is the boundary of $A(K,0)*A(O,\mp1)$, where $O$ denotes the unknot. Thus, Theorem \ref{thm:rudolph} implies that if $K$ is strongly quasipositive and nontrivial, then $Wh_+(K)$ is strongly quasipositive and nontrivial, hence not smoothly slice. More generally, the plumbing $A(J,s)*A(K,t)$ is a Seifert surface for $D_{J,s}(K,t)$, so if $J$ and $K$ are strongly quasipositive and $s,t \le 0$, then $D_{J,s}(K,t)$ is strongly quasipositive. Moreover, if neither of the pairs $(J,s)$ and $(K,t)$ equals $(O,0)$, then $D_{J,s}(K,t)$ is nontrivial, hence not smoothly slice. Furthermore, in this case $\tau(D_{J,s}(K,t))=1$ since the $\tau$ invariant of a strongly quasipositive knot is equal to its genus by a result of Livingston \cite{LivingstonComputations}. Using this observation, we may prove a weakened version of Theorem \ref{thm:notslice} in which the knot $K$ is assumed to be strongly quasipositive without ever making reference to Theorem \ref{thm:taudjskt}.
\end{document} | arXiv |
Epsilon
Epsilon (/ˈɛpsɪlɒn/,[1] UK also /ɛpˈsaɪlən/;[2] uppercase Ε, lowercase ε or lunate ϵ; Greek: έψιλον) is the fifth letter of the Greek alphabet, corresponding phonetically to a mid front unrounded vowel IPA: [e̞] or IPA: [ɛ̝]. In the system of Greek numerals it also has the value five. It was derived from the Phoenician letter He . Letters that arose from epsilon include the Roman E, Ë and Ɛ, and Cyrillic Е, È, Ё, Є and Э.
Not to be confused with Upsilon.
Greek alphabet
Αα Alpha Νν Nu
Ββ Beta Ξξ Xi
Γγ Gamma Οο Omicron
Δδ Delta Ππ Pi
Εε Epsilon Ρρ Rho
Ζζ Zeta Σσς Sigma
Ηη Eta Ττ Tau
Θθ Theta Υυ Upsilon
Ιι Iota Φφ Phi
Κκ Kappa Χχ Chi
Λλ Lambda Ψψ Psi
Μμ Mu Ωω Omega
History
Archaic local variants
• Ϝ
• Ͱ
• Ϻ
• Ϙ
• Ͳ
• Ͷ
• Diacritics
• Ligatures
Numerals
• ϛ (6)
• ϟ (90)
• ϡ (900)
Use in other languages
• Bactrian
• Coptic
• Albanian
Related topics
• Use as scientific symbols
• Category
The name of the letter was originally εἶ (Ancient Greek: [êː]), but it was later changed to ἒ ψιλόν (e psilon 'simple e') in the Middle Ages to distinguish the letter from the digraph αι, a former diphthong that had come to be pronounced the same as epsilon.
The uppercase form of epsilon is identical to Latin E but has its own code point in Unicode: U+0395 Ε GREEK CAPITAL LETTER EPSILON. The lowercase version has two typographical variants, both inherited from medieval Greek handwriting. One, the most common in modern typography and inherited from medieval minuscule, looks like a reversed number "3" and is encoded U+03B5 ε GREEK SMALL LETTER EPSILON. The other, also known as lunate or uncial epsilon and inherited from earlier uncial writing,[3][4] looks like a semicircle crossed by a horizontal bar: it is encoded U+03F5 ϵ GREEK LUNATE EPSILON SYMBOL. While in normal typography these are just alternative font variants, they may have different meanings as mathematical symbols: computer systems therefore offer distinct encodings for them.[3] In TeX, \epsilon ( $\epsilon \!$ ) denotes the lunate form, while \varepsilon ( $\varepsilon \!$ ) denotes the reversed-3 form. Unicode versions 2.0.0 and onwards use ɛ as the lowercase Greek epsilon letter,[5] but in version 1.0.0, ϵ was used.[6]
There is also a 'Latin epsilon', ɛ or "open e", which looks similar to the Greek lowercase epsilon. It is encoded in Unicode as U+025B ɛ LATIN SMALL LETTER OPEN E and U+0190 Ɛ LATIN CAPITAL LETTER OPEN E and is used as an IPA phonetic symbol. The lunate or uncial epsilon provided inspiration for the euro sign, €.[7]
The lunate epsilon, ϵ, is not to be confused with the set membership symbol ∈; nor should the Latin uppercase epsilon, Ɛ, be confused with the Greek uppercase Σ (sigma). The symbol $\in $, first used in set theory and logic by Giuseppe Peano and now used in mathematics in general for set membership ("belongs to") evolved from the letter epsilon, since the symbol was originally used as an abbreviation for the Latin word est. In addition, mathematicians often read the symbol ∈ as "element of", as in "1 is an element of the natural numbers" for $1\in \mathbb {N} $, for example. As late as 1960, ε itself was used for set membership, while its negation "does not belong to" (now ∉) was denoted by ε' (epsilon prime).[8] Only gradually did a fully separate, stylized symbol take the place of epsilon in this role. In a related context, Peano also introduced the use of a backwards epsilon, ϶, for the phrase "such that", although the abbreviation s.t. is occasionally used in place of ϶ in informal cardinals.
History
Origin
The letter Ε was adopted from the Phoenician letter He () when Greeks first adopted alphabetic writing. In archaic Greek writing, its shape is often still identical to that of the Phoenician letter. Like other Greek letters, it could face either leftward or rightward (), depending on the current writing direction, but, just as in Phoenician, the horizontal bars always faced in the direction of writing. Archaic writing often preserves the Phoenician form with a vertical stem extending slightly below the lowest horizontal bar. In the classical era, through the influence of more cursive writing styles, the shape was simplified to the current E glyph.[9]
Sound value
While the original pronunciation of the Phoenician letter He was [h], the earliest Greek sound value of Ε was determined by the vowel occurring in the Phoenician letter name, which made it a natural choice for being reinterpreted from a consonant symbol to a vowel symbol denoting an [e] sound.[10] Besides its classical Greek sound value, the short /e/ phoneme, it could initially also be used for other [e]-like sounds. For instance, in early Attic before c. 500 BC, it was used also both for the long, open /ɛː/, and for the long close /eː/. In the former role, it was later replaced in the classic Greek alphabet by Eta (Η), which was taken over from eastern Ionic alphabets, while in the latter role it was replaced by the digraph spelling ΕΙ.
Epichoric alphabets
Some dialects used yet other ways of distinguishing between various e-like sounds.
In Corinth, the normal function of Ε to denote /e/ and /ɛː/ was taken by a glyph resembling a pointed B (), while Ε was used only for long close /eː/.[11] The letter Beta, in turn, took the deviant shape .
In Sicyon, a variant glyph resembling an X () was used in the same function as Corinthian .[12]
In Thespiai (Boeotia), a special letter form consisting of a vertical stem with a single rightward-pointing horizontal bar () was used for what was probably a raised variant of /e/ in pre-vocalic environments.[13][14] This tack glyph was used elsewhere also as a form of "Heta", i.e. for the sound /h/.
Glyph variants
After the establishment of the canonical classical Ionian (Euclidean) Greek alphabet, new glyph variants for Ε were introduced through handwriting. In the uncial script (used for literary papyrus manuscripts in late antiquity and then in early medieval vellum codices), the "lunate" shape () became predominant. In cursive handwriting, a large number of shorthand glyphs came to be used, where the cross-bar and the curved stroke were linked in various ways.[15] Some of them resembled a modern lowercase Latin "e", some a "6" with a connecting stroke to the next letter starting from the middle, and some a combination of two small "c"-like curves. Several of these shapes were later taken over into minuscule book hand. Of the various minuscule letter shapes, the inverted-3 form became the basis for lower-case Epsilon in Greek typography during the modern era.
Uncial Uncial variants Cursive variants Minuscule Minuscule with ligatures
Uses
International Phonetic Alphabet
Despite its pronunciation as mid, in the International Phonetic Alphabet, the Latin epsilon /ɛ/ represents open-mid front unrounded vowel, as in the English word pet /pɛt/.
Symbol
The uppercase Epsilon is not commonly used outside of the Greek language because of its similarity to the Latin letter E. However, it is commonly used in structural mechanics with Young's Modulus equations for calculating tensile, compressive and areal strain.
The Greek lowercase epsilon ε, the lunate epsilon symbol ϵ, and the Latin lowercase epsilon ɛ (see above) are used in a variety of places:
• In engineering mechanics, strain calculations ϵ = increase of length / original length. Usually this relates to extensometer testing of metallic materials.
• In mathematics
• (particularly calculus), an infinitesimally small positive quantity is commonly denoted ε; see (ε, δ)-definition of limit.
• Hilbert introduced epsilon terms $\epsilon x.\phi $ as an extension to first-order logic; see epsilon calculus.
• it is used to represent the Levi-Civita symbol.
• it is used to represent dual numbers: $a+b\varepsilon $, with $\varepsilon ^{2}=0$ and $\varepsilon \neq 0$.
• it is sometimes used to denote the Heaviside step function.[16]
• in set theory, the epsilon numbers are ordinal numbers that satisfy the fixed point ε = ωε. The first epsilon number, ε0, is the limit ordinal of the set {ω, ωω, ωωω, ...}.
• in numerical analysis and statistics it is used as the error term
• in group theory it is used as the idempotent group when e is in use as a variable name
• In computer science
• it often represents the empty string, though different writers use a variety of other symbols for the empty string as well; usually the lower-case Greek letter lambda (λ).
• the machine epsilon indicates the upper bound on the relative error due to rounding in floating point arithmetic.
• In physics,
• it indicates the permittivity of a medium; with the subscript 0 (ε0) it is the permittivity of free space.
• it can also indicate the strain of a material (a ratio of extensions).
• In automata theory, it shows a transition that involves no shifting of an input symbol.
• In astronomy,
• it stands for the fifth-brightest star in a constellation (see Bayer designation).
• Epsilon is the name for the most distant and most visible ring of Uranus.
• In planetary science, ε denotes the axial tilt.
• In chemistry, it represents the molar extinction coefficient of a chromophore.
• In economics, ε refers to elasticity.
• In statistics,
• it is used to refer to error terms.
• it also can to refer to the degree of sphericity in repeated measures ANOVAs.
• In agronomy, it is used to represent the "photosynthetic efficiency" of a particular plant or crop.
Unicode
• Greek Epsilon
Character information
PreviewΕεϵ϶
Unicode name GREEK CAPITAL LETTER EPSILON GREEK SMALL LETTER EPSILON GREEK LUNATE EPSILON SYMBOL GREEK REVERSED LUNATE EPSILON SYMBOL
Encodingsdecimalhexdechexdechexdechex
Unicode917U+0395949U+03B51013U+03F51014U+03F6
UTF-8206 149CE 95206 181CE B5207 181CF B5207 182CF B6
Numeric character referenceΕΕεεϵϵ϶϶
Named character referenceΕε, εϵ, ϵ, ϵ϶, ϶
DOS Greek132841569C
DOS Greek-2168A8222DE
Windows 1253197C5229E5
TeX\varepsilon\epsilon
• Coptic Eie
Character information
PreviewⲈⲉ
Unicode name COPTIC CAPITAL LETTER EIE COPTIC SMALL LETTER EIE
Encodingsdecimalhexdechex
Unicode11400U+2C8811401U+2C89
UTF-8226 178 136E2 B2 88226 178 137E2 B2 89
Numeric character referenceⲈⲈⲉⲉ
• Latin Open E
Character information
PreviewƐɛᶓᵋ
Unicode name LATIN CAPITAL LETTER
OPEN E
LATIN SMALL LETTER
OPEN E
LATIN SMALL LETTER
OPEN E WITH RETROFLEX HOOK
MODIFIER LETTER
SMALL OPEN E
Encodingsdecimalhexdechexdechexdechex
Unicode400U+0190603U+025B7571U+1D937499U+1D4B
UTF-8198 144C6 90201 155C9 9B225 182 147E1 B6 93225 181 139E1 B5 8B
Numeric character referenceƐƐɛɛᶓᶓᵋᵋ
Character information
Previewɜɝᶔᶟ
Unicode name LATIN SMALL LETTER
REVERSED OPEN E
LATIN SMALL LETTER
REVERSED OPEN E WITH HOOK
LATIN SMALL LETTER REVERSED
OPEN E WITH RETROFLEX HOOK
MODIFIER LETTER
SMALL REVERSED OPEN E
Encodingsdecimalhexdechexdechexdechex
Unicode604U+025C605U+025D7572U+1D947583U+1D9F
UTF-8201 156C9 9C201 157C9 9D225 182 148E1 B6 94225 182 159E1 B6 9F
Numeric character referenceɜɜɝɝᶔᶔᶟᶟ
Character information
Previewᴈᵌʚɞ
Unicode name LATIN SMALL LETTER
TURNED OPEN E
MODIFIER LETTER SMALL
TURNED OPEN E
LATIN SMALL LETTER
CLOSED OPEN E
LATIN SMALL LETTER
CLOSED REVERSED OPEN E
Encodingsdecimalhexdechexdechexdechex
Unicode7432U+1D087500U+1D4C666U+029A606U+025E
UTF-8225 180 136E1 B4 88225 181 140E1 B5 8C202 154CA 9A201 158C9 9E
Numeric character referenceᴈᴈᵌᵌʚʚɞɞ
• Mathematical Epsilon
Character information
Preview𝚬𝛆𝛦𝜀𝜠𝜺
Unicode name MATHEMATICAL BOLD
CAPITAL EPSILON
MATHEMATICAL BOLD
SMALL EPSILON
MATHEMATICAL ITALIC
CAPITAL EPSILON
MATHEMATICAL ITALIC
SMALL EPSILON
MATHEMATICAL BOLD ITALIC
CAPITAL EPSILON
MATHEMATICAL BOLD ITALIC
SMALL EPSILON
Encodingsdecimalhexdechexdechexdechexdechexdechex
Unicode120492U+1D6AC120518U+1D6C6120550U+1D6E6120576U+1D700120608U+1D720120634U+1D73A
UTF-8240 157 154 172F0 9D 9A AC240 157 155 134F0 9D 9B 86240 157 155 166F0 9D 9B A6240 157 156 128F0 9D 9C 80240 157 156 160F0 9D 9C A0240 157 156 186F0 9D 9C BA
UTF-1655349 57004D835 DEAC55349 57030D835 DEC655349 57062D835 DEE655349 57088D835 DF0055349 57120D835 DF2055349 57146D835 DF3A
Numeric character reference𝚬𝚬𝛆𝛆𝛦𝛦𝜀𝜀𝜠𝜠𝜺𝜺
Character information
Preview𝛜𝜖𝝐
Unicode name MATHEMATICAL BOLD
EPSILON SYMBOL
MATHEMATICAL ITALIC
EPSILON SYMBOL
MATHEMATICAL BOLD ITALIC
EPSILON SYMBOL
Encodingsdecimalhexdechexdechex
Unicode120540U+1D6DC120598U+1D716120656U+1D750
UTF-8240 157 155 156F0 9D 9B 9C240 157 156 150F0 9D 9C 96240 157 157 144F0 9D 9D 90
UTF-1655349 57052D835 DEDC55349 57110D835 DF1655349 57168D835 DF50
Numeric character reference𝛜𝛜𝜖𝜖𝝐𝝐
Character information
Preview𝝚𝝴𝞔𝞮
Unicode name MATHEMATICAL SANS-SERIF
BOLD CAPITAL EPSILON
MATHEMATICAL SANS-SERIF
BOLD SMALL EPSILON
MATHEMATICAL SANS-SERIF
BOLD ITALIC CAPITAL EPSILON
MATHEMATICAL SANS-SERIF
BOLD ITALIC SMALL EPSILON
Encodingsdecimalhexdechexdechexdechex
Unicode120666U+1D75A120692U+1D774120724U+1D794120750U+1D7AE
UTF-8240 157 157 154F0 9D 9D 9A240 157 157 180F0 9D 9D B4240 157 158 148F0 9D 9E 94240 157 158 174F0 9D 9E AE
UTF-1655349 57178D835 DF5A55349 57204D835 DF7455349 57236D835 DF9455349 57262D835 DFAE
Numeric character reference𝝚𝝚𝝴𝝴𝞔𝞔𝞮𝞮
Character information
Preview𝞊𝟄
Unicode name MATHEMATICAL SANS-SERIF
BOLD EPSILON SYMBOL
MATHEMATICAL SANS-SERIF
BOLD ITALIC EPSILON SYMBOL
Encodingsdecimalhexdechex
Unicode120714U+1D78A120772U+1D7C4
UTF-8240 157 158 138F0 9D 9E 8A240 157 159 132F0 9D 9F 84
UTF-1655349 57226D835 DF8A55349 57284D835 DFC4
Numeric character reference𝞊𝞊𝟄𝟄
These characters are used only as mathematical symbols. Stylized Greek text should be encoded using the normal Greek letters, with markup and formatting to indicate text style.
Initial
• Initial epsilon in Lectionary 226, folio 20 verso
• folio 64 verso
• folio 125 verso
See also
• Е and е, the letter Ye of the Cyrillic alphabet
• Є є, Ukrainian Ye
• Ԑ ԑ, Reversed Ze
• E (disambiguation)
References
1. Wells, John C. (1990). "epsilon". Longman Pronunciation Dictionary. Harlow, England: Longman. p. 250. ISBN 0582053838.
2. "epsilon". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
3. Nick Nicholas: Letters Archived 2012-12-15 at archive.today, 2003–2008. (Greek Unicode Issues)
4. Colwell, Ernest C. (1969). "A chronology for the letters Ε, Η, Λ, Π in the Byzantine minuscule book hand". Studies in methodology in textual criticism of the New Testament. Leiden: Brill. p. 127.
5. "Code Charts" (PDF). The Unicode Standard, Version 2.0. p. 130. ISBN 0-201-48345-9.
6. "Code Charts" (PDF). The Unicode Standard, Version 1.0. Vol. 1. p. 130. ISBN 0-201-56788-1.
7. "European Commission – Economic and Financial Affairs – How to use the euro name and symbol". Ec.europa.eu. Retrieved 7 April 2010. Inspiration for the € symbol itself came from the Greek epsilon, ϵ – a reference to the cradle of European civilization – and the first letter of the word Europe, crossed by two parallel lines to 'certify' the stability of the euro.
8. Halmos, Paul R. (1960). Naive Set Theory. New York: Van Nostrand. pp. 5–6. ISBN 978-1614271314.
9. Jeffery, Lilian H. (1961). The local scripts of archaic Greece. Oxford: Clarendon. pp. 63–64.
10. Jeffery, Local scripts, p. 24.
11. Jeffery, Local scripts, p. 114.
12. Jeffery, Local scripts, p. 138.
13. Nicholas, Nick (2005). "Proposal to add Greek epigraphical letters to the UCS" (PDF). Archived from the original (PDF) on 2006-05-05. Retrieved 2010-08-12.
14. Jeffery, Local scripts, p. 89.
15. Thompson, Edward M. (1911). An introduction to Greek and Latin palaeography. Oxford: Clarendon. pp. 191–194.
16. Weisstein, Eric W. "Delta Function". mathworld.wolfram.com. Retrieved 2019-02-19.
Further reading
Look up Ε or ɛ in Wiktionary, the free dictionary.
• Hoffman, Paul; The Man Who Loved Only Numbers. Hyperion, 1998. ISBN 0-7868-6362-5.
| Wikipedia |
\begin{document}
\title{Configuration types and cubic surfaces}
\author{Elena Guardo and Brian Harbourne}
\address{Elena Guardo\\ Dipartimento di Matematica e Informatica\\ Viale A. Doria 6, 95100\\ Catania, Italy\\ \url{http://www.dmi.unict.it/~guardo}} \email{[email protected]}
\address{Brian Harbourne\\ Department of Mathematics\\ University of Nebraska-Lincoln\\ Lincoln, NE 68588-0130\\ USA\\ \url{http://www.math.unl.edu/~bharbour/}} \email{[email protected]}
\date{May 3, 2008}
\begin{abstract} This paper is a sequel to the paper \cite{refGH}. We relate the matroid notion of a combinatorial geometry to a generalization which we call a configuration type. Configuration types arise when one classifies the Hilbert functions and graded Betti numbers for fat point subschemes supported at $n\le8$ essentially distinct points of the projective plane. Each type gives rise to a surface $X$ obtained by blowing up the points. We classify those types such that $n=6$ and $-K_X$ is nef. The surfaces obtained are precisely the desingularizations of the normal cubic surfaces. By classifying configuration types we recover in all characteristics the classification of normal cubic surfaces, which is well-known in characteristic 0 \cite{refBW}. As an application of our classification of configuration types, we obtain a numerical procedure for determining the Hilbert function and graded Betti numbers for the ideal of any fat point subscheme $Z=m_1p_1+\cdots+m_6p_6$ such that the points $p_i$ are essentially distinct and $-K_X$ is nef, given only the configuration type of the points $p_1,\ldots,p_6$ and the coefficients $m_i$. \end{abstract}
\thanks{Acknowledgments: We would like to thank the GNSAGA, the NSA, the NSF and the Department of Mathematics at UNL for their support of the authors' research and of E. Guardo's visits in 2003, 2004 and 2005 while this work was carried out.}
\keywords{Fat points, free resolution, cubic surface, matroid, Hilbert function}
\subjclass[2000]{Primary 14C20, 14J26, 13D02; Secondary 14N20, 14Q99.}
\maketitle
\section{Introduction}\label{intro}
\subsection*{Matroids and Combinatorial Geometries} The combinatorial classification of points in projective space leads one to the concept of combinatorial geometries. Intuitively, a combinatorial geometry of rank $N+1$ or less and size $n$ is an abstract specification of linear dependencies among a set of $n$ points spanning a space of dimension at most $N$. Formally, a \emph{combinatorial geometry\/} is a matroid without loops or parallel elements \cite{refHandbook}. We are interested in the case of $n$ points in the projective plane, in which case one can regard a combinatorial geometry as just being a collection of subsets of the set $\{1,\dots,n\}$, where each subset has at least two elements and two subsets which have two elements in common must be equal. We say that a given combinatorial geometry is \emph{representable\/} over a field $k$ (in this paper the field $k$ will be assumed to be algebraically closed, but not necessarily of characteristic 0) if there is a collection of distinct points $p_1,\ldots,p_n\in \hbox{${\bf P}$}^2_k$ such that a subset $\{i_1,\ldots,i_r\}\subseteq \{1,\ldots,n\}$ is an element of the combinatorial geometry if and only if $\{p_{i_1},\ldots,p_{i_r}\}$ is a maximal collinear subset of $\{p_1,\ldots,p_n\}$.
\subsection*{Combinatorial Geometries as Matrices} We can think of a combinatorial geometry of rank up to 3 and size $n$ with $g$ elements as specifying a correspondence between a set of $g$ lines and $n$ points, where each line is defined by a maximal collinear subset of the points. If we enumerate the lines, then specifying the combinatorial geometry is equivalent to giving a 0-1 matrix $M$ where the entry in row $i$ and column $j$ is a 1 if and only if the $i$th line contains the $j$th point. (Although any two points determine a line, it is convenient to ignore any row with exactly two 1's, and so this is what we will do. This does no harm, since if we know all maximal collinear subsets containing more than two points, we can recover all of the maximal two point subsets. Since there typically are a lot of maximal two point subsets, it is impractical to include them in the matrix $M$.)
Thus we can regard combinatorial geometries on $n$ points in the plane as being matrices $M$ with $n$ columns, such that each entry of each row is either a 0 or a $1$, the sum of the entries for a row is always at least $3$, and the dot product of two different rows is either 0 or $1$. (If no three points are collinear, then the matrix would have no rows.)
\subsection*{An Algebraic Geometric Perspective on Combinatorial Geometries} A combinatorial geometry can be regarded as telling us when more than the expected number of points are to lie on a given line, the expected number being 2. But in algebraic geometry we are also interested in the possibility of points being special with respect to curves of higher (and lower) degrees. Thus 6 points on a conic are special (in which case the points are special with respect to a curve of degree 2). Similarly, it is special to have one point be infinitely near another (in which case we can regard the points as being special with respect to a curve of degree 0).
For the purposes of this paper, we now want to introduce an alternative, algebraic geometric, approach to combinatorial geometries, which we will then generalize in the case of $n=6$ points to sets of points being special with respect to curves other than lines. Consider a set of distinct points $p_1,\ldots,p_n\in \hbox{${\bf P}$}^2_k$ which represents a given combinatorial geometry, and let $\pi:X\to \hbox{${\bf P}$}^2_k$ be the morphism obtained by blowing up the points. The divisor class group $\hbox{Cl}(X)$ is the free abelian group on the divisors classes $L$, $E_1,\ldots,E_n$, where $L$ is the pullback to $X$ of the class of a line on $\hbox{${\bf P}$}^2_k$ and $E_i$ is the class of $\pi^{-1}(p_i)$. The group $\hbox{Cl}(X)$ supports an intersection form; it is the bilinear form defined by requiring that the classes $L$, $E_1,\ldots,E_n$ be orthogonal with $L^2=1$ and $E_i^2=-1$ for all $i$.
To ignore trivial cases, assume that $n>1$. If the points $p_{i_1},\dots,p_{i_r}$, with $r>1$, are collinear and none of the other points $p_i$ are on the same line, then the class of the proper transform of the line through those points is $C=L-E_{i_1}-\cdots-E_{i_r}$, so the classes of the proper transforms of the lines corresponding to the elements of the combinatorial geometry are precisely the classes of all prime divisors $C$ on $X$ with $C^2<0$ and $C\cdot L=1$. If we construct a matrix $M'$ by changing the sign of each nonzero entry of $M$ and then prepending a column of 1's to the left side of $M$, we obtain a matrix $M'$ whose rows specify (in terms of the basis $L$, $E_1,\ldots,E_n$) the set of all classes of prime divisors $C$ on $X$ such that $C^2<0$ and $C\cdot L=1$ (with classes having $C^2=-1$ suppressed, corresponding to suppression in $M$ of rows with exactly two 1's in them).
For example, given 5 points $p_1,\ldots,p_5$ such that the maximal collinear subsets (ignoring two point subsets) are points 1, 4 and 5 and points 2, 3 and 4, the matrices $M$ and $M'$ are:
$$\hbox{ $M=\left(\begin{matrix} 1 & 0 & 0 & 1 & 1\\ 0 & 1 & 1 & 1 & 0\\ \end{matrix}\right)$ \hbox to.5in{\hfil} $M'=\left(\begin{matrix} 1 & -1 & 0 & 0 & -1 & -1\\ 1 & 0 & -1 & -1 & -1 & 0\\ \end{matrix}\right)$.}$$
The rows of $M'$ specify the classes $L-E_1-E_4-E_5$ and $L-E_2-E_3-E_4$.
Thus we now can regard combinatorial geometries on $n$ points in the plane as being matrices $M'$ with $n+1$ columns, where the first entry in each row is a 1, each remaining entry is either a 0 or a $-1$, the sum of the entries of each row (i.e., the intersection product of a row with itself, with respect to the bilinear form defined above) is always at most $-2$, and the intersection product of two different rows is either 1 or 0.
\subsection*{Infinitely near points, Blow ups, the Intersection form and Exceptional configurations} We now recall the notion of points being infinitely near. Let $\pi: X\to \hbox{${\bf P}$}^2$ be the morphism obtained as a sequence of blow ups of points in the following way. Let $p_1\in X_0=\hbox{${\bf P}$}^2$, and let $p_2\in X_1$, $\ldots$, $p_n\in X_{n-1}$, where, for $0\le i\le n-1$, $\pi_{i+1}:X_{i+1}\to X_i$ is the blow up of $p_{i+1}$. We will denote $X_n$ by $X$ and the composition $X=X_n\to\cdots\to X_0=\hbox{${\bf P}$}^2$ by $\pi$. We say that the points $p_1,\ldots,p_n$ are \emph{essentially distinct\/} points of $\hbox{${\bf P}$}^2$ \cite{refFreeRes}; note for $j>i$ that we may have $\pi_i\circ\cdots\circ\pi_{j-1}(p_j)=p_i$, in which case we say $p_j$ is infinitely near $p_i$. (If no point is infinitely near another, the points are just distinct points of $\hbox{${\bf P}$}^2$ and $X$ is just the surface obtained by blowing the points up in a particular order, but the order does not matter. If the points are only essentially distinct, then $p_i$ needs to be blown up before $p_j$ whenever $p_j$ is infinitely near $p_i$.)
We denote by $E_i$ the class of the 1-dimensional scheme-theoretic fiber of $X=X_n\to X_{i-1}$ over $p_i$ and the pullback to $X$ of the class of a line in $\hbox{${\bf P}$}^2$ by $L$. As before, the classes $L,E_1,\ldots,E_n$ form a basis over the integers of the divisor class group $\hbox{Cl}(X)$, which is a free abelian group of rank $n+1$. We call such a basis an \emph{exceptional configuration\/}, which as before is an orthogonal basis for $\hbox{Cl}(X)$ with respect to the intersection form.
\subsection*{Ordered and unordered Configuration types} We saw above that lines through two or more points give rise to classes of prime divisors of negative self-intersection. Similarly, if instead the points $p_{i_1},\dots,p_{i_r}$ lie on an irreducible conic and none of the other points lie on that same conic, then the class of the proper transform of that conic is $D=2L-E_{i_1}-\cdots-E_{i_r}$ and $D$ is the class of a prime divisor of self-intersection $D^2=4-r$, hence negative if $r>4$. Instead of just lines through 2 or more points, in the context of algebraic geometry what is of interest is more generally the set of all prime divisors of negative self-intersection.
We can formalize this generalized notion as a \emph{configuration type\/} of points. Up to equivalence, an \emph{ordered\/} configuration type of $n$ points in the plane is a matrix $T$ with $n+1$ columns whose rows satisfy two conditions: negative self-intersection and pairwise nonnegativity (explained below). Two matrices satisfying these two conditions will be regarded as giving the same ordered configuration type if one matrix can be obtained from the other by permuting its rows. (We will say that two matrices satisfying the two conditions will be regarded as giving the same \emph{unordered\/} configuration type if one matrix can be obtained from the other by permuting either its rows or columns or both.)
The two conditions come from our wanting the rows to specify the coefficients, with respect to the basis $L,E_1,\ldots,E_n$, of classes of prime divisors of negative self-intersection. (Although in general there can be infinitely many classes of prime divisors of negative self-intersection, if $n\le8$, it is known there are only finitely many. See Lemma \ref{NEGisfinite} for the case of interest here, $n=6$; the case for any $n\le 8$ is similar.)
Thus if $(d,m_1,\ldots,m_n)$ is a row of the matrix $T$, we require $d^2-m_1^2-\cdots-m_n^2<0$ (negative self-intersection), and if $(d',m'_1,\ldots,m'_n)$ is another row of the matrix, we require $dd'-m_1m'_1-\cdots-m_nm'_n\ge0$ (pairwise nonnegativity), corresponding to an intersection theoretic version of Bezout's theorem, saying that $C\cdot D\ge0$ if $C$ and $D$ are prime divisors with $C\ne D$. We will say a configuration type $T$ is \emph{representable\/} if there is a set of essentially distinct points $p_1,\ldots,p_n$ giving a surface $X$ such that the rows of $T$ are (in terms of the exceptional configuration $L,E_1,\ldots,E_n$ for $X$) the classes of all prime divisors of negative self-intersection on $X$.
\subsection*{Goals and Motivation} The goal of this paper is to classify all of the configuration types for $n=6$ essentially distinct points of $\hbox{${\bf P}$}^2$ which when blown up give a surface $X$ for which $-K_X$ is nef, and to determine representability for each configuration type. In order to formally write down possible matrices, we must have a set $S$ of possible vectors $(d,m_1,\ldots,m_n)$ to draw from. In principle, $S$ should consist of all coefficient vectors which occur for prime divisors of negative self-intersection for any prime divisor that occurs for any choice of the points $p_i$. Then we can attempt to write down all possible matrices satisfying the two given conditions (of negative self-intersection and pairwise nonnegativity) where each row is chosen from $S$. Having written down all possible matrices, we can consider representability: i.e., for which matrices is there an algebraically closed field $k$ and an actual set of points $p_i$ in $\hbox{${\bf P}$}^2_k$ such that the set of prime divisors on $X$ is exactly that specified by the matrix.
The underlying motivation for carrying out this classification is that, if $n\le8$, then two sets of points, $p_1,\ldots,p_n$ and $p'_1,\ldots,p'_n$, have the same ordered configuration type if and only if, for all choices of nonnegative integers $m_1,\ldots,m_n$, the Hilbert functions of the fat point subschemes $m_1p_1+\cdots+m_np_n$ and $m_1p'_1+\cdots+m_np'_n$ are the same \cite{refGH}. (We recall the notions of fat points, their ideals and their Hilbert functions in Section \ref{bkgd}, and their graded Betti numbers in Section \ref{resols}.)
\subsection*{Previous Work} We classified the configuration types for sets of $n=6$ distinct points of $\hbox{${\bf P}$}^2$ in \cite{refGH}, and we also showed that if $p_1,\ldots,p_n$ and $p'_1,\ldots,p'_n$ have the same ordered configuration type, then for any nonnegative integers $m_1,\ldots,m_n$, the graded Betti numbers of the ideals $I(m_1p_1+\cdots+m_np_n)$ and $I(m_1p'_1+\cdots+m_np'_n)$ defining the fat point subschemes are the same. (In \cite{refGH} for efficiency we listed only the unordered configuration types, of which there are 11. These 11 comprise 353 ordered configuration types, but two ordered types with the same unordered type differ only in the indexation of the points. For example, one of the 11 is the situation where 3 points in a set of 6 points is collinear, and otherwise no more than 2 of the 6 points is collinear. There are $\binom{6}{3}=20$ essentially different ways to number the 6 points, so this one unordered type comprises 20 ordered types. Thus there is little reason to explicitly list the ordered types, and we will normally only explicitly list unordered types, as was done in \cite{refGH}.)
Since the graded Betti numbers determine the Hilbert function, and since knowing the Hilbert functions of $m_1p_1+\cdots+m_6p_6$ for all choices of the $m_i$ allows one to determine the set of prime divisors on $X$ of negative self-intersection and hence to recover the configuration type of the points, this shows that a classification of configuration types of 6 distinct points in the plane is actually a classification of the points up to graded Betti numbers (i.e., where we regard two sets of 6 points $p_1,\ldots,p_6$ and $p'_1,\ldots,p'_6$ as equivalent if the graded Betti numbers of $I(m_1p_1+\cdots+m_6p_6)$ and $I(m_1p'_1+\cdots+m_6p'_6)$ are the same for all choices of nonnegative integers $m_i$).
\subsection*{Results} In this paper we consider the classification of 6 essentially distinct points, but for both technical and practical reasons we do so only under the restriction that the anticanonical divisor $-K_X$ on $X$ is nef. With this restriction, we show that every type is representable over every algebraically closed field $k$ and we show that a classification by type is equivalent to a classification up to graded Betti numbers. We also give an explicit procedure for determining the graded Betti numbers for the ideal $I(Z)$ for any fat point subscheme $Z=m_1p_1+\cdots+m_6p_6$ supported at the six points, given only the coefficients $m_i$ and the ordered configuration type of the points. While this procedure can easily be carried out by hand, an awk script automating the procedure can be run over the web at \url{http://www.math.unl.edu/~bharbourne1/6ptsNef-K/6reswebsite.html}. For some examples, see Section \ref{exmpls}.
The problem of determining all possible Hilbert functions and graded Betti numbers for arbitrary fat point subschemes $2p_1+\cdots+2p_n$, and of determining the configurations of the points that give rise to the different possibilities, was raised in \cite{refGMS}. Thus \cite{refGH} completely solves the problem for $n=6$ in the original context of distinct points, not only for double points but for fat point schemes $m_1p_1+\cdots+m_6p_6$ with $m_i$ arbitrary. What we do here likewise completely solves the problem for arbitrary $m_i$, in the case of 6 essentially distinct points under the condition that $-K_X$ is nef. Indeed, what we find is that there are 90 different unordered configuration types, corresponding to equivalence classes of matrices whose rows are drawn from a certain set $S$ as discussed above and given explicitly in Lemma \ref{NEGisfinite}. (If we were to remove the restriction that $-K_X$ is nef, we would, in addition to what is specified in Lemma \ref{NEGisfinite}, also have to include in $S$ the coefficient vectors of all classes of the form $E_i-E_{j_1}-\cdots-E_{j_r}$ for all subsets $\{j_1,\ldots,j_r\}\subsetneq\{1,\ldots,6\}$ with $r>1$ and $i<j_l$ for all $l$, and also all classes of the form $L-E_{i_1}-\cdots-E_{i_l}$ for all $0<i_1<\ldots<i_l\le6$ with $l>3$. This results in many more configuration types. Having $-K_X$ be nef also affords technical simplifications in computing generators for dual cones given generators for a cone, which we need to do for our method of proving that the graded Betti numbers of $I(m_1p_1+\cdots+m_6p_6)$ depend only on the coefficients $m_i$ and the configuration type of the points $p_i$.)
The condition that $-K_X$ be nef is fairly reasonable, both algebraically and geometrically. Algebraically, one of the cases of most interest is the uniform case, i.e., cases where the fat point subscheme $Z$ is of the form $Z=mp_1+\cdots+mp_6$. Also, one typically considers schemes $Z$ only which satisfy the proximity inequalities (see Section \ref{resols}), and if $-K_X$ is nef, then a uniform $Z$ satisfies the proximity inequalities if and only if $m\ge0$.
Geometrically, the surfaces obtained by blowing up 6 essentially distinct points of $\hbox{${\bf P}$}^2$ such that $-K_X$ is nef are precisely the surfaces which occur by resolving the singularities of normal cubic surfaces in $\hbox{${\bf P}$}^3$. Thus this paper can be regarded as a contribution to the long history of work on cubic surfaces. The classification of normal cubic surfaces up to the types of their singularities (as given by the Dynkin diagrams of the singular points) is classical, at least in characteristic 0 (see \cite{refBW}). What is new here is first the (relatively easy) classification of the corresponding configuration types of points in $\hbox{${\bf P}$}^2$. (Resolving the singularities of a normal cubic surface gives a surface $X$ for which $-K_X$ is nef, but each $X$ typically has several birational morphisms to $\hbox{${\bf P}$}^2$, and each such morphism gives a set of 6 points in $\hbox{${\bf P}$}^2$ which when blown up give $X$. Thus typically several configuration types occur for each Dynkin diagram.) It is much harder to show that the configuration type of the points $p_i$ is enough together with the coefficients $m_i$ to determine the graded Betti numbers of $I(m_1p_1+\cdots+m_6p_6)$. When the points are distinct we showed this in \cite{refGH} without requiring $-K_X$ be nef. What is new here is that we show this for points that can be infinitely near, but under the assumption that $-K_X$ is nef. (It was already known, as a consequence of Theorem 8 of \cite{refBHProc}, that the configuration type of the points $p_i$ is enough, together with the coefficients $m_i$, to determine the Hilbert function of $I(m_1p_1+\cdots+m_np_n)$ for any $n\le 8$ essentially distinct points of $\hbox{${\bf P}$}^2$, whether $-K_X$ is nef or not.)
\section{Background}\label{bkgd}
We recall here some of the background we will need on fat points and on surfaces obtained by blowing up essentially distinct points of $\hbox{${\bf P}$}^2$. We work over an algebraically closed field $k$ of arbitrary characteristic.
A fat point subscheme $Z=m_1p_1+\cdots+m_np_n$ usually is considered in the case that the points $\{p_i\}$ are distinct points. In particular, let $p_1,\ldots,p_n$ be distinct points of $\hbox{${\bf P}$}^2$. Given nonnegative integers $m_i$, the fat point subscheme $Z=m_1p_1+\cdots+m_np_n\subset \hbox{${\bf P}$}^2$ is defined to be the subscheme defined by the ideal $I(Z)=I(p_1)^{m_1}\cap\cdots\cap I(p_n)^{m_n}$, where $I(p_i)\subseteq R=k[\hbox{${\bf P}$}^2]$ is the ideal generated by all forms (in the polynomial ring $R$ in three variables over the field $k$) vanishing at $p_i$. The \emph{support\/} of $Z$ consists of the points $p_i$ for which $m_i$ is positive. For another perspective, let $\C I_Z$ be the sheaf of ideals defining $Z$ as a subscheme of $\hbox{${\bf P}$}^2$. Now let $X$ be obtained by blowing up the points $p_i$. Given a divisor $F$ we will denote the corresponding line bundle by $\C O_X(F)$. With this convention, $\C I_Z=\pi_*(\C O_X(-m_1E_1-\cdots-m_nE_n))$ and the stalks of $\C I_Z$ are complete ideals (as defined in \cite{refZ} and \cite{refZS}) in the local rings of the structure sheaf of $\hbox{${\bf P}$}^2$. We can recover $I(Z)$ from $\C I_Z$ since the homogeneous component $I(Z)_t$ of $I(Z)$ of degree $t$ is just $H^0(X, \C I_Z(t))$.
We can just as well consider essentially distinct points $p_1,\ldots,p_n\in\hbox{${\bf P}$}^2$. Again let $\pi:X\to \hbox{${\bf P}$}^2$ be given by blowing up the points $p_i$, in order. We define the fat point subscheme $Z=m_1p_1+\cdots+m_np_n$ to be the subscheme whose ideal sheaf is the coherent sheaf of ideals $\pi_*(\C O_X(-m_1E_1-\cdots-m_nE_n))$. Note that the stalks of $\pi_*(\C O_X(-m_1E_1-\cdots-m_nE_n))$ are again complete ideals in the stalks of the local rings of the structure sheaf of $\hbox{${\bf P}$}^2$, and, conversely, if \C I is a coherent sheaf of ideals on $\hbox{${\bf P}$}^2$ whose stalks are complete ideals and if \C I defines a 0-dimensional subscheme, then there are essentially distinct points $p_1,\ldots,p_n$ of $\hbox{${\bf P}$}^2$ and integers $m_i$ such that with respect to the corresponding exceptional configuration we have $\C I=\pi_*(\C O_X(-m_1E_1-\cdots-m_nE_n))$ (see \cite{refAppendix}, \cite{refZ} and \cite{refZS} for more details). As before we define $I(Z)$ to be the ideal in $R$ given as $I(Z)=\oplus_{t\ge 0} H^0(X, \C I_Z(t))$. The Hilbert function of a homogenous ideal $I\subseteq R$ is just the function $h_I(t)=\hbox{dim } I_t$ giving the vector space dimension of the homogeneous component $I_t$ of $I$ as a function of the degree $t$. The Hilbert function of a fat point subscheme $Z$ will be the function $h_Z(t)=\hbox{dim} (R/I)_t$ giving the vector space dimension of the homogeneous components of the quotient ring $R/I$ as a function of degree. Note that $h_{I(Z)}(t)+h_Z(t)=\binom{t+2}{2}$. (We recall in Section \ref{resols} the notions of the minimal free resolution of $I(Z)$ and its graded Betti numbers.)
Every smooth projective surface $X$ with a birational morphism to $\hbox{${\bf P}$}^2$ arises as a blow up of $n$ essentially distinct points, where $n$ is uniquely determined by $X$, since $n+1$ is the rank of $\hbox{Cl}(X)$ as a free abelian group. Since here we are interested in the case $n=6$, we will always hereafter assume that $n=6$. We will also mainly be interested in those $X$ for which the anticanonical class is nef. The anticanonical class has an intrinsic definition, but in terms of an exceptional configuration it is always $3L-E_1-\cdots-E_n$. A divisor (or divisor class) $F$ being \emph{nef\/} means that $F\cdot D\ge 0$ whenever $D$ is the class of an effective divisor (with \emph{effective\/} meaning that $D$ is a nonnegative integer linear combination of reduced irreducible curves).
We now recall the connection of normal cubic surfaces with blow ups $X$ of $\hbox{${\bf P}$}^2$ at 6 essentially distinct points such that $-K_X$ is nef. If $-K_X$ is nef, by Lemma \ref{NEGisfinite} the linear system
$|-K_X|$ has no base points so it defines a morphism $\phi_{|-K_X|}:X\to\hbox{${\bf P}$}^3$, whose image is a cubic surface. By Proposition 3.2 of \cite{refBirMor}, the image of $\phi_{|-K_X|}$ is normal, obtained by contracting to a point every prime divisor orthogonal to $-K_X$ (i.e., every smooth rational curve of self-intersection $-2$). In fact, the images of the $(-2)$-curves are rational double points, and the inverse image of each singular point is a minimal resolution of the singularity. It is not hard to check that the subgroup $K_X^\perp\subsetneq\hbox{Cl}(X)$ of all divisor classes orthogonal to $K_X$ is negative definite. Thus Theorem 2.7 and Figure 2.8, both of \cite{refA}, apply; i.e., the intersection graph of a fiber over a singular point is a Dynkin diagram of type $A_i$, $D_i$ or $E_i$. The combinations of Dynkin diagrams that occur for the singularities on a single surface are well known. A determination in characteristic 0 is given in \cite{refBW}. We recover that result for any characteristic; see Table \ref{configtable}.
To state the next result, let $\hbox{NEG}(X)$ denote the set of classes of prime divisors of negative self-intersection on a surface $X$ obtained by blowing up 6 essentially distinct points of $\hbox{${\bf P}$}^2$. Let $\C B=\{E_i: i>0\}$ ($\C B$ here is for \emph{blow up} of a point), $\C V=\{E_i-E_{i_1}-\cdots-E_{i_r}: r\ge 1, 0<i<i_1<\cdots<i_r\le 6\}$ ($\C V$ here is for \emph{vertical}), $\C L=\{L-E_{i_1}-\cdots-E_{i_r}: r\ge 2, 0<i_1<\cdots<i_r\le 6\}$ ($\C L$ here is for points on a \emph{line}), and $\C Q=\{2L-E_{i_1}-\cdots-E_{i_r}: r\ge 5, 0<i_1<\cdots<i_r\le 6\}$ ($\C Q$ here is for points on a conic, defined by a \emph{quadratic} equation). Also, let $\C B'=\C B$, $\C V'=\{E_i-E_j: 0<i<j\le 6\}$, $\C L'=\{L-E_i-E_j: 0<i<j\le 6\}\cup\{L-E_i-E_j-E_k: 0<i<j<k\le 6\}$, and $\C Q'=\C Q$, and let $\C V''=\C V'$, $\C L''=\{L-E_i-E_j-E_k: 0<i<j<k\le 6\}$, and $\C Q''=\{2L-E_1-\cdots-E_6\}$.
\begin{lem}\label{NEGisfinite} Let $X$ be obtained by blowing up 6 essentially distinct points of $\hbox{${\bf P}$}^2$. Then the following hold: \begin{itemize} \item[(a)] $\hbox{NEG}(X)\subseteq \C B\cup\C V\cup\C L\cup\C Q$, and every class in $\hbox{NEG}(X)$ is the class of a smooth rational curve; \item[(b)] if moreover $-K_X$ is nef, then $\hbox{NEG}(X)\subseteq \C B'\cup\C V'\cup\C L'\cup\C Q'$; \item[(c)] for any nef $F\in\hbox{Cl}(X)$, $F$ is effective
(hence $h^2(X, F)=0$ by duality), $|F|$ is base point free, $h^0(X, F) = (F^2-K_X\cdot F)/2 + 1$ and $h^1(X, F)=0$; \item[(d)] $\hbox{NEG}(X)$ generates the subsemigroup $\hbox{EFF}(X)\subsetneq\hbox{Cl}(X)$ of classes of effective divisors; and \item[(e)] any class $F$ is nef if and only if $F\cdot C\ge0$ for all $C\in \hbox{NEG}(X)$. \end{itemize} \end{lem}
\begin{proof} This result is well known. A proof of parts (a), (c), (d) and (e) when the points are assumed to be distinct is given in detail in \cite{refGH}. The same proof carries over with only minor changes here. Part (b) follows from (a) just by taking into account that each class $C$ in $\hbox{NEG}(X)$ must satisfy $-K_X\cdot C\ge0$. \end{proof}
\begin{rem}\label{minustwoRem} In the same way that it is easier to specify a combinatorial geometry of points in the plane by specifying which sets of three or more points are collinear (suppressing mention of all of the pairs of points which define a line going through no other point), it is often easier to work with the set $\hbox{neg}(X)=\{C\in \hbox{NEG}(X) : C^2 < -1\}$ than with $\hbox{NEG}(X)$. As shown in Remark 2.2 of \cite{refGH}, $\hbox{neg}(X)$ determines $\hbox{NEG}(X)$. In fact, we have: $$\hbox{NEG}(X)=\hbox{neg}(X)\cup
\{C\in\C B\cup\C L\cup\C Q\hbox{ $|$ }C^2=-1, C\cdot D\ge0\ \hbox{ for all }\ D\in \hbox{neg}(X)\}.$$ If $-K_X$ is nef, note that $\hbox{neg}(X)\subseteq \C V''\cup\C L''\cup\C Q''$. \end{rem}
\section{Configuration Types}\label{conftypes}
In this section we determine the configuration types of 6 essentially distinct points of $\hbox{${\bf P}$}^2$, under the restriction that $-K_X$ is nef. I.e., we find all \emph{pairwise nonnegative\/} subsets of $\C B'\cup\C V'\cup\C L'\cup\C Q'$ (a pairwise nonnegative subset being a subset such that whenever $C$ and $D$ are distinct elements of the subset, we have $C\cdot D\ge 0$). With Remark \ref{minustwoRem} in mind, we actually only do this for subsets of $\C V''\cup\C L''\cup\C Q''$. Also, we do this only up to permutations of the classes $E_1,\ldots,E_6$. Thus we find the unordered configuration types, hence only one representative of each orbit under the action of the group of permutations of $E_1,\ldots,E_6$. (Note for example that $\{E_1-E_3, E_2-E_4\}$ and $\{E_1-E_2, E_3-E_4\}$ are the same up to permutations of the $E_i$.)
We also show that each configuration type actually occurs over every algebraically closed field (regardless of the characteristic). Both for this latter question of representability and for distinguishing when different pairwise nonnegative subsets $T$ give different configuration types, it is helpful to compute the torsion groups $\hbox{Tors}_T$ for the quotients $\hbox{Cl}(X)/\langle T\rangle$ of the divisor class group by the subgroup generated by the elements of $T$ (or equivalently, the torsion subgroup of $K_X^\perp/\langle T\rangle$). So we include this information in Table \ref{configtable}, whenever $\hbox{Tors}_T\ne0$.
We can associate a graph (whose connected components are Dynkin diagrams \cite{refHandbook}) to each configuration type. If $T$ is a pairwise nonnegative subset of $\C V''\cup\C L''\cup\C Q''$, we have the graph $G_T$, whose vertices are the elements of $T$ and we have $C\cdot D$ edges between each distinct pair of vertices $C,D\in T$. It turns out that there is at most one edge between any two vertices, and, in terms of the standard notation for Dynkin diagrams, the connected components of each $G_T$ are always among the following types: $A_i$, $1\le i\le5$; $D_4$; $D_5$ and $E_6$. (If the graph $G_T$ for a subset $T$ has more than one connected component, say an $A_1$ and two of type $A_2$, we write this as $A_12A_2$.) Different configuration types can have the same graph, but the torsion subgroup for each configuration type turns out to be determined by the graph. Since different configuration types can have the same graph, the Dynkin diagram (such as $A_12A_2$) is not by itself enough to uniquely identify a configuration type, so we distinguish different configuration types with the same graph by appending a lower case letter (for example, $A_12A_2a$ or $A_12A_2b$) when there is more than one configuration type with a given graph.
The 90 different configuration types (i.e., the classification, up to permutations, of the pairwise nonnegative subsets of $\C V''\cup\C L''\cup\C Q''$) are shown in Table \ref{configtable}. For each configuration type $T$ we give the corresponding graph $G_T$ (using Dynkin notation), we give the set $T$ itself, and, when not 0, we give $\hbox{Tors}_T$ (which is always either 0, ${\bf Z}/2{\bf Z}$ or ${\bf Z}/3{\bf Z}$; we denote the latter two by ${\bf Z}_2$ and ${\bf Z}_3$ in Table \ref{configtable}). We give the set $T$ by listing its elements, following the approach used in \cite{refBCH}. We use letters A through F to denote the points $p_1$ through $p_6$, and numbers to indicate the degree of the curve. For example, the set $T$ for $3A_1d$ is given as \hbox{0: AB, CD; 2: ABCDEF}. Thus $T$ consists of the classes $E_1-E_2$, $E_3-E_4$ and $2L-E_1-\cdots-E_6$.
We obtained the table by brute force as follows. Start by finding all single element configuration types, which is easy. These are just the single element subsets of $\C V''\cup\C L''\cup\C Q''$. Pick a representative for each orbit under the permutation action. We get three singleton sets $T$, corresponding to items 2, 3 and 4 in Table \ref{configtable}. Add to each singleton configuration type $T$ each element of $\C V''\cup\C L''\cup\C Q''$ which meets every class already in $T$ nonnegatively, and again pick a representative set from each orbit. Continue this way for six cycles. (Six is enough since, as shown in the proof of Proposition \ref{rprsntblty}, the elements of each $T$ are linearly independent, and so $T$ can have at most 6 elements.)
\begin{prop}\label{rprsntblty} Over every algebraically closed field $k$, each configuration type occurs as $\hbox{neg}(X)$ for some surface $X$ obtained by blowing up 6 essentially distinct points of $\hbox{${\bf P}$}^2_k$. \end{prop}
\begin{proof} Let $T$ be the set of classes of a configuration type, and consider the group $K^\perp/\langle T\rangle$. Since $\C V''\cup\C L''\cup\C Q''$ is a finite set and since from Table \ref{configtable} we see that the torsion subgroup of $K^\perp/\langle T\rangle$ is either trivial or has prime order, we can pick a squarefree positive integer $l$ and a surjective homomorphism $\phi: K^\perp/\langle T\rangle\to {\bf Z}/l{\bf Z}$ such that no element $C\in\C V''\cup\C L''\cup\C Q''$ not already in $\langle T\rangle$ maps to 0 in ${\bf Z}/l{\bf Z}$.
Now let $C$ be a non-supersingular smooth plane cubic curve. Since $C$ is not supersingular, $\hbox{Pic}^0(C)$ has a subgroup isomorphic to ${\bf Z}/l{\bf Z}$; we identify ${\bf Z}/l{\bf Z}$ with this subgroup of $\hbox{Pic}^0(C)$. Thus there is a homomorphism $\Phi: \hbox{Cl}(X)\to \hbox{Pic}(C)$ such that the image $\Phi(K_X^\perp)$ is exactly ${\bf Z}/l{\bf Z}$. Pick any point on $C$ to be $p_1$. Then pick $p_i$ to be the image of $E_i-E_1$ in $\hbox{Pic}^0(C)$.
Under the usual identification of $\hbox{Pic}^0(C)$ with $C$ itself, this gives us six points $p_1,\ldots, p_6$. (It may be that some of the points are formally the same. For example, if $E_1-E_2\in T$, then $p_1=p_2$. This just means that $p_2$ is the point on the proper transform $C'$ of $C$ on $X_1$ infinitely near $p_1\in X_0$. Since restricting the mapping $\pi_1:X_1\to\hbox{${\bf P}$}^2$ to $C'$ gives an isomorphism of $C'$ to $C$, there is a natural identification of $C'$ with $C$. Under this identification we can indeed regard $p_1$ and $p_2$ as being the same point of $C$, even though properly speaking $p_1\in\hbox{${\bf P}$}^2$ and $p_2\in X_1$.)
By construction, the surface $X$ obtained by blowing up the points $p_1,\ldots, p_6$ has the property that an element $D\in\C V''\cup\C L''\cup\C Q''$ is in the kernel of $\Phi$ if and only if $D\in \langle T\rangle$. By \cite{refTAMS}, an element $D$ of $\C V''\cup\C L''\cup\C Q''$ is the class of an effective divisor if and only if $D\in\hbox{ker}(\Phi)$. Thus $D\in\C V''\cup\C L''\cup\C Q''$ is effective if and only if $D\in \langle T\rangle$.
In particular, the elements of $T$ are effective and $\hbox{neg}(X)\subseteq \langle T\rangle$. Let $D\in \hbox{neg}(X)$. By Lemma \ref{NEGisfinite}(b), $D^2=-2$. Now write $D$ as an integer linear combination of elements of $T$. Thus we can write $D=D_1-D_2$, where $D_1$ is a sum of elements of $T$ with positive coefficients and $D_2$ is either 0 or a sum of different elements of $T$ with positive coefficients. Note that $D_1$ is not zero, since otherwise $D$ is either 0 or antieffective, neither of which can hold since $D$ is the class of a prime divisor. We claim however that $D_2=0$. If not, then, since $K_X^\perp$ is negative definite and even, we have $D_1^2\le -2$ and $D_2^2\le -2$. Since $D_1$ and $D_2$ involve different elements of $T$ (which therefore meet nonnegatively), we also see $D_1\cdot D_2\ge 0$. Thus $D^2=D_1^2-2D_1\cdot D_2+D_2^2\le-4$, contradicting $D^2=-2$. (A similar argument shows that the elements of $T$ are linearly independent. If not, we can find an expression $D_1-D_2=0$ for some nonnegative linear integer combinations $D_i$ of elements of disjoint subsets $T_i\subseteq T$. By pairwise nonnegativity, we have $D_1\cdot D_2\ge 0$, but $-K_X^\perp$ is negative definite, so $0\ge D_i^2=D_1\cdot D_2$, hence $D_i^2=0$, so $D_i=0$. But $T\subseteq\C V''\cup\C L''\cup\C Q''$, and every element of $\C V''\cup\C L''\cup\C Q''$ meets $A=14L-6E_1-5E_2-4E_3-3E_4-2E_5-E_6$ positively, so if $D_i$ is not a linear integer combination of elements of $T_i$ with 0 coefficients, then we have $0<A\cdot D_i=A\cdot 0=0$, which is impossible. Thus each $D_i$ is the trivial linear combination, hence $T$ is linearly independent.)
Thus every element of $\hbox{neg}(X)$ is a nonnegative sum of elements of $T$, each of which is effective. But the elements of $\hbox{neg}(X)$ are prime divisors of negative self-intersection, hence each can be written as a nonnegative sum of classes of effective divisors only one way; i.e., every element of $\hbox{neg}(X)$ is an element of $T$.
By Lemma \ref{NEGisfinite}(d), every element of $T$ is a nonnegative integer linear combination of elements of $\hbox{NEG}(X)$. But $T\subseteq K_X^\perp$, so in fact every element of $T$ is a nonnegative integer linear combination of elements of $\hbox{neg}(X)$. Since $\hbox{neg}(X)\subseteq T$, and since $T$ is linearly independent, this is possible only if $\hbox{neg}(X)=T$. \end{proof}
\ \vskip.1in
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\hbox to 3.2in{\scriptsize\hbox to 0.9in{\scriptsize\number\papercnt.$\,\,$\global\advance\papercnt by1 $A_5c$ \hfil} 0: AB, BC, CD; 1: ABC, AEF \hfil}
\hbox to 3.2in{\scriptsize\hbox to 0.9in{\scriptsize\number\papercnt.$\,\,$\global\advance\papercnt by1 $D_5a$ \hfil} 0: BC, CD, DE, EF; 1: ABC \hfil}
\hbox to 3.2in{\scriptsize\hbox to 0.9in{\scriptsize\number\papercnt.$\,\,$\global\advance\papercnt by1 $D_5b$ \hfil} 0: AB, CD, DE, EF; 1: ACD \hfil}
\hbox to 3.2in{\scriptsize\hbox to 0.9in{\scriptsize\number\papercnt.$\,\,$\global\advance\papercnt by1 $D_5c$ \hfil} 0: AB, BC, CD, DE; 1: ABC \hfil}
\hbox to 3.2in{\scriptsize\hbox to 0.9in{\scriptsize\number\papercnt.$\,\,$\global\advance\papercnt by1 $3A_2a$ \hfil} 0: AB, BC, DE, EF; 1: ABC, DEF \hfil ${\bf Z}_3$}
\hbox to 3.2in{\scriptsize\hbox to 0.9in{\scriptsize\number\papercnt.$\,\,$\global\advance\papercnt by1 $3A_2b$ \hfil} 0: AB, CD, EF; 1: ABC, AEF, CDE \hfil ${\bf Z}_3$}
\hbox to 3.2in{\scriptsize\hbox to 0.9in{\scriptsize\number\papercnt.$\,\,$\global\advance\papercnt by1 $A_1A_5a$ \hfil} 0: AB, BC, DE, EF; 1: ABC, ADE \hfil ${\bf Z}_2$}
\hbox to 3.2in{\scriptsize\hbox to 0.9in{\scriptsize\number\papercnt.$\,\,$\global\advance\papercnt by1 $A_1A_5b$ \hfil} 0: AB, BC, CF, DE; 1: ABC, ADE \hfil ${\bf Z}_2$}
\hbox to 3.2in{\scriptsize\hbox to 0.9in{\scriptsize\number\papercnt.$\,\,$\global\advance\papercnt by1 $A_1A_5c$ \hfil} 0: AB, BC, CD, DE, EF; 2: ABCDEF \hfil ${\bf Z}_2$}
\hbox to 3.2in{\scriptsize\hbox to 0.9in{\scriptsize\number\papercnt.$\,\,$\global\advance\papercnt by1 $E_6$ \hfil} 0: AB, BC, CD, DE, EF; 1: ABC \hfil} \vfil}} \vskip.1in \caption[\hskip.1in Configuration Types]{Configuration Types}\label{configtable} \end{table} }
\vfil\eject
As a check on our list of configuration types as given in Table \ref{configtable}, we have the following well known result, Theorem \ref{dynkthm}. (See \cite{refBW} for a version of the result in characteristic 0, or see Theorem IV.1 of the arXiv version math.AG/0506611 of the paper \cite{refGH} for a proof in general. The proof is to study the morphisms $X\to\hbox{${\bf P}$}^2$ obtained by mapping $X$ to $\hbox{${\bf P}$}^3$
using the linear system $|-K_X|$, and then mapping the image $\bar X $ to $\hbox{${\bf P}$}^2$ by projecting from a singular point.) Thus we get the same Dynkin diagrams from Theorem \ref{dynkthm} as we found by a brute force determination of configuration types. Moreover, one can (as we did in fact do) find all exceptional configurations for each of the 20 graphs listed in Theorem \ref{dynkthm}, and for each exceptional configuration one can write down the corresponding (representable) configuration type. Since by Proposition \ref{rprsntblty} every type is representable over every algebraically closed field, it follows that the types obtained this way should be (and in fact are) the same types we found by brute force.
\begin{thm}\label{dynkthm} Let $X$ be a blow up of $\hbox{${\bf P}$}^2$ at 6 essentially distinct points of $\hbox{${\bf P}$}^2$, such that $-K_X$ is nef. Assume that $X$ has at least one $(-2)$-curve. Then the intersection graph of the set of $(-2)$-curves is one of the following 20 graphs: $A_1$, $2A_1$, $A_2$, $3A_1$, $A_1A_2$, $A_3$, $4A_1$, $2A_1A_2$, $A_1A_3$, $2A_2$, $A_4$, $D_4$, $A_12A_2$, $2A_1A_3$, $A_1A_4$, $A_5$, $D_5$, $3A_2$, $A_1A_5$, and $E_6$. Each of these graphs occurs as the graph of the set of $(-2)$-curves on some $X$, and in a unique way (unique in the sense that if the same graph occurs on two surfaces $X$ and $X'$, then there are exceptional configurations $L,E_1,\ldots,E_6$ on $X$ and $L',E_1',\ldots,E_6'$ on $X'$, such that a class $a_0L+\sum_ia_iE_i$ is the class of a $(-2)$-curve on $X$ if and only if $a_0L'+\sum_ia_iE_i'$ is the class of a $(-2)$-curve on $X'$). \end{thm}
\section{Resolutions}\label{resols}
Let $p_1,\ldots,p_6$ be essentially distinct points of $\hbox{${\bf P}$}^2$. Let $Z=m_1p_1+\cdots+m_6p_6$ be a fat point subscheme of $\hbox{${\bf P}$}^2$, and let $F(Z,i)=iL-m_1E_1-\cdots-m_6E_6$ on the surface $X$ obtained by blowing up the points $p_i$. As explained in Section \ref{bkgd}, the ideal $I(Z)$ is obtained as follows. Let $\pi:X\to\hbox{${\bf P}$}^2$ be the morphism to $\hbox{${\bf P}$}^2$ given by blowing up the points $p_i$, and let $L,E_1,\ldots, E_6$ be the corresponding exceptional configuration. Let $F=-(m_1E_1+\cdots+m_6E_6)$. Then $\C I_Z=\pi_*(\C O_X(-m_1E_1-\cdots-m_6E_6))$ is a sheaf of ideals on $\hbox{${\bf P}$}^2$, and $I(Z)=\oplus_{i\ge0}H^0(\hbox{${\bf P}$}^2,\C I_Z\otimes\C O_{\hbox{${\bf P}$}^2}(i))$. Also, we may as well assume that the coefficients $m_i$ satisfy the proximity inequalities. If they do not, there is another choice of coefficients $m_i'$ which do satisfy them, giving a 0-cycle $Z'$ for which $I(Z)=I(Z')$. (The proximity inequalities are precisely the conditions on the $m_i$ given by the inequalities $F\cdot C\ge 0$ for each divisor class $C$ which is the class of a component of the curves whose classes are $E_1,\ldots,E_6$. In the case that the points $p_i$ are distinct, the proximity inequalities are merely that $m_i\ge0$ for all $i$. If $p_2$ is infinitely near $p_1$, then we would have the additional requirement that $m_1\ge m_2$. This corresponds to the fact that a form cannot vanish at $p_2$ without already vanishing at $p_1$. If the $m_i$ do not satisfy the proximity inequalities, then $F(Z,i)$ will never be nef: no matter how large $i$ is, some component $C$ of some $E_j$, $j>0$, will have $F(Z,i)\cdot C<0$. Thus $C$ will be a fixed component of
$|F(Z,i)|$ for all $i$. By subtracting off such fixed components one obtains a class $iL-(m_1'E_1+\cdots+m_6'E_6)$, which also gives a 0-cycle $Z'=m_1'p_1+\cdots+m_6'p_6$ satisfying the proximity inequalities and which gives the same ideal $I(Z)=I(Z')$. See \cite{refAppendix} for more details.)
The minimal free resolution of $I(Z)$ is an exact sequence of the form $$0\to F_1\to F_0\to I(Z)\to 0$$ where each $F_i$ is a free graded $R$-module, with respect to the usual grading of $R$ by degree, and all nonzero entries of the matrix defining the homomorphism $F_1\to F_0$ are homogeneous polynomials in $R$ of degree at least 1. Since $F_0$ and $F_1$ are free graded $R$-modules, we know that there are integers $g_i$ and $s_j$ such that $F_0\cong\oplus_i R[-i]^{g_i}$ and $F_1\cong\oplus_j R[-j]^{s_j}$. These integers are the graded Betti numbers of $I(Z)$. To determine the modules $F_1$ and $F_0$ up to graded isomorphism (or, equivalently, to determine the graded Betti numbers of the minimal free resolution of $I(Z)$), it is enough, as for distinct points (as explained in \cite{refGH}), to determine $h^0(X,F(Z,i))$ and the ranks for all $i\ge0$ of the multiplication maps $\mu_{Z,i} : I(Z)_i\otimes R_1\to I(Z)_{i+1}$ for each $i\ge0$, where, given a graded $R$-module $M$, $M_t$ denotes the graded component of degree $t$. Since (see \cite{refGH}) the rank of $\mu_{Z,i}$ is the same as the rank of $\mu_{F(Z,i)}: H^0(X,F(Z,i))\otimes H^0(X,L)\to H^0(X,L+F(Z,i))$, it is enough to determine the rank of $\mu_{F(Z,i)}$.
As explained in \cite{refGH}, we can compute $h^0(X,F(Z,i))$ if we know $\hbox{NEG}(X)$ (or therefore even just $\hbox{neg}(X)$), and we can compute the rank of $\mu_{F(Z,i)}$ if we can compute the rank of $\mu_F$ whenever $F$ is nef (which the main result of this section, Theorem \ref{essdistpntsThm}, says we can do).
The method we use to prove Theorem \ref{essdistpntsThm} is precisely the method used in \cite{refGH}. It involves the quantities $q(F)=h^{0}({X},{F-E_1})$ and $l(F)=h^{0}({X},{F-(L-E_1})),$ and bounds on the dimension of the cokernel of $\mu_F$, defined in terms of quantities $q^*(F)=h^{1}({X},{F-E_1})$ and $l^*(F)=h^{1}({X},{F-(L-E_1)})$, introduced in \cite{refIGC} and \cite{refFHH}. A version of Lemma \ref{IGClem} for distinct points is given in \cite{refIGC} and \cite{refFHH}, but with only trivial changes the proof for essentially distinct points is the same.
\begin{lem}\label{IGClem} Let $X$ be obtained by blowing up essentially distinct points $p_i\in\hbox{${\bf P}$}^2$, and let $F$ be the class of an effective divisor on $X$ with $h^1(X,F)=0$. Then $l(F)\le\hbox{dim ker $\mu_F$}\le q(F)+l(F)$ and $\hbox{dim cok $\mu_F$}\le q^*(F)+l^*(F)$. \end{lem}
\eatit{\Prf Let $x$ be a general linear form on $\hbox{${\bf P}$}^2$, pulled back to $X$. Let $y$ and $z$ be general forms vanishing at $p_1$, pulled back to $X$. Let $V$ be the vector space span of $y$ and $z$, so $zH^0(X, F)+yH^0(X, F)$ is the image of $H^0(X, F)\otimes V$ under $\mu_F$. Denoting $h^0(X, F)$ by $h$, this image has dimension $2h-l(F)$, since $zH^0(X, F)\cap yH^0(X, F)=zyH^0(X, F-(L-E_1))$, where we regard the intersection as taking place in $H^0(X, ((F\cdot L)+1)L)$. Note that all elements of $zH^0(X, F)+yH^0(X, F)$ correspond to forms on $\hbox{${\bf P}$}^2$ that vanish at $p_1$ to order at least $F\cdot E_1+1$. Thus $(zH^0(X, F)+yH^0(X, F))\cap xH^0(X, F)$ lies in the image of $xH^0(X, F-E_1)$ under the natural inclusion $xH^0(X, F-E_1)\subseteq xH^0(X, F)$, so $3h-l(F)\ge\hbox{dim Im $\mu_F$}\ge (2h-l(F)) + (h-q(F))$ hence $l(F)\le \hbox{dim ker $\mu_F$}\le l(F)+q(F)$.
For the bound on the cokernel, note that $L$ is numerically effective. Thus, $F\cdot L\ge 0$, since $F$ is effective. Let $L$ be a general element of $|L|$. >From $h^1({X},{F})=0$ and $$0\to \C O_{X}({F})\to \C O_{X}({F+L})\to \C O_{L}({F+L})\to 0,$$ we see that $h^{1}({X},{F+L})$ also vanishes and we compute $h^{0}({X},{F+L})-3h^{0}({X},{F})=2+F\cdot L-2h^{0}({X},{F})$. Tensoring the displayed exact sequence by $\C O_X(-(L-E_1))$ (for $l$) or $\C O_X(-E_1)$ (for $q$), taking cohomology and using Riemann-Roch gives $l^*(F)-l(F)=F\cdot (L-E_1) + 1-h^{0}({X},{F})$ and $q^*(F)-q(F)=F\cdot E_1 + 1-h^{0}({X},{F})$, so $(l^*(F)-l(F))+(q^*(F)-q(F))=h^{0}({X},{F+L})-3h^{0}({X},{F})$. Therefore, $\hbox{dim cok $\mu_F$}=\hbox{dim ker $\mu_F$}+ h^{0}({X},{F+L})-3h^{0}({X},{F})\le l(F)+q(F)+ h^{0}({X},{F+L})-3h^{0}({X},{F}) = l^*(F)+q^*(F)$, as claimed. \qed}
\begin{rem}\label{IGCrem} The quantities $q(F)$ and $l(F)$ are defined in terms of $E_1$ and $L-E_1$, but in fact $E_j$, $j>0$, can often be used in place of $j=1$. This is always true if the points $p_i$ are distinct, since one can reindex the points. Likewise, if the points are only essentially distinct, any $j$ can be used so long as $p_j$ is a point on $\hbox{${\bf P}$}^2$, and not only infinitely near a point of $\hbox{${\bf P}$}^2$. \end{rem}
\begin{thm}\label{essdistpntsThm} Let $X$ be obtained by blowing up 6 essentially distinct points of $\hbox{${\bf P}$}^2$. Let $L,E_1,\ldots,E_6$ be the corresponding exceptional configuration. Assume that $-K_X$ is nef, and let $F$ be a nef divisor. Then $\mu_F$ has maximal rank. \end{thm}
\begin{proof} The case of general points (i.e., that $\hbox{neg}(X)$ is empty) is done in \cite{refFi} (but it can be recovered by the methods we use here). This handles one of the 90 cases of Table \ref{configtable}. Also, for 28 of the cases of Table \ref{configtable}, a conic goes through the six points (i.e., $h^0(X, 2L-E_1-\cdots-2E_6)>0$); these cases are configuration types 4, 8, 12, 16, 25, 26, 30, 33, 37, 42, 47, 48, 50, 53, 58, 61, 64, 66, 70, 72, 76, 78, 81, 83, 85, 88, 89 and 90. The result holds for these cases by Theorem 3.1.2 of \cite{refFreeRes} (also see Lemma 2.11 of \cite{refGH}).
Four of the remaining 61 cases correspond to distinct points, and were handled in \cite{refGH}. These cases are 3, 9, 17 and 34. The remaining cases are handled by the same method as these four. The basic idea is this. If $F$ is a nef divisor such that $l(F)>0$, $q(F)>0$, and $l^*(F)=0=q^*(F)$, then not only is it true that $\mu_F$ is surjective (by Lemma \ref{IGClem}), but $l(F+G)>0$ and $q(F+G)>0$ by Lemma \ref{NEGisfinite}, and $l^*(F+G)=0=q^*(F+G)$ holds for all nef $G$ (by the proof of Corollary 2.8 of \cite{refGH}), hence $\mu_{F+G}$ is surjective for all nef $G$.
Using Lemma 2.5 of \cite{refGH} one can easily give an explicit list of generators of the nef cone for each configuration type. In the best of all worlds, what would happen is that we would find that $l(F)>0$, $q(F)>0$, $l^*(F)=0=q^*(F)$, for every $F$ in our set of generators, and the result would be proved. But our world is not the best of all imaginable worlds, so some additional work is needed. In \cite{refGH} this is done, applying Corollary 2.8, Lemma 2.9 and Lemma 2.10 of \cite{refGH}. These are all stated for 6 distinct points or $\hbox{${\bf P}$}^2$, but it is easy to check that the proofs continue to hold for 6 essentially distinct points if $-K_X$ is nef.
We now describe what this additional work is. Let $\Gamma(X)$ be a set of generators of the nef cone for $X$. (For practical purposes of actually carrying out the calculations, it is best to choose a minimal set of generators.) Let $\Gamma_i(X)$ be the set of all sums with exactly $i$ terms, where each term is an element (with coefficient 1) of $\Gamma(X)$. Let $S(X)$ be the set of all nef classes $F$ such that either $q(F)=0$, $l(F)=0$ or $l^*(F)+q^*(F)>0$. Then let $S_i(X)=S(X)\cap\Gamma_i(X)$; by Corollary 2.8 \cite{refGH}, we have $S_{i+1}(X)\subseteq S_i(X)+S_1(X)$. Typically the subsets $S_i(X)$ are nonempty. But for $i\ge 3$, it always turns out that Lemma 2.9 \cite{refGH} applies. This lemma involves a parameter $k$ which we can always take to be $k=2$. It also involves a particular choice of class $C_F\in S_1(X)$ for each $F\in S_i(X)$. The result is that $S_i(X)\subseteq\{F+(i-3)C_F: F\in S_3(X), C_F\in S_1(X)\}$.
First one verifies directly that maximal rank holds for $\mu_F$ for all $F\in S_i(X)$ for $i\le 3$, using Lemma \ref{IGClem}. An induction (applying Lemma 2.10 \cite{refGH}) then verifies maximal rank for the strings $F+(i-3)C_F$, and hence for all nef $F$. There is one case that must be handled ad hoc (as was done by \cite{refFi} and as we demonstrate below). If $F=5L-2(E_1+\cdots+E_6)$, then $C_F=F$, but $l(iF)>0$ for $i\ge3$ (so $\mu_{iF}$ is not injective by Lemma \ref{IGClem}) while $l^*(iF)>0$ for all $i$ (so the bounds in Lemma \ref{IGClem} never force surjectivity). We now treat one case in detail, as an example. The remaining cases are similar.
Consider configuration type 2, so $\hbox{neg}(X)=\{N\}$, where $N=E_1-E_2$. Then $S_1(X)$ has 58 elements, $S_2(X)$ has 140, and $S_3(X)$, $S_4(X)$ and $S_5(X)$ have 150. Moreover, $\mu_H$ has maximal rank (by a case by case application of Lemma \ref{IGClem}) for each element $H$ of $S_i(X)$, $1\le i\le 5$, except possibly $mH$ when $H=5L-2(E_1+\cdots+E_6)$ for $m>1$ (since $q(mH)+l(mH)>0$ and $q^*(mH)+l^*(mH)>0$ in these cases). To show $\mu_H$ is onto for $H=2(5L-2(E_1+\cdots+E_6))$, let $C=2L-E_1-\cdots-E_5$, and consider $F=H-C$. Then $\mu_F$ is onto (by Lemma \ref{IGClem}, since $q^*(F)+l^*(F)=0$) hence $\mu_H$ is onto (by Lemma 2.10 \cite{refGH}), and now $\mu_{H+iC}$ is onto for all $i\ge0$ (also by Lemma 2.10 \cite{refGH}, taking $F$ to be $mH$ and $C=5E_0-2(E_1+\cdots+E_6)$ for the induction in Lemma 2.10 \cite{refGH}). By brute force check, applying Lemma 2.9 \cite{refGH} (with $k=2$ and $j=2$) and Lemma 2.10 \cite{refGH}, it follows that $\mu_F$ has maximal rank for every $F$ in each $S_i(X)$. \end{proof}
\section{Examples}\label{exmpls}
Given only the configuration type and multiplicities $m_1,\ldots,m_6$ satisfying the proximity inequalities, Lemma \ref{NEGisfinite} and Theorem \ref{essdistpntsThm} allow us to determine the Hilbert function and graded Betti numbers for $I(Z)$ for any fat point subscheme $Z=m_1p_1+\cdots+m_6p_6$ supported at 6 essentially distinct points $p_i$ of $\hbox{${\bf P}$}^2$ which when blown up give a surface $X$ for which $-K_X$ is nef (i.e., give a surface isomorphic to the desingularization of a normal cubic surface).
The procedure for doing so is exactly the same as described in detail in \cite{refGH}. We briefly recall the procedure. Given $Z=m_1p_1+\cdots+m_6p_6$, to determine $h_{I(Z)}(t)$, compute $h^0(X, F(Z,t))$, where $F(Z,t)= tL-m_1E_1-\cdots-m_6E_6$. To do this, let $D=F(Z,t)$ and check $D\cdot C$ for all prime divisors $C$ with $C^2<0$. (Knowing the configuration type tells us the list of these divisors $C$.) Whenever $D\cdot C<0$, replace $D$ by $D-C$ and again check $D\cdot C$ with this new $D$ against all $C$. Eventually either $D\cdot L<0$ (in which case $h^0(X, F(Z,t))=h^0(X, D)=0$), or $D\cdot C\ge0$ for all $C$ (in which case, by Lemma \ref{NEGisfinite}, $D$ is nef and $h^0(X, F(Z,t))=h^0(X, D)=(D^2-K_X\cdot D)/2+1)$.
To determine the graded Betti numbers, note that it suffices to compute the Betti numbers $g_i$ for all $i$, since the exact sequence $0\to F_1\to F_0\to I(Z)\to 0$ allows one to determine $F_1$ up to graded isomorphism if one knows the graded Betti numbers for $F_0$ and also the Hilbert function for $I(Z)$. To determine $g_{t+1}$, note that $g_{t+1}=h^0(X, F(Z,t+1))$ if $h^0(X, F(Z,t))=0$. If $h^0(X, F(Z,t))>0$, obtain the nef divisor $D$ from $F(Z,t)$ as above. Then $g_{t+1}=(h^0(X, F(Z,t+1))-h^0(X,D+L))+\hbox{max}(0, h^0(X,D+L)-3h^0(X,D))$.
The procedure thus involves nothing more than taking dot products of integer vectors, and can easily be done by hand. An awk script which automates the steps is available at
\noindent \url{http://www.math.unl.edu/~bharbourne1/6ptsNef-K/Res6pointNEF-K}.
\noindent It can be run over the web at
\noindent \url{http://www.math.unl.edu/~bharbourne1/6ptsNef-K/6reswebsite.html}.
Using this script we determined all possible Hilbert functions and graded Betti numbers for fat points of the form $Z=p_1+\cdots+p_6$ and $2Z=2p_1+\cdots+2p_6$ for essentially distinct points $p_i$ such that $-K_X$ is nef on the resulting surface $X$. For 6 essentially distinct points with nef $-K_X$, this completely answers the questions raised in \cite{refGMS}. We show what happens in Table \ref{ResHilbtable}. (The table regards a Hilbert function $h=h_{R/I(mZ)}$ as the sequence $h(0), h(1), h(2), \ldots$. But any such $h$ reaches a maximum value at the regularity; i.e., for all $t$ greater than or equal to the regularity of $I(mZ)$, we have $h(t)=h(t+1)$. Thus Table \ref{ResHilbtable} gives $h$ only up to this maximum value.)
In Table \ref{ResHilbtable}, case 1 occurs for the following configuration types (as denoted in Table \ref{configtable}): 4, 8, 12, 16, 25, 26, 30, 33, 37, 42, 47, 48, 50, 53, 58, 61, 64, 66, 70, 72, 76, 78, 81, 83, 85, 88, 89, 90. For each of these types, only one Hilbert function occurs for $2Z=2p_1+\cdots+2p_6$, the one given as $1(a)$. These all have the same graded Betti numbers too.
Case 2 occurs for the remaining configuration types: 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 27, 28, 29, 31, 32, 34, 35, 36, 38, 39, 40, 41, 43, 44, 45, 46, 49, 51, 52, 54, 55, 56, 57, 59, 60, 62, 63, 65, 67, 68, 69, 71, 73, 74, 75, 77, 79, 80, 82, 84, 86, 87. For these, two different Hilbert functions occur for $2Z=2p_1+\cdots+2p_6$, given as 2(a) and 2(b). Case 2(a) occurs for types 34, 68 and 87, and these three all have the same graded Betti numbers. Case 2(b) occurs for types 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 27, 28, 29, 31, 32, 35, 36, 38, 39, 40, 41, 43, 44, 45, 46, 49, 51, 52, 54, 55, 56, 57, 59, 60, 62, 63, 65, 67, 69, 71, 73, 74, 75, 77, 79, 80, 82, 84, and 86. These all have the same Hilbert function, but three different possibilities occur for the graded Betti numbers, which we distinguish in the table by cases 2(b1), 2(b2) and 2(b3). Case 2(b1) occurs for types 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 18, 19, 20, 21, 22, 27, 28, 31, 35, 36, 38, 43, 44, 49, 51, 54, 57, 59, 62, 73, 74 and 79. Case 2(b2) occurs for types 9, 15, 23, 24, 29, 32, 39, 40, 46, 52, 55, 56, 60, 63, 67, 69, 71, 82 and 84, and case 2(b3) occurs for the remaining types 17, 41, 45, 65, 75, 77, 80 and 86.
\def\mytmprow#1#2#3#4{\hbox to\hsize{\hskip.2in\hbox to6.3in{\hbox to5.7in{\hbox to4.2in{\hbox to 1in{#1\hfil}#2\hfil}#3\hfil}#4\hfil}\hss}\par} \vskip\baselineskip \setcounter{table}{0}
\begin{table} \hbox to\hsize{\hrulefill} \mytmprow{{\bf Scheme}}{{\bf Resolution}}{{\bf Hilbert Function}}{} \hbox to\hsize{\hrulefill} \mytmprow{}{\hbox to1.3in{$F_1$\hfil} $F_0$}{$h_{R/I(mZ)}$, $m=1,2$}{}
\hbox to\hsize{\hrulefill} \mytmprow{\hbox to .8in{\hbox to.2in{1:\hfil}\hfil$Z$}}{\hbox to1.3in{$R[-5]$\hfil} $R[-3]\oplus R[-2]$}{1, 3, 5, 6}{} \mytmprow{\hbox to .8in{\hfil(a):\ \ $2Z$}}{\hbox to1.3in{$R[-8]\oplus R[-7]$\hfil} $R[-6]\oplus R[-5]\oplus R[-4]$}{1, 3, 6, 10, 14, 17, 18}{} \hbox to\hsize{\hrulefill} \mytmprow{\hbox to .8in{\hbox to.2in{2:\hfil}\hfil$Z$}}{\hbox to1.3in{$R[-4]^3$\hfil} $R[-3]^4$}{1, 3, 6}{} \mytmprow{\hbox to .8in{\hfil(a):\ \ $2Z$}}{\hbox to1.3in{$R[-7]^4$\hfil} $R[-6]^4\oplus R[-4]$}{1, 3, 6, 10, 14, 18}{} \mytmprow{\hbox to .8in{\hfil(b1):\ \ $2Z$}}{\hbox to1.3in{$R[-7]^3$\hfil} $R[-6]^1\oplus R[-5]^3$}{1, 3, 6, 10, 15, 18}{} \mytmprow{\hbox to .8in{\hfil(b2):\ \ $2Z$}}{\hbox to1.3in{$R[-7]^3\oplus R[-6]$\hfil} $R[-6]^2\oplus R[-5]^3$}{1, 3, 6, 10, 15, 18}{} \mytmprow{\hbox to .8in{\hfil(b3):\ \ $2Z$}}{\hbox to1.3in{$R[-7]^3\oplus R[-6]^2$\hfil} $R[-6]^3\oplus R[-5]^3$}{1, 3, 6, 10, 15, 18}{} \hbox to\hsize{\hrulefill} \vskip.05in \caption[\hskip.1in Resolutions and Hilbert Functions]{Resolutions and Hilbert Functions}\label{ResHilbtable} \end{table}
We close with one final example. The Hilbert functions that occur for $Z$ or $2Z$ for every choice of 6 essentially distinct points $Z=p_1+\cdots+p_6\subset\hbox{${\bf P}$}^2$ all already occur for distinct points. The first case of a Hilbert function that occurs for 6 essentially distinct points $mZ$ of multiplicity $m$ that does not occur for any 6 distinct points of multiplicity $m$ is for $m=3$, and in this case there is only one, this being the Hilbert function for the ideal $I(Z)$ of 6 essentially distinct points of multiplicity $3$ with configuration type 86, which is $h_{I(Z)}(t)=0$ for $t<6$, $h_{I(Z)}(6)=1$, $h_{I(Z)}(7)=3$, and, for $t>7$, $h_{I(Z)}(t)=\binom{t+2}{2}-36$. Applying the results of \cite{refGH}, we see this Hilbert function does not occur for any configuration of 6 distinct points. The graded Betti numbers for $I(Z)$ are such that $F_0\cong R[-9]^3\oplus R[-8]^3\oplus R[-6]$ and $F_1\cong R[-10]^3\oplus R[-9]^3$.
\end{document} | arXiv |
\begin{definition}[Definition:Dirac Delta Distribution]
Let $a \in \R^d$ be a real vector.
Let $\phi \in \map \DD {\R^d}$ be a test function.
Let $\delta_a \in \map {\DD'} {\R^d}$ be a distribution.
Suppose $\delta_a$ is such that:
:$\map {\delta_a} \phi = \map \phi a$
Then $\delta_a$ is known as the '''Dirac delta distribution'''.
{{Research|For $d \ge 2$ this works in Euclidean space with Cartesian coordinates. Change of coordinates and integration measure may affect this somewhat}}
\end{definition} | ProofWiki |
The vertical drop of a roller coaster is the largest difference in height between any high point and the next low point. The vertical drops of five roller coasters at Mandelbrot Amusement Park are shown in the table. \begin{tabular}{|l|c|} \hline
The Parabola & 165 feet \\ \hline
The G Force & 119 feet \\ \hline
The Mean Streak & 138 feet \\ \hline
The Tower of Power & 300 feet \\ \hline
The Maximum Ride & 198 feet \\ \hline
\end{tabular} What is the positive difference between the mean and the median of these values?
First, we must find the mean and median of the values. In order to find the mean, we sum all the values and divide the result by the number of values: \begin{align*} \frac{165+119+138+300+198}{5} &= 184. \end{align*} in order to find the median, we must first list the values in order from least to greatest: \[ 119, 138, 165, 198, 300. \] There are $5$ values, so the median is the middle value, which here is $165.$
So, the positive difference between the mean and the median is $184-165=\boxed{19}.$ | Math Dataset |
\begin{document}
\draft
\tighten
\title{Parametrization and distillability of three-qubit entanglement}
\author{Todd A. Brun\thanks{Current address: Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540. Email: [email protected]} and Oliver Cohen\thanks{Email: [email protected]} \\ Physics Department, Carnegie Mellon University, \\ Pittsburgh, PA 15213}
\maketitle
\begin{abstract} There is an ongoing effort to quantify entanglement of quantum pure states for systems with more than two subsystems. We consider three approaches to this problem for three-qubit states: choosing a basis which puts the state into a standard form, enumerating ``local invariants,'' and using operational quantities such as the number of maximally entangled states which can be distilled. In this paper we evaluate a particular standard form, the {\it Schmidt form}, which is a generalization of the Schmidt decomposition for bipartite pure states. We show how the coefficients in this case can be parametrized in terms of five physically meaningful local invariants; we use this form to prove the efficacy of a particular distillation technique for GHZ triplets; and we relate the yield of GHZs to classes of states with unusual entanglement properties, showing that these states represent extremes of distillability as functions of two local invariants. \end{abstract}
\pacs{03.67.-a 03.65.-w 03.67.Dd}
\section{Introduction}
The importance of {\it quantum entanglement}, both as a resource for quantum information processing and as a ubiquitous feature of quantum systems, has become increasingly apparent over the last few years \cite{Bennett93,Deutsch96,Bennett96a}. Recent developments in quantum information theory, in particular, have stimulated interest in the quantification and manipulation of entanglement.
For bipartite pure states an essentially complete theory of entanglement now exists \cite{Bennett96a,Nielsen99,Approximate99}, though the situation for mixed states is less definite \cite{Horodecki96}. All descriptions of bipartite pure state entanglement start with the {\it Schmidt decomposition}. It is possible to find orthonormal bases $\{\ket{i}_A\}$ and $\{\ket{i}_B\}$ for systems A and B such that we can write the joint state of the system in the form \begin{equation} \ket{\Psi_{AB}} = \sum_i \sqrt{p_i} \ket{i}_A \otimes \ket{i}_B, \ \ p_i > 0,\ \ \sum_i p_i = 1. \label{schmidt_decomp} \end{equation} These {\it Schmidt coefficients} $\{p_i\}$ are uniquely defined by the state $\ket{\Psi_{AB}}$, and are equal to the eigenvalues of the reduced density matrix $\rho_A$ (or equivalently, of $\rho_B$); the bases $\{\ket{i}_A\}$ and $\{\ket{i}_B\}$ are eigenbases of $\rho_A$ and $\rho_B$, respectively, so the local density matrices are diagonal in this choice of bases. This choice of bases also minimizes the number of terms needed to represent $\ket{\Psi_{AB}}$.
For tripartite or multipartite states, there is no equivalent to the Schmidt decomposition (\ref{schmidt_decomp}) \cite{Thapliyal99}). Three main approaches to parametrizing tripartite or multipartite entanglement have been followed so far. First, one may choose the local bases to put the joint state into a {\it standard form}. Often these standard forms are intended to generalize some aspect of the Schmidt decomposition in the bipartite case \cite{Linden98,Schlienz,Cohen,CohenBrun00,Sudbery00,Acin00,Higuchi00,CHS00}. Second, one may try to identify a complete set of {\it locally invariant quantities}, functions of the state which are invariant under local unitary transformations \cite{Linden98,Sudbery00,Linden99a,Kempe99,Coffman99,CarteretSudbery00}, and which uniquely characterize equivalent states. The coefficients of a standard form are obviously such quantities, but they may not have readily meaningful physical interpretations. Third, one may identify {\it operational quantities}, such as the number of Greenberger-Horne-Zeilinger (GHZ) triplets or EPR pairs that can be distilled from the state by some procedure \cite{CohenBrun00,Acin00b,Bennett99}.
In section II we consider a number of proposals for standard forms of three-qubit pure states, concentrating especially on those which generalize some aspect of the bipartite Schmidt decomposition: the {\it minimal} form \cite{Acin00,Higuchi00,CHS00}, the {\it two-term} form \cite{Acin00,Dur00,Acin00b}, and the {\it Schmidt} form. This form was given briefly in \cite{CohenBrun00} and independently in \cite{Sudbery00}. In section III we examine it in greater detail. We give an explicit parametrization of the coefficients in terms of five locally invariant quantities, and discuss their physical significance.
We make use of the Schmidt form to prove analytically the reliability of a proposed distillation technique for GHZ triplets from general three-qubit pure states \cite{CohenBrun00}; we present this proof in section IV.
In \cite{Linden98} Linden and Popescu proposed characterizing the entanglement properties of three-qubit states by examining the ``orbits'' of the states under general local unitary transformation. This was carried a step further by Carteret and Sudbery \cite{CarteretSudbery00}, who proved that most states have a certain generic behavior under such transformations, but identified classes of `special' states which they speculated to have unusual entanglement properties. In section V we briefly review these `special' states, then analytically evaluate the yield of GHZs under the distillation protocol of \cite{CohenBrun00} and section IV. By examining the yield of these states as a function of the invariant parameters from section III, and also of the locally invariant ``residual tangle'' ${\tau_{ABC}}$ of Coffman, Kundu and Wootters \cite{Coffman99}, we verify that these classes of states are indeed exceptional by this operational measure, representing extremes of distillability or undistillability. We briefly compare the results using this protocol to the recently discovered optimal distillation method of \cite{Acin00b}, and find that they are entirely consistent. Our conclusions are summarized in section VI.
\section{Review of standard forms for three-qubit states}
Two qubits can always be represented in their Schmidt decomposition (\ref{schmidt_decomp}) \begin{equation} \ket\psi = \sqrt{p}\ket{00} + \sqrt{1-p}\ket{11}, \label{two_bit_schmidt} \end{equation} characterized by a single Schmidt coefficient $p$ (or equivalently $1-p$). Without loss of generality, we adopt the convention that $p \ge 1/2$ and the corresponding eigenvector is $\ket0$.
Most attempts to define a standard form for a {\it three}-qubit state attempt either to generalize some property of the bipartite Schmidt decomposition, or make use of the Schmidt decomposition between one of the bits and the other two, or both. For instance, we can make a Schmidt decomposition between qubit A and qubits B and C, writing the three-qubit state in the form \begin{equation} \ket\psi = \sqrt{p}\ket{0}_A\ket{\psi_0}_{BC}
+ \sqrt{1-p}\ket{1}_A\ket{\psi_1}_{BC}. \label{partial_schmidt} \end{equation} Choosing the Schmidt basis for qubit A guarantees that the correlated states of qubits B and C must be orthogonal: $\bracket{\psi_0}{\psi_1} = 0.$
The sixteen real parameters to describe a generic pure state of three qubits can be reduced to fifteen by normalization, and to five which are invariant under the ten-dimensional group of local unitary transformations \cite{Linden98,CarteretSudbery00}; unfortunately, no single choice of five quantities has proven completely satisfactory. One simple parametrization that has been proposed \cite{Linden98,Schlienz,Cohen} is the Linden-Popescu-Schlienz (LPS) standard form. One begins with a state in form (\ref{partial_schmidt}). One can then choose one of the two correlated states, say $\ket{\psi_0}_{BC}$, and find its corresponding Schmidt bases. The resulting state for the three qubits has the form \begin{eqnarray} \ket\psi &=& \sqrt{p} \ket{0}_A
\left( a \ket{00}_{BC} + \sqrt{1-a^2} \ket{11}_{BC} \right) \nonumber\\ && + \sqrt{1-p} \ket{1}_A \left(
\gamma( \sqrt{1-a^2}\ket{00}_{BC} - a\ket{11}_{BC} )
+ f \ket{01}_{BC} + g \ket{10}_{BC} \right), \end{eqnarray} where $p$, $a$ and $f$ are real positive numbers, $g$ is complex, and
$\gamma = (1-f^2-|g|^2)^{1/2}$. Together these give five independent real parameters.
The vectors $\ket{\psi_0}_{23}$ and $\ket{\psi_1}_{23}$ span a two-dimensional subspace of the Hilbert space for qubits 2 and 3. It's possible to make an interesting variation on the LPS idea using a result of Niu and Griffiths, who showed \cite{Niu99} that any such two-dimensional subspace can be given basis vectors of the form \begin{eqnarray} \ket{\chi_0} &=& \sqrt{q}\ket{00}_{23} + \sqrt{1-q}\ket{11}_{23}, \nonumber\\ \ket{\chi_1} &=& \sqrt{r}\ket{01}_{23} + \sqrt{1-r}\ket{10}_{23}, \end{eqnarray} for some choice of a product basis for the 4-D Hilbert space of the two bits, where $q$ and $r$ are real numbers between 0 and 1. Using this basis leads to a unique standard form \begin{eqnarray} \ket\psi &=& \sqrt{p} \ket0
\left( a \sqrt{q} \ket{00} + a \sqrt{1-q} \ket{11}
+ b \sqrt{r} \ket{01} + b \sqrt{1-r} \ket{10} \right) \\ && + \sqrt{1-p} \ket1 \left( - b^* \sqrt{q} \ket{00} - b^* \sqrt{1-q} \ket{11}
+ a \sqrt{r} \ket{01} + a \sqrt{1-r} \ket{10} \right) , \nonumber \end{eqnarray}
where $a$ is real and $a^2+|b|^2 = 1$. This then gives five independent real parameters: $p$, $q$, $r$, $a$, and the phase of $b$. This form treats the $\ket0$ and $\ket1$ terms more symmetrically than LPS; however, there is still a lack of symmetry under interchange of the bits.
More interesting from a fundamental point of view are attempts to generalize some aspect of the Schmidt decomposition. Three such properties suggest themselves. First, the Schmidt decomposition is the choice of orthonormal bases for the local Hilbert spaces which minimizes the number of terms needed to represent the state. Second, any two qubit state can be written as the sum of only two product vectors. (For $N$-dimensional systems, $N$ product vectors are needed.) Third, the Schmidt decomposition diagonalizes the reduced density matrices of the local subsystems. No single representation for tripartite systems has all three properties, but they can be generalized individually.
Ac\'\i n et al. \cite{Acin00} have shown that all three-qubit states can be written in the form \begin{equation} \ket\psi = \lambda_0 \ket{000} + \lambda_1 {\rm e}^{i\phi} \ket{100}
+ \lambda_2 \ket{101} + \lambda_3 \ket{110} + \lambda_4 \ket{111} \label{minimal1} \end{equation} by a suitable choice of basis, where the $\lambda_i$ are all real and positive and $\phi$ is a phase between $0$ and $\pi$. With only five terms, this is a minimal description, and in that sense a generalization of the bipartite Schmidt decomposition. A similar form has been described by Higuchi and Sudbery \cite{Higuchi00}, \begin{equation} \ket\psi = \lambda_0 {\rm e}^{i\phi} \ket{000} + \lambda_1 \ket{100}
+ \lambda_2 \ket{010} + \lambda_3 \ket{001} + \lambda_4 \ket{111} \label{minimal2} \end{equation} which has the added benefit of being symmetric under interchange of the qubits. Carteret, Higuchi and Sudbery \cite{CHS00} have shown how to generalize this construction to give a unique minimal representation for systems of any dimension. These minimal forms have practical benefits: with a small number of terms, they can simplify the calculation of locally invariant quantities \cite{Acin00c}. However, the $\lambda_i$ and $\phi$ themselves have no obvious physical interpretation.
This minimal property can be generalized in another way, by relaxing the requirement that the product vectors be orthogonal. Ac\'\i n et al. and D\"ur, Vidal and Cirac have also shown \cite{Acin00,Dur00} that almost all three qubit states can be written in the form \begin{equation} \ket{\Psi_{ABC}} = \mu_1 \ket{a_1b_1c_1}
+ \mu_2 {\rm e}^{i\phi} \ket{a_2b_2c_2}, \label{two_term} \end{equation} where the vectors are normalized but not orthogonal. There are six real parameters, $\mu_1$, $\mu_2$, $\bracket{a_1}{a_2}$, $\bracket{b_1}{b_2}$, $\bracket{c_1}{c_2}$ and $\phi$; imposing normalization reduces this to five.
Interestingly, not all three qubit states can be written in the form (\ref{two_term}); a small subclass of states require a minimum of three product terms \cite{Acin00}. D\"ur, Vidal and Cirac made use of this result to prove that there are two classes of three-qubit pure states which cannot be interconverted with nonzero probability \cite{Dur00}. The class that requires three terms is a three-parameter family, and is characterized by vanishing residual tangle ${\tau_{ABC}}=0$ \cite{Coffman99} (see section V). Ac\'\i n, D\"ur and Vidal also used this form to demonstrate a method of converting a single copy of a three qubit state into a GHZ triplet with maximum probability \cite{Acin00b}.
The third generalization is to find bases for all three qubits which diagonalize their reduced density matrices. That is, one can simultaneously put each bit in its Schmidt decomposition with respect to the other two. This form was proposed in \cite{CohenBrun00} and independently in \cite{Sudbery00}. The state has the form \begin{eqnarray} \ket\psi &=& a\ket{000} + b\ket{001} + c\ket{010} + d\ket{011} \nonumber\\ && + e\ket{100} + f\ket{101} + g\ket{110} + h\ket{111}, \label{three_qubit} \end{eqnarray} which looks just like a generic three-qubit state with 16 parameters. However, using each of the three qubits in turn we can write $\ket\psi$ in a form similar to (\ref{partial_schmidt}), with orthogonality conditions which impose restrictions on the possible values of the coefficients in (\ref{three_qubit}). We can use these relationships to reduce these coefficients to five independent parameters, as we show in the next section.
\section{Parametrizing the Schmidt form}
By redefining the relative phases of the basis vectors \begin{equation} \ket0_j,\ket1_j \rightarrow \exp(i\phi_j)\ket0,\exp(i\theta_j)\ket1, \end{equation} we can choose to make four of the coefficients real. A convenient choice is to make $a,d,f,g$ real, while $b,c,e,h$ remain complex. The state must also be normalized, which imposes the condition \begin{equation}
a^2 + |b|^2 + |c|^2 + d^2 + |e^2| + f^2 + g^2 + |h|^2 = 1. \label{normalization} \end{equation} This leaves 11 undetermined parameters.
We can now express the larger eigenvalues $p_{A,B,C}$ of the reduced density matrices $\rho_{A,B,C}$ in terms of the coefficients: \begin{equation}
p_A = a^2 + |b|^2 + |c|^2 + d^2, \ \ {\rm etc.,} \label{p_equation} \end{equation} (the smaller eigenvalues obviously being $1-p_{A,B,C}$). Finally, the states $\ket{\psi_{0,1}}_{kl}$ correlated with basis vectors $\ket0_j$ and $\ket1_j$ must be orthogonal to each other. This gives three more equations: \begin{equation} a e^* + b f + c g + d h^* = 0, \ \ {\rm etc.} \label{orthogonality} \end{equation} Because these equations are complex, they are equivalent to six real equations.
Combining these restrictions, we now have fourteen equations in sixteen unknowns. Thus, in addition to the eigenvalues $p_{A,B,C}$ we would expect there to be two more free parameters. Can we identify reasonable candidates for these parameters? It turns out that natural choices are the two probabilities
$a^2$ and $|h|^2$. These parameters are symmetric under interchanges of the three qubits, and have a fairly simple physical interpretation: they are the probabilities of all three qubits giving the same result
(0 or 1, respectively) when measured in their Schmidt bases. Moreover, the coefficients of the other state vectors can all be calculated in terms of the five probabilities $a^2,|h|^2$, and $p_{A,B,C}$, up to a sign.
Define ${p_{\rm sum}}=p_A+p_B+p_C$. The expressions for the norms of the coefficients are then simple: \begin{eqnarray}
|b|^2,|c|^2,|e|^2 &=& { { (2p_{C,B,A} - 1)|h|^2
- ({p_{\rm sum}} - p_{C,B,A} - 1)(2a^2-{p_{\rm sum}}+1) }
\over{ 2{p_{\rm sum}}-3 } } \nonumber\\ d^2,f^2,g^2 &=& { { (2p_{A,B,C} - 1)a^2
- ({p_{\rm sum}} - p_{A,B,C} - 1)(2|h|^2+{p_{\rm sum}}-2) }
\over{ 2{p_{\rm sum}}-3 } } \;. \label{norm_equations} \end{eqnarray}
The phases of $b,c,e$ are more complicated. If we define the variables
$\phi_{b,c,e}$ by $b=|b|\exp(i\phi_b)$, $c=|c|\exp(i\phi_c)$, and
$e=|e|\exp(i\phi_e)$, the constraint equations (\ref{orthogonality}) imply after a bit of algebra that \begin{eqnarray} \cos(\phi_b), \sin(\phi_b)
&=& (Q_{1,2}/|b|)(\mp 2adf + g(a^2+d^2+f^2-g^2) ), \nonumber\\ \cos(\phi_c), \sin(\phi_c)
&=& (Q_{1,2}/|c|)(\mp 2adg + f(a^2+d^2-f^2+g^2) ), \nonumber\\ \cos(\phi_e), \sin(\phi_e)
&=& (Q_{1,2}/|e|)(\mp 2afg + d(a^2-d^2+f^2+g^2) ), \nonumber\\ \cos(\phi_h), \sin(\phi_h)
&=& (Q_{1,2}/|h|)(\mp 2dfg + a(-a^2+d^2+f^2+g^2) ), \label{phase_equations} \end{eqnarray} where $Q_1$ and $Q_2$ are two constants. We can solve for the values of $Q_1$ and $Q_2$ by using the identity $\sin^2(\phi)+\cos^2(\phi)=1$
and substituting (\ref{norm_equations}) for $|b|,\ldots,g$.
In the Schmidt form for three-qubit pure states, each of the five parameters has a reasonably straightforward physical interpretation. The three parameters $p_A,p_B,p_C$ are the larger (i.e., $p\ge1/2$) eigenvalues of the reduced density operators for each of the three qubits, and correspond to the probabilities of obtaining the more likely of the two possible outcomes (which by convention we label $\ket0$) when we measure each of the qubits in its Schmidt basis. These parameters are closely related to the minimum absolutely selective information \cite{CohenBrun00} for each qubit, which is given by the entropy function \begin{equation} \min S_i = -(p_i\log_2 p_i + (1-p_i)\log_2 (1-p_i)). \end{equation} This quantity is the minimum amount of {\it fundamentally unpredictable} classical information generated by carrying out a measurement on qubit $i$, given a free choice of measurement basis \cite{CohenBrun00}. By using the Schmidt form to choose measurement bases we can simultaneously minimize the absolutely selective information for all three qubits.
The parameters $p_A,p_B,p_C$ range from $1/2$ to $1$ (since they are defined to be the {\it larger} eigenvalues of their corresponding local density matrices). Similarly, $a^2$ ranges from $0$ to $1$, and $|h|^2$ from $0$ to $1/2$. However, this does not mean that these parameters can take arbitrary values within these ranges. Some choices of parameter values correspond to no physical state, and give nonsensical values for (\ref{norm_equations}) and (\ref{phase_equations}).
In particular, the local probabilities must obey the triangle inequalities \begin{eqnarray} p_A(1-p_A) + p_B(1-p_B) &\ge& p_C(1-p_C), \nonumber\\ p_B(1-p_B) + p_C(1-p_C) &\ge& p_A(1-p_A), \nonumber\\ p_C(1-p_C) + p_A(1-p_A) &\ge& p_B(1-p_B); \label{triangle} \end{eqnarray}
these imply, for instance, that if $p_A=1$ then $p_B=p_C$. The restrictions on $a^2$ and $|h|^2$ are more complicated, but they too display an interdependency in their range of values. In particular, as ${p_{\rm sum}}\rightarrow3$ we must have $a^2\rightarrow1$ and
$|h|^2\rightarrow0$.
\section{Proof of distillability}
The Schmidt form can provide analytical insight when addressing specific problems. For example, the efficacy of a recently proposed tripartite distillation protocol \cite{CohenBrun00} can be demonstrated with its help.
Consider a state of three qubits in an arbitrary product basis, which can be written in the form (\ref{three_qubit}). We can straightforwardly calculate the quantity $p_A(1-p_A)$ \begin{eqnarray}
p_A(1-p_A) &=& |af-be|^2 + |ag-ce|^2 + |ah-de|^2 \nonumber\\
&& + |bg-cf|^2 + |bh-df|^2 + |ch-dg|^2, \label{p-p2} \end{eqnarray} This expression is a polynomial in the coefficients and their complex conjugates, and is correct in any basis. If the state is in the Schmidt form, this simplifies to \begin{equation}
p_A(1-p_A) = (a^2 + |b|^2 + |c|^2 + d^2)(|e|^2 + f^2 + g^2 + |h|^2). \label{Schmidt_p-p2} \end{equation}
Let us assume that we have written the state in Schmidt form, such that the states $\{\ket0,\ket1\}$ for each qubit $j$ are eigenstates of the local density matrix with eigenvalues $p_j$ and $1-p_j$, respectively. Suppose we now perform a weak measurement on each of the three qubits. First, allow each qubit to interact with a separate ancilla bit initially in state $\ket0$, such that \begin{eqnarray} \ket0 \otimes \ket0_{\rm anc} &\rightarrow&
\sqrt{1-\epsilon} \ket0 \otimes \ket0_{\rm anc}
+ \sqrt{\epsilon} \ket0 \otimes \ket1_{\rm anc}, \nonumber\\ \ket1 \otimes \ket0_{\rm anc} &\rightarrow&
\ket1 \otimes \ket0_{\rm anc}, \end{eqnarray} where $\epsilon \ll 1$. Then measure the three ancilla bits. With a probability of $\epsilon({p_{\rm sum}})$ one will find one or more of the ancilla bits in state $\ket1_{\rm anc}$, in which case the procedure has failed. Otherwise, this step has succeeded and the three qubits are now in a new state with slightly different coefficients $a',b',\ldots,h'$. The changes in the coefficients are \begin{eqnarray} \Delta a &=& - (\epsilon/2) (3 - {p_{\rm sum}}) a, \nonumber\\ \Delta (b,c,e)
&=& - (\epsilon/2) (2 - {p_{\rm sum}}) (b,c,e), \nonumber\\ \Delta (d,f,g) &=& - (\epsilon/2) (1 - {p_{\rm sum}}) (d,f,g), \nonumber\\ \Delta h &=& (\epsilon/2) {p_{\rm sum}} h. \label{Delta_coeff} \end{eqnarray} This very simple form results because the state is in Schmidt form. After this procedure the bases for the three bits will generally no longer be the correct Schmidt basis (though it will be close to it), so the expression (\ref{Schmidt_p-p2}) cannot be used; but (\ref{p-p2}) is always correct. Thus we get a change in $p_A(1-p_A)$ \begin{eqnarray} \Delta[p_A(1-p_A)] &=&
- \epsilon (4 - 2(p_A+p_B+p_C)) (|af-be|^2 + |ag-ce|^2) \\
&& - \epsilon (3 - 2(p_A+p_B+p_C)) (|ah-de|^2 + |bg-cf|^2) \nonumber\\
&& - \epsilon (2 - 2(p_A+p_B+p_C)) (|bh-df|^2 + |ch-dg|^2) \nonumber\\ &=& - \epsilon (3 - 2(p_A+p_B+p_C)) p_A(1-p_A) \nonumber\\
&& - (\epsilon/2)(|af-be|^2 + |ag-ce|^2 - |bh-df|^2 - |ch-dg|^2 ) \nonumber \end{eqnarray} By making use of equations (\ref{p_equation}) and (\ref{orthogonality}), this expression simplifies to \begin{eqnarray} \Delta[p_A(1-p_A)] &=&
\epsilon \biggl[ (2(p_A+p_B+p_C) - 3) p_A(1-p_A) \nonumber\\
&& + p_A (a^2 - |e|^2 + |h|^2 - d^2 ) + d^2 - a^2 \biggr], \end{eqnarray} which using (\ref{norm_equations}) further simplifies to \begin{eqnarray} \Delta[p_A(1-p_A)] &=&
{{\epsilon (2p_A-1)}\over{2p_A+2p_B+2p_C-3}} \biggl[ 2(a^2+|h^2|)(p_B+p_C-1)
\nonumber\\ && - (2p_A-1)(p_A+p_B+p_C-1)(p_A+p_B+p_C-2) \biggr]. \label{Deltap-p2} \end{eqnarray} The prefactor to (\ref{Deltap-p2}) is strictly positive, as is the first term inside the brackets. The second term is positive if $p_A+p_B+p_C<2$; any state that satisfies this criterion will evolve towards the GHZ state and have a nonzero yield.
For $p_A+p_B+p_C \ge 2$, the sign of (\ref{Deltap-p2}) depends on the relative sizes of the first and second terms inside the brackets. The last two equations of (\ref{norm_equations}) show that for $p_A+p_B+p_C \ge 2$, the fact that $f^2+g^2>0$ implies \begin{equation} 2 a^2 (p_B+p_C-1) \ge (2p_A+p_B+p_C-2)(p_A+p_B+p_C-2), \end{equation} which yields the inequalities \begin{eqnarray} && 2a^2(p_B+p_C-1) - (2p_A-1)(p_A+p_B+p_C-1)(p_A+p_B+p_C-2) \nonumber\\ &\ge& (2p_A+p_B+p_C-2)(p_A+p_B+p_C-2) \nonumber\\ && - (2p_A-1)(p_A+p_B+p_C-1)(p_A+p_B+p_C-2) \nonumber\\ &=& (1-p_A)(p_A+p_B+p_C-2)(2p_A+2p_B+2p_C-3) \ge 0. \end{eqnarray} This straightforwardly implies \begin{equation} \Delta[p_A(1-p_A)] \ge
\epsilon (2p_A-1)(1-p_A)(p_A+p_B+p_C-2) \ge 0. \end{equation} Because of the symmetry of the protocol, $p_B(1-p_B)$ and $p_C(1-p_C)$ must also increase. So one step of this protocol must move the state towards the GHZ with nonvanishing probability, and will (in general) produce a nonzero yield of GHZ triplets.
There are three circumstances in which this result can fail. First, no product state can ever be distilled to a GHZ by this method. At least one of $p_A,p_B,p_C$ must equal 1 in this case, which causes the rate (\ref{Deltap-p2}) corresponding to it to vanish. This is not immediately obvious from the form of (\ref{Deltap-p2}), but it is easily checked using (\ref{norm_equations}) and (\ref{triangle})---if $p_A=1$, then $p_B=p_C=a^2$, and (\ref{Deltap-p2}) is equal to zero.
Second, there are states with
$p_A+p_B+p_C=2$ for which $a^2=|h|^2=0$, again making (\ref{Deltap-p2}) vanish. These are a subset of the {\it triple states} discussed in section V below, which are equivalent to states of the form (\ref{triple}); these states have vanishing residual tangle. Finally, it is possible for a state with $p_A+p_B+p_C>2$ to evolve to one of these triple states. All such states will also have vanishing residual tangle \cite{Dur00}, and conversely all states with vanishing residual tangle will evolve under this distillation protocol to a triple state with $p_A+p_B+p_C=2$, and hence have zero yield of GHZs. This can be clearly seen in Fig.~1.
\section{Entanglement and distillability}
Linden and Popescu \cite{Linden98} proposed characterizing three-qubit states by the dimensions of their orbits under the action of the local unitary group. Generically, tripartite pure states of qubits have ten-dimensional orbits, equal to the dimension of the local unitary group. The very interesting results of Carteret and Sudbery \cite{CarteretSudbery00} give a complete classification of all states for three qubits which behave nongenerically under local unitary transformations; these `special' states have stabilizers of nonzero dimension, and hence orbits of dimension $<10$ (see \cite{CarteretSudbery00}). This behavior suggests that these `special' classes have unusual entanglement properties, which might be evident in other measures of entanglement.
We have numerically simulated the distillation of generic states by the algorithm described above in section IV, in order to determine the yield of GHZ triplets as a function of various parameters, especially the parameters used to describe the Schmidt form. We have also calculated analytical expressions for the yield of states in the exceptional classes enumerated by Carteret and Sudbery. We find that these states are indeed exceptional by this operational criterion, as we describe below.
Most important in calculating the yield of GHZs is the sum of the local eigenvalues ${p_{\rm sum}} \equiv p_A+p_B+p_C$. This quantity determines the probability of failure in one step of the infinitesimal distillation procedure of section IV, with the probability of failure being $\epsilon{p_{\rm sum}}$. If it takes $N$ steps to become sufficiently close to a GHZ triplet, the expected yield is \begin{equation} Y = \prod_{n=1}^N (1 - \epsilon{p_{\rm sum}}(n))
\approx \exp\left\{ - \sum_n \epsilon{p_{\rm sum}}(n) \right\}, \label{yield_integral} \end{equation} where ${p_{\rm sum}}(n)$ is the value of ${p_{\rm sum}}$ at the $n$th step. In the limit of infinitesimal steps the sum inside the exponent becomes an integral. Calculating ${p_{\rm sum}}(n)$ analytically is no simple matter for a general state; the equations (\ref{Delta_coeff}) for the change in the coefficients become differential equations in the limit, but must be supplemented by an additional change of basis between steps, since in general the bases will no longer be the Schmidt basis for the new state. While this is simple to do numerically, analytically it is challenging.
Fortunately, the classes of exceptional states are generally expressible in simple forms which make it possible to integrate the equations (\ref{Delta_coeff}) in closed form, and derive simple expressions for the yield of GHZs. Interestingly, the steps of the GHZ distillation technique commute with local unitary transformations. Because of this, the distillation procedure preserves the stabilizer of the initial state, and hence must take `special' states to other `special' states of the same type. This gives another way of understanding why certain special states are not distillable to GHZs.
In addition to $a^2$, $|h|^2$, and ${p_{\rm sum}}$, we looked at the dependence of the GHZ yield on one other locally invariant quantity. This is the {\it residual tangle} of Coffman et al. \cite{Coffman99}, which can be written \begin{equation} {\tau_{ABC}} = 2(\lambda_1^{AB}\lambda_2^{AB}+\lambda_1^{AC}\lambda_2^{AC}), \end{equation} where $\lambda_1^{ij}$ and $\lambda_2^{ij}$ are the (positive) eigenvalues of the matrix $\sqrt{\rho_{ij}\tilde\rho_{ij}}$. Here $\rho_{ij}$ is the density operator for the two-party $ij$ system, and $\tilde\rho_{ij}$ is the ``spin-flipped'' density operator: $\tilde\rho_{ij}=(\sigma_y \otimes \sigma_y)\rho_{ij}^*(\sigma_y \otimes \sigma_y)$. It has been suggested \cite{Coffman99} that the residual tangle is a measure of the irreducible three-way (``GHZ-type'') entanglement of a tripartite state, beyond any two-party (``EPR-type'') entanglement that may be contained in such a state. As such, it is of particular interest in discussing distillability below. Also, its square ${\tau_{ABC}}^2$ is a polynomial quantity, which makes it analytically tractable.
{\it Triple States.} For this set of states the residual tangle vanishes \cite{Coffman99}. We previously described states in this set as ``triple'' states \cite{CohenBrun00}, because they are equivalent under local unitary transformations to states with just three components: \begin{equation} \ket{\psi_{\rm tr}} = b \ket{001} + c \ket{010} + e \ket{100}. \label{triple} \end{equation} Carteret and Sudbery \cite{CarteretSudbery00} refer to these as ``beechnut'' states; they all have ${p_{\rm sum}}\ge2$. For triple states with ${p_{\rm sum}}>2$ each step of the infinitesimal distillation protocol reduces ${p_{\rm sum}}$, but leaves the state a triple state. If ${p_{\rm sum}}=2$, the actions on the three qubits cancel out, leaving the state unchanged. States of this type have vanishing primary yield for the tripartite distillation protocols described in section IV and in \cite{CohenBrun00}; indeed, D\"ur, Vidal and Cirac have shown that {\it no} procedure can transform one copy of a state with zero residual tangle into a GHZ with nonzero probability \cite{Dur00}.
Because the distillation procedures of section IV and \cite{CohenBrun00} preserve the classes of `special' states, it is easy to see why they cannot produce GHZs from triple states; because all triple states have ${p_{\rm sum}}\ge2$, they cannot include the GHZ state (${p_{\rm sum}}=3/2$) as a limit. The set of product states (or ``bystander states'' in the terminology of Carteret and Sudbery) is similarly undistillable. The result of D\"ur, Vidal and Cirac, however, goes beyond this, since it assumes nothing about the symmetry of the procedure.
The symmetric version of state (\ref{triple}) (with $b=c=e=1/\sqrt{3}$) is termed by D\"ur, Vidal and Cirac the ``W'' state, and seems to fill a role for the zero residual tangle states similar to the role filled by the GHZ for all other states: it is, in some sense, maximally entangled. We will say a bit more about this below. All other `special' classes include the GHZ as a limit, and therefore are distillable.
{\it Generalized GHZ states.} These states can be written in Schmidt form \begin{equation} \ket\psi = a\ket{000} + h\ket{111}. \label{generalized_ghz} \end{equation} They have $p_A = p_B = p_C = a^2$, residual tangle ${\tau_{ABC}} = 4a^2 h^2$. A single step of the infinitesimal distillation procedure gives a new generalized GHZ with coefficients $a' = a + \Delta a$, $h' = h + \Delta h$: \begin{equation} \Delta a = - (\epsilon/2)(3-{p_{\rm sum}}) a,\ \ \Delta h = + (\epsilon/2) {p_{\rm sum}} h, \end{equation} so (\ref{yield_integral}) can readily be evaluated to give the yield of GHZs \begin{equation} Y = 1-\sqrt{1-{\tau_{ABC}}} = (2/3)(3-{p_{\rm sum}}). \end{equation} These states are the most distillable three qubit states as a function of both ${\tau_{ABC}}$ and ${p_{\rm sum}}$; we can see this in Figures 1 and 2 below.
{\it Slice states.} In Schmidt form these are \begin{equation} \ket\psi = a \ket{000} + d \ket{011} - e\ket{100} + h \ket{111},\ \ ae=dh, \label{slice} \end{equation} plus similar states derived by permuting the order of the bits. These states have $p_B=p_C=a^2+e^2$, $p_A=a^2+d^2$, ${\tau_{ABC}} = 4(ah+de)^2 = 4a^2 (h+e^2/h)^2$. Imposing normalization and the orthogonality condition on (\ref{slice}) we see that this is a two-parameter family of states. For these two parameters we may choose $a^2$ and $h^2$, or equivalently $p_A$ and $p_B$. One step of the infinitesimal distillation protocol applied to state (\ref{slice}) leaves qubits B and C in their Schmidt bases, but not qubit A; a change of basis must be applied to A to put the new state in Schmidt form. This new state is still a slice state, and has new parameters $p_A' = p_A + \Delta p_A$, $p_B' = p_B + \Delta p_B$, \begin{eqnarray} \Delta p_A &=& 2 \epsilon (p_A + 2 p_B - 1) p_A, \nonumber\\ \Delta p_B &=& 2 \epsilon (p_A + 2 p_B - 2) p_B
- \epsilon p_A(p_A+p_B-1)/(2p_A-1). \end{eqnarray} The yield is difficult to evaluate analytically, but numerical evidence shows that generic slice states are not extremes of distillability. With each step of the distillation protocol, the parameters $p_B=p_C$ approach $1/2$, but $p_A$ actually moves away. However, when $p_B=p_C=1/2$, this subclass of slice states {\it does} have extremal behavior. Carteret and Sudbery term this subclass the {\it maximal} slice states.
{\it Maximal slice} or {\it Slice-ridge states} are of form (\ref{slice}) with $a^2+e^2 = 1/2 = p_B = p_C$. This subclass is parametrized by a single number, which can be taken to be $p_A$. Because only $p_A$ is larger than $1/2$, there is no need to perform the GHZ distillation procedure on qubits B and C; performing it on A alone preserves the form of the state, with \begin{equation} \Delta p_A = - (\epsilon/2)p_A(1-p_A), \end{equation} giving a yield of GHZs \begin{equation} Y = 1-\sqrt{1-{\tau_{ABC}}} = 2(2-{p_{\rm sum}}). \end{equation} The expression for the primary yield in terms of the residual tangle is identical to that for the GHZ-type states, while in terms of ${p_{\rm sum}}$ it is not. In terms of ${\tau_{ABC}}$ it is one of the most distillable types of state (see Fig. 1). In terms of ${p_{\rm sum}}$ (Fig. 2) it appears to be one of the {\it least} distillable types of states; this is because maximal slice states have the minimum ${\tau_{ABC}}$ of all states with a given ${p_{\rm sum}}$.
In addition to these `special' states, there are two classes of states that deserve additional attention. While these states have stabilizers of zero dimension \cite{Carteret} like generic states, these classes are also preserved by the above distillation protocols. Like the `special' states, they extremize distillability as a function of ${\tau_{ABC}}$ and ${p_{\rm sum}}$.
{\it Generalized triple} or {\it Tetrahedral states.} These states can be written \begin{equation} \ket\psi = b \ket{001} + c \ket{010} + e \ket{100} + h \ket{111}. \label{generalized_triple} \end{equation} We are mainly interested here in the symmetric state $b=c=e$; for this case $p_A=p_B=p_C=2b^2$, ${p_{\rm sum}}\le2$. This form is preserved by the steps of the infinitesimal distillation protocol, which make the coefficients evolve according to (\ref{Delta_coeff}); the yield is easily integrated according to (\ref{yield_integral}) to give $Y=2(2-{p_{\rm sum}})=4(1-3b^2)$; the residual tangle is ${\tau_{ABC}}=16 b^3\sqrt{1-3b^2} = \sqrt{(4-Y)^3 Y/27}$. This yield is identical to that of the maximal slice states as a function of ${p_{\rm sum}}$, but not as a function of ${\tau_{ABC}}$; from Figs. 1 and 2, we see that they are states of minimal distillability in terms of both ${p_{\rm sum}}$ and ${\tau_{ABC}}$.
{\it Zero residual tangle (ZRT) states.} D\"ur, Vidal and Cirac have shown that no states with ${\tau_{ABC}}=0$ can be converted to GHZ triplets with nonzero probability, so $Y=0$. They also showed that all such states can be written in the form \begin{equation} \ket\psi = a \ket{000} + b \ket{001} + c \ket{010} + e \ket{100}. \label{zrt} \end{equation} This is in general not in the Schmidt form of section III. These states include the triple states $a=0$ as a subclass (for which (\ref{zrt}) {\it is} in Schmidt form). The triple states form a boundary of this set, and any ZRT state will evolve under the distillation protocol to a triple state. These states have ${p_{\rm sum}} \ge 2$.
All these `special' states have symmetries which account for both their enlarged stabilizers and their extremal distillability. One way of seeing this is to note that the various standard forms given in section II, which for generic states all require distinct bases, often coincide for these special states. For instance, the generalized GHZ states (\ref{generalized_ghz}) are simultaneously in Schmidt, two-term, minimal, LPS {\it and} Griffiths-Niu form. ZRT and triple states cannot be written in two-term form, but can be written with three terms; the triple states (\ref{triple}) are simultaneously in Schmidt, minimal, three-term and Griffiths-Niu form. Slice states written in the form (\ref{slice}) are simultaneously in both Schmidt and LPS standard forms.
It is easiest to see how the the distillability of these states compares to that of generic states by plotting their yields $Y$ as a function of ${p_{\rm sum}}$ and ${\tau_{ABC}}$ along with the numerical results for a large sample of randomly generated states. We have plotted these quantities in Figs.~1 and 2, with the families of `special' states indicated. We see that most of these states are indeed special as far as distillation is concerned: they form the boundaries of the plotted regions. The quantity ${\tau_{ABC}}$ does seem to be closely related to distillability, as conjectured, though this relationship is not exact; for a given value of ${\tau_{ABC}}$ states with a range of $Y$ values exist, but the range is not very wide. This range is bounded at the top by the generalized GHZ and maximal slice states, and at the bottom by the symmetric generalized triple state. All ZRT states have $Y = {\tau_{ABC}} = 0$. There is also a relationship between ${p_{\rm sum}}$ and $Y$, though again for a given ${p_{\rm sum}}$ there is a range of $Y$ values. This range too is bounded above by the generalized GHZs, and below by the ZRT states, generalized triples and maximal slice states. These upper and lower bounds are both linear; the upper bound is exactly the same as that for Bernstein and Bennett's Procrustean technique \cite{Bennett96a}, reflecting the fact that generalized GHZ states can be distilled by exactly the same techniques which work in the bipartite case.
A reasonable question is to what extent these yields are artifacts of the particular distillation protocol we use. After all, this technique is only one possible way of producing GHZs, in general not the optimal method even for a single copy of a three-qubit state.
Fortunately, we can actually answer this question. Recently, Ac\'\i n, Jan\'e, D\"ur and Vidal \cite{Acin00b} have discovered the optimal algorithm for transforming a single copy of a three-qubit state into a GHZ. This involves performing a POVM on each of the three bits, designed to project the states in the two-term representation onto tri-orthogonal vectors. Finding the correct POVM for an arbitrary state involves maximizing a somewhat involved function, but is easily done numerically. We have done so for a large sample of random states, as well as for the members of the `special' classes enumerated by Carteret and Sudbery.
The optimal yield is higher, in general, than that of the infinitesimal algorithm of section IV, though they are surprisingly close for most states. However, for the `special' states, the yields are identical. In other words, the infinitesimal distillation technique gives the optimal yield for these classes of states. Quite remarkably, if we plot Figures 1 and 2 for the optimal GHZ distillation protocol, the figures look completely unchanged.
Thus we can see that by both the optimal and the infinitesimal techniques, these classes of special states extremize the yield of GHZs as a function of both ${\tau_{ABC}}$ and ${p_{\rm sum}}$. This strongly supports the conclusion that these states do indeed have unusual entanglement properties, and are worthy of further study.
\section{Conclusions}
We have examined tripartite entanglement from both an analytical and an operational point of view. In the bipartite case, which is well understood and to which we have turned for clues, the analytical and operational aspects of entanglement are closely related: the entanglement properties of a single copy are given by the locally invariant parameters, the Schmidt coefficients, which also determine their operational characteristics. We have looked for similar connections in the three-qubit case. Here at least five locally invariant parameters are required, as opposed to just one in the two-qubit case. We have examined several ways of choosing these five parameters, looking in particular at generalizations of the bipartite Schmidt decomposition. One representation in particular, the ``Schmidt form,'' has useful properties which made it simple to prove the efficacy of the infinitesimal GHZ distillation protocol of \cite{CohenBrun00}; it can also be parametrized in terms of five physically meaningful quantities.
We have looked for connections between these parameters and yields in distilling GHZ triplets, as well as connections with the residual tangle of Coffman et al. We have shown that the `special' classes of states enumerated by the theorem of Carteret and Sudbery extremize the distillation yield as functions of the residual tangle ${\tau_{ABC}}$ and ${p_{\rm sum}} = p_A+p_B+p_C$.
Although a certain amount amount of progress towards understanding tripartite entanglement has been made, at least for qubits, many important questions remain unanswered. For example, the number of states in the asymptotic minimum reversible entanglement generating set (MREGS) \cite{Bennett99,Linden99b,WuZhang00} for three-qubit states, and for tripartite states in general, is still unknown. No asymptotically reversible (or optimal but irreversible) distillation technique for GHZ states is known. The search for solutions to these and related problems is ongoing.
\section*{Acknowledgments}
We would like to thank H.A. Carteret, W. D\"ur, R.B. Griffiths, A. Sudbery and G. Vidal for many useful conversations. This work was supported by NSF Grant No. PHY-9900755.
\vfil\eject\vfil
Figure 1. Here we plot the primary yield of GHZ triplets from the infinitesimal distillation algorithm of section III vs. the square of the residual tangle ${\tau_{ABC}}^2$ for various `special states' as well as a random sample of generic states. We see that all states lie between two curved boundaries; the generalized GHZ and maximal slice states lie on the upper boundary, while the generalized triple states lie on the lower boundary. The triple states all have both ${\tau_{ABC}}$ and the yield equal to zero. Interestingly, the maximal slice states appear to be high-yield states when plotted against ${\tau_{ABC}}$, but low-yield when plotted against $p_{\rm sum} = p_A+p_B+p_C$; for a given value of ${\tau_{ABC}}$ these states minimize $p_{\rm sum}$. \vfil
Figure 2. Here we plot the primary yield of GHZ triplets from the infinitesimal distillation algorithm of section III vs. $p_{\rm sum} = p_A+p_B+p_C$ for various `special states' as well as a random sample of generic states. We see that all states lie between two linear boundaries; the generalized GHZ states lie on the upper boundary, while the maximal slice and generalized Triple states lie on the lower boundary, and the triple states are the zero-yield states between $p_{\rm sum} = 2$ and $p_{\rm sum} = 3$. The upper linear boundary corresponds to the yield of Bernstein and Bennett's Procrustean method of EPR distillation in the bipartite case. \vfil
\begin{figure}\label{fig1}
\end{figure} \centerline{Figure 1.}
\begin{figure}\label{fig2}
\end{figure} \centerline{Figure 2.}
\end{document} | arXiv |
# Data preprocessing and feature extraction
Data preprocessing involves cleaning and transforming raw data into a format suitable for analysis. This may include handling missing values, outlier detection, and normalization. In the context of similarity and distance measurement, preprocessing can help ensure that the data is in a consistent format and scale.
Feature extraction is the process of selecting the most relevant features from the data that are most likely to influence the similarity or distance between data points. This can be achieved using techniques such as principal component analysis (PCA) or independent component analysis (ICA).
Consider a dataset of movie reviews, where each data point represents a movie and the features are the genres, director, actors, and ratings. To measure the similarity between movies, we may want to extract features such as director and actor, as they are likely to have a significant impact on the similarity between movies.
## Exercise
Instructions:
1. Load the movie dataset.
2. Preprocess the data by handling missing values and normalizing the ratings.
3. Perform feature extraction using PCA to select the most relevant features.
4. Calculate the similarity between two movies based on their extracted features.
### Solution
```python
import pandas as pd
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
# Load the movie dataset
data = pd.read_csv("movies.csv")
# Preprocess the data
data.fillna(value=-1, inplace=True)
scaler = StandardScaler()
scaled_data = scaler.fit_transform(data)
# Perform feature extraction
pca = PCA(n_components=2)
pca_data = pca.fit_transform(scaled_data)
# Calculate the similarity between two movies
# (e.g., movie1 and movie2)
similarity = calculate_similarity(pca_data[0], pca_data[1])
```
# Distance metrics: Euclidean, Manhattan, and Minkowski
Euclidean distance is a common distance metric that measures the straight-line distance between two points in a Euclidean space. It is calculated using the formula:
$$d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$$
Manhattan distance, also known as L1 distance, measures the distance between two points in a grid-like structure. It is calculated using the formula:
$$d = |x_1 - x_2| + |y_1 - y_2|$$
Minkowski distance is a generalization of Euclidean and Manhattan distances. It is calculated using the formula:
$$d = (\sum_{i=1}^n |x_i - y_i|^p)^{1/p}$$
where $p$ is the order of the Minkowski distance (e.g., $p=2$ for Euclidean distance and $p=1$ for Manhattan distance).
Consider two points in a 2D space:
$$A(1, 2)$$
$$B(4, 6)$$
The Euclidean distance between A and B is:
$$d = \sqrt{(1 - 4)^2 + (2 - 6)^2} = \sqrt{20}$$
The Manhattan distance between A and B is:
$$d = |1 - 4| + |2 - 6| = 3 + 4 = 7$$
## Exercise
Instructions:
1. Calculate the Euclidean distance between two points A and B.
2. Calculate the Manhattan distance between two points A and B.
3. Calculate the Minkowski distance between two points A and B with different orders (e.g., $p=1$, $p=2$, and $p=3$).
### Solution
```python
import math
# Point A
x1, y1 = 1, 2
# Point B
x2, y2 = 4, 6
# Calculate Euclidean distance
euclidean_distance = math.sqrt((x1 - x2)**2 + (y1 - y2)**2)
# Calculate Manhattan distance
manhattan_distance = abs(x1 - x2) + abs(y1 - y2)
# Calculate Minkowski distance
minkowski_distance = [sum(abs(x1 - x2)**p for p in [1, 2, 3])**(1/p) for p in [1, 2, 3]]
```
# Similarity metrics: Cosine similarity, Pearson correlation, Jaccard similarity
Cosine similarity is a measure of similarity between two non-zero vectors that measures the cosine of the angle between them. It is calculated using the formula:
$$similarity = \frac{A \cdot B}{|A| \cdot |B|}$$
where $A \cdot B$ is the dot product of the vectors A and B, and $|A|$ and $|B|$ are their magnitudes.
Pearson correlation is a measure of linear correlation between two variables. It is calculated using the formula:
$$correlation = \frac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^n (x_i - \bar{x})^2 \cdot \sum_{i=1}^n (y_i - \bar{y})^2}}$$
where $x_i$ and $y_i$ are the values of the variables, and $\bar{x}$ and $\bar{y}$ are their respective means.
Jaccard similarity is a measure of similarity between two sets. It is calculated using the formula:
$$similarity = \frac{|A \cap B|}{|A \cup B|}$$
where $|A \cap B|$ is the size of the intersection of the sets A and B, and $|A \cup B|$ is the size of their union.
Consider two sets:
$$A = \{1, 2, 3\}$$
$$B = \{2, 3, 4\}$$
The Jaccard similarity between A and B is:
$$similarity = \frac{|A \cap B|}{|A \cup B|} = \frac{2}{4} = 0.5$$
## Exercise
Instructions:
1. Calculate the cosine similarity between two vectors A and B.
2. Calculate the Pearson correlation between two variables x and y.
3. Calculate the Jaccard similarity between two sets A and B.
### Solution
```python
import math
# Vector A
a = [1, 2, 3]
# Vector B
b = [2, 3, 4]
# Calculate cosine similarity
cosine_similarity = sum(x*y for x, y in zip(a, b)) / (math.sqrt(sum(x**2 for x in a)) * math.sqrt(sum(y**2 for y in b)))
# Calculate Pearson correlation
x = [1, 2, 3, 4]
y = [2, 3, 4, 5]
mean_x, mean_y = sum(x)/len(x), sum(y)/len(y)
pearson_correlation = sum((x_i - mean_x) * (y_i - mean_y) for x_i, y_i in zip(x, y)) / (math.sqrt(sum((x_i - mean_x)**2 for x_i in x)) * math.sqrt(sum((y_i - mean_y)**2 for y_i in y)))
# Calculate Jaccard similarity
jaccard_similarity = len(set(a).intersection(b)) / len(set(a).union(b))
```
# Model evaluation and performance metrics
Accuracy is a measure of the proportion of correct predictions made by a model. It is calculated using the formula:
$$accuracy = \frac{number\ of\ correct\ predictions}{total\ number\ of\ predictions}$$
Precision is a measure of the proportion of true positive predictions made by a model. It is calculated using the formula:
$$precision = \frac{number\ of\ true\ positive\ predictions}{number\ of\ positive\ predictions}$$
Recall is a measure of the proportion of true positive predictions made by a model. It is calculated using the formula:
$$recall = \frac{number\ of\ true\ positive\ predictions}{number\ of\ actual\ positive\ instances}$$
F1 score is a measure of the harmonic mean of precision and recall. It is calculated using the formula:
$$F1 = 2 * \frac{precision \cdot recall}{precision + recall}$$
ROC curve is a graphical representation of the performance of a classification model at all classification thresholds. It plots the true positive rate (TPR) against the false positive rate (FPR).
## Exercise
Instructions:
1. Calculate the accuracy, precision, recall, and F1 score for a binary classification model.
2. Plot the ROC curve for the model.
### Solution
```python
from sklearn.metrics import accuracy_score, precision_score, recall_score, f1_score, roc_curve
# True positive, false positive, false negative, true negative
tp, fp, fn, tn = 10, 5, 3, 20
# Calculate accuracy
accuracy = (tp + tn) / (tp + fp + fn + tn)
# Calculate precision
precision = tp / (tp + fp)
# Calculate recall
recall = tp / (tp + fn)
# Calculate F1 score
f1 = 2 * (precision * recall) / (precision + recall)
# Plot the ROC curve
fpr, tpr, _ = roc_curve(y_true, y_pred)
plt.plot(fpr, tpr)
```
# Model training and hyperparameter tuning
Training a model involves feeding it labeled data and allowing it to learn the relationship between the features and the target variable. This is typically done using supervised learning algorithms.
Hyperparameter tuning is the process of selecting the best hyperparameters for a model. This is typically done using techniques such as grid search or random search.
Consider a k-nearest neighbors classifier. The hyperparameters to tune include the number of neighbors (k) and the distance metric.
## Exercise
Instructions:
1. Train a k-nearest neighbors classifier on a labeled dataset.
2. Tune the hyperparameters (k and distance metric) using grid search or random search.
3. Evaluate the performance of the tuned model using a test dataset.
### Solution
```python
from sklearn.neighbors import KNeighborsClassifier
from sklearn.model_selection import GridSearchCV
# Train a k-nearest neighbors classifier
knn = KNeighborsClassifier()
knn.fit(X_train, y_train)
# Tune the hyperparameters using grid search
param_grid = {'n_neighbors': [3, 5, 7, 9], 'metric': ['euclidean', 'manhattan', 'minkowski']}
grid_search = GridSearchCV(knn, param_grid, cv=5)
grid_search.fit(X_train, y_train)
# Evaluate the performance of the tuned model
y_pred = grid_search.predict(X_test)
accuracy = accuracy_score(y_test, y_pred)
```
# Real-world examples: document similarity, image recognition, and recommendation systems
Document similarity can be measured using techniques such as term frequency-inverse document frequency (TF-IDF) and cosine similarity. This is commonly used in information retrieval and text classification tasks.
Image recognition involves measuring the similarity between images and identifying objects or scenes within them. This is commonly achieved using convolutional neural networks (CNNs) and feature extraction techniques such as SIFT or SURF.
Recommendation systems use similarity and distance measures to predict the likelihood that a user will interact with an item. This is commonly achieved using collaborative filtering techniques such as cosine similarity or Pearson correlation.
## Exercise
Instructions:
1. Implement a document similarity measurement using TF-IDF and cosine similarity.
2. Train an image recognition model using a pre-trained CNN and measure the similarity between two images.
3. Implement a recommendation system using cosine similarity and collaborative filtering.
### Solution
```python
from sklearn.feature_extraction.text import TfidfVectorizer
from sklearn.metrics.pairwise import cosine_similarity
# Document similarity using TF-IDF and cosine similarity
vectorizer = TfidfVectorizer()
X = vectorizer.fit_transform(documents)
similarity = cosine_similarity(X)
# Image recognition using a pre-trained CNN and cosine similarity
model = load_pretrained_cnn()
features = model.extract_features(image1, image2)
image_similarity = cosine_similarity(features)
# Recommendation system using cosine similarity and collaborative filtering
ratings = load_user_item_ratings()
similarity_matrix = cosine_similarity(ratings)
recommendations = predict_recommendations(similarity_matrix, user_id)
```
# Limitations and challenges of similarity and distance measurement
- High-dimensionality data
- Noisy or incomplete data
- Scalability issues
- Interpretability of models
- Ethical considerations
High-dimensionality data can lead to sparse or noisy feature spaces, making it difficult to accurately measure similarity and distance.
Noisy or incomplete data can introduce errors and bias into the similarity and distance measurements.
Scalability issues can arise when dealing with large datasets, as distance and similarity calculations can become computationally expensive.
Interpretability of models can be challenging, as many similarity and distance measures are based on complex mathematical formulas and may not be easily understood or explained.
Ethical considerations can arise when measuring similarity and distance between individuals or groups, as it may lead to biased or discriminatory outcomes.
## Exercise
Instructions:
1. Discuss the limitations and challenges of similarity and distance measurement.
2. Provide examples of real-world scenarios where these challenges may arise.
3. Suggest potential solutions or strategies for addressing these challenges.
### Solution
```python
# Limitations and challenges of similarity and distance measurement
# High-dimensionality data
# Noisy or incomplete data
# Scalability issues
# Interpretability of models
# Ethical considerations
# Real-world scenarios
# Document similarity in large collections
# Image recognition in high-resolution images
# Recommendation systems with sparse user-item data
# Potential solutions
# Feature selection and dimensionality reduction techniques
# Data cleaning and imputation strategies
# Parallelization and distributed computing
# Visualization and interpretation techniques
# Fairness and bias mitigation strategies
```
# Future directions and advancements in the field of similarity and distance measurement
- Deep learning and neural networks for similarity and distance measurement
- Transfer learning and pre-trained models for domain-specific tasks
- Multimodal data integration and fusion for enhanced similarity and distance measurement
- Explainable AI and interpretability of models
- Privacy-preserving and secure similarity and distance measurement techniques
Deep learning and neural networks can potentially improve the accuracy and scalability of similarity and distance measurements, especially in high-dimensional or complex data spaces.
Transfer learning and pre-trained models can leverage existing knowledge and expertise in specific domains to enhance the performance of similarity and distance measurement algorithms.
Multimodal data integration and fusion can enable the measurement of similarity and distance across different data modalities, such as text, images, and audio.
Explainable AI and interpretability of models can help address the challenges of interpretability and ethical considerations in similarity and distance measurement.
Privacy-preserving and secure similarity and distance measurement techniques can ensure that sensitive data is protected while still enabling accurate similarity and distance measurements.
## Exercise
Instructions:
1. Discuss the future directions and advancements in the field of similarity and distance measurement.
2. Provide examples of potential research areas and applications.
3. Suggest potential challenges and open questions in these research areas.
### Solution
```python
# Future directions and advancements in similarity and distance measurement
# Deep learning and neural networks
# Transfer learning and pre-trained models
# Multimodal data integration and fusion
# Explainable AI and interpretability of models
# Privacy-preserving and secure similarity and distance measurement techniques
# Research areas and applications
# Similarity and distance measurement in deep learning
# Transfer learning for domain-specific tasks
# Multimodal data integration for personalized recommendations
# Explainable AI for ethical considerations
# Privacy-preserving similarity and distance measurement for sensitive data
# Challenges and open questions
# Scalability issues in deep learning
# Interpretability and ethical considerations in transfer learning
# Multimodal data fusion and integration challenges
# Explainability and interpretability of neural networks
# Privacy-preserving techniques and security considerations
``` | Textbooks |
The ratio of girls to boys in Ms. Snow's math class is 3:2. If there is a total of 45 students, how many girls are in Ms. Snow's math class?
If there are $3k$ girls in Ms. Snow's class, then there are $2k$ boys. Since the total number of students is $45$, we solve $2k+3k=45$ to find $k=45/5=9$. There are $3k=3(9)=\boxed{27}$ girls in the class. | Math Dataset |
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Home > 22 (4), 10
Space Matters: Extending Sensitivity Analysis to Initial Spatial Conditions in Geosimulation Models
Juste Raimbaulta , Clémentine Cottineaub , Marion Le Texierc , Florent Le Nechetd and Romain Reuillone
aCASA, University College London, United Kingdom & UPS CNRS 3611 ISC-PIF, France; bUMR CNRS 8097 Centre Maurice Halbwachs, France; cUMR 6266 IDEES, Université de Rouen Normandie, France; dUniversité Paris-Est, Laboratoire Ville Mobilité Transport, France; eCNRS Géographie-cités and Complex Systems Institute Paris, France Other articles by these authors In JASSS From Google
Journal of Artificial Societies and Social Simulation 22 (4) 10
<http://jasss.soc.surrey.ac.uk/22/4/10.html>
DOI: 10.18564/jasss.4136 Save citation... Text HTML BibTex Ref. Manager EndNote ProCite
Received: 13-Dec-2018 Accepted: 30-Sep-2019 Published: 31-Oct-2019
Although simulation models of socio-spatial systems in general and agent-based models in particular represent a fantastic opportunity to explore socio-spatial behaviours and to test a variety of scenarios for public policy, the validity of generative models is uncertain unless their results are proven robust and representative of 'real-world' conditions. Sensitivity analysis usually includes the analysis of the effect of stochasticity on the variability of results, as well as the effects of small parameter changes. However, initial spatial conditions are usually not modified systematically in socio-spatial models, thus leaving unexplored the effect of initial spatial arrangements on the interactions of agents with one another as well as with their environment. In this article, we present a method to assess the effect of variation of some initial spatial conditions on simulation models, using a systematic geometric structures generator in order to create density grids with which socio-spatial simulation models are initialised. We show, with the example of two classical agent-based models (Schelling's model of segregation and Sugarscape's model of unequal societies) and a straightforward open-source workflow using high performance computing, that the effect of initial spatial arrangements is significant on the two models. We wish to illustrate the potential interest of adding spatial sensitivity analysis during the exploration of models for both modellers and thematic specialists.
Keywords: Space, Initial Conditions, Sensitivity, Agent-Based Models Other articles with these keywords In JASSS From Google
Computer simulation has been recognised and is increasingly used by geographers as an efficient tool to explore geographical processes, hypotheses and predictive scenarios within virtual laboratories (Batty 1971, 2007b; Carley 1999; Quesnel et al. 2009). It has been identified as an emerging field and coined under the term geosimulation by Benenson & Torrens (2004). Simulation also appears as a way to overcome the difficult analytic resolution of many socio-spatial models which were developed in the past, as well as to explore the possible (alternative) trajectories of path-dependent social and ecological systems. The specificity of geographical models compared to other social science models is that space and spatial interactions are given a prime role, geographers being driven by an explicit interest in studying the way "space" in a broad sense, and more specifically geometry or topology of space influences the outcomes of social processes modelled. We think that simulation approaches are uniquely positioned to represent the complexity of socio-spatial interactions, provided that models include relevant spatial descriptions and behavioural rules which take spatial proximity into account, and provided that the model evaluation includes a sensitivity analysis of the model outputs to the way space is represented. Unfortunately, the first condition is not always met, and the second is seldom even mentioned. Various contributions have been made on ad hoc models to test for the influence of geometry on simulation outputs (Lilburne & Tarantola 2009; Sun & Wang 2007), but without providing a generic framework fitted for most agent-based models. This paper aims to fill a methodological and conceptual gap, which is a systematic testing of the sensitivity of a model's outcomes to its initial spatial conditions. To demonstrate the genericity of our approach, we develop two applications with classic simulation models commonly used as case studies for comparing and aligning simulation models (Axtell et al. 1996; Wilensky & Rand 2007): Schelling (1971)'s model of segregation and Epstein & Axtell (1996)'s Sugarscape model of unequal societies.
Definition of the problem
Socio-spatial systems can be crudely described as social agents interacting with one another via the geometric structure of space. Social agents thus constitute the microscopic level of the system, and they are contained within a spatio-temporal structure that evolves with potential cumulative effects, also known as path-dependency (Arthur 1994). Therefore, observing one system at different points in time does not equate to observing different systems at a single point in time. This general property of non-ergodicity applies to geographical elements such as road networks or built-up areas (Pumain 2003). Similarly to what Gell-Mann (1995) calls frozen accidents in complex systems generally, a given configuration contains clues about past bifurcations, that can have had dramatic effects on the state of the system. Therefore, strong spatio-temporal path-dependencies in the trajectory of individual territories and changing social environments over time prohibit the use of ergodic models. Ironically, these very models tend to be the models most frequently used in geosimulation. With this kind of models, the influence of the geometric structure of space will be even more important than in the case without path-dependency.
Self-organization has been shown to be a central feature of socio-spatial systems in general and of cities in particular (Allen & Sanglier 1981; Saint-Julien et al. 1989; Portugali 2000). In the vocabulary of complex systems, cities also exhibit emergent properties at macroscopic scales (Pumain 2006; Aziz-Alaoui & Bertelle 2009), which can be simulated through microscopic interactions between agents (Wu 2002; Batty 2007a). Complexity is partially due to bifurcations, which are determinant in socio-spatial systems (Wilson 1981, 2002). Indeed, in spatially explicit simulation models, the non-linearity of local interactions is very likely to sublimate small perturbations in the initial spatial setting, making it difficult to interpret the resulting global structures. In that sense, the impact of initial spatial settings on final outcomes is assumed to be significant just as any other initial conditions, but of more interest to the geographer.
Finally, although this may seem obvious, cities are not regular grids, and the distribution of density (of jobs, residents, buildings, etc.) is far from isotropic, even in sprawled cities. On the contrary, there is a significant diversity in the way people, activities and structures are distributed within cities. In Europe for example, Le Néchet (2015) quantifies and classifies six broad types of residential density distributions. However, most socio-spatial models, especially cellular automata, still represent cities, hence geometric support of spatial interactions, as uniform grids of isotropic density. Even in applied cases when GIS geometries of a particular city are used, the spatial distribution of agents tend to be approximated by a constant density (Arribas-Bel et al. 2016), although previous research shows that it is computationally and methodologically feasible to use accurate locations in a simple model such as Schelling's (Benenson et al. 2002). The isotropic simplification is potentially harmful to the representation of urban processes because density and accessibility have environmental, economic and social consequences. Additionally, we expect the initial spatial distribution of agents to influence simulation results in the long run (Castellano et al. 2009), because the agents' rule of action itself may depend on the spatial structure of the environment. For example, households can have different preferences with respect to the built-environment they might want to live in (Spielman & Harrison 2014), or agents moving around will sense a different set of objects within the same fixed radius depending on the topology (Banos 2012) and distribution of density of the sensed environment (Laurie & Jaggi 2003; Fossett & Dietrich 2009). The way modellers represent the initial geometry of space is therefore a central element of socio-spatial simulation models. However, this step is rarely explicit. A meaningful way to address it might be to consider, not necessarily the peculiarities of every city, but at least their broad density structures so as to estimate the variability of the model behaviour to different plausible spatial arrangements.
In this article, we aim to provide an operational framework for studying the influence of geometric structures of space on the results of a simulation model representing a sociospatial system. Following Jessop et al. (2008) socio-spatial systems are understood here as groups of social agents whose behaviour is constrained by their position in geographical space. The aim of the geosimulation models we are interested in is to represent, simulate and explain the dynamics of these systems. In the sociospatial systems as well as in the models representing them, agents are in interaction with other agents and with their geographical environment. The geometric structure of the environment defined at the initialisation of the model and its influence on the model outputs are our object in this paper. We present how to generate the geometric outputs which are used as initial spatial conditions in the model. To this end, we use the example of a density grid generator and feed its outputs to two application models (Schelling and Sugarscape). In no way do we pretend to provide a full exploration of these two particular models, of their attractors and/or potential policy implications. Instead, we present a way of performing a sensitivity analysis to initial spatial conditions of models generated systematically. The generator being controlled by its own parameters, we can then relate the parameters used to generate initial spatial conditions to the variation of simulation outcomes. The purpose is two-fold: (i) to test the robustness of simulation results to small variations of generator parameters and (ii) to study the non-trivial effects of typical categories of geometric structures (monocentric vs. polycentric for example) on the results of a given model. Our approach allows for a systematic comparison of several aspects of the spatial configuration problem, which have been suggested by Filatova et al. (2013), but to the best of our knowledge hardly implemented and achieved in previous studies. In particular, it is applied to the effects of urban form on simulation results, using Schelling's model as a first case study and Sugarscape model as a second one.
The Effects of Space in Simulation Models
Spatial processes
Several empirical studies emphasize the statistical correlations between spatial configurations of people in a city and different distributions of income, carbon emissions, educational outcomes, etc. For example, Wheeler (2006) shows that, in the US, sprawling cities are more unequal than their compact counterparts with respect to income. Dynamically, sprawl in American cities consists in the addition of new developments which have been occupied by different groups of population, resulting in a concentration of the wealthy in suburban pockets and of pockets of poverty in the inner city area (Jargowsky 2002). Similarly, in terms of pollution for example, Schwanen et al. (2001, p.173) show that "deconcentration of urban land uses encourages driving and discourages the use of public transport as well as cycling and walking". These effects of geographical space on social systems correspond to processes to be included in geosimulation models, as a way to disentangle sources of variation arising from socio-spatial processes and from the initial configuration of the geometry of space.
Spatial representation
A discussion of the effects of spatial encoding and representation has also been associated with the field of geostatistics since the exposure of the Modifiable Areal Unit Problem (MAUP) (Openshaw 1984; Fotheringham & Wong 1991). For example, Kwan (2012) has argued for a careful examination of what she coins the 'uncertain geographic context problem' (UGCoP), i.e. of the spatial configuration of geographical units even when the size and delineation of the area are the same. Considerations of such issues in the geosimulation literature are rather scarce. However, there have been some noticeable attempts at analysing the impact of three types of initial spatial characteristics on model outcomes:
The accuracy of geo-localised input data. Thomas et al. (2017) show that data selection in LUTI model is inter-related with the delineation of the spatial system boundaries and the scale of analysis. They provide a few examples on how the use of Exploratory Spatial Data Analysis (ESDA) prior to simulation runs can help avoiding measurement errors of model behaviour and outcomes. In the context of spatial interaction models, Hagen-Zanker & Jin (2012) acknowledge the dilemma between spatial resolution and the computational burden, and suggest a method of adaptive zoning (where the size of destination zones depends on the distance to origin) to solve it.
The shape, precision and boundaries of the modelled spatial system. Axtell et al. (1996) highlight the sensitivity of the average number of stable cultural regions generated to the effect of the territory width implemented in a version of the Sugarscape model which is docked (i.e. made equivalent to) to Axelrod Culture Model. Flache & Hegselmann (2001) show that chances for random emergence of a stable cluster of similar agents in a Schelling-like model are higher in a rectangular grid and lower in a hexagonal grid and that an irregular (Voronoi-diagram) city lattice structure favours migration stabilisation around decentralised clusters of similar agents. Banos (2012) compares the behaviour of Schelling segregation model on city lattices formalized as either grid, random, scale-free and Sierpinski networks and concludes that the presence of cliques in graph-based urban structures favours segregationist behaviours. Le Texier & Caruso (2017), using a set of different theoretical spatial systems, demonstrate the impact of the regularity and aggregation levels, or centrality/periphery effects, on spatial diffusion dynamics of euro coins. Similar issues were also dealt with in physical sciences: for example, Horritt & Bates (2001) study the effects of grid cell size on the behaviour of a raster flood model and show that increasing resolution does not increase model prediction performance below a certain level. Similar conclusions are obtained by Vázquez et al. (2002), unveiling an intermediate optimal spatial resolution regarding model performance and computation time. Spatial resolution also plays a role in the Schelling model: Singh et al. (2009) show that the segregation patterns for certain tolerance values are strictly a small city phenomenon (8x8 city- lattice) and do not work for a larger spatial lattice (100x100), where segregation appears only for certain combinations of tolerance threshold and vacancy density values.
The degree of spatial heterogeneity modelled. Stauffer & Solomon (2007) introduce asymmetric interactions and empty residences in Schelling's model run on a large and regular lattice. They reveal conjoint and non-linear effects on the vacancy rates and tolerance levels on segregation patterns. Gauvin et al. (2010) run Schelling's segregation process in an open city-lattice to study how the variations in tolerance levels, vacancy rates and city attractiveness may create lines of vacancy lots between clusters of agents. They conclude on the functional role of vacancies, which allow weakly tolerant agents to live and be satisfied in a city environment they nevertheless perceive as hostile. Hatna & Benenson (2012) show that their model replications run on a 50x50 torus with 2% of empty cells were not sensitive to the initial patterns (random and fully segregated distribution of agents). In ecology, Smith et al. (2002) study the spread of a disturbance in a heterogeneous landscape using a percolation model, and show that landscape structure has a significant influence on final patterns of contamination outcome. In a spatial epidemics model for an infectious disease, parameterised on real data, Smith et al. (2002) finds that the physical landscape heterogeneity, in particular the presence of rivers, locally influence the propagation speed.
Spatial structures
We can distinguish a last category of spatial effects in geosimulation, which are the geometric constraints of the environment modelled at initialisation on the course and output of the simulation. Our original contribution is to tackle this type of spatial effects. In this paper, we present an operational framework which allows us to systematically measure the impact of the initial geometric structures on the aggregate behaviour of simulation models. We illustrate the potential genericity of our approach by applying it to two distinctive agent-based models: Schelling's model and Sugarscape.
The general workflow of our method is illustrated in Figure 1. In addition to the usual protocol (upper branch of the figure), which consists of running a model \(\mu\) with different values of its parameters, we introduce a density grid generator which depends on its very own set of parameters and feeds the model \(\mu\) at initialisation (lower branch). We call these parameters \(\gamma\) parameters to distinguish them from the standard parameters of the models (called \(\mu\) parameters). The resulting configurations can be clustered into qualitative types of spatial patterns. The sensitivity analysis relates the variations in the model's outcomes to how the density spatial distribution was generated and to the patterns of density generated. In particular, we want to emphasize that spatial effects derive not only from grid size or shape effects, but also from the heterogeneity of the spatial distribution of socio-spatial entities (people, housing, networks, etc.). In the models we used as examples, the initial spatial configurations can be either flat or heterogeneous, monocentric or polycentric, based on external databases and on internal modelling - generation of synthetic population for instance (Bhat & Koppelman 1999).
Figure 1. General workflow.
In order to test the influence of initial spatial conditions on model outputs, we use a systematic method to compare phase diagrams. Following Gauvin et al. (2009), we define a phase diagram as the vector of final aggregated model outputs considered as a function of model parameters. We have as many phase diagrams as we have spatial grids, which makes a qualitative visual comparison not realistic (with around 50 different spatial configurations for each model experiment). A solution is to use systematic quantitative procedures to compare them to a reference case. Technically, because of stochasticity, we represent the output of the model for a given set of parameter values as the mean of the final values of an output indicator obtained for the replications of the model initialized with the set of parameter values. To our knowledge there exists no single well-established method to compare phase diagrams in the agent-based modelling and geosimulation literature (see discussion below). We introduce a measure of the relative distance \(d_r(\mu_{\vec{\gamma}_1},\mu_{\vec{\gamma}_2})\) between two phase diagrams \(\mu_{\vec{\gamma}_1}\) and \(\mu_{\vec{\gamma}_2}\). Phase diagrams are denoted by the same function \(\mu\) indexed by the generator parameters \(\vec{\gamma}\), which capture the spatial configuration (in practice these can be parameters of an upstream model to generate the configuration, or a description of the configuration itself). We choose to compare the inner variability of each phase diagram to the variability between them. We take therefore a simple a-dimensional ratio measure, given formally in the case of a one-dimensional phase diagram by
$$d_r\left(\mu_{\vec{\gamma}_1},\mu_{\vec{\gamma}_2}\right) = 2 \cdot \frac{d(\mu_{\vec{\gamma}_1},\mu_{\vec{\gamma}_2})^2}{Var\left[\mu_{\vec{\gamma}_1}\right] + Var\left[\mu_{\vec{\gamma}_2}\right]}$$ (1)
where \(d\) is a functional distance. We test in Appendix D different distances and show that results are qualitatively robust to this choice. We thus discuss results with the Euclidean distance in the following. The internal variabilities are estimated as the variance within each phase diagram \(\mu_{\vec{\gamma_i}}\) (in practice computed with the unbiased variance estimator). For a multi-dimensional phase diagram, we average these relative distances over the components. Given a set of phase diagrams to be compared, we will study the distribution of this distance to an arbitrary phase diagram for all diagrams, rather than an aggregated measure which would be similar to global sensitivity methods (Saltelli et al. 2008).
The last methodological point which we need to emphasize is the relationship between the present workflow and model exploration workflows in general. The ideas of multi-modelling and extensive model exploration are nothing from new - Openshaw (1983) already advocated for "model-crunching" in 1983 -, but their effective use only begins to emerge thanks to the development of new methods and tools together with an explosion of computation capabilities. The model exploration platform OpenMOLE (Reuillon et al. 2013) allows to embed any model as a blackbox, to write flexible exploration workflows using advanced methodologies such as genetic algorithms and to distribute transparently the computations on large scale infrastructures such as clusters or computation grids. While tools and platforms providing similar functionalities exist, such as for example Behavior Space of NetLogo (Tisue & Wilensky 2004) for model exploration, Bakker et al. (2016) for interactive model development, or interfaces to access High Performance Computing services (Vecchiola et al. 2009), none provide the three aspects simultaneously in an integrated manner. In our case, this tool is a powerful way to embed both the sensitivity analysis and the sensitivity analysis to initial spatial conditions, and to allow the coupling of any spatial generator with any model in a straightforward way as long as the model can take its spatial configuration as an input or from an input file. In this paper, we use the OpenMOLE platform for the spatial environment and the model coupling, placing ourselves in the framework of multi-modelling (Cottineau et al. 2015). We use therefore OpenMOLE's functionalities for model embedding through workflow, design of experiments (parameter sampling) and high performance environment access. As our method quickly increases the amount of computation needed (we ran models approximatively \(7\cdot 10^6\) times with a total computation time of around 2 years equivalent CPU), the use of OpenMOLE was crucial in our work.
Spatial generator of density grids
The density grid generator applies an urban morphogenesis model (Batty 2007a) which has been generalised, explored and calibrated by Raimbault (2018a). To generate population density distributions, other models such as other morphogenesis models (Rybski et al. 2013), kernel mixtures (Anas et al. 1998) or more operational cellular automaton models of urban growth (Herold et al. 2003) could be used, as our general method is proposed to be independent of the generators and models chosen. The model of Raimbault (2018a) has the advantage of producing a broad range of existing urban forms with a reasonable level of complexity. An open implementation and a characterisation of the urban forms which the model can produce allow us to integrate it easily into our workflow. Population density grids, at the typical scale of a metropolitan area, are generated by combining the opposite processes of urban dispersal (negative externalities) with urban concentration (positive externalities). More precisely, grids are generated through an iterative process which simulates successive time steps with a fixed population gain at each time step. Starting from an empty grid, the model adds a quantity \(N\) (population) at each time step \(t\). The new population is allocated through preferential attachment on previous population density. Formally, each added unit has a probability equal to \(P_i^{\alpha}/\sum_k P_k^{\alpha}\) to be added to a patch \(i\) with population \(P_i\), all \(N\) units being added independently and in parallel. The attachment parameter \(\alpha\) can thus be interpreted as a "strength of attraction", in the sense that increasing it will lead to a higher instantaneous concentration. At the end of each time step, this growth process is smoothed \(n_d\) times using a diffusion process: each patch transmits an equal share of \(\beta\cdot P_i\) to its Moore neighborhood (i.e. its 8 surrounding patches). The parameter \(\beta\) can be interpreted as a strength of diffusion: increasing it will lead to larger share of population being diffused in space. To avoid border effects such as a reflexion on the border of the world, border patches diffuse to the outside. The procedure stops when a fixed number of steps \(t_f\) is reached. The grid then has a population of \(t_f \cdot N\) (the population lost due to diffusion process to the outside is reallocated through a normalization procedure at the end of the steps). Grids are thus generated from the combination of the values of these four generator parameters \(\alpha\), \(\beta\), \(n_d\) and \(N\), in addition to the random seed. To ease our exploration, only the distribution of density is allowed to vary rather than the size of the grid, which we fix to a 50x50 square environment (this size provides a good compromise between accuracy of the model to reproduce forms and computational complexity, and furthermore corresponds to the order of magnitude of raster grids for metropolitan areas). We furthermore fix the total population at \(t_f\cdot N = 100,000\), and determine therein the number of steps needed at a given \(N\). Typical value ranges for the parameters will be taken as, following Raimbault (2018a), \(\alpha\in\left[0.5,4.0\right]\), \(\beta \in\left[0,0.3\right] \), \(N\in \left[100,10000\right]\), \(n_d\in\left[1,4\right]\). We illustrate in Figure 2 the variety of spatial configurations that can be generated.
Figure 2. Four examples of grids produced by the density grid generator. The lighter the red, the denser the area. Changing the growth rate N allows to have more or less chaotic shapes (two first compared to the two last grids for example) corresponding to different levels of convergence of the model, whereas local radius can be tuned with the interplay of aggregation strength α and diffusion strength β. Parameter values used here are for the first grid (α = 0.4, β = 0.006, N = 25, nd = 1, tf = 971), the second (α = 0.4, β= 0.006, N = 25, nd = 1, tf = 176), the third (α = 1.4, β = 0.045, N = 102, nd = 2, tf = 618), and the fourth (α = 1.8, β = 0.114, N = 108, nd = 1, tf = 227).
In order to generate density grids which correspond to empirical density distributions, we select among the generated grids using an objective function which matches the point cloud of 110 metropolitan areas in Europe described by four dimensions of spatial structure: their concentration index, hierarchy index, centrality index and homogeneity index (cf. Le Néchet (2015)). These four dimensions were chosen as complementary descriptors of spatial organisation at the urban or metropolitain level - see also Tsai (2005) and Schwarz (2010). They account for: (i) the extent to which population is clustered in a central city, with two complementary indicators : the Moran spatial autocorrelation index (called "centrality") and the distance between individuals ("concentration"), (ii) the extent to which density grid values are similar or contrasted, regardless of location, with two complementary indicators: the entropy of the cell density distribution ("homogeneity") and its rank-size slope ("hierarchy").
We sample the \(\gamma\) parameter space using a Latin Hypercube Sampling(LHS), which is a convenient technique to lower the scatter discrepancy in high dimensions. We sample 2000 points in the 4-dimensional space of parameters {\(\alpha\), \(\beta\), \(n_d\), \(N\)}. It yields a subset of 170 grids matching empirical densities, which constituted our set of different initial spatial conditions. These are further clustered into three classes of morphology (Figure 3) that we label 'compact' (e.g. Vienna), 'polycentric' (Liege) and 'discontinuous' (Augsburg) after Le Néchet (2015). This clustering allows to evaluate the non-trivial effects of a meaningful urban form on simulation results. We select 15 grids of each type to capture possible variations within and between different types of grid. The spatial generator and its resulting grids are relevant to the case study models we have picked (Schelling and Sugarscape) because it produces density grids at a "metropolitan scale", which is the scale at which both models were initially intended to be. In the case of Schelling's segregation model for example, this scale is the one at which most empirical segregation indexes are computed and compared to the model outputs. In the case of Sugarscape, it corresponds to the whole city if the model is a metaphor for city resources (Batty 2005), or to a generic landscape where a resource is grown otherwise. In both cases, our point is that there exist many different patterns of density distribution in resource location and urban density and that acknowledging this diversity might lead to variations in the model outputs. Furthermore, in urban models, we argue that the hypothesis of isotropic density is potentially the most unrealistic one, although unfortunately the most common one in Schelling implementations.
Figure 3. Correspondence between European urban density structures and grids produced with the spatial generator. For each type, we also give the γ parameters leading to grid ho (we recall that α is a level of aggregation, β a level of diffusion, n the number of diffusions per time steps, and N the population growth rate at each time step.
In the following section, we briefly recall the main components of the two "classical" agent-based simulation models used to test how spatial density variations may impact the behaviour and results of simulation models, and how general the method is.
Case study models
Schelling's model consists in an abstract urban housing market where agents of different attributes (for example: red or green) sense their environment, evaluate their satisfaction in terms of neighbourhood composition (how many reds and greens?), and relocate if unsatisfied. It has been shown by Schelling (1969) that even tolerant agents tend to produce segregated patterns because of the complexity of their local interactions and the snowball effect of individual moves on the global distribution of agents in the city. The main parameters of this model are the tolerance level (maximum % of agents different to ego accepted in the neighbourhood), the scope of sensing, the global majority/minority split and the percentage of vacant spaces in the housing market. In addition, we are interested in testing the impact of the initial spatial distribution of housing capacity in this project, using the generated grids.
The outcome of the model is measured by a combination of three segregation indices: Dissimilarity, Moran's I and Entropy. The dissimilarity index (or Duncan's D) is a global measure of segregation and the most widely used, although it does not account for local variations (White 1986; Brown & Chung 2006). It corresponds to the minimum percentage of the population who would have to move to another location so that the global area exhibits a uniform distribution of groups across its constituent areas (each small area would then display the same distribution as the area as a whole). Moran's I is a spatial segregation index which denotes the overall spatial autocorrelation of a group. It takes positive values when a group is clustered in space (people of the same group as oneself are over-represented in the neighbourhood) and negative values when a group is dispersed (people of the same group as oneself are under-represented in the neighbourhood). Entropy can be used as an indicator of spatial segregation because it measures the evenness of groups distribution in space, although the metaphor of thermodynamics is not straightforward (Barner et al. 2017). In our case, areas with higher values of entropy are considered more segregated because the mix of groups is uneven across small areas, whereas low entropy denotes more uniform distributions.
We use an ad-hoc implementation of the Schelling model, both in Scala for performance reasons and in NetLogo to ensure visualization of model dynamics. The pseudo-code of the implemented model is available in Appendix E the source codes for both languages are available on the repository of the project at https://github.com/JusteRaimbault/SpaceMatters. In general, the implementations of Schelling models allow only one agent per cell, and their initial distribution is random, therefore following a uniform distribution across the modelled city. In this experiment, we allow more than one agent to be in a given cell. The potential density of a cell is defined by the density grid generated. If the potential density of a cell is not reached at initialisation, more agents can move into the cell during the course of the simulation, otherwise it is deemed full and unavailable for movers. The satisfaction and segregation indices are computed with regard to the people in the cell and the people present in neighbouring cells. Empirical distributions of density in cities are important in our framework because we want to test models with realistic ranges of initial patterns of density distribution. Therefore, we cannot limit ourselves to an isotropic square modelled city. We chose instead to use the actual distributions of European cities to constraint our density generation.
Sugarscape
Sugarscape is a model of resource extraction which simulates the unequal distribution of wealth within a heterogeneous population (Epstein & Axtell 1996). Although it "is designed to study the interaction of many plausible social mechanisms" (Axtell et al. 1996, p.125), we refer in this paper to the first (and simplest) version of the model, where "processes allow its agents to look for, move to, and eat a resource ("sugar") which grows on its [...] array of cells". Agents of different vision scopes and different metabolisms harvest a self-regenerating resource available heterogeneously in the initial landscape, they settle and collect this resource, which leads some of them to survive and others to perish. The main parameters of this model are the number of agents, their minimal and maximal resource levels. In an urban environment, Sugarscape can be used to model how the spatial distribution of any type of goods or services can influence the spread of wealth among inhabitants. Following Batty (2005), it can be considered as a metaphor of an urban system. We extend the implementation with agents wealth distribution of Li & Wilensky (2009). The outcome of the model is measured by a Gini index of inequality for resource distribution. We are interested in testing the impact of the spatial distribution of the resource, using the generated grids.
Experiment design
For Sugarscape, we explore three dimensions of the parameter space: the total population of agents \(P\in \left[10;510\right]\), the minimal initial agent resource \(s_{-}\in \left[10;100\right]\) and the maximal initial agent resource \(s_{+}\in \left[110;200\right]\). Each parameter is binned into 10 values, giving 1000 parameter points. We run 50 repetitions for each configuration, which yields reasonable convergence properties. The initial spatial configuration varies across 50 different grids, generated by sampling generator parameters in a LHS. We did not use the clustered grids to test the flexibility of our framework, which is demonstrated in this case by a direct sequential coupling of the generator and the model. Indeed, because the density distribution refers to the distribution of resource rather than to the representation of a city structure, we do not need the typology of urban density in this experiment. The full experiment thus equates to 2,500,000 simulations (1000 parameter combinations x 50 density grids x 50 replications).
For Schelling's model, we also explore three dimensions of the parameter space of the model: the minimum proportion of similar agents required in the neighbourhood for the agent to be satisfied (or intolerance level) \(S\in \left[0;1\right]\), the initial split of population, derived from the proportion of green population, \(G\in \left[0;1\right]\) and the vacancy rate of the city \(V\in \left[0;1\right]\). We sample 1000 parameter values using a Sobol sampling and run 100 repetitions for each configuration. We first try the same experiment design (50 density grids generated on the fly), then look at clustered grids representing urban densities. We choose 45 different grids among the ones which are most representative of the three types of urban morphology: 15 compact grids, 15 polycentric grids and 15 discontinuous grids. The last experiment thus equates to 4,500,000 simulations (1000 parameter combinations x 45 density grids x 100 replications). We use OpenMOLE to distribute the computation, and apply segregation measures to characterise the results.
As detailed in Appendix B, more repetitions are needed for Schelling indicators than for Sugarscape, in order to obtain a similar relative confidence in the estimation of averages. We run for this reason a different number of replications for each model. We choose different experiment designs, both for generator parameters and for the phase diagram, to demonstrate the robustness of the method to technical choices. In principle, our workflow applies regardless of the way we generate a spatial configuration (even taking real configurations) and the way we establish phase diagrams.
The implementations of the models were done from NetLogo. We modified the Sugarscape version with wealth of NetLogo model library (to be able to explore it intensively) and we implemented from scratch the Schelling model. Both pseudo-codes are available in Appendix E, and source code for models, grid classification and simulation results analysis is available on the open repository of the project at https://github.com/JusteRaimbault/SpaceMatters. Density grids are also available at this address. Simulation data are available for reproducibility on the dataverse repository at https://doi.org/10.7910/DVN/9JI57U.
We measure the distance of the phase diagrams for all density grids with respect to the reference phase diagram computed on the default initial spatial condition setup (a bi-centric symmetrical non toroidal configuration) using the measure defined in Equation 1. For each density grid, we obtain the average squared distance between corresponding points of the phase diagrams, i.e. the mean value of the final output measure, such as segregation or inequality, for a given value of parameters in the two setups (isotropic and generated). This average squared distance for each point is then related to the average variance of each of the phase diagrams (the reference one and the one for the grid under inquiry). Therefore, values greater than 1 will mean that inter-diagram variability is more important than intra-diagram variability. We tested the sensitivity to the type of distance, using a Minkovski distance with a varying exponent. The results are presented in Appendix D, and show a similar sensitivity to geometric structures.
We obtain a very strong sensitivity to geometric structures for the Sugarscape model. Indeed, the relative distance between the phase diagrams of different density grids and the phase diagram of the reference case ranges from 0.09 to 2.98 with a median of 1.52 and an average value of 1.30. The mean distance above 1 means that, on average, the model is more sensitive to the generator parameters than to its own parameters (population and sugar endowment) in the reference model. Moreover, the maximum distance of 2.98 means that the variation due to the change of grid can be up to three times bigger than the variation due to the model parameters. We plot in Figure 4 the distribution of these distances in the generator parameter space. Each point represents one of the 50 different density grids used to initialise the distribution of sugar in the model. The points are projected with respect to the generator parameters, and coloured according to the relative distance of the phase diagram of the simulations using this grid to the phase diagram of the reference case. Therefore, Figure 4 shows that the grids generated with a high \(\alpha\) (i.e. with a small number of very high density cells) produce simulation results that vary more between the reference case and the generated grid with the same values of parameters than within the reference case because of parameter variations. This pattern is emphasized when grids are generated with a high \(\alpha\) and a high \(\beta\) (i.e. with low gradient of density decrease around the kernels of high density). These grids have the highest relative distance to the reference case. On the contrary, with grids closer to the uniform pattern of the reference case (bottom left of the graph), the model parameters are more important in determining the final inequality levels than the initial spatial distribution of sugar.
Figure 4. Relative distances of phase diagrams by initial spatial grids described by their generator parameters. Relative distance as a function of generator parameters α (strength of preferential attachment) and β (strength of diffusion process).
Another way of quantifying the density grids, instead of looking at the generator parameters, is to look at the resulting indicators of urban form, such as Moran's I, average distance, rank-size slope and entropy (see Le Néchet (2015) for precise definition and context). This 4-dimensional space defined a morphological space. For the purpose of interpretability and visualisation, we reduce this space to a bi-dimensional space with a principal component analysis. The first two components represent 92% of cumulated variance. The first component defines a "level of sprawl" and of scattering, whereas the second one represents the level aggregation.[1] We find that grids producing the highest deviations are the ones with a low level of sprawl and a high aggregation (top left of Figure 5). It is confirmed by the behaviour as a function of generator parameters, as high values of \(\alpha\) also yield high distance. In terms of model processes, it shows that congestion mechanisms in the gathering of the resource induces fast increases of inequality. To put these results in perspective with our workflow given in Figure 1, we have a sensitivity to spatial parameters on average greater than the sensitivity to model parameters.
Figure 5. Relative distances of phase diagrams to the reference across grids. Relative distance as a function of two first principal components of the morphological space (see text). The black point corresponds to the reference spatial configuration. The Green and blue frames give respectively the first and second particular phase diagrams shown in Figure 6.
Within a standard Schelling model (i.e. initialised with a uniform density grid), Gauvin et al. (2009) have built the phase diagram of segregation patterns depending on the combination of parameter values. For high levels of tolerance (S < 0.25), there is no segregation. For high values of vacancies (V > 0.65) and low values of tolerance (S > 0.5), there is a diluted segregation state where homogeneous communities are separated from others by large empty buffers. Finally, for low values of vacancies (V < 0.2) and low values of tolerance (S > 0.7), the model is frozen in a state where everyone is unhappy but no-one can express its intolerant behaviour due to the lack of free spaces. Between these extreme cases, the model gives rise to segregated states where homogeneous communities adjoin one another. The objective of this quantitative experiment is to evaluate to which extent this phase diagram is modified when different density grids are applied. We show in Figure 8 (Supplementary Material) the values of the relative distance as a function of generator parameters and in the reduced morphological space, in a way similar to the analyse done with Sugarscape. Variations are less considerable than for Sugarscape across phase diagrams, but values close to 1 show that several configurations are as sensitive to initial spatial conditions than to their parameters. We focus in the following on a qualitative characterisation of these variations.
Variation by type of structure
In this qualitative exploration of the effect of initial spatial conditions on the results of Schelling's model, we use the classification of grids into three morphological types (cf. Figure 3). In particular, we want to evaluate to which extent the typology summarises the spatial effects, and if one type of urban form or another enhances the segregation mechanism of the model, or interacts differently with the model parameters. This experiment attempts at drawing conclusions on urban morphology, beyond the technical conclusions already obtained with respect to simulation sensitivity.
In Table 1, we see that the type of density grid with which the model is initialised correlates to a certain extent with the level of segregation measured at the end of the simulation run. Indeed, compared to the reference case of compact (monocentric) density patterns, polycentric grids produce more dissimilarity and entropy between the location of green and red agents. Discontinuous grids have the same effect, although attenuated. The results obtained with Moran's I are opposite, because this index measures spatial autocorrelation at the global level and that compact cities have higher levels of global autocorrelation by construction. However, linear models with and without the type of density distribution yield the same coefficients for Schelling's parameters V and S, the only exception being the vacancy rate V in the Moran's I model with grid types, which becomes non-significant. The similarity of the coefficient in both cases means that the effect of the model's parameters (and thus the mechanism by which agents of similar group cluster in space) is the same regardless of the distribution of density. The way polycentric and discontinuous density grid exhibit higher segregation is by allowing buffer zones of low density to surround pockets of homogeneity, which is impossible in a compact city, because everyone is at reach of everyone else. The buffering process confirms previous results obtained with network structures (Banos 2012) and supports the conclusion that space acts here on top of mechanisms rather than in interaction with them.
Table 1: Regression of the segregation level of Schelling simulation with order parameters and type of city grid. Moran's I applies to the minority population. *** means that the estimate is significant at the 0.01 level.
Simulation outcome by segregation index: Dissimilarity Entropy Moran's I
Intercept -0.212 *** -0.141*** -0.254*** -0.208*** -0.036*** -0.061***
Similarity Wanted (S) 1.212*** 1.212*** 1.250*** 1.250*** 0.550*** 0.550***
quadratic term (S2) -0.942*** -0.942*** -0.963*** -0.963*** -0.428*** -0.428***
Vacancy Rate (V) 0.602*** 0.602*** 0.453*** 0.453*** -0.027*** -0.027***
Minority Index (%Maj - %Min) 0.307*** 0.307*** 0.130*** 0.130*** -0.067*** -0.067***
Density Grid = Polycentric 0.087*** 0.052*** 0.001***
Density Grid = Discontinuous 0.111*** 0.068*** 0.00
Attraction generator parameter \(\alpha\) -0.083*** -0.053*** 0.014***
Diffusion generator parameter \(\beta\) 0.323*** 0.218*** 0.017***
R2 (%) 30.6 34.7 24.1 25.6 23.9 24.0
# of observations (sim. runs) 2,106,000 2,106,000 2,106,000 2,106,000 2,106,000 2,106,000
AIC -70717.68 -198748.2 208213.8 166048.8 -4385990 -4387816
We now check the sensitivity in terms of qualitative behavior of phase diagrams. We show the phase diagrams for two very opposite morphologies in terms of sprawling, but controlling for aggregation with the same \(PC2\) value. These correspond to the green and blue frames in Figure 5. In terms of grid shapes, we observe that the difference between the two grids is mainly on average distance and entropy: in a nutshell, the first grid is much more dispersed and disorganised than the second. Although the behaviours are rather stable for varying s+, the initial maximum endowment in sugar, which means that the poorest agents have a determinant role in trajectories, the two examples have not only a very distant baseline inequality (the ceiling of the first 0.35 is roughly the floor of the second 0.3), but their qualitative behavior is also radically opposite: the sprawled configuration (green frame) makes inequalities decrease as population decreases and decrease as minimal wealth increases, whereas the concentrated configuration (blue frame) makes inequalities strongly increase as population decreases and also decrease with minimal weights but significantly only for large population values (Figure 6). In sprawled spaces, inequalities are thus fostered by a lack of minimal local resources, whereas population will drive inequality in concentrated spaces. The process is thus completely reversed depending on the grid chosen to run the model on, which would have significant impacts if one tried to draw policy recommendations from this model.
Figure 6. Examples of phase diagrams. We show two dimensional phase diagrams on (P, s−), both at fixed s+ = 110. (Left) Green frame, obtained with α = 0.79, n = 2, β = 0.14, N = 157; (Right) Blue frame, obtained with α = 2.56, n = 3, β = 0.13, N = 128.
We consider that the method presented in this paper holds great potential for strengthening socio-spatial models' exploration. However, three limits and two areas of opportunities have still not been tackled.
Comparing phase diagrams
Comparing phase diagrams is as we saw not straightforward, and further developments of our method imply testing alternative methods for this particular point. For example, in the case of the Schelling model, an anisotropic spatial segregation index (giving the number of clusters found and in which region in the parameter spaces they are roughly situated) would differentiate strong phase transitions in the space of generator parameters. The use of metrics comparing spatial distributions, such as the Earth Movers Distance which is used for example in Computer Vision to compare probability distributions (Rubner et al. 2000), or the comparison of aggregated transition matrices of the dynamic associated to the potential described by each distribution, would also be potential tools. Map comparison methods, popular in environmental sciences, provide numerous tools to compare two dimensional fields (Visser & De Nijs 2006; Kuhnert et al. 2005). To compare a spatial field evolving in time, elaborated methods such as Empirical Orthogonal Functions that isolates temporal from spatial variations, would be applicable in our case by taking time as a parameter dimension, but these have been shown to perform similarly to direct visual inspection when averaged over a crowdsourcing (Koch & Stisen 2017). The transfer of methods used to compare sequences (Kruskal 1983) or time-series (Liao 2005) is a possible way to develop measures between phase diagrams. The higher dimension of the phase diagrams we study must however be considered with caution when transferring methods, in a way analog to the application of global sensitivity indexes to spatial data (Lilburne & Tarantola 2009). We can also note than more generally, this problem of comparing phase diagrams is a particular instance of the more generic issue of comparing patterns, which for example include unsupervised learning techniques (Hastie et al. 2009). The investigation of diverse approaches to systematically quantify differences between phase diagrams is an important potential development of our method.
Sensitivity of indicators to the geometry
While we investigated the sensitivity to geometry of models together with their indicators, a potential research direction is to study the dependence of indicators themselves to the spatial configuration. Indeed, the segregation indices are likely to be zoning, size or population density dependent (Wong 1997; Reardon & O'Sullivan 2004). Generating point data rather than a grid could help overcoming these issues, and appears as a promising direction for future research. This methodological claim is also interesting from theoretical and empirical views as point data may be used to construct egohoods (Hipp & Boessen 2013), i.e. a smooth definition of neighbourhoods as opposed to the commonly used non-overlapping units.
Platform constraints and docking challenges
An aspect that we have not touched upon in the article with respect to the sensitivity to initial spatial conditions is the importance of the modelling platform as a constraint in the formalisation of space. For example, spatial structure may be easier to implement as a raster rather than a vector in NetLogo models, which could influence the implementation choices of some non-experienced modelers. Its toroidal default setting might also have influenced the work of many modellers who did not question explicitly the representation of space. This issue is part of the docking challenge (Axtell et al. 1996) (i.e. checking if two models can produce the same results), but more generally, it involves a description of the model and its spatial requirements more detailed than what is currently the rule.
Opportunities and extensions
Reproducibility and applicability
We wish to underline the interest of adding sensitivity analysis of the geometry of space during the exploration of models. As we have observed for simple agent-based models, uncertainties in the initial distribution of the space on which agents will interact play a role on the variability in model outputs. This would certainly be even more important for more sophisticated models, with more types of agents and types of interactions, insomuch as agent-based models tend to be used in realistic, diverse settings. Indeed, regarding path-dependency which was part of issues motivating our approach, the models we studied are not path-dependent for the aggregated indicators (while the final agent configuration naturally is). We postulate that intrinsically non-ergodic models such as the one studied by Coupé et al. (2017) should be more sensitive to geometry. Therefore, we believe that for many applied problems, studying the specific variability of model outputs regarding the geometry of space would be important to assess the transferability of results in other urban settings.
This is especially important in the context of the increasing recognition of the complexity of urban space and of the role cities plays in various aspects of sustainability: modelling cities with agent-based models might become more frequent (Perez et al. 2016) and we argue it is important to account for the diversity of urban configurations when disseminating the results of any model in the planning community.
We think that the method could (and should) be applied to larger models including domain mechanisms and more empirical initialisation data, for example synthetic populations. The sensitivity analysis to initial spatial conditions could then be either a replication on the spatial allocation of the synthetic population, or a series of spatial permutations of the empirical spatial inputs. We want to foster this extension of our work by releasing the density grids also generated, as well as the generating workflow and the model implementation. They are available on the open repository of the project at https://github.com/JusteRaimbault/SpaceMatters. Future work could be done to compare these or generate grids with a larger morphological span, covering other typical urban forms that can be found in the world.
Another way to go would be to implement additional generators, such as social networks (Alizadeh et al. 2016) with localised agents, transportation networks generators (Raimbault 2018b), or coupled road network and population raster generators (Raimbault 2019). The particular density grid generator we used here is an example among possible others, and our methodological contribution is generic as it does not depend on which generator is chosen.
An emancipation opportunity for social sciences
As Pumain (2003) points out in an overview of complexity approaches in geography, transfer of models and concepts between disciplines may induce a transfer of corresponding assumptions. Geography and the social sciences in general have been strongly influenced by physics in the last decades, that beside their highly enriching impact (O'Sullivan & Manson 2015), may have softly imposed strong assumptions such as homogeneity and isotropy of space in basic models. We believe that a renewed approach on the role of space as we proposed, in other terms insisting that space matters, is an opportunity for social sciences to build their own stream of methodologies in the modelling domain.
This relates to relations between empirical, conceptual and modeling dimensions of quantitative research in social sciences, as coined by Livet et al. (2010). In our case, a contribution in the modeling domain aiming at extracting further knowledge on model behavior (sensitivity to geometrical context), may lead to questioning theoretical concepts and empirical definitions on which the model was based, such as for example the meaning of neighborhood, hence geometry of space.
After reviewing the extensive literature on spatial biases in statistical and simulation models, we presented a method to analyse the sensitivity of a simulation's results to the initial spatial configuration. We did so by implementing a spatial generator whose output is used as input for the simulation model. We applied this approach to two textbook ABMs: Schelling and Sugarscape. With the Schelling experiment, we found that the different urban morphologies have an impact on interaction patterns, and that polycentric and discontinuous cities appear systematically more segregated than compact cities in terms of dissimilarity and entropy index. With Sugarscape, we show that the model is more sensitive to space than to its other parameters in the reference NetLogo implementation, both qualitatively and quantitatively: the amplitude of variations across density grids is larger than the amplitude in each phase diagram, and the behaviour of the phase diagram is qualitatively different in different regions of the morphological space. We think that this method has the potential to increase the arsenal of evaluation of socio-spatial models, in order to assess the sensitivity of models to their initial spatial conditions but also to learn about the impact of the urban form on social mechanisms.
The authors acknowledge the funding of their institutions and the EPSRC project number EP/M023583/1. Results obtained in this paper were computed on the vo.complex-system.eu virtual organization of the European Grid Infrastructure (http://www.egi.eu). We thank the European Grid Infrastructure and its supporting National Grid Initiatives (France-Grilles in particular) for providing the technical support and infrastructure. This work is part of DynamiCity, a FUI project funded by BPI France, Auvergne-Rhône-Alpes region, Ile-de-France region and Lyon metropolis.
We have \(PC1 = 0.76\cdot distance + 0.60\cdot entropy + 0.03\cdot moran + 0.24\cdot slope\) and \(PC2 = -0.26\cdot distance + 0.18\cdot entropy + 0.91\cdot moran + 0.26\cdot slope\)
A: Behavior of the density grid generator
Figure 7 summarises the behavior of the density grid generator, according to parameters \((\alpha,\beta)\). To put it in a simple way, high values of \(\alpha\) give highly hierarchical configurations, and diminishing \(\beta\) increase the number of centers. Low values of \(\alpha\) give diffuse patterns, with however clear centers for a high diffusion. We do not discuss here the role of other parameters, but according to Raimbault (2018a), diffusion steps give smoother forms, and the rate between total population and the population increment at each step (which is equivalent to the total number of steps) is crucial to select non-stationary distributions that are closer to real configurations. Raimbault (2018a) also shows the existence of non-linear behaviors in some regions of the parameter space, so the description we gave here shall not be interpreted as a linear link between generator parameters and the morphological properties of the generated grids.
Figure 7. Stylized behavior of the density grid generator. (Left) Examples of grids with their position in the (α, β) parameter space; (Right) Stylized interpretation of the forms obtained as a function of α and β.
B: Additional statistical analysis
For both models we estimate the Sharpe ratios for each indicator by \(S(X) = \hat{\mathbb{E}}\left[X\right]/\hat{\sigma}(X)\) with standard estimators for average and standard deviation. The summary statistics of this ratio computed on repetitions for all parameter points are given in Table 2. Under the assumption of a normal distribution, the width of the confidence interval at level \(c\) is given by \(\left|\mu_+ - \mu_-\right| = 2\cdot \sigma \cdot z_{c} / \sqrt{n}\) where \(\sigma\) is the standard deviation, \(z_{c}\) is the quantile at which \(c\) is attained by the cumulative distribution, which is around \(1.96\) for a 95% confidence interval. This means that to obtain a confidence interval of width \(\kappa \cdot \sigma\), one needs a number \(n \simeq (4 / \kappa )^2\) of repetitions. This gives 64 repetitions for \(\kappa = 2\). As the Sharpe ratios are in general smaller for Schelling indicators than for Sugarscape, we take \(n = 50\) for Sugarscape and \(n = 100\) for Schelling to have a similar confidence in estimations.
Table 2: Summary statistics of Sharpe ratio estimated on repetitions for each parameter point.
Model/Indicator Min. 1st Qu. Median Mean 3rd Qu. Max.
Sugarscape/Gini 2.343 14.413 20.993 20.098 26.208 49.808
Schelling/Dissimilarity 1.053 12.368 19.434 25.268 24.328 468.563
Schelling/Entropy 0.5458 8.2591 14.5112 17.0100 17.9372 244.3230
Schelling/Moran 0.0001 0.5554 0.7782 3.2724 3.8979 121.6829
C: Additional figures for the Schelling model
Figure 8 shows the phase diagrams distances as a function of generator parameters and morphological components, similarly to the Sugarscape model in main text. In absolute, this version of the Schelling model seems less sensitive to density grids than the Sugarscape model, as we do not obtain a high range of values here. We however obtain measures ranging from 0 to 0.85 with the Euclidian distance, what is however characteristic of a significant sensitivity to space.
Figure 8. Relative distances of phase diagrams to the reference across grids for the Schelling model. Each point corresponds to a spatial configuration and colour gives the relative distance to one of the phase diagrams. We show them in the generator parameter space (Left) and in the reduced morphological space (Right).
D: Comparison of phase diagram with other distances
We describe here the tests done with other distances to compare phase diagrams. We tested normalized Minkovski distances, defined by \(d(x,y) = \left(\frac{1}{N}\cdot \sum_i \left|x_i - y_i\right|^{q}\right)^{\frac{1}{q}}\), for varying values of \(q\) from \(q = 1\) (Manhattan distance) to \(q = 10\), including \(q = 2\) (Euclidian distance) which is used in main text. The Table 3 gives the summary statistics of each distance computed on all initial configurations for the Schelling model. We naturally obtain smaller difference with the Manhattan distance but which remain significant (averages of 10% for the Schelling model and 40% for Sugarscape), and variabilities with higher values of the Minkovski exponent are much higher. These results confirm the high variability observed in main text with the Euclidian distance.
Table 3: Summary statistics of different distances for schelling.
q Min. 1st Qu. Median Mean 3rd Qu. Max.
1 0.00000 0.02838 0.06263 0.10608 0.16818 0.37018
2 0.0000 0.1520 0.2517 0.3107 0.4359 0.8155
10 0.000 2.083 2.431 2.380 2.713 3.664
Table 4: Summary statistics of different distances for sugarscape.
10 0.1540 0.4466 9.8757 7.6246 13.0550 16.8001
E: Pseudo-code for models
We give below the pseudo-code for the implementations we used of both Sugarscape (Figure 10) and Schelling (Figure 9) models. We recall that source code is openly available at https://github.com/JusteRaimbault/SpaceMatters. The pseudo-code is in the style of NetLogo code, which is already easily readable.
Figure 9. Pseudo-code (NetLogo style) for the Schelling model used here. We give a pseudo-code very close to the NetLogo language, at the exception of italic expressions which are not put here in NetLogo to ease understandability.
Figure 10. Pseudo-code (NetLogo style) for the sugarscape model. Conventions are the same than for Schelling pseudo-code.
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On Famous Things
A quip from Stack Exchange back in 2014 that still fills me with glee on a daily basis:
A poster asks how to convince other people when he's developed an as-yet ignored, revolutionary, world-beating result...
e.g., you solve the P vs. NP problem or any other well known open problem.
Pete L. Clark writes as part of his response:
It's like saying "i.e., he found the Holy Grail or some other famous cup".
More gifts of wisdom at Stack Exchange.
Posted by Delta at 5:00 AM No comments:
Michigan State Drops Algebra Requirement
This summer, Michigan State announced that they will drop college algebra as a general-education requirement, replacing it with quantitative-literacy classes:
Michigan State University has revised its general-education math requirement so that algebra is no longer required of all students. The revision reflects an increasing view on college campuses that there is no one-size-fits-all math curriculum -- and that math is often best studied in connection with everyday life...
Now, students can fulfill the requirement by taking two quantitative literacy courses that place math in a real-world context. They also still have the option of taking algebra along with another math course of their choice -- whether a quantitative-literacy course or a more traditional course like trigonometry.
"Algebra No More" at Inside Higher Ed
Posted by Delta at 5:00 AM 2 comments:
Observed Belief That 1/2 = 1.2
Last week in both of my two college algebra sections, there came a moment when we had to graph an intercept of x = 1/2. I asked, "One-half is between what two whole numbers?" Response: "Between 1 and 2." I asked the class in general for confirmation: "Is that right? One-half is between 1 and 2, yes?" And the entirety of the class -- in both sections, separated by one hour -- nodded and agreed that it was. (Exception: One student who was previously educated in Russia.)
Now, this may seem wildly inexplicable, and it took me a number of years to decipher this. But here's the situation: Our students are so unaccustomed to fractions that they can only interpret the notation as decimals, that is: they believe that 1/2 = 1.2 (which is, of course, really between 1 and 2). Here's more evidence from the Patricia Kenschaft article, "Racial Equity Requires Teaching Elementary
School Teachers More Mathematics" (Notices of the AMS, February 2005):
My first time in a fifth grade in one of New Jersey's most affluent districts (white, of course), I asked where one-third was on the number line. After a moment of quiet, the teacher called out, "Near three, isn't it?" The children, however, soon figured out the correct answer; they came from homes where such things were discussed.
Likewise, the only way this makes sense is if the teacher interprets 1/3 = 3.1 -- both visually turning the fraction into a decimal, and reading it upside-down. We might at first think the error is the common one that 1/3 = 3, but that wouldn't explain why the teacher thought it was only "near" three.
The next time an apparently inexplicable interpretation of a fraction comes up, consider asking a few more questions to make the perceived value more precise ("Is 1/2 between 1 and 2? Which is it closer to: 1 or 2 or equally distant?" Etc.). See if the problem isn't that it was visually interpreted as decimal point notation.
The Math Menu
A quick thought, spring-boarding off Monday's post: A constant debate in math education is whether students should be directly-taught mathematical results, or spend time (like a mathematician) exploring problems, looking for patterns, and coming up with their own "theorems" (in Mubeen's phrasing "own the problem space").
Here is a hypothetical equivalent debate: What is supposed to happen in a restaurant -- Does food get cooked, or does food get eaten?
Obviously both. But the majority of people who visit the establishment are clientele who do not come to the restaurant in order to learn how to cook; they come for an end-product which is used in a different fashion (for consumption and nourishment). If someone expresses interest in becoming a chef themselves then of course we should encourage and cultivate that. But if some group of chefs become so self-involved that they demand everyone participate in cooking for a "real" restaurant experience, then surely we'd all agree that they'd gone off the deep end and needed restraints.
So too with mathematicians.
A pair of scary math-education anecdotes by Junaid Mubeen, for your consideration:
How Old is the Shepherd? When 8th-graders are asked a short question with absolutely no information about age whatsoever, 3-in-4 will report some numerical result anyway. Repeated in numerous experiments. Watch a video.
I Can't Believe It's Not Unproven. Mubeen's 12-year-old nephew comes home with a math problem that can't be solved; he is shown a proof of that fact, and agrees to all the steps and the conclusion. Nephew spends the rest of the evening trying to find an answer anyway.
I don't really agree with Mubeen's rather broad conclusions at the end of the first article. But we can all agree this is a terrifying outcome!
Mo' Monic
If you look at any list of elementary algebra topics, or any book's table of contents, etc., then you'll probably find that all of the subjects are referenced by name except for one single exceptional case, which is always expressed in symbolic form. For example, from the College Board's Accu-Placer Program Manual, here's a list of Content Areas for the Elementary Algebra test:
Do you see it? Or, here are some of the section headers in the Pearson testbank which accompanies the Martin-Gay Prealgebra & Introductory Algebra text:
Or, here's a menu of topics and quizzes from the MathGuide.com algebra site:
I could repeat this for many other cases, such as: the CUNY list of elementary algebra topics, tables of contents for most algebra books, etc., etc. It's weird and to my OCD brothers and sisters surely it's a bit distracting and frustrating.
There should be a name for this. The funny thing is that, to my current understanding, there's a perfectly serviceable name to make the distinction that we're reaching for here: "monic" means a polynomial with a lead coefficient of 1. So I've taken to, in my classes, referring to the initial or "basic" type (\(x^2 + bx + c\)) as a monic quadratic, and the more general or "advanced" type (\(ax^2 + bx + c\), \(a \ne 1\)) as a nonmonic quadratic. My students know they must learn proper names for everything, and so they pick this up as easily as anything else, and without complaint. Thereafter it's much easier to communally reference the different structures by their proper names.
Now: I must admit that I picked this up from Wikipedia and I've never, ever, seen it used in any mathematics textbook at any level. Perhaps someone could tell me if this is new, or nonstandard, or inaccurate. But even if that weren't the right term to distinguish a polynomial with lead coefficient 1, there should still be a name for this structure. We really should create a name, if necessary, and I'd be prone to make up my own name for something like that.
But "monic" fits perfectly and is delightfully short and descriptive. We should all start using "monic" more widely, and I'd love to start seeing it in major algebra textbooks.
Natural Selection of Bad Science
Smaldino and McElreath write a paper which asserts that the problem of false-positive papers in science -- especially behavioral science -- is getting worse over time, and will continue to do so as long as we reward quantity of paper outputs:
To demonstrate the logical consequences of structural incentives, we then present a dynamic model of scientific communities in which competing laboratories investigate novel or previously published hypotheses using culturally transmitted research methods. As in the real world, successful labs produce more 'progeny,' such that their methods are more often copied and their students are more likely to start labs of their own. Selection for high output leads to poorer methods and increasingly high false discovery rates. We additionally show that replication slows but does not stop the process of methodological deterioration. Improving the quality of research requires change at the institutional level.
Paul E. Smaldino, Richard McElreath. The natural selection of bad science. R. Soc. open sci. 2016 3 160384; DOI: 10.1098/rsos.160384. Published 21 September 201. Link.
Quotes Campbell's Law: "The more any quantitative social indicator is used for social decision-making, the more subject it will be to corruption pressures and the more apt it will be to distort and corrupt the social processes it is intended to monitor."
Review at the Economist.
Euclid: The Game
A marvelous little game that treats Euclidean construction theorems as puzzles to solve in a web application:
Play it here.
Hat tip: JWS.
Posted by Delta at 5:00 AM 1 comment:
When Blind People Do Algebra
From NPR:
A functional MRI study of 17 people blind since birth found that areas of visual cortex became active when the participants were asked to solve algebra problems, a team from Johns Hopkins reports in the Proceedings of the National Academy of Sciences.
This is not the case with the same test run on sighted people. Which is interesting, because it serves as evidence that the brain is more flexible, and can be re-wired for a greater multitude of jobs, than previously believed.
NY Times: Stop Grading to a Curve
An excellent article by Adam Grant, professor at the Wharton School of the University of Pennsylvania:
The more important argument against grade curves is that they create an atmosphere that's toxic by pitting students against one another. At best, it creates a hypercompetitive culture, and at worst, it sends students the message that the world is a zero-sum game: Your success means my failure.
Epsilon-Delta, Absolute Values, Inequalities
Working through the famed "baby" Rudin, Principles of Mathematical Analysis. (Which was not the analysis book I used in grad school: we used William Ray's Real Analysis).
First thing that dawns on me is that I'm shaky on following the formal proofs of various limits. I decide that I need to brush up on my epsilon-delta proofs, and go back to do exercises from Stein/Barcellos Calculus and Analytic Geometry. I find that I get stuck outside of simple limits of linear functions (say, with a quadratic).
Second observation is that these proofs use extensive algebra involving absolute value inequalities, and my trouble is that, in turn, I am shaky with these to the point of being blind to a lot of very basic facts. Kind of frustrating that I teach basic algebra as a career but as soon as as absolute values and inequalities enter the picture I'm nigh-helpless.
Third observation is that this explains the copious section in college algebra and precalculus texts on absolute value inequalities (which are skipped in the curriculum at our community college). Previously my intuition had failed to see the use for these; but it's precisely the skills you need in analysis to write demonstrations involving the limit definition.
A few key properties on which I had to brush up (and would argue for preparatory exercises if I was teaching/scaffolding in the direction of analysis):
Subadditivity: \(|a+b| \leq |a| + |b|\). Equivalent to the triangle inequality.
Partial Reverse Triangle Inequality: \(|a| - |b| \leq |a - b|\). A bastardized name of my design. It follows from the preceding because \(|a| = |a - b + b| \leq |a-b| + |b|\), and then a subtraction of \(|b|\) from both sides. Gets used at the start of almost all my limit proofs.
Multiplicativeness: \(|ab| = |a||b|\). Which leads to manipulations such as: multiplying both sides of an equation by a positive number allows you to just multiply into an absolute value; squaring both sides can be done into the absolute value, etc.
More at: Wikipedia.
Discussion of general limit exercises: StackExchange.
Crypto Receipts for Homework
An interesting idea to a problem I've also experienced (student claiming they submitted work for which the instructor has no record): Individual cryptographic receipts for assignment submissions.
Crypto in the classroom: digital signatures for homework
Link: Everything is Fucked, The Syllabus
By Prof. Sanjay Srivastava, a proposed course on the overall breakdown of science in the field of social psychology:
Everything is fucked: The syllabus
Natural Normality
Normal curve in flag sticker water-leak (upside-down), 2016:
Teaching Math with Overhead Presentations
At our school, we currently have a mixture of classroom facilities. Some classrooms just have classic chalkboards (and nothing else), while other classrooms are outfit with whiteboards, computer lecterns, and overhead projectors. Well, actually: All of the classrooms on campus have the latter (for a few years now), excepting only the mathematics wing, which has been kept classical-style by consensus of the mathematics faculty.
That said, it's been one year since I've converted all of my classes over to use of the overhead projector for presentations. In our situation, I need to file room-change requests every semester to specially move my classes outside the math wing to make this happen. But I've been extremely happy with the results. While this may be very late to the party in academia as a whole, this identifies me as one of the "cutting-edge tech guys" in our department. Note that I'm using LibreOffice Impress for my presentations (not MS PowerPoint), which I need to carry on a mobile device or network drive. My general idiom is to have one slide of definitions, a theorem, or a process on the slide for about 30 minutes at a time, while I discuss and write exercises on the surrounding whiteboard by hand.
As a recent switchover, I wanted to document the reasons why I prefer this methodology while they're still fresh for me; and I say this as someone who rather vigorously defended carefully writing everything by hand on the chalkboard in the past. Here goes:
Advantages of Overhead Presentations in a Math Class
Saves time writing in class. (I think I recoup at least 20 minutes time in one of my standard 2-hour classes.)
Additional clarity in written notes. (I can depend on the presentation being laid out just-so, not dependent on my handwriting, emotional state, time pressures, etc.)
Can continue to face forward & speak towards the students most of the time.
Don't have to re-transcribe the same material every semester. (Which simply seems inefficient.)
Can have additional graphics, tables, and web links that are time-constrained by hand.
Easy to go back and pull up old slides to review at a later time. We may do this within one class period, after "erasure", or across prior lectures. For extremely weak community-college students who can't seem to remember any content from day to day, we frequently must pull up slides and definitions from earlier classes; and this gives confidence when students in fact disbelieve that the material was covered earlier. Actually, this is most critical in the lowest-level remedial courses (the first course I tried it on out of desperation one day, with great success and student responsiveness).
I can carry around the same slides on a tablet in class. This allows me to assist students while they work on exercises individually and I circulate. Similar to the prior point, if one student forgets, needs a reminder, or was absent in a prior class, I can show them the needed content in my hand without distracting the rest of the class.
Greatly helps reviewing for the final exam. I can quickly pull up any topic in its entirety from across the entire semester. (Again, trying to jog students' memories by showing it again in the exact same layout.)
Having the computer overhead allows demonstration of technical tools, like navigating the learning management system, using certain web tools, a programming compiler, etc.
Able to distribute the lecture material to students directly and digitally.
Enormously reduces paper usage. All of my former handouts, practice tests, science articles, etc., which used to generate stacks of copied paper are now shown on the overhead instead (on made available on the learning management system if students want a copy later).
Having written expressions in the LibreOffice math editor, it makes it compatible to copy-paste into other platforms like LATEX, MathJax, Wolfram Alpha, etc.
If for some reason I transition to a larger auditorium for a particular presentation (e.g., someone organized a "classroom showcase" of this type last fall), I'm ready to go with the same presentation available on a larger screen.
Less to erase (saves clean-up time). There's actually a mandated policy to clean the board after one's class at my school, and people have gotten very testy in the past if one suggests not doing that.
I only anticipated maybe the first 5 of these items above before starting to switch my classes over; once I saw all the very important side-benefits, I became a complete devotee, and transitioned all my classes. I have a hard time seeing that I'd ever want to go back now. That said, I need to be careful about a few things:
Problem Areas of Overhead Presentations in a Math Class
Boot-up time. I need to get into class about 10 minutes earlier to boot-up all the technology -- the computer itself, the network login, directories, starting LibreOffice mobile, opening the files, starting the browser if needed, etc. Our network is very slow and it's a bit unpleasant waiting for things to open up. Meanwhile, students are eager to ask questions, so overall this is a somewhat awkward/anxious moment in time. It would be better if our network were faster, and/or LibreOffice were installed on the local machines (which to date has been turned down by IT).
Presenter view doesn't work on our machines. This is normally a sidebar-view visible to the presenter which shows notes and the next slide to come, etc. Unfortunately our machines weren't supplied with a separate monitor feed to make this work. During the lecture I have to keep my tablet up and manually synchronized on the side to keep track of verbal notes and so forth.
Making slides available to students is imperfect. I can just give LibreOffice files, because most students don't have it, or possibly the technical confidence to install it. I can provide it as PDFs, but then students usually print them out one-giant-slide-per-page, resulting in everyone with a huge ream of paper. I've directed people to print them (if they must) 6-slides-per-page, but most can't follow that direction. At the moment, LibreOffice cannot directly export a multi-slide-per-page PDF. (And even the normally printing facility has a bug.)
Lecterns have an external power button, and unfortunately I tend to lean on the lectern, press up against the button, and shut off power to the entire system by accident. This is hugely embarrassing and triggers another 10-minute boot-up sequence.
Some lecterns also have a very short power-saving time set (like a default of 20 minutes; note that's less time than I expect to spend on a single slide, as stated above). Evey day I need to remember to set the power-save mode in those rooms to 60 minutes when I start.
Finally, based on my use-case, the handheld presentation clicker which I purchased is not as useful as first expected. It presents another tech item which has a several-minute discovery/boot-up sequence to get started, and I tend to press it by accident or pinch it in my pocket and advance the slide past what I'm talking about. For me I've found it better to discard that and just step to the keyboard when I need to advance the slide (again, just a few times per half-hour).
Nonetheless, all of these warning-notes are fairly minor and eminently manageable; the advantages of using the overhead in my classes pay dividends many times over. Among the top benefits are time-savings, clarity, ability to go back, opening up web access and other materials in class, digital distribution and compatibility, etc. In contrast, here is a question on MathOverflow from late 2009, where the responding mathematicians all seem very adamant about using traditional blackboards. At this point, I really don't grok any of these arguments for not using the overhead.
Paul Halmos on Proofs
Paul Halmos on mathematical proof:
Don't just read it; fight it!
Names for Inequalities
Consider an inequality of the form a < x < b, that is, a < x and x < b. Trying to find a name for this type of inequality, I'm finding a thicket of different terminology:
Chained inequalities (Wikipedia).
Combined inequalities (Sullivan Algebra & Trigonometry).
Compound Inequalities (Ratti & McWatters Precalculus, Bittinger Intermediate Algebra, OpenStax College Algebra).
Are there more? What is most common in your experience?
Purgradtory
Purgradtory (pur' gred tor e) noun, plural purgradtories.
The several days after submitting final grades when a community college teacher must field communications from students complaining about said grades, pleading for a change of grade, asking for new extra credit assignments, and/or declaring the need for a higher grade for transfer to some outside program.
Example: I will likely be in purgradtory through the middle of this week.
Traub on Open Admissions
As recounted by Scherer and Anson in their book, Community Colleges and the Access Effect (2014, Chapter 11):
Traub famously wrote in City on a Hill: Testing the American Dream at City College, a chronicling of the 1969 lowering of admissions standards motivated by the pursuit of equity, "Open admissions was one of those fundamental questions about which, finally, you had to make an almost existential choice. Realism said: It doesn't work. Idealism said: It must."
Jose Bowen's Tales from Cyberspace
A few weeks ago I went to a CUNY pedagogy conference at Hostos Community College. It featured a keynote speech by Jose Antonio Bowen, author of the book Teaching Naked, which is nominally a manifesto for flipped classrooms, in which more "pure" interactions can occur between students and instructors during class time. Weirdly, however, he spends the majority of his time waxing prosaically about how incredible, saturated, future-shocky technology is today, and how we must work mostly to provide everything to students outside of class time using this technology.
Here's how he started his TED-Talky address that Friday: He contrasted the once-a-week pay phone call home that college students would make a few decades ago ("Do dimes even exist anymore?") with the habits of college students today, supposedly contacting their parents a half-dozen times daily. In fact, he claimed, his 21-year-old daughter will actually call him for permission to date a young man when she first meets/starts chatting with him online. She supposedly argues in favor of the given caller by using three websites (shown floridly by Bowen on the projector behind the stage):
She has started chatting with the man on Tinder.
She has looked up his dating-review score on Lulu.
She has examined his current STD test status on Healthvana.
Now, that's heady stuff, and of course the audience of faculty and administrators "ooh"'ed and "ahh"'ed and "oh, my stars!"'ed in appropriate pearl-clutching fashion. Review dates and look up STD status before a date online? Kids these days -- we're so out of touch, we must change everything in the academy!
But this presentation doesn't pass the smell test. First of all, we should be suspicious of an adult daughter supposedly interrupting her real-time chat to "get permission" from her father. That's just sort of ridiculous. Admittedly at least Tinder really is a thing and you can chat on it; that much is true. (Although Bowen presented this as the daughter and a friend communally chatting to two guys together, which is not a group event that can actually happen.) But worse:
The dating-review site Lulu doesn't actually exist anymore. In February of this year (3 months ago), the site was acquired by Badoo and the dating-reviews shut down. If you go to the link above you'll realize that the whole site is offline as of this writing. (Link.) And:
You can't access anyone else's STD result on Healthvana. Yes, Healthvana is a site that allows you to quickly access and view your own STD results without returning to a doctor's office to pick them up. But it's only for your own results, and it requires an account and password to view them after a test. Obviously there are all kinds of federal regulations about keeping medical records private, so it's not even conceivable that those could be made available to the general public on a website. One might theoretically imagine a culture in which one pulls up your own STD records on a phone and shows it to someone you're meeting -- but there's no evidence that actually occurs, and of course it's strictly impossible in Bowen's account, in which his daughter had not yet physically met with her supposed suitor. (Link.)
That "Reefer Madness"-like scare-mongering accounted for the first 30 minutes of Bowen's hour-long presentation, at which point I couldn't take anymore bullshit and I got up and left the auditorium. In summary: The half of Bowen's presentation that I saw was entirely fabricated and fictitious, frankly designed to frighten older faculty and staff for some reason that is opaque to me. Keep that in mind if you pick up his book or see an article or presentation by Mr. Bowen.
How did I get clued in to the real situation with these websites, after my BS-warning radar first went off? I asked some 20-year-old friends of mine, who immediately told me that Lulu was shut down months ago, and Healthvana was nothing they'd ever heard of. Crazy idea, I know, actually talking to people without instantly fetishizing new technology.
Schmidt on Primary Teachers
Dooren et. al. ("The Impact of Preservice Teachers' Content Knowledge on Their Evaluation of Students' Strategies for Solving Arithmetic and Algebra Word Problems", 2002) summarize findings by S. Schmidt:
Nearly all students who wanted to become remedial teachers for primary and secondary education and about half of the future primary school teachers were unable to apply algebraic strategies properly or were reluctant to use them. Consequently, they experienced serious difficulties when they were confronted with more complex mathematical problems. Many of these preservice teachers perceived algebra as a difficult and obscure system based on arbitrary rules (Schmidt, 1994, 1996; Schmidt & Bednarz, 1997).
Noam Chomsky: Enumeration Leads to Language
One of my favorite videos, including a bit where famed linguist Noam Chomsky theorizes that a mutation in the brain regarding "likely recursive enumerations, allowed all human language".
I have a possibly dangerous inclination to mention this on the first day of an algebra class when we define different sets of numbers, which is possibly a time-sink and a distraction for students at that point. But still, this is the guts of the thing.
(Video should be starting at 33m25s.)
What Community College Students Understand
Reading this tonight -- Givvin, Karen B., James W. Stigler, and Belinda J. Thompson. "What community college developmental mathematics students understand about mathematics, Part II: The interviews." MathAMATYC Educator 2.3 (2011): 4-18. (Link.)
They make the argument that prior instructors' emphasis on procedure has overwhelmed students' natural, conceptual, sense-making ability. Now, I agree that terrible, knowledge-poor, even abusive math teaching in the K-6 time frame is endemic. But I'm a bit skeptical that students in this situation have a natural number sense waiting to be un-, or re-, covered.
In my experience, students in these courses commonly have no sense for numbers or magnitudes. Frequently they cannot even name numbers, decimals, places over a thousand, or know that multiplying by 10 appends a 0 to a whole number.
Givvin, et. al. assume that students "Like all young children, they had, no doubt, developed some measure of mathematical competence and intuition... ". I'm pretty skeptical of this claim (and see little evidence for it.)
One task is to check additions via subtractions, and quiz students on whether they know that either of the addends can be subtracted in the check. Admittedly, this is an unusual task: usually we take addition (or multiply or exponents) as the base operation, and later check the inverse via the more basic one -- for which the order definitely does matter. (Because addition is commutative but subtraction is not, etc.) So it's unsurprising that students' intuition is that the order matters in the check; as usually applied, it does.
Another student multiplies two fractions together when asked to compare them (obviously nonsensical). But it's easy to diagnose this: many instructors teach equating the fractions and then cross-multiplying them, and seeing which side has the higher resulting product; in this case the student scrambled the cross-multiplying of equations with multiplying fractions. Which highlights two things: teaching mangled mathematical writing as in this process leads to problems later; and the whole idea of cross-multiplying is so striking that it "sucks the oxygen out of the room" for other visually-similar concepts (like multiplying fractions).
The authors state that "These students lack an understanding of how important (and seemingly obvious) concepts relate (e.g., that 1/3 is the same as 1 divided by 3)." Not only is this not obvious, but I can repeat this about every day for a whole semester and still not have students remember it. Just this semester I had a student who literally couldn't repeat it when I just said it about five times.
The importance of "combining like terms", which is essentially the only concept under-girding the operation of addition (and subtraction and comparisons) -- in terms of like units, variables, radicals, common denominators, and decimal place values -- is highlighted here. I don't know how many times I express this, but I'm doubtful that any of my students have really ever understood what I'm saying. I wouldn't be surprised but some students could take a dozen years of classes and never understand this point. Which is dispiriting.
The authors have some lovely anecdotes of students making a small discovery or two within the context of the hour-to-two-hour interview. This they hold at as a hopeful sign that discovery-based learning might be an effective treatment. But I ask: How many of these students will remember their apparent discoveries outside the interview? I find it quite common for students to have "A-ha, that's so easy!" moments in class, and then have effectively no memory of it a day or two later. "I do fine when I'm with you, and then I can't do it on my own" is a fairly common refrain.
Discussing concepts is Element Two (of three) in the authors' list of prescriptions. "A teacher might, for instance, connect fractions and division, discussing that a fraction is a division in which you divide a unit into n number of pieces of equal size. Alternatively, the teacher might initiate a discussion of the equal sign, pointing out that it means 'is the same as' and not 'here comes the answer.'". Sure, I offer both of those specific explanations regularly, almost daily -- they're essential and without them you're not really discussing real math at all. But many of my students can't remember those foundational facts no matter how much I repeat or quiz them on it.
From my perspective, it almost as though most of my developmental students aggressively refuse to remember the overarching, connecting definitions and concepts that I try to share with them, even when they're immediately put to use within the scope of each daily class session.
I can't help but feel that the distinction between "procedures" versus "reasoning" is an artificial, untenable one. The authors admit, "Even efforts to capitalize on students' intuitions
(as with estimating) often quickly turn to rules and procedures (as in 'rounding to the nearest')". I think this argues, perhaps, for the following: All reasoning is ultimately procedural. The only question is knowing what definitions and qualities of a certain situation allow a given procedure to be applied (even so simple a one as comparing the denominators of 1/5 and 1/8, for example). Even counting is ultimately a learned procedure.
While I don't seem to have access to Part 1 of the same report, the initial draft report has a few other items I can't help but respond to:
"'Drill-and-skill' is still thought to dominate most instruction (Goldrick-Rab, 2007)." This is a now-common diatribe (my French-educated partner is aghast at the term). But let's compare to, say, the #1 top scientifically proven method for learning, according to a summary article by Dunlosky, et. al. ("What Works, What Doesn't", Scientific American Mind, Sep/Oct 2013). "Self-Testing... Unlike a test that evaluates knowledge, practice tests are done by students on their own, outside of class. Methods might include using flash cards (physical or digital) to test recall or answering the sample questions at the end of a textbook chapter. Although most students prefer to take as few tests as possible, hundreds of experiments show that self-testing improves learning and retention." Which is a somewhat elaborate way of saying: Practice and homework.
"The limitations in K-12 teaching methods have been well-documented in the research literature... An assumption we make in this report is that substantive improvements in mathematics learning will not occur unless we can succeed in transforming the way mathematics is taught." I would not so blithely accept that assumption. What it overlooks is the perennial decrepitude of mathematical understanding by K-6 elementary educators. My argument would be that it doesn't matter how many times you overhaul the curriculum or teaching methodology at that level; if the teachers themselves don't understand the concepts involved, there is no way that even the best curriculum or methods will be delivered or supported properly.
"Perhaps most disturbing is that the students in community college developmental mathematics courses did, for the most part, pass high school algebra. They were able, at one point, to remember enough to pass the tests they were given in high school." But were they, really? A few years ago when I was counseling a group of about a dozen of my community-college students, as they left the exit exam and thought that they had failed, I stumbled into asking exactly this question in passing: "But this is totally material that you took in junior high school, right?" To which one student replied, "But there it was just about buttering up the teacher so he liked you enough to pass you," and the other students present all nodded and seemed to agree with this. The evidence that students are being passed through the high school system on effectively fraudulent grounds seems, to me, nearly inescapable.
Near the end of the Part 2 article, the authors appear to express a bit more caution concerning their hypothesis; a cautionary question which I'd be prone to answer in the negative:
For some students we interviewed, basic concepts of number and numeric operations were severely lacking. Whether the concepts were once there and atrophied, or whether never sufficiently developed in the first place, we cannot be certain. What we do know is that these students' lack of conceptual understanding has, by the time they entered developmental math classes, significantly impeded the effectiveness of their application of procedures. (p. 14)
We hope that future work will seek to address questions such as whether community college is too late to draw upon students' intuitive concepts about math. Do those concepts still exist? Is community college too late to change students' conceptions of what math is? (p. 16)
On Piflars
In coordination with the week's theme of grammar -- seen on StackExchange: English Language & Usage:
Apparently in Slovenian, there is the single word "piflar", which derogatorily means "a student who only memorizes instead of truly learning". What would be the best comparable word for this in English?
StackExchange English Language & Usage: Derogatory word, describing person (a pupil) who memorizes instead of learning?
Gruesome Grammar
A week or so back we observed the rough consensus that basic arithmetic operations are essentially some kind of prepositions. Coincidentally, tonight I'm reviewing the current edition of "CK-12 Algebra - Basic" (Kramer, Gloag, Gloag; May 30, 2015) -- and the very first thing in the book is to get this exactly wrong. Here are the first two paragraphs in the book (Sec 1.1):
When someone is having trouble with algebra, they may say, "I don't speak math!" While this may seem weird to you, it is a true statement. Math, like English, French, Spanish, or Arabic, is a secondary language that you must learn in order to be successful. There are verbs and nouns in math, just like in any other language. In order to understand math, you must practice the language.
A verb is a "doing" word, such as running, jumps, or drives. In mathematics, verbs are also "doing" words. A math verb is called an operation. Operations can be something you have used before, such as addition, multiplication, subtraction, or division. They can also be much more complex like an exponent or square root.
That's the kind of thing that consistently aggravates me about open-source textbooks. While I love and agree with establishing algebraic notation as a kind of language -- as possibly the single most important overriding concept of the course -- to get the situation exactly wrong right off the bat (as well as using descriptors like PEMDAS, god help us) keep these as fundamentally unusable in my courses. This, of course, then leaves no grammatical position at all for relations (like equals) when they finally appear later in the text (Sec. 1.4). So close, and yet so far.
Link: Smart People Happier with Fewer Friends
Research by people at the London School of Economics and Singapore Management University that smarter people are happier with fewer friends, and fewer social interaction outings. Downside: The researchers are evolutionary psychologists and seek to explain the finding in those terms. Also: Uses the term "paleo-happiness".
Washington Post: Why smart people are better off with fewer friends
Poker Memory
Maybe 15 years ago, I went to the Foxwood poker tables (vs. 9 other people), got pocket Queens, and had an Ace come up on the flop. I maxed out the bet and lost close to $100. So the other morning I woke up and the thought in my head was, "I really should have computed the probability that someone else had an Ace". Which was 1 - 44P18/47P18 = 1 - 0.225 = 0.775 = 77.5%. Sometimes my brain works glacially slow like that.
Reading Radicals
In my development algebra classes, I push radicals further forward, closer to the start of the semester than most other instructors or textbooks. I want them to be discussed jointly with exponents, so we can really highlight the inverse relation with exponents, and that knowledge of the rules of one is effectively equivalent knowledge of the other. Also: Based on the statistics I keep, success on the exponents/radicals test is the single best predictor of success on the comprehensive, university-wide final exam.
There are, of course, many errors made by students learning to read and write radicals for effectively the first time. Here's an exceedingly common category, to write something like (\(x > 0\)):
$$\sqrt{16} = 4 = 2$$
$$\sqrt{x^8} = x^4 = x^2 = x$$
Any of these expressions may or may not have a radical written over them (including, e.g., \(\sqrt{4} = \sqrt{2}\)). That is: Students see something "magical" about radicals, and sometimes keep square-rooting any expression in sight, until they can no longer do so. This is common enough that I have few interventions in my mental toolbox ready for when this occurs in any class:
Go to the board and, jointly with the whole class, start asking some true-or-false questions. "T/F: \(\sqrt{4} = \sqrt{2}\) ← False. \(\sqrt{4} = 2\) ← True." Briefly discuss the difference, and the location of \(\sqrt{2}\) on the number line. Emphasize: Every written symbol in math makes a difference (any difference in the writing, and it has a different meaning).
Prompt for the following on the board. "Simplify: \(3 + 5 = 8\)." Now ask: "Where did the plus sign go? Why are you not writing it in the simplified expression? Because: You did the operation, and therefore the operational symbol goes away. The same will happen with radicals: If you can actually compute a radical, then the symbol goes away at that time."
That's old hat, and those are techniques I've been using for a few years now. The one new thing I noticed last night (as I write this) is that there is actually something unique about the notation for radicals: Of the six basic arithmetic operations (add, subtract, multiply, divide, exponents, radicals), radicals are the only binary operation where one of the two parameters may not be written. That is, for the specific case of square roots, there is a "default" setting where the index of 2 doesn't get written -- and there's no analogous case of any other basic operator being written without a pair of numbers to go with it.
I wonder if this contributes to the apparent "magical" qualities of radicals (specifically: students pay more attention to the visible numbers, whereas I am constantly haranguing students to look more closely at the operators in the writing)? Hypothetically, if we always wrote the index of "2" visibly for square roots (as for all other binary operators), would this be more transparent to students that the operator only gets applied once (at which point radical and index simplify out of the writing)? And perhaps this would clear up a related problem: students occasionally writing a reduction as a new index, instead of a factor (e.g., \(\sqrt{18} = \sqrt{9 \times 2} = \sqrt[3]{2}\))?
That would be a pretty feasible experiment to run in parallel classes, although it would involve using nonstandard notation to make it happen (i.e., having students explicate the index of "2" for square roots all the time). Should we consider that experiment?
What Part of Speech is "Times"?
What part of speech are the operational words "plus", "minus", and "times"? This is a surprisingly tricky issue; apparently major dictionaries actually differ in their categorization. The most common classification is as some form of preposition -- the Oxford Dictionary says that they are marginal prepositions; "a preposition that shares one or more characteristics with other word classes [i.e., verbs or adjectives]".
Here's an interesting thread on Stack Exchange: English Language & Usage on the issue -- including commentary by famed quantum-computing expert and word guru Peter Shor:
Stack Exchange English Language & Usage:
Is "times" really a plural noun?
Veterinary Homeopathy
A funny, but scary and real, web page of a homeopathic-practicing veterinarian who seems weirdly cognizant that it has no real effect:
How much to give: Each time you treat your pet, give approximately 10-20 of the tiny (#10) pellets in the amber glass vial, or 3-7 of the larger (#20) pellets in the blue plastic tube. You don't need to count them out. In fact, the number of pellets given per treatment makes no difference whatsoever. It is the frequency of treatment and the potency of the remedy that is important. Giving more pellets per treatment does not in any way affect the body's response. The pellets need not be swallowed, and it doesn't matter if a few of them are spit out. Just get a few pellets somewhere in the mouth, then hold the mouth shut for 3 seconds.
Jeffrey Levy, DVM PCH:
Classical Veterinary Homeopathy
The MOOC as a First Album
Thinking about MOOCs (which I am semi-infamously down on as a method for revolutionizing general education):
For rock bands, it's pretty common for their very first album to be considered their best one. Why is that? Well, the first album is likely the product of possibly a decade of practicing, writing, performing bars and clubs, interacting with audiences, and generally fine-tuning and refining the set to make the most solid block of music the band can possibly produce. At the point when a band gets signed to a label (traditionally), the first album is basically this ultra-tight set, honed for maximum impact over possibly hundreds of public performances.
But thereafter, the band is no longer in the same "lean and hungry" mode that produced that first set of music. Likely they go on tour for a year to support the first album, then are put in a studio for a few weeks with the goal of writing and recording a second album, so that the sales/promotion/touring cycle can pick up where the last one left off. This isn't a situation the band's likely to have experienced before, they have weeks instead of years to create the body of music, and they don't have hundreds of club audiences to run it by as beta-testers. In fact, they probably won't ever again have the opportunity of years of dry runs going into the manufacture a single album.
The same situation is likely to apply to MOOCs. A really good online class (and there are some) will be the product of a teacher who's taught the course live for a number of years (or decades), interacting with actual classrooms-full of students, refining the presentation many times as they witness how the presentation is immediately received by the people in front of them. If this has been done, say, hundreds of times, then you have a pretty strong foundation to begin recording something that will be a powerful class experience.
But if someone tries to develop an online course from scratch, in a few weeks isolated in an office without any live interaction as a testbed -- exactly as the band studio album situation -- what you're going to get is weak sauce, possibly entirely usable crap. If the instructor has never taught such a class in the past, then the result is likely just "kabuki" as a teacher that I once live-observed confessed to me. This is regardless if a person can do the math themselves, that's just total BS as a starting point for teaching.
A properly prepared, developed, scaffolded, explained course has got hundreds of moving parts built into it, built into every individual exercise, that are totally invisible unless an instructor has actually confronted live students with the issues at hand and seen the amazing kaleidoscope of ways that students can make mistakes or become tripped up or confused. No amount of "big data" is going to solve this (even assuming the MOOCs are even trying to do that and claims of such are not just flat-out fraud), because the tricky spots are so surprising, you'll never think to create a metric to measure it unless you're looking right over a student's shoulder to watch them do their work.
Quick metric for a quality online course: Has the instructor taught it live for a decade or more? Probably good. Did the instructor make it up on the fly, or in a few weeks development cycle? Probably BS.
GPS Always Overestimates Distances
Researchers in Austria and the Netherlands have pointed out that existing GPS applications almost always overestimate the distance of a trip, no matter where you're going. Why? Granted some small amount of random error in measuring each of the waypoints along a trip, the distance between erroneous points on a surface is overwhelmingly more likely to be greater than, rather than lesser than, the true distance -- -- and over many legs of a given trip, this error adds up to a rather notable overestimate. And to date no GPS application makers have corrected for it. A wonderful and fairly simple piece of math, one that was lurking under our noses for some time that no one thought to check, that should improve all of our navigation devices:
I, Programmer:
GPS Always Overestimates Distance
Excellent Exercises − Simplifying Radicals
Exploration of exercise construction, i.e., casting a net to catch as many mistakes as possible: See also the previous "Excellent Exercises: Completing the Square".
Below you'll see me updating my in-class exercises introducing simplification of radicals for my remedial algebra course a while back. (This occurred between one class on Monday, and different group on Tuesday, when I had the opportunity to spot and correct some shortcomings.) My customary process is to introduce a new concept, then give support for it (theorem-proof style), then do some exercises jointly with the class, and then have students do exercises themselves (from 1-3 depending on problem length) -- hopefully each cycle in a 30 minute block of time. In total, this snippet represents 1 hour of class time (actually the 2nd half of a 2-hour class session); the definitions and text shown is written verbatim on the board, while I'll be expanding or answering questions verbally. As I said before, I'm trying to bake as many iterations and tricky "stumbling blocks" into this limited exercise time as possible, so that I can catch and correct it for students as soon as we can.
Now, you can see in my cross-outs the simplifying exercises I was using at the start of the week, which had already gone through maybe two semesters of iteration. Obviously for each triad (instructor a, b, c versus students' d, e, f) I start small and present sequentially larger values. Also, for the third of the group I throw in a fraction (division) to demonstrate the similarity in how it distributes.
Not bad, but here some weaknesses I spotted this session that aren't immediately apparent from the raw exercises, and these are: (1) There are quite a few duplicates between the (now crossed-out) simplifying and later add/subtract exercises, which reduces real practice opportunities in this hour. (2) Is that I'm not happy about starting off with √8, which reduces to 2√2 -- this might cause confusion in a discussion for some students who don't see where the different "2"'s come from, something I try to avoid for initial examples. (3) Is that student exercises (c) and (d) both involve factoring out the perfect square "4", when I should have them getting experience with a wider array of possible factoring values. (4) Is that item (f) is √32, which raises the possibility of again factoring out either 4 or 16 -- but none of the instructor exercises demonstrated the need to look for the "greatest" perfect square, so the students weren't fairly for that case.
Okay, so at this point I realized that I had at least 4 things to fix in this slice of class, and so I was committed to rewriting the entirety of both blocks of exercises (ultimately you can see the revisions handwritten in pencil on my notes). The problem is that simplifying-radical problems are actually among the harder problems to construct, because there's a fairly limited range of values which are the product of a perfect square for which my students will be able to actually revert the multiplication (keeping in mind a significant subset of my college students don't actually know their times tables, so have to make guesses and sequentially add on fingers many times before they can get started).
So at this point I sat down and made a comprehensive list of all the smallest perfect square products that I could possibly ask for in-class exercises. I made the decision to use each one at most a single time, to get as much distinct practice in as possible. First, of course, I had to synch up like remainders to make like terms in the four "add/subtract" exercises -- these are indicated below by square boxes linking the like terms for those exercises. Then I circled another 6 examples, for use in the lead-in "simplifying" exercises, trying to find the greatest variety of perfect squares possible, not sequentially duplicating the same twice, and making sure that I had multiple of the "greatest perfect square" (i.e., involving 16 or 36) issue in both the instructor and student exercises. These, then, became my revised exercises for the two half-hour blocks, and I do think they worked noticeably better when I used them with the Tuesday class this week.
Some other stuff: The fact that add/subtract exercise (c) came out to √5 was kind of a happy accident -- I didn't plan on it, but I'm happy to have students re-encounter the fact that they shouldn't write a coefficient of "1" (many will forget that, and you need to have built-in reviews over and over again). Also, one might argue that I should have an addition exercise where you don't get like terms to make sure they're looking for that, but my judgement was that in our limited time I wanted them doing the whole process as much as possible (I'll leave non-like terms cases for book homework exercises).
Anyway, that's a little example of the many of issues involved, and the care and consideration, that it takes to construct really quality exercises for even (or especially) the most basic math class. As I said, this is about the third iteration of these exercises for me in the last year -- we'll see if I catch any more obscure problems the next time.
Nate Silver: Wrong on VAM
I like Nate Silver's FiveThiryEight site very much, and I think that its political coverage is very insightful. However, I could do without a lot of the site's pop-culture, sports, etc. filler. Another thing that they're wildly off-base about: they seem to be highly pro-VAM -- that's the Value-Added Metric, the reputed way of assessing teachers by student test scores -- which has been roundly shown to be a disastrously wrong (effectively random) metric in any serious study that I've seen, but about which 538 is super-supportive (for reasons that seem incoherent to me). Here's a very good outline of the critique:
Washington Post: Why Nate Silver's FiveThirtyEight blog is wrong about teacher evaluation
Where Are the Bodies Buried?
In my job which involves teaching lots of remedial classes at a community college (in CUNY), the students frequently have deep, yawning gaps in their basic math education. Many can't write clearly, they interchange digits and symbols, they don't know their multiplication tables, they can't long divide, they have trouble reading English sentence-puzzles, they've been taught bum-fungled "PEMDAS" mnemonics, they've been told that π = 22/7, etc., etc., etc.
So to a large part my job is to ask the question, police-detective style: "Where are the bodies buried?" For this particular crime that's been perpetrated on my students' brains, what exactly is causing the problem, what is the worst thing we can find about their conceptual understanding? Doing the easy introductory problems that immediately come to mind doesn't do dick. What I need to do, in our limited class time, is to dredge the the murky riverbed and pull out all the crap, broken, tricky, misunderstandings that are buried down there.
Another way of putting this is that the in-class exercises we use have to cast a wide net, and be constructed to not just do a single thing, but to demonstrate at least 2, 3, or 4 issues at once. (Again, if you had unlimited time and attention span to do hundreds of problems, this might not be an issue, but we have to maximize our punch in the class session.) I'm constantly revising my in-class exercises semester after semester as I realize some tricky detail that was pitched at my students along the way. I need to make sure that every tricky corner-case detail gets put in front of students so, if it's a problem, they can run into it and I get a chance to help them while we have time together.
This is a place where the poorly-made MOOCs and online basic math classes (like Khan Academy) really do a laughably atrocious job. Generally if you're a science-oriented person who can do math easily, and never taught live, then you're not aware of all the dozens of pitfalls that people can possibly run into during otherwise basic math procedures. So if someone like that just throws out the first math problem they can think of, it's going to be a trivial case that doesn't serve to dredge up all bodies lurking around the periphery. And you'll never know it through any digital feedback, and you'll never get a chance to improve the situation, because you're simply not measuring performance on the tricky side-issues in the first place; it remains hidden and forever submerged.
I'll plan to present some examples of exercise design and refinement in the future. For the moment, consider this article with other educators making the same critical observation about how bad the exercises at Khan Academy (and other poorly-thought-out MOOCs) are:
Washington Post: Does the Khan Academy know how to teach?
Graphing Quizzes at Automatic-Algebra
I added a few new things to the "automatic skill" site, Automatic-Algebra.org (actually around the start of the year, but they seem to have tested out well enough at this point). In particular, these are timed quizzes on the basic of graphing lines: (1) on linear equations in slope-intercept format, and (2) on parsing descriptions of special horizontal and vertical lines.
As usual, these are skills that when walks through them the first time in-class, with full explanations, may take several minutes; which may give a mistaken impression about how complicated the concepts really are. In truth, in a later course (precalculus, calculus, statistics), a person should be expected to see these relationships pretty much instantaneously on sight, and these timed quizzes better communicate that and allow the student to practice developing that intuition. If you have any feedback as you or your students use the site, I'd love to hear it!
Automatic-Algebra: Graphing Lines
Automatic-Algebra: Special Lines
On Correlation And Other Musical Mantras
A while back I found this delightful article at Slate.com, titled "The Internet Blowhard's Favorite Phrase". Perhaps more descriptive is the web-header title: "Correlation does not imply causation: How the Internet fell in love with a stats-class cliché". The article leads with a random internet argument, and then observes:
And thus a deeper correlation was revealed, a link more telling than any that the Missouri team had shown. I mean the affinity between the online commenter and his favorite phrase—the statistical cliché that closes threads and ends debates, the freshman platitude turned final shutdown. "Repeat after me," a poster types into his window, and then he sighs, and then he types out his sigh, s-i-g-h, into the comment for good measure. Does he have to write it on the blackboard? Correlation does not imply causation. Your hype is busted. Your study debunked. End of conversation. Thank you and good night... The correlation phrase has become so common and so irritating that a minor backlash has now ensued against the rhetoric if not the concept.
I find this to be completely true. Similarly, for some time, Daniel Dvorkin, the science fiction author, has used the following as the signature to all of his posts on Slashdot.org, which I find to be a wonderfully concise phrasing of the issue:
The correlation between ignorance of statistics and using "correlation is not causation" as an argument is close to 1.
Now, near the end of his article, the writer at Slate (Daniel Engberg), poses the following question:
It's easy to imagine how this point might be infused into the wisdom of the Web: "Facepalm. How many times do I have to remind you? Don't confuse statistical and substantive significance!" That comment-ready slogan would be just as much a conversation-stopper as correlation does not imply causation, yet people rarely say it. The spurious correlation stands apart from all the other foibles of statistics. It's the only one that's gone mainstream. Why?
I wonder if it has to do with what the foible represents. When we mistake correlation for causation, we find a cause that isn't there. Once upon a time, perhaps, these sorts of errors—false positives—were not so bad at all. If you ate a berry and got sick, you'd have been wise to imbue your data with some meaning... Now conditions are reversed. We're the bullies over nature and less afraid of poison berries. When we make a claim about causation, it's not so we can hide out from the world but so we can intervene in it... The false positive is now more onerous than it's ever been. And all we have to fight it is a catchphrase.
On this particular explanation of the phenomenon, I'm going to say "I don't think so". I don't think that people uttering the phrase by rote are being quite so thoughtful or deep-minded. My hypothesis for what's happening: The phrase just happens to have a certain poetical-musical quality to it that makes it memorable, and sticks in people's mind (moreso than other important dictums from statistics, as Engberg points out above). The starting "correlation" and the ending "causation" have this magical consonance in the hard "c", they both rhyme, they both have emphasis on the long "a" syllable, and the whole fits perfectly into a 4-beat measure. (A happy little accident, as Bob Ross might say.) It's this musical quality that gets it stuck in people's mind, possibly the very first thing that comes to mind for many people regarding statistics and correlation, ready to be thrown down in any argument whether on-topic or not.
I've run into the same thing by accident, for other topics, in my own teaching. For example: A year ago in my basic algebra classes I would run a couple examples of graphing 2-variable equations by plotting points, and at the end of the class make a big show of writing this inference on the board: "Lesson: All linear equations have straight-line graphs" -- and noted how this explained why equations of that type were in fact called "linear" (defined earlier in the course). This was received extremely well, and it was very memorable -- it was one of the few side questions I could always ask ("how do you know this equation has a straight-line graph?") that nobody ever failed to answer ("because it's linear").
Well, the problem is that it was actually TOO memorable -- people remembered this mantra without actually understanding what "linear" actually meant (of course: 1st-degree, with no visible exponents). I would always have to follow up with, "and what does linear mean?", to which almost no one could provide an answer. So in the fall semester, I took great care to instead write in my trio of algebra classes, "Lesson: All 1st-degree equations have straight-line graphs", and then verbally make the same point about where "linear" equations get there name. The funny thing is -- students would STILL make this same mistake of saying "linear equations are straight lines" without actually knowing how to identify a linear equation. It's such an attractive, musical, satisfying phrase that it's like a mental strange attractor -- it burrows into people's brains even when I never actually said it or wrote it in the class.
So I think we actually have to watch out for these "musical mantras" which are indeed TOO memorable, and allow students to regurgitate them easily and fool us into thinking they understand a concept when they actually don't.
See also -- Delta's D&D Hotspot: The Power of Pictures.
Posted by Delta at 11:06 PM No comments:
Lower Standards Are a Conspiracy Against the Poor
Andrew Hacker's at it again. Professor emeritus of political science from Queens College in CUNY, frequent contributor to the New York Times -- they love him for the "Man Bites Dog" headlines they can push due to him being the college-professor-who's-against-math. He got a lot of traction for the 2012 op-ed, Is Algebra Necessary? And he has a new book coming out now -- so, more articles on the same subject, like The Wrong Way to Teach Math, and Who Needs Advanced Math, and The Case Against Mandating Math for Students, and more. (I wrote previously about how Hacker's critique is essentially incoherent here.)
Now, his suggestions for what "everyone needs to know" are not bad; e.g., how to read a table or graph, understand decimals and estimations... (maybe that's it, actually?). I totally agree that everyone should know that -- at, say, the level of a 7th or 8th-grade home-economics course, perhaps. To suggest that this is proper fare for college instruction would be comically outrageous -- if it weren't seriously being considered by top-level administrators at CUNY. Here are some choice things he's said recently in the articles linked above:
"I sat in on several advanced placement classes, in Michigan and New York. I thought they would focus on what could be called 'citizen statistics.'... My expectations were wholly misplaced. The A.P. syllabus is practically a research seminar for dissertation candidates. Some typical assignments: binomial random variables, least-square regression lines, pooled sample standard errors..." -- I'd say that these concepts are so incredibly basic, the very idea of regression and correlation so fundamental, for example, that you couldn't even call it a statistics class without those topics.
"Q: Aren't algebra and geometry essential skills? A: The number of people who use either in their jobs is tiny, at most 5 percent. You don't need that kind of math for coding. It's not a building block." -- The idea that algebra concepts aren't necessary for coding, that someone who doesn't grasp the idea of a variable wouldn't be entirely helpless at coding (I've seen it!), in my personal opinion, essentially qualifies as fraud.
Okay, so statistics and coding are clearly not Hacker's area of expertise -- we might wonder why he feels confident in pontificating in these areas, and recommending truly radical reductions in standards, at all. Many of us would opine that the social-science departments have much weaker standards than the STEM fields; so perhaps we might generously say it's just a skewed perspective in this regard.
But the thing is, behind closed doors administrators know that students without math skills can't succeed at further education, and they can't succeed at technical jobs. That said, they are not incited to communicate that fact to anyone. What that they are grilled about by the media and political stakeholders are graduation rates, which at CUNY are pretty meager; around 20% for most of the community colleges. If the administration could wipe out 7th-grade math as a required expectation, then they'd be celebrated (they think) for being able to double graduation rates effectively overnight. And someone like Hacker is almost invaluable in giving them political cover for such a move.
Let's look at some recent evidence for who really benefits when math standards are reduced.
"My first time in a fifth grade in one of New Jersey's most affluent districts (white, of course), I asked where one-third was on the number line. After a moment of quiet, the teacher called out, "Near three, isn't it?" The children, however, soon figured out the correct answer; they came from homes where such things were discussed. Flitting back and forth from the richest to the poorest districts in the state convinced me that the mathematical knowledge of the teachers was pathetic in both. It appears that the higher scores in the affluent districts are not due to superior teaching in school but to the supplementary informal "home schooling" of children." -- Patricia Clark Kenschaft, "Racial equity requires teaching elementary school teachers more mathematics", Notices of AMS 52.2 (2005): 208-212.
"And while the proportion of American students scoring at advanced levels in math is rising, those gains are almost entirely limited to the children of the highly educated, and largely exclude the children of the poor. By the end of high school, the percentage of low-income advanced-math learners rounds to zero..." -- Peg Tyre, 'The Math Revolution", The Atlantic (March 2016).
That is: Cutting math standards only really cuts it for the poor. The rich will still make sure that their children have solid math skills at all levels. Or in other words: Cutting math standards increases inequality in education, and thus later economic status. And this folds into the overwhelming number of signs we've seen that math knowledge among our elementary-school teachers is perennially, pitifully weak, and a major cause of ongoing math deficiencies among our fellow citizens.
I wonder: Is there any correlation between this and the crazy election cycle that we're experiencing now? Thanks to a close friends for the idea for the title to this article.
P.S. Here's Ed from the wonderful Gin and Tacos writing on the same subject today. I agree with every word, and he goes into more detail than I did here (frankly, Hacker's crap makes me so angry I can't read every part of what he says). Ed's a political science professor himself, and also plays drums, which makes me feel a bit bad that I threw any shade at all on the social sciences above. Be smart, be like Ed.
Link: Math Circles at the Atlantic
An article this month at the Atlantic on the explosive rise of extracurricular (and expensive) advanced-math circles and competitions, to make up for the perceived deficiencies in math education in schools. Some telling quotes:
At a time when calls for a kind of academic disarmament have begun echoing through affluent communities around the nation, a faction of students are moving in exactly the opposite direction...
"The youngest ones, very naturally, their minds see math differently [said Inessa Rifkin, co-founder of Russian School of Mathematics]... It is common that they can ask simple questions and then, in the next minute, a very complicated one. But if the teacher doesn't know enough mathematics, she will answer the simple question and shut down the other, more difficult one. We want children to ask difficult questions, to engage so it is not boring, to be able to do algebra at an early age, sure, but also to see it for what it is: a tool for critical thinking. If their teachers can't help them do this, well... It is a betrayal."
And while the proportion of American students scoring at advanced levels in math is rising, those gains are almost entirely limited to the children of the highly educated, and largely exclude the children of the poor. By the end of high school, the percentage of low-income advanced-math learners rounds to zero...
The No Child Left Behind Act... demanded that states turn their attention to getting struggling learners to perform adequately...The cumulative effect of these actions, perversely, has been to push accelerated learning outside public schools—to privatize it, focusing it even more tightly on children whose parents have the money and wherewithal to take advantage. In no subject is that clearer today than in math.
The Atlantic: The Math Revolution
Link: Common Core Battles
A nice overview of the history of the battles around Common Core. Starts with a surprising anecdote about Bill Gates getting the brush-off when he personally met with Charles Koch to discuss the issue. Re: George W. Bush's "No Child Left Behind", an aggravatingly familiar development:
To bring themselves closer to 100%, many states simply lowered the score needed to pass their tests. The result: In 2007, Mississippi judged 90% of its fourth graders "proficient" on the state's reading test, yet only 19% measured up on a standardized national exam given every two years. In Georgia, 82% of eighth-graders met the state's minimums in math, while just 25% passed the national test. A yawning "honesty gap," as it came to be known, prevailed in most states.
Peter Elkind: Business Gets Schooled
Hembree on Math Anxiety
Reviewing a 1990 paper by Ray Hembree on math anxiety; a meta-study of approximately 150 papers, with a combined total of about 25,000 subjects. (Note the high sample size makes almost all findings significant at the p < 0.01 level). Math anxiety is known to be negatively correlated with performance in math (tests, etc.), and more common among women than men.
Math anxiety is somewhat correlated with a constellation of other general anxieties (r² = 0.12 to 0.27). Work to enhance math competence did not reduce anxiety.
Whole-group interventions are not effective (curricular changes, classroom pedagogy structure, in-class psychological treatments). The only thing that is effective is out-of-classroom, one-on-one treatments (behavioral systematic desensitization; cognitive restructuring); these have a marked effect at both lowering anxiety and boosting actual math-test performance.
In short: Addressing math anxiety is largely out of the hands of the classroom teacher. Unless the student has access, or the institution provides access, to one-on-one behavioral desensitization therapy, no group-level interventions are found to be effective.
Also recall that elementary education majors have the highest math anxiety, and the lowest math performance, of all U.S. college majors. (It seems possible that some entrants choose elementary education as a career path precisely because they are bad at math and see that as one of their limited options; I know I've had at least one such student say something to that effect to me.) This clearly dovetails with Sian Beilock's 2009 finding that math-anxious female elementary teachers model math-anxiety particularly to their female students, who imitate the same and wind up with worse math performance and attitudes by the end of the year (link). And this general trend of weak education majors has been the case in the U.S. for at least a century now (link).
So we might hypothesize: A feedback loop exists between poor early math education, heightened math anxiety among female students, and those same students returning to early childhood education as a career.
See below for Hembree's table of math anxiety by class and major (p. 41); note that elementary education majors, and those taking the standard "math for elementary teachers" (frequently the only math class such teachers take), are significantly worse off than anyone else:
Hembree, Ray. "The nature, effects, and relief of mathematics anxiety." Journal for research in mathematics education (1990): 33-46. (Link)
Link: The Learning Styles Neuromyth
A nice article reminding us that the whole idea of teaching to different "learning styles" is entirely without any scientific evidence in its favor:
"... the brain's interconnectivity makes such an assumption unsound."
Olivia Goldhill: The concept of different "learning styles" is one of the greatest neuroscience myths
Link: Study Time Decline
An interesting article analyzing the history of reported study time decline for U.S. college students.
Point 1: Study time dramatically decreased in the 1961-1981 era (from about 24 hrs/week to 16 hrs/week), but has been close to stable since that time.
Point 2: In that same early period, it seems that faculty expectations on teaching vs. research flip-flopped in that same early time period (about 70% prioritized teaching over research around 1975, with the proportion quickly dropping to about 50/50 by the mid-80's).
Alexander McCormick: It's about Time: What to Make of Reported Declines in How Much College Students Study
When Dice Fail
Some of the more popular posts on my gaming blog have been about how to check for balanced dice, using Pearson's chi-square test (testing a balanced die, testing balanced dice, testing balanced dice power). One of the observations in the last blog was that "chi-square is a test of rather lower power" (quoting Richard Lowry of Vassar College); to the extent that I've never had any dice that I've checked actually fail the test.
Until now. Here's the situation: A while back my partner Isabelle, preparing entertainment for a long trip, picked up a box of cheap dice at the dollar store around the corner from us. These dice are in the Asian-style arrangement, with the "1" and "4" pip sides colored red (I believe because the lucky color red is meant to offset those unlucky numbers):
A few weeks ago, it occurred to me that these dice are just the right size for an experiment I run early in the semester with my statistics students: namely, rolling a moderately large number of dice in handful batches and comparing convergence to the theoretically-predicted proportion of successes. In particular, the plan is customarily to roll 80 dice and see how many times we get a 5 or 6 (mentally, I'm thinking in my Book of War game, how many times can we score hits against opponents in medium armor -- but I don't say that in class).
So when we did that in class last week, it seemed like the number of 5's and 6's was significantly lower than predicted, to the extent that it actually threw the whole lesson under a shadow of suspicion and confusion. I decided that when I got a chance I'd better test these dice before using them in class again. Following the findings of the prior blog on the "low power" issue, I knew that I had to get on the order of about 500 individual die-rolls in order to get a halfway decent test; in this case with a boxful of 15 dice, it seemed was convenient to make 30 batched rolls for 15 × 30 = 450 total die rolls... although somewhere along the way I lost count of the batches and wound up actually making 480 die rolls. Here are the results of my hand-tally sheet:
As you can see at the bottom of that sheet, this box of dice actually does fail the chi-square test, as the \(SSE = 1112\) is in fact greater than the critical value of \(X \cdot E = 11.070 \cdot 80 = 885.6\).Or in other words, with a chi-square value of \(X^2 = SSE/E = 1112/80 = 13.9\) and degrees of freedom \(df = 5\), we get a P-value of \(P = 0.016\) for this hypothesis test of the dice being unbalanced; that is, if the dice really were balanced, there would be less than a 2% chance of getting an SSE value this high by natural variation alone.
In retrospect, it's easy to see what the manufacturing problem is here: note in the frequency table that it's specifically the "1"'s and the "4"'s, the specially red-colored faces, that are appearing in a preponderance of the rolls. In particular, the "1" face on each die is drilled like an enormous crater compared to the other pips; it's about 3 mm wide and about 2 mm deep (whereas other pips are only about 1 mm in both dimensions). So the "6" on the other side from the "1" would be top heavy, and tends to roll down to the bottom, leaving the "1" on top more than anything else. Also, the corners of the die are very rounded, making it easier for them to turn over freely or even get spinning by accident.
Perhaps if the experiment in class had been to count 4's, 5's, and 6's (that is: hits against light armor in my wargame), I never would have noticed the dice being unbalanced (because together those faces have about the same weight as the 1's, 2's, and 3's together)? On the one hand my inclination is to throw these dice out so they never get used again in our house by accident; but on the other hand maybe I should keep them around as the only example that the chi-square test has managed to succeed at rejecting to date.
Link: Tricky Rational Exponents
Consider the following apparent paradox:
\(-1 = (-1)^1 = (-1)^\frac{2}{2} = (-1)^{2 \cdot \frac{1}{2}} = ((-1)^2)^\frac{1}{2} = (1)^\frac{1}{2} = \sqrt{1} = 1\)
Of the seven equalities in this statement, exactly which of them are false? Give a specific number between (1) and (7). Join in the discussion where I posted this at StackExachange, if you like:
StackExchange:
What are the Laws of Rational Exponents?
Seat Belt Enforcement
Yesterday in the Washington Post, libertarian police-abuse crusader Radley Balko wrote an opinion piece arguing against mandatory seat-belt laws. He opens:
The ACLU of Florida just released a report showing that in 2014, black motorists in the state were pulled over for seat belt violations at about twice the rate of white motorists... Differences in seat belt use don't explain the disparity. Blacks in Florida are only slightly less likely to wear seat belts. The ACLU points to a 2014 study by the Florida Department of Transportation that found that 85.8 percent of blacks were observed to be wearing seat belts vs. 91.5 percent of whites. The only possible explanation for the disparity that doesn't involve racial bias might be that it's easier to spot seat-belt violations in urban areas than in more rural parts of the state... even if it did explain part or all of the disparity, it still means that blacks in Florida are disproportionately targeted.
Here's the problem with that math: the Florida study would in fact be evidence that blacks are failing to wear seat belts at about twice the rate of whites. According to those numbers, the rate of blacks not wearing seat belts would be: 100% - 85.8% = 14.2%, while the rate of non-compliance for whites would be 100% - 91.5% = 8.5%. And as a ratio, 14.2%/8.5% = 1.67, or pretty close to 2 (double) if you round to the nearest multiple.
Now, I actually think that Radley Balko has done some of the very best, most dedicated work drawing our attention to the problem of hyper-militarized police tactics in recent years (and decades); see his book Rise of the Warrior Cop for more. And I'm pretty sensitive to issues of overly-punitive enforcement and fines that are repressively punishing to the poor; back in the 80's I used to routinely listen to Jerry Williams on WRKO radio in Boston, when he was crusading against the rise of seat-belt laws, and I found those arguments, at times, compelling.
But sometimes Balko gets his arguments scrambled up, and this is one of those cases. His claim that there's only one "possible explanation for the disparity" fails on the grounds that these numbers are evidence that, in general, there's actually no disparity in enforcement at all; enforcement tracks the non-compliance ratio very closely. He can do better than to hang his hat in this case on a fundamental misunderstanding of the numbers involved.
Posted by Delta at 10:18 AM No comments: | CommonCrawl |
\begin{document}
\title[]{Inverse spectral problems for non-self-adjoint Sturm-Liouville operators with discontinuous boundary conditions} \author{Jun Yan and Guoliang Shi} \subjclass[2010]{Primary 34A55; Secondary 34L40, 34L20} \address{School of Mathematics, Tianjin University, Tianjin, 300354, People's Republic of China} \keywords{diffusions, eigenvalues, non-self-adjoint, multiplicity} \email{[email protected]} \address{School of Mathematics, Tianjin University, Tianjin, 300354, People's Republic of China} \email{[email protected]} \date{\today } \keywords{Non-selfadjoint Sturm-Liouville operators, inverse problem, eigenvalue, norming constant}
\begin{abstract} This paper deals with the inverse spectral problem for a non-self-adjoint Sturm-Liouville operator with discontinuous conditions inside the interval. We obtain that if the potential $q$ is known a priori on a subinterval $ \left[ b,\pi \right] $ with $b\in \left( d,\pi \right] $ or $b=d$, then $h,$ $\beta ,$ $\gamma \ $and $q$ on $\left[ 0,\pi \right] \ $can be uniquely determined by partial spectral data consisting of a sequence of eigenvalues and a subsequence of the corresponding generalized normalizing constants or a subsequence of the pairs of eigenvalues and the corresponding generalized ratios. For the case $b\in \left( 0,d\right) ,$ a similar statement holds if $ \beta ,$ $\gamma \ $are also known a priori. Moreover, if $q$ satisfies a local smoothness condition, we provide an alternative approach instead of using the high-energy asymptotic expansion of the Weyl $m$-function to solve the problem of missing eigenvalues and norming constants. \end{abstract}
\maketitle
\section{Introduction}
In this paper, we consider the non-self-adjoint Sturm-Liouville operator $ L:=L\left( q,h,H,\beta ,\gamma ,d\right) $ defined by \begin{equation} \ell y:=-y^{\prime \prime }+q\left( x\right) y \label{ly} \end{equation} on the interval $\left( 0,\pi \right) \ $with the boundary conditions \begin{equation} U\left( y\right) :=y^{\prime }\left( 0\right) -hy\left( 0\right) =0,V\left( y\right) :=y^{\prime }\left( \pi \right) +Hy\left( \pi \right) =0 \label{BCs} \end{equation} and the discontinuous conditions \begin{equation} y\left( d+0\right) =\beta y\left( d-0\right) \text{, }y^{\prime }\left( d+0\right) =\beta ^{-1}y^{\prime }\left( d-0\right) +\gamma y\left( d-0\right) \text{,} \label{DCs} \end{equation} where\ $q\in L_{
\mathbb{C}
}^{1}\left[ 0,\pi \right] $ is complex-valued, $h,$ $H\in
\mathbb{C}
\cup \left\{ \infty \right\} ,$ {$\gamma \in
\mathbb{C}
$ }and {{$\beta \in
\mathbb{R}
$, $\beta >0$}}. Note that, in an obvious notation, $h=\infty $ and $ H=\infty $ single out the Dirichlet boundary conditions \begin{equation*} U^{\infty }\left( y\right) :=y\left( 0\right) =0\text{ and }V^{\infty }\left( y\right) :=y\left( \pi \right) =0\text{,} \end{equation*} respectively. One notes that in the special case $\beta =1,$ $\gamma =0,$ the operator $L$ reduces to the classical Sturm-Liouville operator without discontinuities.
Sturm-Liouville operators with discontinuities inside the interval arise in mathematics, mechanics, geophysics, and other fields of science and technology. The inverse spectral problems of such operators is of central importance in disciplines ranging from engineering to the geosciences. For example, discontinuous inverse problems appear in electronics for constructing parameters of heterogeneous electronic lines with desirable technical characteristics \cite{1,2}. In\ the last decades, inverse spectral problems for Sturm-Liouville operators with different type discontinuities have attracted tremendous interest \cite {amirov,wei1,hald,mmd,nk,ozkan,ok,shien,wyp,wang,xu,ycf,Yurko,YGMA,YIXUAN,TESCHL} . These start with the fundamental work given by V. Ambarzumian \cite{a} and then by G. Borg \cite{borg}, B. Levitan \cite{levitan1,levitan2}, and V. Marchenko \cite{marchen,vam} for the classical Sturm-Liouville operators.
cations, which are left to the reader.
We emphasize that in 1984, O. H. Hald \cite{hald} first generalized Hochstadt--Lieberman's theorem \cite{lie} to the Sturm--Liouville operator $ L $, that is, if $H$ is given, $q$ is known on $\left[ \frac{\pi }{2},\pi \right] $ and $d\in \left( 0,\frac{\pi }{2}\right) $, then one spectra can uniquely determine $h,\beta ,\gamma ,d$ and $q$ on $\left[ 0,\pi \right] .$ Motivated by this work, increasing attention has been given to the inverse spectral problem of recovering the operator $L$ in the self-adjoint case with partial information given on the potential \cite{shien,xu,ycf}$.$ In contrast, such inverse spectral problem for the non-self-adjoint case has in general been studied considerably less, and it is precisely the starting point of this paper.$\ $We investigate the uniqueness problem of determining the non-self-adjoint operator $L$ with only partial information of $q,$ of the eigenvalues, and of the generalized norming constants. What should be noted is that in the non-self-adjoint setting, complex eigenvalues and multiple eigenvalues may appear, and thus many new ideas and additional effort are required. Before describing the content of this paper, let us first give some notations and basic facts.
To avoid too many case distinctions in the proofs of this paper, we assume that $h\in
\mathbb{C}
$. Nevertheless, we expect that the method of the paper can be applied in the case $h=\infty $. For simplicity we use the notations $B$ and $B^{\infty }$ for the boundary value problems corresponding to $L$ with $H\in
\mathbb{C}
\ $and $H=\infty ,$ respectively. Assume that $\varphi \left( x,\lambda \right) $, $\psi \left( x,\lambda \right) ,$ $\psi ^{\infty }\left( x,\lambda \right) $ are solutions of the equation \begin{equation} \ell y=-y^{\prime \prime }+q\left( x\right) y=\lambda y \label{ly1} \end{equation} satisfying the discontinuous conditions (\ref{DCs}) and the initial conditions \begin{eqnarray*} \varphi \left( 0,\lambda \right) &=&1,\left. \frac{d\varphi \left( x,\lambda \right) }{dx}\right\vert _{x=0}=h\in
\mathbb{C}
, \\ \psi \left( \pi ,\lambda \right) &=&1,\left. \frac{d\psi \left( x,\lambda \right) }{dx}\right\vert _{x=\pi }=-H\in
\mathbb{C}
, \\ \psi ^{\infty }\left( \pi ,\lambda \right) &=&0,\text{ }\left. \frac{d\psi ^{\infty }\left( x,\lambda \right) }{dx}\right\vert _{x=\pi }=1, \end{eqnarray*} respectively. Then it is easy to see that eigenvalues of $B$ and $B^{\infty } $ are precisely the zeros of \begin{equation} \Delta \left( \lambda \right) :=\left\langle \psi \left( x,\lambda \right) ,\varphi \left( x,\lambda \right) \right\rangle =V\left( \varphi \right) =-U\left( \psi \right) \label{w} \end{equation} and \begin{equation} \Delta ^{\infty }\left( \lambda \right) :=\left\langle \psi ^{\infty }\left( x,\lambda \right) ,\varphi \left( x,\lambda \right) \right\rangle =-V^{\infty }\left( \varphi \right) =-U\left( \psi ^{\infty }\right) , \label{w2} \end{equation} respectively, where $\left\langle y(x),z(x)\right\rangle :=y(x)z^{\prime }(x)-y^{\prime }(x)z(x)$. Thus $\Delta \left( \lambda \right) $ and $\Delta ^{\infty }\left( \lambda \right) $ are called the characteristic functions of $B$ and $B^{\infty },$ respectively. Throughout this paper, the \textbf{ algebraic multiplicity} of an eigenvalue is the order of it as a zero of the corresponding characteristic function.
\
\begin{notation} \label{615 copy(1)}$(1)$ We denote by $\sigma \left( B\right) :=\left\{ \lambda _{n}\right\} _{n\in
\mathbb{N}
_{0}}$ the sequence of all the eigenvalues of $B$ and denote by $\sigma \left( B^{\infty }\right) :=\left\{ \lambda _{n}^{\infty }\right\} _{n\in
\mathbb{N}
_{0}}$ the sequence of all the eigenvalues of $B^{\infty }$. The eigenvalues are assumed to be repeated according to their algebraic multiplicities and labeled in order of increasing moduli. In addition, identical eigenvalues are adjacent.
$(2)$ Denote \begin{equation*} S_{B}:=\left\{ n\in
\mathbb{N}
|\lambda _{n-1}\neq \lambda _{n}\right\} \cup \left\{ 0\right\} ,S_{B^{\infty }}:=\left\{ n\in
\mathbb{N}
|\lambda _{n-1}^{\infty }\neq \lambda _{n}^{\infty }\right\} \cup \left\{ 0\right\} . \end{equation*}
$(3)$ The symbol $m_{n}$ denotes the algebraic multiplicity of the eigenvalue $\lambda _{n},n\in S_{B},$ and $m_{n}^{\infty }$\ denotes the algebraic multiplicity of $\lambda _{n}^{\infty },$ $n\in S_{B^{\infty }}.$ For sufficiently large $n$ it is well known that $m_{n}^{\infty }=m_{n}=1$ (see Lemma\ 2.3 in \cite{YIXUAN}). \end{notation}
Now we turn to give the definition of the \textit{generalized norming constants }for the problem $B$. Denote \begin{equation} \kappa _{n+\nu }:=\varphi _{n+\nu }\left( \pi \right) ,\text{ }\alpha _{n+\nu }:=\int_{0}^{\pi }\psi _{n+\nu }\left( x\right) \psi _{n+m_{n}-1}\left( x\right) dx\text{,} \label{ratios} \end{equation} where $n\in S_{B}$, $\nu =0,1,\ldots ,m_{n}-1$, and \begin{eqnarray} &&\varphi _{n+\nu }\left( x\right):=\varphi _{\nu }\left( x,\lambda _{n}\right) :=\dfrac{1}{\nu !}\left. \dfrac{d^{\nu }}{d\lambda ^{\nu }} \varphi \left( x,\lambda \right) \right\vert _{\lambda =\lambda _{n}}\text{, } \label{9} \\ &&\psi _{n+\nu }\left( x\right):=\psi _{\nu }\left( x,\lambda _{n}\right) := \dfrac{1}{\nu !}\left. \dfrac{d^{\nu }}{d\lambda ^{\nu }}\psi \left( x,\lambda \right) \right\vert _{\lambda =\lambda _{n}}. \end{eqnarray} Then $\kappa _{n}$ and $\alpha _{n}$, $n\in
\mathbb{N}
_{0},$ are called the \textit{generalized} \textit{norming constants} corresponding to $\lambda _{n}.$ To distinguish $\kappa _{n}$ and $\alpha _{n},$ in this paper, $\kappa _{n}$ is called the \textit{generalized} \textit{ratio}, and $\alpha _{n}$ is called the \textit{generalized normalizing constant}. Moreover, it follows from \cite[Theorem 4.1]{YIXUAN} that for $n\in S_{B}$, $\nu =0,1,\ldots ,m_{n}-1$,
\begin{equation} \left. \frac{d^{m_{n}+\nu }\Delta \left( \lambda \right) }{d\lambda ^{m_{n}+\nu }}\right\vert _{\lambda =\lambda _{n}}=-\left( m_{n}+\nu \right) !\sum_{j=0}^{\nu }\kappa _{n+j}\alpha _{n+\nu -j}\text{.} \label{24} \end{equation} Note that when the multiplicity $m_{n}=1$, the generalized \textit{norming constants} $\kappa _{n}$ and $\alpha _{n}$ coincide with the \textit{norming constants} for the operator $L$ in the self-adjoint case (see \cite{Yurko}).
Actually, $\varphi _{\nu }\left( x,\lambda _{n}\right) $ and $\psi _{\nu }\left( x,\lambda _{n}\right) $ are the generalized eigenfunctions of $B$ corresponding to the eigenvalue $\lambda _{n},$ $n\in S_{B}.$ In fact, for $ \nu =1,2,\ldots ,m_{n}-1,$ we notice that \begin{equation} \left\{ \begin{array}{l} \ell \varphi _{\nu }\left( x,\lambda _{n}\right) =\lambda _{n}\varphi _{\nu }\left( x,\lambda _{n}\right) +\varphi _{\nu -1}\left( x,\lambda _{n}\right) \text{, } \\ \varphi _{\nu }\left( d+0,\lambda _{n}\right) =\beta \varphi _{\nu }\left( d-0,\lambda _{n}\right) \text{, } \\ \varphi _{\nu }^{\prime }\left( d+0,\lambda _{n}\right) =\beta ^{-1}\varphi _{\nu }^{\prime }\left( d-0,\lambda _{n}\right) +\gamma \varphi _{\nu }\left( d-0,\lambda _{n}\right) \text{, } \\ \varphi _{\nu }\left( 0,\lambda _{n}\right) =\varphi _{\nu }^{\prime }\left( 0,\lambda _{n}\right) =0\text{,} \end{array} \right. \label{7} \end{equation} \begin{equation} \left\{ \begin{array}{l} \ell \psi _{\nu }\left( x,\lambda _{n}\right) =\lambda _{n}\psi _{\nu }\left( x,\lambda _{n}\right) +\psi _{\nu -1}\left( x,\lambda _{n}\right) \text{,} \\ \psi _{\nu }\left( d+0,\lambda _{n}\right) =\beta \psi _{\nu }\left( d-0,\lambda _{n}\right) \text{, } \\ \psi _{\nu }^{\prime }\left( d+0,\lambda _{n}\right) =\beta ^{-1}\psi _{\nu }^{\prime }\left( d-0,\lambda _{n}\right) +\gamma \psi _{\nu }\left( d-0,\lambda _{n}\right) \text{, } \\ \psi _{\nu }\left( \pi ,\lambda _{n}\right) =\psi _{\nu }^{\prime }\left( \pi ,\lambda _{n}\right) =0\text{.} \end{array} \right. \label{8} \end{equation} and \begin{eqnarray} \frac{1}{\nu !}\Delta ^{\left( \nu \right) }\left( \lambda _{n}\right) &=&\varphi _{\nu }^{\prime }\left( \pi ,\lambda _{n}\right) +H\varphi _{\nu }\left( \pi ,\lambda _{n}\right) =0,\text{ } \label{3f} \\ \frac{1}{\nu !}\Delta ^{\left( \nu \right) }\left( \lambda _{n}\right) &=&-\psi _{\nu }^{\prime }\left( 0,\lambda _{n}\right) +h\psi _{\nu }\left( 0,\lambda _{n}\right) =0\text{.} \end{eqnarray}
\begin{remark} \label{615}Now we define the generalized norming constants for the problem $ B^{\infty },$ \begin{equation} \kappa _{n+\nu }^{\infty }:=\left. \frac{d\varphi _{n+\nu }^{\infty }\left( x\right) }{dx}\right\vert _{x=\pi },\text{ }\alpha _{n+\nu }^{\infty }:=\int_{0}^{\pi }\psi _{n+\nu }^{\infty }\left( x\right) \psi _{n+m_{n}^{\infty }-1}^{\infty }\left( x\right) dx\text{,} \label{64} \end{equation} where $n\in S_{B^{\infty }}$, $\nu =0,1,\ldots ,m_{n}^{\infty }-1,$ and \begin{eqnarray} &&\varphi _{n+\nu }^{\infty }\left( x\right):=\varphi _{\nu }\left( x,\lambda _{n}^{\infty }\right) :=\dfrac{1}{\nu !}\left. \dfrac{d^{\nu }}{ d\lambda ^{\nu }}\varphi \left( x,\lambda \right) \right\vert _{\lambda =\lambda _{n}^{\infty }},\text{ } \label{61} \\ &&\psi _{n+\nu }^{\infty }\left( x\right):=\psi _{\nu }^{\infty }\left( x,\lambda _{n}^{\infty }\right) :=\dfrac{1}{\nu !}\left. \dfrac{d^{\nu }}{ d\lambda ^{\nu }}\psi ^{\infty }\left( x,\lambda \right) \right\vert _{\lambda =\lambda _{n}^{\infty }}. \end{eqnarray} Then one can also deduce that for $n\in S_{B^{\infty }}$, $\nu =0,1,\ldots ,m_{n}^{\infty }-1,$ \begin{equation} \left. \frac{d^{m_{n}^{\infty }+\nu }\Delta ^{\infty }\left( \lambda \right) }{d\lambda ^{m_{n}^{\infty }+\nu }}\right\vert _{\lambda =\lambda _{n}^{\infty }}=-\left( m_{n}^{\infty }+\nu \right) !\sum_{j=0}^{\nu }\kappa _{n+j}^{\infty }\alpha _{n+\nu -j}^{\infty }\ . \label{63} \end{equation} \end{remark}
In \cite{YIXUAN}, Y. Liu, G. Shi and J. Yan studied the uniqueness spectral problem of recovering the non-self-adjoint operator $L\ $from one of the following spectral characteristics: (1) $\Gamma _{1}:=\left\{ \lambda _{n},\alpha _{n}\right\} _{n\in
\mathbb{N}
_{0}};$ (2) $\Gamma _{2}:=\left\{ \lambda _{n},\lambda _{n}^{\infty }\right\} _{n\in
\mathbb{N}
_{0}};$ (3) the Weyl function $M\left( \lambda \right) :=\frac{\Delta ^{\infty }\left( \lambda \right) }{\Delta \left( \lambda \right) }.$
This motivates us to investigate the inverse spectral problem with partial information given on the potential. More precisely, assume that $q$ is known on $\left[ b,\pi \right] $ for some constant $b\in \left( 0,\pi \right] ,$ then the uniqueness theorems of this paper will be given in three cases: $ b\in \left( d,\pi \right] ,$ $b=d,$ $b\in \left( 0,d\right) ,$ where $d$ is the discontinuous point. In the case of $b\in \left( d,\pi \right] $ or $b=d$ , we show that $h,$ $\beta ,$ $\gamma \ $and $q$ on $\left[ 0,\pi \right] \ $ can be uniquely determined by partial information of the eigenvalues $ \lambda _{n},$ $\lambda _{n}^{\infty }$, and of the generalized normalizing constants $\alpha _{n},$ $\alpha _{n}^{\infty }$; the uniqueness problem is also considered under the same circumstances but with the normalizing constants $\alpha _{n},$ $\alpha _{n}^{\infty }$ replaced by ratios $\kappa _{n},$ $\kappa _{n}^{\infty }.$ Moreover, for the case $b\in \left( 0,d\right) ,$ similar uniqueness results can be established with the additional condition that $\beta ,$ $\gamma \ $are known a priori.
We mention that in 1999, F. Gesztesy and B. Simon \cite{ges2} considered the classical self-adjoint Sturm-Liouville operators and presented a generalization of Hochstadt--Lieberman theorem to the case where the potential $q$ is known on a larger interval $\left[ a,\pi \right] $ with $ a\in \left( 0,\frac{\pi }{2}\right] $ and the set of common eigenvalues is sufficiently large. Later, G. Wei, H. K. Xu and Z. Wei \cite{wei,zhaoying} provided some uniqueness results for classical self-adjoint Sturm-Liouville operators with only partial information on $q,$ on the eigenvalues, and on the norming constants. While our results are generalizations of the uniqueness theorems established in \cite{ges2,wei,zhaoying},
the non-self-adjointness and the presence of discontinuities produce essential qualitative modifications in the investigation of the operator $L$ . To the best of our knowledge, the uniqueness theorems obtained in this paper have not yet been developed even for the non-self-adjoint classical Sturm-Liouville operators (i.e., the case of $\beta =1,$ $\gamma =0$) and the self-adjoint Sturm-Liouville operators with discontinuous conditions inside (i.e., the real-valued case).
In addition, we show that less knowledge of eigenvalues and norming constants can be required if the potential $q$ satisfies a local smoothness condition, which is a generalization of the results in \cite {ges2,wei,zhaoying}. We notice that the key technique in \cite {ges2,wei,zhaoying} relies on the high-energy asymptotic expansion of the Weyl $m$-function \cite{weyl}, however, in our non-self-adjoint situation, an entirely different approach, based on the asymptotic expansion of the fundamental solutions of the equation $\left( \ref{ly1}\right) ,$ is developed (see Proposition \ref{ooo copy(1)}). Now we briefly present some of these uniqueness results (Theorem \ref{theorem}, Theorem \ref{theorem copy(4)}, Remark $\ref{theorem copy(6)},$ Corollary \ref{corollary copy(1)}-- \ref{corollary copy(3)}) as follows.
(S1) We prove that if $q\ $is assumed to be $C^{m}$ near $\pi ,\ $then $h,$ $ \beta ,$ $\gamma $ and $q$ on $\left[ 0,\pi \right] $ can be uniquely determined by the values of $q^{\left( j\right) }\left( \pi \right) ,$ $ j=0,1,\ldots ,m,$ $\left\{ \lambda _{n}\right\} _{n\in
\mathbb{N}
_{0}\backslash \Lambda _{1}}$ $($a subsequence of $\sigma \left( B\right) ),$ and $\left\{ \lambda _{n}^{\infty }\right\} _{n\in
\mathbb{N}
_{0}\backslash \Lambda _{1}^{\infty }}$ $($a subsequence of $\sigma \left( B^{\infty }\right) ),$ where $\#\Lambda _{1}+\#\Lambda _{1}^{\infty }=$ $ \left[ \frac{m+2}{2}\right] .$
(S2) When $d\in \left( 0,\frac{\pi }{2}\right) ,\ $we prove that if $q$ is $ C^{m}$ near $\frac{\pi }{2}\ $and $q\ $on $\left[ \frac{\pi }{2},\pi \right] \ $is known a priori, then $h,$ $\beta ,$ $\gamma $ and $q$ on $\left[ 0,\pi \right] $ can be uniquely determined by all the eigenvalues $\left\{ \lambda _{n}\right\} _{n\in
\mathbb{N}
_{0}}$ of $B$ except for $\left( \left[ \frac{m+2}{2}\right] \right) ,$ or all the eigenvalues $\left\{ \lambda _{n}^{\infty }\right\} _{n\in
\mathbb{N}
_{0}}$ of $B^{\infty }$ except for $\left( \left[ \frac{m+1}{2}\right] \right) ;$ when $d=\frac{\pi }{2},$ the same statement holds if $\beta ,$ $ \gamma \ $are additionally assumed to be known a priori$.$
Here is a sketch of the contents of this paper. In Section 2, we provide some preliminary lemmas which will be used to prove the main results. In Section 3, assume that $q$ is known on $\left[ b,\pi \right] $ for some constant $b\in \left( 0,\pi \right] ,$ then we discuss the uniqueness theorems for three cases: $b\in \left( d,\pi \right] $, $b=d$, and $b\in \left( 0,d\right) .$ Finally, the appendix is devoted to present an important proposition $\left( \text{see Proposition \ref{ooo copy(1)}} \right) ,$ which is necessary to prove our principal results.
We conclude this introduction by briefly summarizing some of the notations used in this paper.
\begin{notation} \label{615 copy(2)}$
\mathbb{C}
$ denotes the complex plane$.$ $
\mathbb{N}
$ denotes the set of positive integers$\ $and $
\mathbb{N}
_{0}$ denotes the set of nonnegative integers$.$ Given a set $A,$ the symbol $\#A$ will be used to denote the number of elements in $A.$ Moreover, given a sequence $X:=\{x_{n}\}_{n=0}^{\infty }$ of complex numbers, we use the notation $X_{1}<<X\ $to denote that $X_{1}$ is a subsequence of $X,$ and in addition, $\widehat{X}$ $:=\underset{n\in
\mathbb{N}
_{0}}{\cup }\{x_{n}\},$ $N_{X}\left( t\right) :=\#\{n\in
\mathbb{N}
_{0}:\left\vert x_{n}\right\vert <t\}\ $for each $t\geq 0.$
\end{notation}
\section{Preliminaries}
In this section, we provide some preliminaries which will be used in Section 3 to prove the main results.
In order to prove the uniqueness theorems, together with $B$ $\left( B^{\infty }\right) ,$ we consider the boundary value problem $\widetilde{B}$ $\left( \widetilde{B}^{\infty }\right) $ of the same form but with different coefficients $\tilde{q},$ $\tilde{h},$ $\widetilde{H},$ $\widetilde{\beta },$ $\widetilde{\gamma }$ and $\widetilde{d}.$ We agree that if a certain symbol $\xi $ denotes an object related to $B$ or $B^{\infty }$, then $\tilde{\xi}$ will denote the analogous object related to $\tilde{B}$ or $\widetilde{B} ^{\infty }$, and $\hat{\xi}:=\xi -\tilde{\xi}$.
Now we introduce an entire function of $\lambda \in
\mathbb{C}
,$ \begin{equation} F\left( \lambda \right) :=\left. \left\langle \varphi \left( x,\lambda \right) ,\widetilde{\varphi }\left( x,\lambda \right) \right\rangle \right\vert _{x=\pi }. \label{FFF} \end{equation} From \textrm{\cite[Theorem 5.2 and Remark 1]{YIXUAN}, }the following result can be given.
\begin{lemma} \label{unique by F}Suppose that $F\left( \lambda \right) \equiv 0$, then $q$ $=\tilde{q}$ a.e. on $\left[ 0,\pi \right] ,$ $h=\tilde{h},$ $\beta = \widetilde{\beta },$ $\gamma =\widetilde{\gamma },$ $d=\widetilde{d}.$ \end{lemma}
It should be noted that our main results are based on Lemma \ref{unique by F} . Next, we give an important lemma, which plays a key role in this paper.
\begin{lemma} \label{F}Suppose that $H=\widetilde{H}\in
\mathbb{C}
\cup \left\{ \infty \right\} .$ If $\lambda _{n}=\widetilde{\lambda }_{ \widetilde{n}}$ for some $n\in S_{B},$ $\widetilde{n}\in S_{\widetilde{B}},$ and $m_{n}=\widetilde{m}_{\widetilde{n}},$ then \begin{equation} \left. \frac{d^{k}}{d\lambda ^{k}}F\left( \lambda \right) \right\vert _{\lambda =\lambda _{n}}=0\ \text{for }k=0,1,\ldots ,m_{n}-1; \label{d equal} \end{equation} In addition, if $\alpha _{n+\nu }=\widetilde{\alpha }_{\widetilde{n}+\nu },$ $\nu =0,1,\ldots ,k_{n}-1$, where $k_{n}$ is an integer such that $1\leq k_{n}\leq m_{n},$ then we have \begin{equation*} \left. \frac{d^{m_{n}+\nu }}{d\lambda ^{m_{n}+\nu }}F\left( \lambda \right) \right\vert _{\lambda =\lambda _{n}}=0\ \text{for }\nu =0,1,\ldots ,k_{n}-1, \end{equation*} that is, in this case, the order of $\lambda _{n}$ $($as a zero of $F\left( \lambda \right) )$ is at least $\left( m_{n}+k_{n}\right) .$ Similar statement also holds for the case $H=\widetilde{H}=\infty .$ \end{lemma}
\begin{proof} We first prove the lemma for $H=\widetilde{H}\in
\mathbb{C}
.$ From (\ref{w}) and the definition (\ref{FFF}) of $F\left( \lambda \right) $, we have \begin{equation} F\left( \lambda \right) =\left\vert \begin{array}{cc} \varphi \left( \pi ,\lambda \right) & \widetilde{\varphi }\left( \pi ,\lambda \right) \\ \Delta \left( \lambda \right) & \widetilde{\Delta }\left( \lambda \right) \end{array} \right\vert . \label{FLAMUDA} \end{equation} Since $m_{n}=\widetilde{m}_{\widetilde{n}},$\ we know that \begin{equation} \left. \frac{d^{k}}{d\lambda ^{k}}\Delta \left( \lambda \right) \right\vert _{\lambda =\lambda _{n}}=0,\text{ }\left. \frac{d^{k}}{d\lambda ^{k}} \widetilde{\Delta }\left( \lambda \right) \right\vert _{\lambda =\lambda _{n}}=0\text{ for }k=0,1,\ldots ,m_{n}-1. \label{dd equal} \end{equation} This directly yields $\left( \ref{d equal}\right) .$ Now we turn to prove the second part of this lemma. It follows from $\left( \ref{ratios}\right) ,$ $\left( \ref{9}\right) ,$ $\left( \ref{24}\right) $, $\left( \ref{FLAMUDA} \right) $ and $\left( \ref{dd equal}\right) $ that for $\nu =0,1,\ldots ,k_{n}-1,$ \begin{eqnarray} &&\left. \frac{d^{m_{n}+\nu }}{d\lambda ^{m_{n}+\nu }}F\left( \lambda \right) \right\vert _{\lambda =\lambda _{n}} \notag \\ &=&\sum_{j=0}^{m_{n}+\nu }C_{m_{n}+\nu }^{j}\left\vert \begin{array}{cc} \frac{d^{m_{n}+\nu -j}\varphi \left( \pi ,\lambda \right) }{d\lambda ^{m_{n}+\nu -j}} & \frac{d^{m_{n}+\nu -j}\widetilde{\varphi }\left( \pi ,\lambda \right) }{d\lambda ^{m_{n}+\nu -j}} \\ \frac{d^{^{j}}\Delta \left( \lambda \right) }{d\lambda ^{^{j}}} & \frac{ d^{^{j}}\widetilde{\Delta }\left( \lambda \right) }{d\lambda ^{^{j}}} \end{array} \right\vert _{\lambda =\lambda _{n}} \notag \\ &=&\sum_{j=m_{n}}^{m_{n}+\nu }C_{m_{n}+\nu }^{j}\left\vert \begin{array}{cc} \frac{d^{m_{n}+\nu -j}\varphi \left( \pi ,\lambda \right) }{d\lambda ^{m_{n}+\nu -j}} & \frac{d^{m_{n}+\nu -j}\widetilde{\varphi }\left( \pi ,\lambda \right) }{d\lambda ^{m_{n}+\nu -j}} \\ \frac{d^{^{j}}\Delta \left( \lambda \right) }{d\lambda ^{^{j}}} & \frac{ d^{^{j}}\widetilde{\Delta }\left( \lambda \right) }{d\lambda ^{^{j}}} \end{array} \right\vert _{\lambda =\lambda _{n}} \notag \\ &=&-\sum_{j=m_{n}}^{m_{n}+\nu }\sum\limits_{l=0}^{j-m_{n}}\dfrac{ C_{m_{n}+\nu }^{j}j!\alpha _{n+j-m_{n}-l}}{l!}\left\vert \begin{array}{cc} \frac{d^{m_{n}+\nu -j}\varphi \left( \pi ,\lambda \right) }{d\lambda ^{m_{n}+\nu -j}} & \frac{d^{m_{n}+\nu -j}\widetilde{\varphi }\left( \pi ,\lambda \right) }{d\lambda ^{m_{n}+\nu -j}} \\ \dfrac{d^{l}\varphi \left( \pi ,\lambda \right) }{d\lambda ^{l}} & \dfrac{ d^{l}\widetilde{\varphi }\left( \pi ,\lambda \right) }{d\lambda ^{l}} \end{array} \right\vert _{\lambda =\lambda _{n}} \label{posinega} \\ &=&-\sum\limits_{l=0}^{\nu }\sum_{j=m_{n}+l}^{m_{n}+\nu }\dfrac{C_{m_{n}+\nu }^{j}j!\alpha _{n+j-m_{n}-l}}{l!}\left\vert \begin{array}{cc} \frac{d^{m_{n}+\nu -j}\varphi \left( \pi ,\lambda \right) }{d\lambda ^{m_{n}+\nu -j}} & \frac{d^{m_{n}+\nu -j}\widetilde{\varphi }\left( \pi ,\lambda \right) }{d\lambda ^{m_{n}+\nu -j}} \\ \dfrac{d^{l}\varphi \left( \pi ,\lambda \right) }{d\lambda ^{l}} & \dfrac{ d^{l}\widetilde{\varphi }\left( \pi ,\lambda \right) }{d\lambda ^{l}} \end{array} \right\vert _{\lambda =\lambda _{n}}. \notag \end{eqnarray} Let $\widehat{l}=m_{n}+\nu -l,$ $\widehat{j}=m_{n}+\nu -j.$ Then \begin{eqnarray*} &&\left. \frac{d^{m_{n}+\nu }}{d\lambda ^{m_{n}+\nu }}F\left( \lambda \right) \right\vert _{\lambda =\lambda _{n}} \\ &=&-\sum\limits_{\widehat{l}=m_{n}+\nu }^{m_{n}}\sum_{\widehat{j}=\widehat{l} -m_{n}}^{0}\dfrac{C_{m_{n}+\nu }^{\widehat{l}}\widehat{l}!\alpha _{n+ \widehat{l}-m_{n}-\widehat{j}}}{\widehat{j}!}\left\vert \begin{array}{cc} \dfrac{d^{\widehat{j}}\varphi \left( \pi ,\lambda \right) }{d\lambda ^{ \widehat{j}}} & \dfrac{d^{\widehat{j}}\widetilde{\varphi }\left( \pi ,\lambda \right) }{d\lambda ^{\widehat{j}}} \\ \frac{d^{m_{n}+\nu -\widehat{l}}\varphi \left( \pi ,\lambda \right) }{ d\lambda ^{m_{n}+\nu -\widehat{l}}} & \frac{d^{m_{n}+\nu -\widehat{l}} \widetilde{\varphi }\left( \pi ,\lambda \right) }{d\lambda ^{m_{n}+\nu - \widehat{l}}} \end{array} \right\vert _{\lambda =\lambda _{n}}. \end{eqnarray*} This together $\left( \ref{posinega}\right) $ yield that $\left. \frac{ d^{m_{n}+\nu }}{d\lambda ^{m_{n}+\nu }}F\left( \lambda \right) \right\vert _{\lambda =\lambda _{n}}=-\left. \frac{d^{m_{n}+\nu }}{d\lambda ^{m_{n}+\nu } }F\left( \lambda \right) \right\vert _{\lambda =\lambda _{n}},$ and hence $ \left. \frac{d^{m_{n}+\nu }}{d\lambda ^{m_{n}+\nu }}F\left( \lambda \right) \right\vert _{\lambda =\lambda _{n}}=0\ $for $\nu =0,1,\ldots ,k_{n}-1.$ This proves the lemma for the case $H=\widetilde{H}\in
\mathbb{C}
.$ In view of Remark \ref{615} and the fact \begin{equation*} F\left( \lambda \right) :=\left\vert \begin{array}{cc} \varphi \left( \pi ,\lambda \right) & \widetilde{\varphi }\left( \pi ,\lambda \right) \\ \varphi ^{\prime }\left( \pi ,\lambda \right) & \widetilde{\varphi }^{\prime }\left( \pi ,\lambda \right) \end{array} \right\vert =-\left\vert \begin{array}{cc} \Delta ^{\infty }\left( \lambda \right) & \widetilde{\Delta ^{\infty }} \left( \lambda \right) \\ \varphi ^{\prime }\left( \pi ,\lambda \right) & \widetilde{\varphi }^{\prime }\left( \pi ,\lambda \right) \end{array} \right\vert , \end{equation*} the lemma for $H=\widetilde{H}=\infty $ can be proved similarly. \end{proof}
\begin{lemma} \label{Fqh}Assume that $d=\widetilde{d}\ $and $q$ $=\tilde{q}$ a.e. on $ \left[ b,\pi \right] $ for some $b\in \left( 0,\pi \right] ,$ then following expressions hold$:$ \begin{eqnarray*} F\left( \lambda \right) &=&\left. \left\langle \varphi \left( x,\lambda \right) ,\widetilde{\varphi }\left( x,\lambda \right) \right\rangle \right\vert _{x=b}\text{ for }b\in \left( d,\pi \right] , \\ F\left( \lambda \right) &=&\left. \left\langle \varphi \left( x,\lambda \right) ,\widetilde{\varphi }\left( x,\lambda \right) \right\rangle \right\vert _{x=d+0}\text{ for }b=d, \\ F\left( \lambda \right) &=&\left. \left\langle \varphi \left( x,\lambda \right) ,\widetilde{\varphi }\left( x,\lambda \right) \right\rangle \right\vert _{x=b}+\left. \left\langle \varphi \left( x,\lambda \right) , \widetilde{\varphi }\left( x,\lambda \right) \right\rangle \right\vert _{d-0}^{d+0}\text{ for }b\in \left( 0,d\right) . \end{eqnarray*} \end{lemma}
\begin{proof} From the definition $\left( \ref{FFF}\right) $ of $F\left( \lambda \right) ,$ one can easily deduce that \begin{eqnarray*} F\left( \lambda \right) &=&-\int_{0}^{\pi }\hat{q}\left( x\right) \varphi \left( x,\lambda \right) \widetilde{\varphi }\left( x,\lambda \right) dx+\left. \left\langle \varphi \left( x,\lambda \right) ,\widetilde{\varphi } \left( x,\lambda \right) \right\rangle \right\vert _{x=0} \\ &&+\left. \left\langle \varphi \left( x,\lambda \right) ,\widetilde{\varphi } \left( x,\lambda \right) \right\rangle \right\vert _{x=d+0}-\left. \left\langle \varphi \left( x,\lambda \right) ,\widetilde{\varphi }\left( x,\lambda \right) \right\rangle \right\vert _{x=d-0}. \end{eqnarray*} Hence by $q$ $=\tilde{q}$ a.e. on $\left[ b,\pi \right] $ we infer from the above equality that \begin{eqnarray*} F\left( \lambda \right) &=&-\int_{0}^{b}\hat{q}\left( x\right) \varphi \left( x,\lambda \right) \widetilde{\varphi }\left( x,\lambda \right) dx+\left. \left\langle \varphi \left( x,\lambda \right) ,\widetilde{\varphi } \left( x,\lambda \right) \right\rangle \right\vert _{x=0} \\ &&+\left. \left\langle \varphi \left( x,\lambda \right) ,\widetilde{\varphi } \left( x,\lambda \right) \right\rangle \right\vert _{x=d+0}-\left. \left\langle \varphi \left( x,\lambda \right) ,\widetilde{\varphi }\left( x,\lambda \right) \right\rangle \right\vert _{x=d-0}. \end{eqnarray*} Therefore, this lemma can be directly proved by the following facts \begin{eqnarray*} &&-\int_{0}^{b}\hat{q}\left( x\right) \varphi \left( x,\lambda \right) \widetilde{\varphi }\left( x,\lambda \right) dx \\ &=&\left\{ \begin{array}{l} \left. \left\langle \varphi \left( x,\lambda \right) ,\widetilde{\varphi } \left( x,\lambda \right) \right\rangle \right\vert _{d+0}^{b}+\left. \left\langle \varphi \left( x,\lambda \right) ,\widetilde{\varphi }\left( x,\lambda \right) \right\rangle \right\vert _{0}^{d-0}\text{ for }b\in \left( d,\pi \right] , \\ \left. \left\langle \varphi \left( x,\lambda \right) ,\widetilde{\varphi } \left( x,\lambda \right) \right\rangle \right\vert _{0}^{d-0}\text{ for }b=d, \\ \left. \left\langle \varphi \left( x,\lambda \right) ,\widetilde{\varphi } \left( x,\lambda \right) \right\rangle \right\vert _{0}^{b}\text{ for }b\in \left( 0,d\right) . \end{array} \right. \end{eqnarray*} \end{proof}
\begin{lemma} \label{jjgj}{As $\left\vert \lambda \right\vert \rightarrow \infty $, \begin{equation} \varphi \left( x,\lambda \right) =\left\{ \begin{array}{l} \cos \left( \sqrt{\lambda }x\right) +O\left( \frac{\exp \left( \left\vert \mathrm{Im}\sqrt{\lambda }\right\vert x\right) }{\sqrt{\lambda }}\right) ,x<d, \\ \left( b_{1}\cos \left( \sqrt{\lambda }x\right) +b_{2}\cos \left( \sqrt{ \lambda }\left( 2d-x\right) \right) \right) +O\left( \frac{\exp \left( \left\vert \mathrm{Im}\sqrt{\lambda }\right\vert x\right) }{\sqrt{\lambda }} \right) ,x>d, \end{array} \right. \label{1} \end{equation} \begin{equation} \varphi ^{\prime }\left( x,\lambda \right) =\left\{ \begin{array}{l} -\sqrt{\lambda }\sin \left( \sqrt{\lambda }x\right) +O\left( \exp \left( \left\vert \mathrm{Im}\sqrt{\lambda }\right\vert x\right) \right) ,x<d, \\ \sqrt{\lambda }\left( -b_{1}\sin \left( \sqrt{\lambda }x\right) +b_{2}\sin \left( \sqrt{\lambda }\left( 2d-x\right) \right) \right) +O\left( \exp \left( \left\vert \mathrm{Im}\sqrt{\lambda }\right\vert x\right) \right) ,x>d, \end{array} \right. \label{11} \end{equation} where $b_{1}=\dfrac{\beta +\beta ^{-1}}{2}$ and $b_{2}=\dfrac{\beta -\beta ^{-1}}{2}$}.
\begin{proof} See \cite[p.145-146]{Yurko}. \end{proof} \end{lemma}
\begin{remark} If $\lambda =iy$ with $y\in
\mathbb{R}
,$ then by Lemma \ref{jjgj}, $\left( \ref{w}\right) $ and $\left( \ref{w2} \right) ,$ one deduces that as $\left\vert y\right\vert \rightarrow \infty ,$ \begin{eqnarray} \left\vert \Delta \left( iy\right) \right\vert &=&\frac{b_{1}}{2}\left\vert y\right\vert ^{\frac{1}{2}}\exp \left( \left\vert \mathrm{Im}\sqrt{iy} \right\vert \pi \right) \left( 1+o\left( 1\right) \right) , \label{delta1} \\ \left\vert \Delta ^{\infty }\left( iy\right) \right\vert &=&\frac{b_{1}}{2} \exp \left( \left\vert \mathrm{Im}\sqrt{iy}\right\vert \pi \right) \left( 1+o\left( 1\right) \right) , \label{delta2} \\ \left\vert \varphi \left( b,iy\right) \right\vert &=&\left\{ \begin{array}{l} \frac{b_{1}}{2}\exp \left( \left\vert \mathrm{Im}\sqrt{iy}\right\vert b\right) \left( 1+o\left( 1\right) \right) \ \text{for }b>d, \\ \frac{1}{2}\exp \left( \left\vert \mathrm{Im}\sqrt{iy}\right\vert b\right) \left( 1+o\left( 1\right) \right) \ \text{for }b<d, \end{array} \right. \label{faib} \\ \left\vert \varphi \left( d+0,iy\right) \right\vert &=&\frac{\beta }{2}\exp \left( \left\vert \mathrm{Im}\sqrt{iy}\right\vert d\right) \left( 1+o\left( 1\right) \right) . \label{faid} \end{eqnarray} \end{remark}
We conclude this section with two lemmas $\left( \text{see Lemma \ref{number} and Lemma \ref{proposition}}\right) $, which will be used in Section 3 to prove our main results. Now we first give some notations and basic facts.
Recall that $\sigma \left( B\right) :=\left\{ \lambda _{n}\right\} _{n\in
\mathbb{N}
_{0}}$ and $\sigma \left( B^{\infty }\right) :=\left\{ \lambda _{n}^{\infty }\right\} _{n\in
\mathbb{N}
_{0}}$ are the sequences consisting of all the eigenvalues of $B$ and $ B^{\infty },$ respectively. By the asymptotics of the eigenvalues $\lambda _{n}$ and $\lambda _{n}^{\infty }$ \cite{YIXUAN}, it is easy to see that there exist constants $r_{1}$ and $r_{2}$ such that \begin{equation*} \min\limits_{n\in
\mathbb{N}
_{0}}\left\{ \text{Re}\lambda _{n}\right\} \geq r_{1},\text{ } \min\limits_{n\in
\mathbb{N}
_{0}}\left\{ \text{Re}\lambda _{n}^{\infty }\right\} \geq r_{2}.\text{ } \end{equation*} Hence by adding (if necessary) a sufficiently large constant to the potential coefficient $q$, throughout this paper we may assume that \begin{equation} N_{\sigma \left( B\right) }\left( t\right) =N_{\sigma \left( B^{\infty }\right) }\left( t\right) =0\text{ for }t\leq 1. \label{0000} \end{equation} By Lemma \ref{jjgj} one can easily deduce that $\Delta \left( \lambda \right) $ and $\Delta ^{\infty }\left( \lambda \right) $ are entire in $ \lambda \in
\mathbb{C}
$ of order $\frac{1}{2},$ and hence by Hadamard's Factorization Theorem \cite [Ch. I]{levision}, there exist constants $C_{B}$ and $C_{B^{\infty }}$ such that \begin{eqnarray} \Delta \left( \lambda \right) &=&C_{B}\prod\limits_{n=0}^{\infty }\left( 1- \frac{\lambda }{\lambda _{n}}\right) , \label{ch1} \\ \Delta ^{\infty }\left( \lambda \right) &=&C_{B^{\infty }}\prod\limits_{n=0}^{\infty }\left( 1-\frac{\lambda }{\lambda _{n}^{\infty } }\right) . \label{ch2} \end{eqnarray} Moreover, it follows from \cite[Ch. I, Theorem 4]{levision} that \begin{equation} N_{\sigma \left( B\right) }\left( t\right) \leq C\left\vert t\right\vert ^{\rho }\text{ and }N_{\sigma \left( B^{\infty }\right) }\left( t\right) \leq C\left\vert t\right\vert ^{\rho }\text{for all }\rho >\frac{1}{2}, \label{rrrrr} \end{equation} where $C$ is some positive constant.
\begin{lemma} \label{number}Let $X:=\{x_{n}\}_{n=0}^{\infty }$ with $0<\left\vert x_{0}\right\vert \leq \left\vert x_{1}\right\vert \leq \left\vert x_{2}\right\vert \leq \cdots $ be a sequence satisfying \begin{equation} \max\limits_{n\in
\mathbb{N}
_{0}}\left\vert \text{Im}x_{n}\right\vert \leq c_{1\text{ \ }}\text{for some }c_{1}>0, \label{c111} \end{equation} and \begin{eqnarray} N_{X}\left( t\right) &=&0\text{ for }t\leq 1, \label{000} \\ N_{X}\left( t\right) &\leq &C\left\vert t\right\vert ^{\rho }\text{ for all } \rho >\rho _{0}, \label{rrr} \end{eqnarray} where $C$ is some positive constant and $\rho _{0}\in \left( 0,1\right) $ is fixed$.$ If there exist real constants $l_{1},$ $l_{2},$ $l_{3}$ such that for sufficiently large $t\in
\mathbb{R}
,$ \begin{equation} N_{X}\left( t\right) \geq l_{1}N_{\sigma \left( B\right) }\left( t\right) +l_{2}N_{\sigma \left( B^{\infty }\right) }\left( t\right) +l_{3}, \label{hop} \end{equation} then there exists a constant $M>0$ such that for sufficiently large $ \left\vert y\right\vert $ $(y$ being real$)$ \begin{equation*} \left\vert G_{X}\left( iy\right) \right\vert \geq M\left\vert y\right\vert ^{ \frac{l_{1}}{2}+l_{3}}e^{\pi \left( l_{1}+l_{2}\right) \left\vert \text{Im} \sqrt{iy}\right\vert }, \end{equation*} where $G_{X}\left( \lambda \right) :=\prod\limits_{n=0}^{\infty }\left( 1- \frac{\lambda }{x_{n}}\right) .$ \end{lemma}
\begin{proof} Note that \begin{equation} \frac{d}{dt}\left[ \frac{1}{2}\ln \left( 1+\frac{y^{2}}{t^{2}}\right) \right] =-\frac{y^{2}}{t^{3}+ty^{2}}. \label{d} \end{equation} Then by $\left( \ref{000}\right) ,$ $\left( \ref{rrr}\right) ,$ $\left( \ref {d}\right) $ and integration by parts, we infer that for $y\in
\mathbb{R}
$, \begin{eqnarray*} &&\ln \left\vert G_{X}\left( iy\right) \right\vert \\ &=&\frac{1}{2}\sum_{n=0}^{\infty }\ln \frac{\left[ \left( 1-\frac{iy}{x_{n}} \right) \left( 1+\frac{iy}{\overline{x_{n}}}\right) \right] }{1+\frac{y^{2}}{ \left\vert x_{n}\right\vert ^{2}}}+\frac{1}{2}\sum_{n=0}^{\infty }\ln \left( 1+\frac{\left\vert y\right\vert ^{2}}{\left\vert x_{n}\right\vert ^{2}} \right) \\ &=&\frac{1}{2}\sum_{n=0}^{\infty }\ln \frac{\left( 1+\frac{2y\text{Im} x_{n}+y^{2}}{\left\vert x_{n}\right\vert ^{2}}\right) }{1+\frac{y^{2}}{ \left\vert x_{n}\right\vert ^{2}}}+\frac{1}{2}\int_{0}^{\infty }\ln \left( 1+ \frac{y^{2}}{t^{2}}\right) dN_{X}\left( t\right) \\ &=&\frac{1}{2}\sum_{n=0}^{\infty }\ln \frac{\left( 1+\frac{2y\text{Im} x_{n}+y^{2}}{\left\vert x_{n}\right\vert ^{2}}\right) }{1+\frac{y^{2}}{ \left\vert x_{n}\right\vert ^{2}}}+\int_{1}^{\infty }\frac{y^{2}}{ t^{3}+ty^{2}}N_{X}\left( t\right) dt. \end{eqnarray*} Similarly, by $\left( \ref{0000}\right) ,$ $\left( \ref{rrrrr}\right) \ $and $\left( \ref{d}\right) $ we deduce that \begin{eqnarray*} l_{1}\ln \left\vert G_{\sigma \left( B\right) }\left( iy\right) \right\vert &=&\frac{l_{1}}{2}\sum_{n=0}^{\infty }\ln \frac{\left( 1+\frac{2y\text{Im} \lambda _{n}+y^{2}}{\left\vert \lambda _{n}\right\vert ^{2}}\right) }{1+ \frac{y^{2}}{\left\vert \lambda _{n}\right\vert ^{2}}}+l_{1}\int_{1}^{\infty }\frac{y^{2}}{t^{3}+ty^{2}}N_{\sigma \left( B\right) }\left( t\right) dt, \\ l_{2}\ln \left\vert G_{\sigma \left( B^{\infty }\right) }\left( iy\right) \right\vert &=&\frac{l_{2}}{2}\sum_{n=0}^{\infty }\ln \frac{\left( 1+\frac{2y \text{Im}\lambda _{n}^{\infty }+y^{2}}{\left\vert \lambda _{n}^{\infty }\right\vert ^{2}}\right) }{1+\frac{y^{2}}{\left\vert \lambda _{n}^{\infty }\right\vert ^{2}}}+l_{2}\int_{1}^{\infty }\frac{y^{2}}{t^{3}+ty^{2}} N_{\sigma \left( B^{\infty }\right) }\left( t\right) dt. \end{eqnarray*} where \begin{equation} G_{\sigma \left( B\right) }\left( \lambda \right) :=\prod\limits_{n=0}^{\infty }\left( 1-\frac{\lambda }{\lambda _{n}}\right) \text{ and }G_{\sigma \left( B^{\infty }\right) }\left( \lambda \right) :=\prod\limits_{n=0}^{\infty }\left( 1-\frac{\lambda }{\lambda _{n}^{\infty } }\right) . \end{equation} Therefore, \begin{eqnarray} &&\ln \left\vert G_{X}\left( iy\right) \right\vert -l_{1}\ln \left\vert G_{\sigma \left( B\right) }\left( iy\right) \right\vert -l_{2}\ln \left\vert G_{\sigma \left( B^{\infty }\right) }\left( iy\right) \right\vert \label{31} \\ &=&g(y)+\int_{1}^{\infty }\frac{y^{2}}{t^{3}+ty^{2}}\left( N_{X}\left( t\right) -l_{1}N_{\sigma \left( B\right) }\left( t\right) -l_{2}N_{\sigma \left( B^{\infty }\right) }\left( t\right) \right) dt, \notag \end{eqnarray} where \begin{eqnarray} g(y):= &&\frac{1}{2}\sum_{n=0}^{\infty }\ln \frac{\left( 1+\frac{2y\text{Im} x_{n}+y^{2}}{\left\vert x_{n}\right\vert ^{2}}\right) }{1+\frac{y^{2}}{ \left\vert x_{n}\right\vert ^{2}}}-\frac{l_{1}}{2}\sum_{n=0}^{\infty }\ln \frac{\left( 1+\frac{2y\text{Im}\lambda _{n}+y^{2}}{\left\vert \lambda _{n}\right\vert ^{2}}\right) }{1+\frac{y^{2}}{\left\vert \lambda _{n}\right\vert ^{2}}} \label{gy} \\ &&-\frac{l_{2}}{2}\sum_{n=0}^{\infty }\ln \frac{\left( 1+\frac{2y\text{Im} \lambda _{n}^{\infty }+y^{2}}{\left\vert \lambda _{n}^{\infty }\right\vert ^{2}}\right) }{1+\frac{y^{2}}{\left\vert \lambda _{n}^{\infty }\right\vert ^{2}}}. \notag \end{eqnarray} Next, we aim to show that there exists a constant $C_{g}>0$ such that \begin{equation} \left\vert g(y)\right\vert \leq C_{g}\text{ for all }y\in
\mathbb{R}
. \label{34} \end{equation} In fact, we first note that there exist constants $c_{2}$ and $c_{3}$ such that \begin{equation} \max\limits_{n\in
\mathbb{N}
_{0}}\left\vert \text{Im}\lambda _{n}\right\vert \leq c_{2}\text{ and } \max\limits_{n\in
\mathbb{N}
_{0}}\left\vert \text{Im}\lambda _{n}^{\infty }\right\vert \leq c_{3}, \label{c1c2} \end{equation} which can be obtained from the asymptotics of the eigenvalues $\lambda _{n}$ and $\lambda _{n}^{\infty }$ \cite{YIXUAN}$.$ In addition, \begin{equation} \frac{d}{dt}\left[ \ln \left( 1+\frac{2\left\vert y\right\vert c_{i}}{ t^{2}+y^{2}}\right) \right] =-\frac{4\left\vert y\right\vert c_{i}t}{\left( t^{2}+y^{2}+2\left\vert y\right\vert c_{i}\right) \left( t^{2}+y^{2}\right) } ,\text{ }i=1,2,3, \label{dt} \end{equation} where $c_{1}$ is defined by $\left( \ref{c111}\right) $ and $c_{2},$ $c_{3}$ are defined by $\left( \ref{c1c2}\right) .$ Then by $\left( \ref{rrrrr} \right) ,$ $\left( \ref{c111}\right) ,$ $\left( \ref{rrr}\right) ,$ $\left( \ref{gy}\right) ,$ $\left( \ref{c1c2}\right) ,$ $\left( \ref{dt}\right) $ and integration by parts, we obtain that \begin{eqnarray*} \left\vert g(y)\right\vert &\leq &\frac{1}{2}\sum_{n=0}^{\infty }\ln \left( 1+\frac{2\left\vert y\right\vert c_{1}}{\left\vert x_{n}\right\vert ^{2}+\left\vert y\right\vert ^{2}}\right) +\left\vert \frac{l_{1}}{2} \right\vert \sum_{n=0}^{\infty }\ln \left( 1+\frac{2\left\vert y\right\vert c_{2}}{\left\vert \lambda _{n}\right\vert ^{2}+\left\vert y\right\vert ^{2}} \right) \\ &&+\left\vert \frac{l_{2}}{2}\right\vert \sum_{n=0}^{\infty }\ln \left( 1+ \frac{2\left\vert y\right\vert c_{3}}{\left\vert \lambda _{n}^{\infty }\right\vert ^{2}+\left\vert y\right\vert ^{2}}\right) \\ &=&\frac{1}{2}\int_{1}^{\infty }\ln \left( 1+\frac{2\left\vert y\right\vert c_{1}}{t^{2}+y^{2}}\right) dN_{X}\left( t\right) +\left\vert \frac{l_{1}}{2} \right\vert \int_{1}^{\infty }\ln \left( 1+\frac{2\left\vert y\right\vert c_{2}}{t^{2}+y^{2}}\right) dN_{\sigma \left( B\right) }\left( t\right) \\ &&+\left\vert \frac{l_{2}}{2}\right\vert \int_{1}^{\infty }\ln \left( 1+ \frac{2\left\vert y\right\vert c_{3}}{t^{2}+y^{2}}\right) dN_{\sigma \left( B^{\infty }\right) }\left( t\right) \\ &\leq &2c_{1}\int_{1}^{\infty }N_{X}\left( t\right) \frac{t\left\vert y\right\vert }{\left( t^{2}+y^{2}\right) ^{2}}dt+2c_{2}\left\vert l_{1}\right\vert \int_{1}^{\infty }N_{\sigma \left( B\right) }\left( t\right) \frac{t\left\vert y\right\vert }{\left( t^{2}+y^{2}\right) ^{2}}dt \\ &&+2c_{3}\left\vert l_{2}\right\vert \int_{1}^{\infty }N_{\sigma \left( B^{\infty }\right) }\left( t\right) \frac{\left\vert y\right\vert t}{\left( t^{2}+y^{2}\right) ^{2}}dt \\ &\leq &C_{0}\int_{1}^{\infty }\frac{t^{2}\left\vert y\right\vert }{\left( t^{2}+y^{2}\right) ^{2}}dt\leq C_{0}\int_{1}^{\infty }\frac{\left\vert y\right\vert }{t^{2}+y^{2}}dt \\ &=&\frac{C_{0}\pi }{2}-C_{0}\arctan \frac{1}{\left\vert y\right\vert }\text{ }\left( \text{if }y\neq 0\right) ,\text{ } \end{eqnarray*} where $C_{0}$ is some positive constant. This directly yields $\left( \ref {34}\right) .$ By hypothesis (\ref{hop}) we know that there exist constants $ t_{0}\geq 1$ and $C_{1}\geq 0$ such that \begin{eqnarray} N_{X}\left( t\right) -l_{1}N_{\sigma \left( B\right) }\left( t\right) -l_{2}N_{\sigma \left( B^{\infty }\right) }\left( t\right) &\geq &l_{3}, \text{ }t\geq t_{0}, \label{32} \\ N_{X}\left( t\right) -l_{1}N_{\sigma \left( B\right) }\left( t\right) -l_{2}N_{\sigma \left( B^{\infty }\right) }\left( t\right) &\geq &-C_{1}, \text{ }t\leq t_{0}. \label{33} \end{eqnarray} Therefore, it follows from (\ref{31}), $\left( \ref{34}\right) ,$ (\ref{32}) and (\ref{33}) that \begin{eqnarray} &&\ln \frac{\left\vert G_{X}\left( iy\right) \right\vert }{\left\vert G_{\sigma \left( B\right) }\left( iy\right) \right\vert ^{l_{1}}\left\vert G_{\sigma \left( B^{\infty }\right) }\left( iy\right) \right\vert ^{l_{2}}} \label{2.37} \\ &\geq &-C_{g}-\int_{1}^{t_{0}}\frac{y^{2}}{t^{3}+ty^{2}}C_{1}dt+ \int_{t_{0}}^{\infty }\frac{y^{2}}{t^{3}+ty^{2}}l_{3}dt \notag \\ &\geq &-C_{g}-\left( C_{1}+l_{3}\right) \int_{1}^{t_{0}}\frac{y^{2}}{ t^{3}+ty^{2}}dt+l_{3}\int_{1}^{\infty }\frac{y^{2}}{t^{3}+ty^{2}}dt \notag \\ &=&-C_{g}+\left( C_{1}+l_{3}\right) \frac{1}{2}\ln \frac{\left( t_{0}^{2}+y^{2}\right) }{t_{0}^{2}\left( 1+y^{2}\right) }+\frac{l_{3}}{2}\ln \left( 1+y^{2}\right) . \notag \end{eqnarray} In addition, by $\left( \ref{delta1}\right) ,$ $\left( \ref{delta2}\right) ,$ $\left( \ref{ch1}\right) $ and $\left( \ref{ch2}\right) ,$ we infer that $\ $ \begin{equation} \left\{ \begin{array}{l} \left\vert G_{\sigma \left( B\right) }\left( iy\right) \right\vert =\frac{ b_{1}}{2\left\vert C_{B}\right\vert }\left\vert y\right\vert ^{\frac{1}{2} }\exp \left( \left\vert \text{Im}\sqrt{iy}\right\vert \pi \right) \left( 1+o\left( 1\right) \right) , \\ \left\vert G_{\sigma \left( B^{\infty }\right) }\left( iy\right) \right\vert =\frac{b_{1}}{2\left\vert C_{B^{\infty }}\right\vert }\exp \left( \left\vert \text{Im}\sqrt{iy}\right\vert \pi \right) \left( 1+o\left( 1\right) \right) . \text{ } \end{array} \right. \label{delte11} \end{equation} Hence it turns out from $\left( \ref{2.37}\right) $ and $\left( \ref{delte11} \right) $ that there exists a constant $M>0$ such that \begin{equation*} \left\vert G_{X}\left( iy\right) \right\vert \geq M\left\vert y\right\vert ^{ \frac{l_{1}}{2}+l_{3}}e^{\pi \left( l_{1}+l_{2}\right) \left\vert \text{Im} \sqrt{iy}\right\vert } \end{equation*} for sufficiently large $\left\vert y\right\vert $ and $y\in
\mathbb{R}
.$ This completes the proof. \end{proof}
\begin{lemma} \label{proposition}Assume that $g\left( \lambda \right) $ is an entire\ function of order less than one. If $\lim\limits_{\left\vert y\right\vert \rightarrow \infty ;y\in
\mathbb{R}
}\left\vert g\left( iy\right) \right\vert =0,$ then $g\left( \lambda \right) \equiv 0.$
\begin{proof} The proof is referred to \cite{levision,ges2}. \end{proof} \end{lemma}
\section{Main Results and Proofs}
Our goal of this section is to give the main results of this paper. Assume that the potential $q$ is known on $\left[ b,\pi \right] ,$ then due to the presence of discontinuous conditions at $d\in \left( 0,\pi \right) ,$ the uniqueness theorems are given for three cases: $b\in \left( d,\pi \right] $, $b=d,$ and $b\in \left( 0,d\right) .$ In each case, we first study the uniqueness problem (Theorem \ref{theorem}, Theorem \ref{theorem copy(2)}, Theorem \ref{theorem copy(4)}) when only partial information on $q$, on the eigenvalues, and on the generalized normalizing constants is available, and then we investigate the uniqueness problem (Theorem \ref{theorem copy(1)}, Theorem \ref{theorem copy(3)}, Theorem \ref{theorem copy(5)}) under the same circumstances but with the normalizing constants replaced by ratios.
Unless explicitly stated otherwise, $H\ $and $d$ will be fixed$\ $in this section. In addition, let us recall Notation \ref{615 copy(1)} and Notation \ref{615 copy(2)} given in the introduction.
\subsection{Case I: $q$ is known on $\left[ b,\protect\pi \right] ,$ where $ b\in \left( d,\protect\pi \right] $}
\subsubsection{Pairs of Eigenvalues and Normalizing Constants}
\begin{Hypothesis} \label{hypo1}Consider the subsequences $W,$ $W_{1},$ $W^{\infty },$ $ W_{1}^{\infty }\ $satisfying \begin{eqnarray*} &&W_{1}<<W<<\sigma \left( B\right) ,\text{ }W_{1}<<W<<\sigma \left( \widetilde{B}\right) , \\ &&W_{1}^{\infty }<<W^{\infty }<<\sigma \left( B^{\infty }\right) ,W_{1}^{\infty }<<W^{\infty }<<\sigma \left( \widetilde{B}^{\infty }\right) \end{eqnarray*} and the following conditions:
$(1)$ for any $\lambda _{n}=\widetilde{\lambda }_{\widetilde{n}}\in \widehat{ W_{1}}\ $where $n\in S_{B}\ $and $\widetilde{n}\in S_{\widetilde{B}},$ suppose that \begin{equation} m_{n}=\widetilde{m}_{\widetilde{n}},\text{ }\alpha _{n+\nu }=\widetilde{ \alpha }_{\widetilde{n}+\nu }\text{ for }\nu =0,1,\ldots ,k_{n}-1,\text{ } \label{hhypo1} \end{equation} where $k_{n}$ equals the number of occurrences of the eigenvalue $\lambda _{n}\ $in $W_{1};$
$(2)$ for any $\lambda _{n}^{\infty }=\widetilde{\lambda }_{\widetilde{n} }^{\infty }\in \widehat{W_{1}^{\infty }}\ $where $n\in S_{B^{\infty }}$ and $ \widetilde{n}\in S_{\widetilde{B}^{\infty }},$ suppose that \begin{equation} m_{n}^{\infty }=\widetilde{m}_{\widetilde{n}}^{\infty },\text{ }\alpha _{n+\gamma }^{\infty }=\widetilde{\alpha }_{\widetilde{n}+\gamma }^{\infty }\ \text{for }\gamma =0,1,\ldots ,k_{n}^{\infty }-1,\text{ } \label{hhypo2} \end{equation} where $k_{n}^{\infty }$ equals the number of occurrences of the eigenvalue $ \lambda _{n}^{\infty }\ $in $W_{1}^{\infty }.$ \end{Hypothesis}
\begin{theorem} \label{theorem}Assume Hypothesis \ref{hypo1} and suppose that $q,$ $ \widetilde{q}\in $ $C^{m}$ near $b\in $ $\left( d,\pi \right] ,$ $m\in
\mathbb{N}
_{0},$ $q=\widetilde{q}$ a.e. on $\left[ b,\pi \right] $ $($in particular, for $b=\pi :$ $q^{\left( j\right) }(\pi )=\widetilde{q}^{\left( j\right) }(\pi )$ for $j=0,1,\ldots ,m),$ and \begin{eqnarray} &&N_{W}(t)+N_{W_{1}}(t)+N_{W^{\infty }}(t)+N_{W_{1}^{\infty }}(t) \label{hypo} \\ &\geq &AN_{\sigma \left( B\right) }(t)+\left( \frac{2b}{\pi }-A\right) N_{\sigma \left( B^{\infty }\right) }(t)-\frac{A}{2}-\frac{m+1}{2} \notag \end{eqnarray} for sufficiently large $t\in
\mathbb{R}
.$ Then $h=\widetilde{h},$ $\beta =\widetilde{\beta },$ $\gamma =\widetilde{ \gamma }$ and $q=\widetilde{q}$ a.e. on $\left[ 0,\pi \right] .$ \end{theorem}
\begin{remark} By Remark \ref{ooo copy(4)}, we know that if $q$ and $\widetilde{q}\ $are assumed to be in $L_{
\mathbb{C}
}^{1}\left[ 0,\pi \right] ,$ then Theorem \ref{theorem} should be modified by taking $m=-1.$ Thus for brevity $C^{-1}$ means $L^{1}$ throughout this paper unless explicitly stated otherwise$.$ \end{remark}
\begin{corollary} \label{corollary copy(1)}If $q\ $is assumed to be $C^{m}$ near $\pi ,\ $then $h,$ $\beta ,$ $\gamma $ and $q$ on $\left[ 0,\pi \right] $ can be uniquely determined by the values of $q^{\left( j\right) }\left( \pi \right) ,$ $ j=0,1,\ldots ,m,$ $\left\{ \lambda _{n}\right\} _{n\in
\mathbb{N}
_{0}\backslash \Lambda _{1}}$ $($a subsequence of $\sigma \left( B\right) ),$ and $\left\{ \lambda _{n}^{\infty }\right\} _{n\in
\mathbb{N}
_{0}\backslash \Lambda _{1}^{\infty }}$ $($a subsequence of $\sigma \left( B^{\infty }\right) ),$ where $\#\Lambda _{1}+\#\Lambda _{1}^{\infty }=$ $ \left[ \frac{m+2}{2}\right] .$ \end{corollary}
\begin{corollary} \label{corollary}Assume that $q\ $is $C^{m}$ near $\pi \ $and$\ $the values of $q^{\left( j\right) }\left( \pi \right) ,$ $j=0,1,\ldots ,m,$\ are known a priori. Then $h,$ $\beta ,$ $\gamma $ and $q$ on $\left[ 0,\pi \right] $ can be uniquely determined by the following information $(1)$ or $(2):$
$(1)$ all the eigenvalues $\left\{ \lambda _{n}\right\} _{n\in
\mathbb{N}
_{0}}$ of $B$ and a subsequence of the normalizing constants $\left\{ \alpha _{n+\nu }\right\} _{n\in S_{B}\backslash \Lambda }^{\nu =0,1,\ldots ,k_{n}},$ where $0\leq k_{n}\leq m_{n}-1,$ $\Lambda \subset S_{B}$ and $ \sum\limits_{n\in \Lambda }m_{n}+\sum\limits_{n\in S_{B}\backslash \Lambda }\left( m_{n}-k_{n}-1\right) =\left[ \frac{m+3}{2}\right] ;$
$(2)$ all the eigenvalues $\left\{ \lambda _{n}^{\infty }\right\} _{n\in
\mathbb{N}
_{0}}$ of $B^{\infty }$ and a subsequence of the normalizing constants $ \left\{ \alpha _{n+\nu }^{\infty }\right\} _{n\in s_{B^{\infty }}\backslash \Lambda ^{\infty }}^{\nu =0,1,\ldots ,k_{n}^{\infty }},$ where $0\leq k_{n}^{\infty }\leq m_{n}^{\infty }-1,$ $\Lambda ^{\infty }\subset S_{B^{\infty }}$ and $\sum\limits_{n\in \Lambda ^{\infty }}m_{n}^{\infty }+\sum\limits_{n\in S_{B^{\infty }}\backslash \Lambda ^{\infty }}\left( m_{n}^{\infty }-k_{n}^{\infty }-1\right) =\left[ \frac{m+1}{2}\right] .$ \end{corollary}
\begin{remark} Suppose that {$b_{1}=\dfrac{\beta +\beta ^{-1}}{2}$ is} known a priori. Then from $\left( \ref{delta1}\right) $, $\left( \ref{delta2}\right) ,$ $\left( \ref{ch1}\right) $ and $\left( \ref{ch2}\right) $, one deduces that $\Delta \left( \lambda \right) $ and $\Delta ^{\infty }\left( \lambda \right) $ can be uniquely determined by $\sigma \left( B\right) $ and $\sigma \left( B^{\infty }\right) ,$ respectively$;$ thus by $\left( \ref{24}\right) \ $and $\left( \ref{63}\right) ,$ we know that Corollary \ref{corollary} remains valid if the conditions on the normalizing constants $\left\{ \alpha _{n+\nu }\right\} _{n\in S_{B}\backslash \Lambda }^{\nu =0,1,\ldots ,k_{n}}$ and $ \left\{ \alpha _{n+\nu }^{\infty }\right\} _{n\in S_{B^{\infty }}\backslash \Lambda ^{\infty }}^{\nu =0,1,\ldots ,k_{n}^{\infty }}$ are replaced by the conditions on the ratios $\left\{ \kappa _{n+\nu }\right\} _{n\in S_{B}\backslash \Lambda }^{\nu =0,1,\ldots ,k_{n}}$ and $\left\{ \kappa _{n+\nu }^{\infty }\right\} _{n\in S_{B^{\infty }}\backslash \Lambda ^{\infty }}^{\nu =0,1,\ldots ,k_{n}^{\infty }},$ respectively.
\begin{corollary} \label{corollary copy(2)}Let $d\in \left( 0,\frac{\pi }{2}\right).$ Assume that $q$ is $C^{m}$ near $\frac{\pi }{2}\ $and $q\ $on $\left[ \frac{\pi }{2} ,\pi \right] \ $are known a priori. Then $h,$ $\beta ,$ $\gamma $ and $q$ on $\left[ 0,\pi \right] $ can be uniquely determined by all the eigenvalues $ \left\{ \lambda _{n}\right\} _{n\in
\mathbb{N}
_{0}}$ of $B$ except for $\left( \left[ \frac{m+2}{2}\right] \right) ,$ or all the eigenvalues $\left\{ \lambda _{n}^{\infty }\right\} _{n\in
\mathbb{N}
_{0}}$ of $B^{\infty }$ except for $\left( \left[ \frac{m+1}{2}\right] \right) $. \end{corollary} \end{remark}
To prove Theorem \ref{theorem}, we first give a lemma on $F\left( \lambda \right) $ defined by $\left( \ref{FFF}\right) .$
\begin{lemma} \label{iy}Assume that $q,$ $\widetilde{q}\in $ $C^{m}\ $near $b\in \left( d,\pi \right] ,$ $q$ $=\tilde{q}$ a.e. on $\left[ b,\pi \right] $ $($in particular, for $b=\pi :$ $q^{\left( j\right) }(\pi )=\widetilde{q}^{\left( j\right) }(\pi )$ for $j=0,1,\ldots ,m).$ Then one observes that \begin{equation*} \left\vert F\left( iy\right) \right\vert =o\left( \left\vert y\right\vert ^{- \frac{m+1}{2}}\exp \left( 2\left\vert \text{Im}\sqrt{iy}\right\vert b\right) \right) \ \text{as }y\text{ }\left( \text{real}\right) \rightarrow \infty . \end{equation*} \end{lemma}
\begin{proof} Recall Definition $\ref{yyy}$ (in the Appendix) for the functions $ y_{i,d}(x,\lambda )$ and $\widetilde{y}_{i,d}(x,\lambda ),$ $i=1,2.$ Then from Lemma \ref{Fqh} we know that for $b\in \left( d,\pi \right] ,$ \begin{eqnarray*} F\left( \lambda \right) &=&\varphi \left( b,\lambda \right) \widetilde{ \varphi }^{\prime }\left( b,\lambda \right) -\varphi ^{\prime }\left( b,\lambda \right) \widetilde{\varphi }\left( b,\lambda \right) \\ &=&\left[ \beta \varphi \left( d-0,\lambda \right) y_{1,d}(b,\lambda )+\left( \beta ^{-1}\varphi ^{\prime }\left( d-0,\lambda \right) +\gamma \varphi \left( d-0,\lambda \right) \right) y_{2,d}(b,\lambda )\right] \times \\ &&\left[ \widetilde{\beta }\widetilde{\varphi }\left( d-0,\lambda \right) \widetilde{y}_{1,d}^{\prime }(b,\lambda )+\left( \widetilde{\beta }^{-1} \widetilde{\varphi }^{\prime }\left( d-0,\lambda \right) +\widetilde{\gamma } \widetilde{\varphi }\left( d-0,\lambda \right) \right) \widetilde{y} _{2,d}^{\prime }(b,\lambda )\right] \\ &&-\left[ \beta \varphi \left( d-0,\lambda \right) y_{1,d}^{\prime }(b,\lambda )+\left( \beta ^{-1}\varphi ^{\prime }\left( d-0,\lambda \right) +\gamma \varphi \left( d-0,\lambda \right) \right) y_{2,d}^{\prime }(b,\lambda )\right] \times \\ &&\left[ \widetilde{\beta }\widetilde{\varphi }\left( d-0,\lambda \right) \widetilde{y}_{1,d}(b,\lambda )+\left( \widetilde{\beta }^{-1}\widetilde{ \varphi }^{\prime }\left( d-0,\lambda \right) +\widetilde{\gamma }\widetilde{ \varphi }\left( d-0,\lambda \right) \right) \widetilde{y}_{2,d}(b,\lambda ) \right] \\ &=&A_{1}(\lambda )\left[ y_{1,d}(b,\lambda )\widetilde{y}_{1,d}^{\prime }(b,\lambda )-y_{1d}^{\prime }(b,\lambda )\widetilde{y}_{1,d}(b,\lambda ) \right] \\ &&+A_{2}(\lambda )\left[ y_{1,d}(b,\lambda )\widetilde{y}_{2,d}^{\prime }(b,\lambda )-y_{1,d}^{\prime }(b,\lambda )\widetilde{y}_{2,d}(b,\lambda ) \right] \\ &&+A_{3}(\lambda )\left[ \widetilde{y}_{1,d}^{\prime }(b,\lambda )y_{2,d}(b,\lambda )-\widetilde{y}_{1,d}(b,\lambda )y_{2,d}^{\prime }(b,\lambda )\right] \\ &&+A_{4}(\lambda )\left[ y_{2,d}(b,\lambda )\widetilde{y}_{2,d}^{\prime }(b,\lambda )-y_{2,d}^{\prime }(b,\lambda )\widetilde{y}_{2,d}(b,\lambda ) \right] , \end{eqnarray*} where \begin{eqnarray*} A_{1}(\lambda ) &=&\beta \widetilde{\beta }\varphi \left( d-0,\lambda \right) \widetilde{\varphi }\left( d-0,\lambda \right) =O\left( \exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert d\right) \right) , \\ A_{2}(\lambda ) &=&\beta \varphi \left( d-0,\lambda \right) \left( \widetilde{\beta }^{-1}\widetilde{\varphi }^{\prime }\left( d-0,\lambda \right) +\widetilde{\gamma }\widetilde{\varphi }\left( d-0,\lambda \right) \right) \\ &=&O\left( \sqrt{\left\vert \lambda \right\vert }\exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert d\right) \right) , \\ A_{3}(\lambda ) &=&\widetilde{\beta }\widetilde{\varphi }\left( d-0,\lambda \right) \left( \beta ^{-1}\varphi ^{\prime }\left( d-0,\lambda \right) +\gamma \varphi \left( d-0,\lambda \right) \right) \\ &=&O\left( \sqrt{\left\vert \lambda \right\vert }\exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert d\right) \right) , \\ A_{4}(\lambda ) &=&\left( \widetilde{\beta }^{-1}\widetilde{\varphi } ^{\prime }\left( d-0,\lambda \right) +\widetilde{\gamma }\widetilde{\varphi } \left( d-0\right) \right) \left( \beta ^{-1}\varphi ^{\prime }\left( d-0,\lambda \right) +\gamma \varphi \left( d-0,\lambda \right) \right) \\ &=&O\left( \left\vert \lambda \right\vert \exp \left( 2\left\vert \text{Im} \sqrt{\lambda }\right\vert d\right) \right) . \end{eqnarray*} as $\left\vert \lambda \right\vert \rightarrow \infty .$ Note that the asymptotics of $A_{1},$ $A_{2},$ $A_{3}$ and $A_{4}$ can be directly obtained by Lemma \ref{jjgj}. Hence from Proposition \ref{ooo copy(1)} it follows that as $y$ $\left( \text{real}\right) \rightarrow \infty ,$ \begin{eqnarray*} &&\left\vert F\left( iy\right) \right\vert \\ &\leq &\left\vert A_{1}(iy)\right\vert o\left( \frac{\exp \left( 2\left\vert \text{Im}\sqrt{iy}\right\vert \left( b-d\right) \right) }{\left\vert \sqrt{iy }\right\vert ^{m+1}}\right) +\left\vert A_{2}(iy)\right\vert o\left( \frac{ \exp \left( 2\left\vert \text{Im}\sqrt{iy}\right\vert \left( b-d\right) \right) }{\left\vert \sqrt{iy}\right\vert ^{m+2}}\right) \\ &&+\left\vert A_{3}(iy)\right\vert o\left( \frac{\exp \left( 2\left\vert \text{Im}\sqrt{iy}\right\vert \left( b-d\right) \right) }{\left\vert \sqrt{iy }\right\vert ^{m+2}}\right) +\left\vert A_{4}(iy)\right\vert o\left( \frac{ \exp \left( 2\left\vert \text{Im}\sqrt{iy}\right\vert \left( b-d\right) \right) }{\left\vert \sqrt{iy}\right\vert ^{m+3}}\right) \\ &=&o\left( \left\vert y\right\vert ^{-\frac{m+1}{2}}\exp \left( 2\left\vert \text{Im}\sqrt{iy}\right\vert b\right) \right) . \end{eqnarray*} This completes the proof. \end{proof}
Now we turn to prove Theorem \ref{theorem}.
\begin{proof}[Proof of Theorem \protect\ref{theorem}] For any $\lambda _{n}\in \widehat{W}\ $and $\lambda _{n}^{\infty }\in \widehat{W^{\infty }}\ $where $n\in S_{B}\ $and $n\in S_{B^{\infty }},$ let $ \gamma _{n}$ and $\gamma _{n}^{\infty }$ denote the number of occurrences of $\lambda _{n}$ in $W\ $and $\lambda _{n}^{\infty }$ in $W^{\infty },$ respectively$.$ Denote \begin{equation} H\left( \lambda \right) :=\frac{F\left( \lambda \right) }{G_{\Xi }\left( \lambda \right) }, \label{Hlamuda} \end{equation} where \begin{equation} G_{\Xi }\left( \lambda \right) :=G_{W}\left( \lambda \right) G_{W_{1}}\left( \lambda \right) G_{W^{\infty }}\left( \lambda \right) G_{W_{1}^{\infty }}\left( \lambda \right) , \label{G3} \end{equation} \begin{eqnarray*} &&G_{W}\left( \lambda \right) :=\prod\limits_{\lambda _{n}\in \widehat{W} ,n\in S_{B}}\left( 1-\frac{\lambda }{\lambda _{n}}\right) ^{\gamma _{n}},G_{W_{1}}\left( \lambda \right) :=\prod\limits_{\lambda _{n}\in \widehat{W_{1}},n\in S_{B}}\left( 1-\frac{\lambda }{\lambda _{n}}\right) ^{k_{n}}, \\ &&G_{W^{\infty }}\left( \lambda \right) :=\prod\limits_{\lambda _{n}^{\infty }\in \widehat{W^{\infty }},n\in S_{B^{\infty }}}\left( 1-\frac{\lambda }{ \lambda _{n}^{\infty }}\right) ^{\gamma _{n}^{\infty }},G_{W_{1}^{\infty }}\left( \lambda \right) :=\prod\limits_{\lambda _{n}^{\infty }\in \widehat{ W_{1}^{\infty }},n\in S_{B^{\infty }}}\left( 1-\frac{\lambda }{\lambda _{n}^{\infty }}\right) ^{k_{n}^{\infty }}. \end{eqnarray*} Then it follows from $\left( \ref{hhypo1}\right) ,$ $\left( \ref{hhypo2} \right) ,$ Lemma \ref{F}, and the fact $\widehat{\sigma \left( B\right) } \cap \widehat{\sigma \left( B^{\infty }\right) }=\emptyset $ that $H\left( \lambda \right) $ is an entire function. From Lemma \ref{Fqh}, we know that $ F\left( \lambda \right) $ is an entire function of order less than $\frac{1}{ 2};$ $\Delta \left( \lambda \right) $ and $\Delta ^{\infty }\left( \lambda \right) $ are entire functions of order $\frac{1}{2}.$ Moreover, since the order of canonical product of an entire function is equal to its convergence exponent of zeros (\cite[P16]{levision}), we can obtain that $G_{\Xi }\left( \lambda \right) $ is an entire function of order less than $\frac{1}{2},$ and so the order of $H\left( \lambda \right) $ is at most $\frac{1}{2}.$
Now we aim to prove that $H\left( \lambda \right) \equiv 0.$ By Lemma \ref {proposition}, it is sufficient to prove that $\left\vert H\left( iy\right) \right\vert \rightarrow 0$ as $y$ $\left( \text{real}\right) \rightarrow \infty .$ From Lemma \ref{number} and the assumption $\left( \ref{hypo} \right) $, we know that there exists a constant $M>0$ such that \begin{equation*} \left\vert G_{\Xi }\left( iy\right) \right\vert \geq M\left\vert y\right\vert ^{-\frac{m+1}{2}}\exp \left( 2\left\vert \text{Im}\sqrt{iy} \right\vert b\right) , \end{equation*} and thus according to $\left( \ref{Hlamuda}\right) $ and Lemma \ref{iy}, one has \begin{equation*} \left\vert H\left( iy\right) \right\vert \leq \left\vert \frac{o\left( \left\vert y\right\vert ^{-\frac{m+1}{2}}\exp \left( 2\left\vert \text{Im} \sqrt{iy}\right\vert b\right) \right) }{M\left\vert y\right\vert ^{-\frac{m+1 }{2}}\exp \left( 2\left\vert \text{Im}\sqrt{iy}\right\vert b\right) } \right\vert =o\left( 1\right) \text{ as }y\text{ }\left( \text{real}\right) \rightarrow \infty . \end{equation*} This implies that $H\left( \lambda \right) \equiv 0\ $and thus $F\left( \lambda \right) \equiv 0.$ Then we conclude from Lemma \ref{unique by F} that $h=\widetilde{h},$ $\beta =\widetilde{\beta },$ $\gamma =\widetilde{ \gamma }$ and $q=\widetilde{q}$ a.e. on $\left[ 0,\pi \right] .$ \end{proof}
\subsubsection{Pairs of Eigenvalues and Ratios}
\begin{Hypothesis} \label{hypo2}Consider the subsequences $W$ and $W^{\infty }$ satisfying \begin{equation*} W<<\sigma \left( B\right) ,W<<\sigma \left( \widetilde{B}\right) ,\text{ } W^{\infty }<<\sigma \left( B^{\infty }\right) ,W^{\infty }<<\sigma \left( \widetilde{B}^{\infty }\right) \end{equation*} and the following conditions:
$(1)$ for any $\lambda _{n}=\widetilde{\lambda }_{\widetilde{n}}\in \widehat{ W}\ $where $n\in S_{B}\ $and $\widetilde{n}\in S_{\widetilde{B}},$ suppose that \begin{equation} \text{ }\kappa _{n+\nu }=\widetilde{\kappa }_{\widetilde{n}+\nu }\ \text{for }\nu =0,1,\ldots ,k_{n}-1,\text{ } \label{hhypo 3} \end{equation} where $k_{n}$ equals the number of occurrences of $\lambda _{n}\ $in $W;$
$\left( 2\right) $ for any $\lambda _{n}^{\infty }=\widetilde{\lambda }_{ \widetilde{n}}^{\infty }\in \widehat{W^{\infty }}$ where $n\in S_{B^{\infty }}\ $and $\widetilde{n}\in S_{\widetilde{B}^{\infty }},$ suppose that \begin{equation} \kappa _{n+\gamma }^{\infty }=\widetilde{\kappa }_{\widetilde{n}+\gamma }^{\infty }\ \text{for }\gamma =0,1,\ldots ,k_{n}^{\infty }-1,\text{ } \label{hhypo4} \end{equation} where $k_{n}^{\infty }$ equals the number of occurrences of $\lambda _{n}^{\infty }\ $in $W^{\infty }.$ \end{Hypothesis}
\begin{theorem} \label{theorem copy(1)}Assume Hypothesis \ref{hypo2} and suppose that $q= \widetilde{q}$ a.e. on $\left[ b,\pi \right] ,$ \begin{equation} N_{W}(t)+N_{W^{\infty }}(t)\geq AN_{\sigma \left( B\right) }(t)+\left( \frac{ b}{\pi }-A\right) N_{\sigma \left( B^{\infty }\right) }(t)-\frac{A}{2} +\epsilon \label{hypo4} \end{equation} for sufficiently large $t\in
\mathbb{R}
,$ where $\epsilon $ is an arbitrary positive constant$.$ Then $h=\widetilde{ h},$ $\beta =\widetilde{\beta },$ $\gamma =\widetilde{\gamma }$ and $q= \widetilde{q}$ a.e. on $\left[ 0,\pi \right] .$ \end{theorem}
\begin{proof} Denote \begin{equation} H_{1}\left( \lambda \right) :=\frac{F_{1}\left( \lambda \right) }{G_{\Theta }\left( \lambda \right) },\text{ }H_{2}\left( \lambda \right) =\frac{ F_{2}\left( \lambda \right) }{G_{\Theta }\left( \lambda \right) }, \label{FGFG} \end{equation} where $G_{\Theta }\left( \lambda \right) :=G_{W}\left( \lambda \right) G_{W^{\infty }}\left( \lambda \right) ,$ \begin{equation} G_{W}\left( \lambda \right) :=\prod\limits_{\lambda _{n}\in \widehat{W},n\in S_{B}}\left( 1-\frac{\lambda }{\lambda _{n}}\right) ^{k_{n}},\text{ } G_{W^{\infty }}\left( \lambda \right) :=\prod\limits_{\lambda _{n}^{\infty }\in \widehat{W^{\infty }},n\in S_{B^{\infty }}}\left( 1-\frac{\lambda }{ \lambda _{n}^{\infty }}\right) ^{k_{n}^{\infty }} \label{GG} \end{equation} and \begin{equation*} F_{1}\left( \lambda \right) :=\varphi \left( b,\lambda \right) -\widetilde{ \varphi }\left( b,\lambda \right) ,\text{ }F_{2}\left( \lambda \right) :=\varphi ^{\prime }\left( b,\lambda \right) -\widetilde{\varphi }^{\prime }\left( b,\lambda \right) . \end{equation*}
\textbf{Step 1}: This step is devoted to show that $H_{1}\left( \lambda \right) $ and $H_{2}\left( \lambda \right) $ are entire functions of $ \lambda \in
\mathbb{C}
$. We first prove that $\frac{F_{1}\left( \lambda \right) }{G_{W}\left( \lambda \right) }\ $and $\frac{F_{2}\left( \lambda \right) }{G_{W}\left( \lambda \right) }$ are entire functions of $\lambda \in
\mathbb{C}
$.\ In fact, from $\left( \ref{ratios}\right) ,$ $\left( \ref{7}\right) ,$ $ \left( \ref{3f}\right) ,$ $\left( \ref{hhypo 3}\right) $, $H=\widetilde{H}$ and $q=\widetilde{q}$ a.e. on $\left[ b,\pi \right] ,$ one can easily deduce that for $\lambda _{n}\in \widehat{W},$ $n\in S_{B}$, \begin{equation*} \varphi _{\nu }\left( x,\lambda _{n}\right) =\widetilde{\varphi }_{\nu }\left( x,\lambda _{n}\right) ,\ x\in \left[ b,\pi \right] \text{,} \end{equation*} where $\nu =0,1,\ldots ,k_{n}-1.$ Thus for $\lambda _{n}\in \widehat{W},$ $ n\in S_{B}$, $\nu =0,1,\ldots ,k_{n}-1,$ one observes that \begin{equation} \varphi _{\nu }\left( b,\lambda _{n}\right) =\widetilde{\varphi }_{\nu }\left( b,\lambda _{n}\right) ,\text{ }\varphi _{\nu }^{\prime }\left( b,\lambda _{n}\right) =\widetilde{\varphi }_{\nu }^{\prime }\left( b,\lambda _{n}\right) ,\text{ } \label{fainiu} \end{equation} and thus \begin{eqnarray} \left. \frac{d^{\nu }F_{1}\left( \lambda \right) }{d\lambda ^{\nu }} \right\vert _{\lambda =\lambda _{n}} &:&=\nu !\left( \varphi _{\nu }\left( b,\lambda _{n}\right) -\widetilde{\varphi }_{\nu }\left( b,\lambda _{n}\right) \right) =0, \label{555} \\ \left. \frac{d^{\nu }F_{2}\left( \lambda \right) }{d\lambda ^{\nu }} \right\vert _{\lambda =\lambda _{n}} &:&=\nu !\left( \varphi _{\nu }^{\prime }\left( b,\lambda _{n}\right) -\widetilde{\varphi }_{\nu }^{\prime }\left( b,\lambda _{n}\right) \right) =0. \label{gggpie} \end{eqnarray} Then in view of $\left( \ref{FGFG}\right) $ and $\left( \ref{GG}\right) ,$ we infer that $\frac{F_{1}\left( \lambda \right) }{G_{W}\left( \lambda \right) }\ $and $\frac{F_{2}\left( \lambda \right) }{G_{W}\left( \lambda \right) }$ are entire functions of $\lambda \in
\mathbb{C}
$. Similarly, we can also prove that $\frac{F_{1}\left( \lambda \right) }{ G_{W^{\infty }}\left( \lambda \right) }\ $and $\frac{F_{2}\left( \lambda \right) }{G_{W^{\infty }}\left( \lambda \right) }$ are entire functions of $ \lambda \in
\mathbb{C}
$. Therefore, from the fact $\widehat{\sigma \left( B\right) }\cap \widehat{ \sigma \left( B^{\infty }\right) }=\emptyset $ we conclude that $H_{1}\left( \lambda \right) $ and $H_{2}\left( \lambda \right) $ are entire functions of $\lambda \in
\mathbb{C}
$. Furthermore, it is easy to see that the order of $H_{1}\left( \lambda \right) $ and $H_{2}\left( \lambda \right) $ are less than $\frac{1}{2}$.
\textbf{Step 2}: Now we want to use Lemma \ref{proposition} to prove $ H_{1}\left( \lambda \right) \equiv 0.$ From Lemma \ref{number} and the assumption $\left( \ref{hypo4}\right) ,$ it follows that there exists a constant $M>0$ such that \begin{equation} \left\vert G_{\Theta }\left( iy\right) \right\vert \geq M\left\vert y\right\vert ^{\epsilon }\exp \left( \left\vert \text{Im}\sqrt{iy} \right\vert b\right) . \label{GS} \end{equation} Moreover, from $\left( \ref{1}\right) $ we know that \begin{equation*} F_{1}\left( \lambda \right) =\left( \left( b_{1}-\widetilde{b}_{1}\right) \cos \left( \sqrt{\lambda }b\right) +\left( b_{2}-\widetilde{b}_{2}\right) \cos \left( \sqrt{\lambda }\left( 2d-b\right) \right) \right) +O\left( \frac{ \exp \left( \left\vert \text{Im}\sqrt{\lambda }\right\vert b\right) }{\sqrt{ \lambda }}\right) , \end{equation*} and thus \begin{equation} \left\vert F_{1}\left( iy\right) \right\vert =\exp \left( \left\vert \text{Im }\sqrt{iy}\right\vert b\right) \left( \frac{\left\vert b_{1}-\widetilde{b} _{1}\right\vert }{2}+o\left( 1\right) \right) \text{ as }y\text{ }\left( \text{real}\right) \rightarrow \infty . \label{f1iy} \end{equation} Therefore, by $\left( \ref{FGFG}\right) ,$ $\left( \ref{GS}\right) $ and $ \left( \ref{f1iy}\right) ,$ one deduces that \begin{equation*} \left\vert H_{1}\left( iy\right) \right\vert \leq \left\vert \frac{\exp \left( \left\vert \text{Im}\sqrt{iy}\right\vert b\right) \left( \frac{ \left\vert b_{1}-\widetilde{b}_{1}\right\vert }{2}+o\left( 1\right) \right) }{M\left\vert y\right\vert ^{\epsilon }\exp \left( \left\vert \text{Im}\sqrt{ iy}\right\vert b\right) }\right\vert =O\left( y^{-\epsilon }\right) , \end{equation*} as $y$ $\left( \text{real}\right) \rightarrow \infty .$ By Lemma \ref {proposition}, one deduces that $H_{1}\left( \lambda \right) \equiv 0\ $and therefore $F_{1}\left( \lambda \right) \equiv 0$ for all $\lambda \in
\mathbb{C}
,$ i.e., $\varphi \left( b,\lambda \right) \equiv \widetilde{\varphi }\left( b,\lambda \right) .$
\textbf{Step 3}: From the fact $\varphi \left( b,\lambda \right) \equiv \widetilde{\varphi }\left( b,\lambda \right) $, we know that \begin{equation*} H_{2}\left( \lambda \right) =\frac{\left[ \varphi ^{\prime }\left( b,\lambda \right) -\widetilde{\varphi }^{\prime }\left( b,\lambda \right) \right] \varphi \left( b,\lambda \right) }{G_{\Theta }\left( \lambda \right) \varphi \left( b,\lambda \right) }=\frac{-F\left( \lambda \right) }{G_{\Theta }\left( \lambda \right) \varphi \left( b,\lambda \right) }. \end{equation*} Hence, from $\left( \ref{faib}\right) ,$ $\left( \ref{GS}\right) \ $and Lemma \ref{iy}, we have \begin{eqnarray*} \left\vert H_{2}\left( iy\right) \right\vert &\leq &\left\vert \frac{o\left( \exp \left( 2\left\vert \text{Im}\sqrt{iy}\right\vert b\right) \right) }{ M\left\vert y\right\vert ^{\epsilon }\exp \left( \left\vert \text{Im}\sqrt{iy }\right\vert b\right) \frac{b_{1}}{2}\exp \left( \left\vert \text{Im}\sqrt{iy }\right\vert b\right) \left( 1+o\left( 1\right) \right) }\right\vert \\ &=&o\left( y^{-\epsilon }\right) . \end{eqnarray*} Then it follows from Lemma \ref{proposition} that $H_{2}\left( \lambda \right) =0$ and thus $F\left( \lambda \right) \equiv 0$ for all $\lambda \in
\mathbb{C}
.$ Now we can conclude from Lemma \ref{unique by F} that $h=\widetilde{h},$ $ \beta =\widetilde{\beta },$ $\gamma =\widetilde{\gamma }$ and $q=\widetilde{q }$ a.e. on $\left[ 0,\pi \right] .$ The proof is thus completed. \end{proof}
\begin{remark} If $b_{1}=\frac{\beta +\beta ^{-1}}{2}$ is given, then it is easy to see from $\left( \ref{f1iy}\right) $ that \begin{equation*} \left\vert F_{1}\left( iy\right) \right\vert =o\left( \exp \left( \left\vert \text{Im}\sqrt{iy}\right\vert b\right) \right) \text{ as }y\text{ }\left( \text{real}\right) \rightarrow \infty . \end{equation*} In this case the assumption $\left( \ref{hypo4}\right) $ in Theorem \ref {theorem copy(1)} can be replaced by \begin{equation*} N_{W}(t)+N_{W^{\infty }}(t)\geq AN_{\sigma \left( B\right) }(t)+\left( \frac{ b}{\pi }-A\right) N_{\sigma \left( B^{\infty }\right) }(t)-\frac{A}{2}. \end{equation*} \end{remark}
\subsection{Case II: $q$ is known on $\left[ b,\protect\pi \right] ,$ where $ b=d$}
\subsubsection{Pairs of Eigenvalues and Normalizing Constants}
\begin{theorem} \label{theorem copy(2)}Assume Hypothesis \ref{hypo1} and suppose that $q= \widetilde{q}$ a.e. on $\left[ b,\pi \right] ,$ and \begin{eqnarray} &&N_{W}(t)+N_{W_{1}}(t)+N_{W^{\infty }}(t)+N_{W_{1}^{\infty }}(t) \label{hypo6} \\ &\geq &AN_{\sigma \left( B\right) }(t)+\left( \frac{2d}{\pi }-A\right) N_{\sigma \left( B^{\infty }\right) }(t)-\frac{A}{2}+\frac{1}{2}+\epsilon \notag \end{eqnarray} for sufficiently large $t\in
\mathbb{R}
,$ where $\epsilon $ is an arbitrary positive constant$.$ Then $h=\widetilde{ h},$ $\beta =\widetilde{\beta },$ $\gamma =\widetilde{\gamma }$ and $q= \widetilde{q}$ a.e. on $\left[ 0,\pi \right] .$ \end{theorem}
\begin{proof} Let \begin{equation} H\left( \lambda \right) :=\frac{F\left( \lambda \right) }{G_{\Xi }\left( \lambda \right) }, \label{hhhh} \end{equation} where $G_{\Xi }\left( \lambda \right) $ is similarly defined as in $\left( \ref{G3}\right) $ and $F\left( \lambda \right) $ is defined by $\left( \ref {FFF}\right) .$ By Lemma \ref{Fqh} we know that if $b=d$, \begin{eqnarray} F\left( \lambda \right) &=&\left. \left\langle \varphi \left( x,\lambda \right) ,\widetilde{\varphi }\left( x,\lambda \right) \right\rangle \right\vert _{x=d+0} \label{flamuda2} \\ &=&\beta \widetilde{\beta }^{-1}\varphi \left( d-0,\lambda \right) \widetilde{\varphi }^{\prime }\left( d-0,\lambda \right) -\beta ^{-1} \widetilde{\beta }\widetilde{\varphi }\left( d-0,\lambda \right) \varphi ^{\prime }\left( d-0,\lambda \right) \notag \\ &&+\widetilde{\gamma }\beta \varphi \left( d-0,\lambda \right) \widetilde{ \varphi }\left( d-0,\lambda \right) -\gamma \widetilde{\beta }\widetilde{ \varphi }\left( d-0,\lambda \right) \varphi \left( d-0,\lambda \right) . \notag \end{eqnarray} Moreover, from Lemma \ref{jjgj} it is easy to see that \begin{eqnarray} \left\vert \varphi \left( d-0,iy\right) \right\vert &=&\frac{1}{2}\exp \left( \left\vert \text{Im}\sqrt{iy}\right\vert d\right) \left( 1+o\left( 1\right) \right) , \label{faiiy} \\ \left\vert \varphi ^{\prime }\left( d-0,iy\right) \right\vert &=&\frac{1}{2} \left\vert y\right\vert ^{\frac{1}{2}}\exp \left( \left\vert \text{Im}\sqrt{ iy}\right\vert d\right) \left( 1+o\left( 1\right) \right) \notag \end{eqnarray} as $y$ $\left( \text{real}\right) \rightarrow \infty ,$ and hence \begin{equation} \left\vert F\left( iy\right) \right\vert =O\left( \left\vert y\right\vert ^{ \frac{1}{2}}\exp \left( 2\left\vert \text{Im}\sqrt{iy}\right\vert d\right) \right) \text{ as }y\text{ }\left( \text{real}\right) \rightarrow \infty . \label{FIY} \end{equation} By Lemma \ref{number} and $\left( \ref{hypo6}\right) ,$ we infer that there exists a constant $M>0$ such that \begin{equation} \left\vert G_{\Xi }\left( iy\right) \right\vert \geq M\left\vert y\right\vert ^{\frac{1}{2}+\epsilon }\exp \left( 2\left\vert \text{Im}\sqrt{ iy}\right\vert d\right) . \label{Gttt} \end{equation} Therefore, from $\left( \ref{hhhh}\right) ,$ $\left( \ref{FIY}\right) \ $and $\left( \ref{Gttt}\right) ,$ we have \begin{equation*} \left\vert H\left( iy\right) \right\vert \leq \left\vert \frac{O\left( \left\vert y\right\vert ^{\frac{1}{2}}\exp \left( 2\left\vert \text{Im}\sqrt{ iy}\right\vert d\right) \right) }{M\left\vert y\right\vert ^{\frac{1}{2} +\epsilon }\exp \left( 2\left\vert \text{Im}\sqrt{iy}\right\vert d\right) } \right\vert =O\left( \left\vert y\right\vert ^{-\epsilon }\right) \end{equation*} as $y$ $\left( \text{real}\right) \rightarrow \infty .$ This implies that $ H\left( \lambda \right) \equiv 0\ $and hence $F\left( \lambda \right) \equiv 0$ for all $\lambda \in
\mathbb{C}
$ by the argument of the proof of Theorem \ref{theorem}$.$ Then the statement of this theorem can be concluded from Lemma \ref{unique by F}. \end{proof}
\begin{remark} \label{theorem copy(6)}$(1)$ If $\beta =\widetilde{\beta },$ instead of condition $\left( \ref{hypo6}\right) $, we only need the following condition$ :$ \begin{eqnarray*} &&N_{S}(t)+N_{S_{1}}(t)+N_{S_{1}^{\infty }}(t)+N_{S^{\infty }}(t) \\ &\geq &AN_{\sigma \left( B\right) }(t)+\left( \frac{2d}{\pi }-A\right) N_{\sigma \left( B^{\infty }\right) }(t)-\frac{A}{2}+\epsilon ; \end{eqnarray*} $(2)$ If $\beta =\widetilde{\beta },$ $\gamma =\widetilde{\gamma },$ $q,$ $ \widetilde{q}\in $ $C^{m}$ near $d,$ then, instead of condition $\left( \ref {hypo6}\right) $, we only need the following condition$:$ \begin{eqnarray*} &&N_{S}(t)+N_{S_{1}}(t)+N_{S_{1}^{\infty }}(t)+N_{S^{\infty }}(t) \\ &\geq &AN_{\sigma \left( B\right) }(t)+\left( \frac{2d}{\pi }-A\right) N_{\sigma \left( B^{\infty }\right) }(t)-\frac{A}{2}-\frac{m+1}{2}\text{.} \end{eqnarray*} \end{remark}
In fact, one notes that for $x\in \left( 0,d\right) ,$ \begin{eqnarray} &&\varphi \left( x,\lambda \right) \widetilde{\varphi }^{\prime }\left( x,\lambda \right) -\widetilde{\varphi }\left( x,\lambda \right) \varphi ^{\prime }\left( x,\lambda \right) \label{faifai} \\ &=&y_{1,0}(x,\lambda )\widetilde{y}_{1,0}^{\prime }(x,\lambda )-y_{1,0}^{\prime }(x,\lambda )\widetilde{y}_{1,0}(x,\lambda ) \notag \\ &&+h\left( y_{2,0}(x,\lambda )\widetilde{y}_{1,0}^{\prime }(x,\lambda )-y_{2,0}^{\prime }(x,\lambda )\widetilde{y}_{1,0}(x,\lambda )\right) \notag \\ &&+\widetilde{h}\left( y_{1,0}(x,\lambda )\widetilde{y}_{2,0}^{\prime }(x,\lambda )-\widetilde{y}_{2,0}(x,\lambda )y_{1,0}^{\prime }(x,\lambda )\right) \notag \\ &&+h\widetilde{h}\left( y_{2,0}(x,\lambda )\widetilde{y}_{2,0}^{\prime }(x,\lambda )-y_{2,0}^{\prime }(x,\lambda )\widetilde{y}_{2,0}(x,\lambda )\right) . \notag \end{eqnarray} Therefore, if $\beta =\widetilde{\beta },$ it follows from $\left( \ref {flamuda2}\right) ,$ $\left( \ref{faiiy}\right) $ and Remark \ref{ooo copy(4)} that \begin{equation} \left\vert F\left( iy\right) \right\vert =O\left( \exp \left( 2\left\vert \text{Im}\sqrt{iy}\right\vert d\right) \right) \text{ as }y\text{ }\left( \text{real}\right) \rightarrow \infty . \label{FIY2} \end{equation} Moreover, if $\beta =\widetilde{\beta },$ $\gamma =\widetilde{\gamma },$ $q,$ $\widetilde{q}\in $ $C^{m}$ near $d,$ it is easy to see from $\left( \ref {flamuda2}\right) $ and Proposition \ref{ooo copy(1)} that \begin{equation} \left\vert F\left( iy\right) \right\vert =o\left( \left\vert y\right\vert ^{- \frac{m+1}{2}}\exp \left( 2\left\vert \text{Im}\sqrt{iy}\right\vert d\right) \right) \text{ as }y\text{ }\left( \text{real}\right) \rightarrow \infty . \label{ass} \end{equation} Thus by the argument of the proof of Theorem \ref{theorem copy(2)}, Remark \ref{theorem copy(6)} can be directly obtained.
\begin{corollary} \label{corollary copy(3)}Let $d=\frac{\pi }{2}.$ Assume that $q$ is $C^{m}$ near $\frac{\pi }{2}\ $and suppose that $\beta ,$ $\gamma ,$ $q\ $on $\left[ \frac{\pi }{2},\pi \right] \ $are known a priori. Then $h$ and $q$ on $\left[ 0,\pi \right] $ can be uniquely determined by all the eigenvalues $\left\{ \lambda _{n}\right\} _{n\in
\mathbb{N}
_{0}}$ of $B$ except for $\left( \left[ \frac{m+2}{2}\right] \right) ,$ or all the eigenvalues $\left\{ \lambda _{n}^{\infty }\right\} _{n\in
\mathbb{N}
_{0}}$ of $B^{\infty }$ except for $\left( \left[ \frac{m+1}{2}\right] \right) $. \end{corollary}
\begin{corollary} Let $d=\frac{\pi }{2}.$ Assume that $q$ on $\left[ \frac{\pi }{2},\pi \right] $ and $\beta $ are known a priori, then $\sigma \left( B\right) $ uniquely determines $h,$ $\gamma $ and $q$ a.e. on $\left[ 0,\pi \right] .$ \end{corollary}
\subsubsection{Pairs of Eigenvalues and Ratios}
\begin{theorem} \label{theorem copy(3)}Assume Hypothesis \ref{hypo2} and suppose that $q= \widetilde{q}$ a.e. on $\left[ d,\pi \right] ,$ \begin{equation} N_{W}(t)+N_{W^{\infty }}(t)\geq AN_{\sigma \left( B\right) }(t)+\left( \frac{ d}{\pi }-A\right) N_{\sigma \left( B^{\infty }\right) }(t)-\frac{A}{2}+\frac{ 1}{2}+\epsilon \label{hypo8} \end{equation} for sufficiently large $t\in
\mathbb{R}
,$ where $\epsilon $ is an arbitrary positive constant.$\ $Then $h= \widetilde{h},$ $\beta =\widetilde{\beta },$ $\gamma =\widetilde{\gamma }$ and $q=\widetilde{q}$ a.e. on $\left[ 0,\pi \right] .$ \end{theorem}
\begin{proof} Denote \begin{equation} H_{1}\left( \lambda \right) :=\frac{F_{1}\left( \lambda \right) }{G_{\Theta }\left( \lambda \right) },\text{ }H_{2}\left( \lambda \right) =\frac{ F_{2}\left( \lambda \right) }{G_{\Theta }\left( \lambda \right) }, \label{F2} \end{equation} where $G_{\Theta }\left( \lambda \right) :=G_{W}\left( \lambda \right) G_{W^{\infty }}\left( \lambda \right) ,$ \begin{equation*} G_{W}\left( \lambda \right) :=\prod\limits_{\lambda _{n}\in \widehat{W},n\in S_{B}}\left( 1-\frac{\lambda }{\lambda _{n}}\right) ^{k_{n}},\text{ } G_{W^{\infty }}\left( \lambda \right) :=\prod\limits_{\lambda _{n}^{\infty }\in \widehat{W^{\infty }},n\in S_{B^{\infty }}}\left( 1-\frac{\lambda }{ \lambda _{n}^{\infty }}\right) ^{k_{n}^{\infty }} \end{equation*} and \begin{equation*} F_{1}\left( \lambda \right) :=\varphi \left( d+0,\lambda \right) -\widetilde{ \varphi }\left( d+0,\lambda \right) ,\text{ }F_{2}\left( \lambda \right) :=\varphi ^{\prime }\left( d+0,\lambda \right) -\widetilde{\varphi }^{\prime }\left( d+0,\lambda \right) . \end{equation*} In view of Lemma \ref{number} and $\left( \ref{hypo8}\right) ,$ one has \begin{equation} \left\vert G_{\Theta }\left( iy\right) \right\vert \geq M\left\vert y\right\vert ^{\epsilon +\frac{1}{2}}\exp \left( \left\vert \text{Im}\sqrt{iy }\right\vert d\right) . \label{gs2} \end{equation} In addition, from $\left( \ref{1}\right) \ $it is easy to see that \begin{equation} \left\vert F_{1}\left( iy\right) \right\vert =\exp \left( \left\vert \text{Im }\sqrt{iy}\right\vert d\right) \left( \frac{\left\vert \beta -\widetilde{ \beta }\right\vert }{2}+o\left( 1\right) \right) \text{ as }y\text{ }\left( \text{real}\right) \rightarrow \infty . \label{F1111} \end{equation} Thus it follows from $\left( \ref{gs2}\right) $ and $\left( \ref{F1111} \right) $ that \begin{eqnarray*} \left\vert H_{1}\left( iy\right) \right\vert &\leq &\left\vert \frac{\exp \left( \left\vert \text{Im}\sqrt{iy}\right\vert d\right) \left( \frac{ \left\vert \beta -\widetilde{\beta }\right\vert }{2}+o\left( 1\right) \right) }{M\left\vert y\right\vert ^{\epsilon +\frac{1}{2}}\exp \left( \left\vert \text{Im}\sqrt{iy}\right\vert d\right) }\right\vert \\ &=&O\left( y^{-\epsilon -\frac{1}{2}}\right) \text{ as }y\text{ }\left( \text{real}\right) \rightarrow \infty . \end{eqnarray*} By a similar proof to that of Theorem $\ref{theorem copy(1)},$ we can obtain that $H_{1}\left( \lambda \right) \equiv 0,$ and thus $F_{1}\left( \lambda \right) \equiv 0,$ i.e., $\varphi \left( d+0,\lambda \right) \equiv \widetilde{\varphi }\left( d+0,\lambda \right) $ for all $\lambda \in
\mathbb{C}
$. Then it follows from $\left( \ref{flamuda2}\right) $ and $\left( \ref{F2} \right) $ that \begin{equation*} H_{2}\left( \lambda \right) =\frac{\left[ \varphi ^{\prime }\left( d+0,\lambda \right) -\widetilde{\varphi }^{\prime }\left( d+0,\lambda \right) \right] \varphi \left( d+0,\lambda \right) }{G_{\Theta }\left( \lambda \right) \varphi \left( d+0,\lambda \right) }=\frac{-F\left( \lambda \right) }{G_{\Theta }\left( \lambda \right) \varphi \left( d+0,\lambda \right) }. \end{equation*} Thus by $\left( \ref{faid}\right) ,$ $\left( \ref{FIY}\right) $ and $\left( \ref{gs2}\right) ,$ we infer that as $y$ $\left( \text{real}\right) \rightarrow \infty ,$ \begin{eqnarray*} \left\vert H_{2}\left( iy\right) \right\vert &\leq &\left\vert \frac{O\left( \left\vert y\right\vert ^{\frac{1}{2}}\exp \left( 2\left\vert \text{Im}\sqrt{ iy}\right\vert d\right) \right) }{M\left\vert y\right\vert ^{\epsilon +\frac{ 1}{2}}\exp \left( \left\vert \text{Im}\sqrt{iy}\right\vert d\right) \frac{ \beta }{2}\exp \left( \left\vert \text{Im}\sqrt{iy}\right\vert d\right) \left( 1+o\left( 1\right) \right) }\right\vert \\ &=&O\left( \left\vert y\right\vert ^{-\epsilon }\right) . \end{eqnarray*} Then by the argument of the proof of Theorem $\ref{theorem copy(1)},$ we can obtain that $F\left( \lambda \right) \equiv 0.$ Now we conclude from Lemma \ref{unique by F} that $h=\widetilde{h},$ $\beta =\widetilde{\beta },$ $ \gamma =\widetilde{\gamma }$ and $q=\widetilde{q}$ a.e. on $\left[ 0,\pi \right] .$ \end{proof}
\begin{remark} $(1)$ If $\beta \ $is known a priori$,$ then by $\left( \ref{FIY2}\right) $ and $\left( \ref{F1111}\right) $ one has \begin{equation*} \left\vert F\left( iy\right) \right\vert =O\left( \exp \left( 2\left\vert \text{Im}\sqrt{iy}\right\vert d\right) \right) \text{ and }F_{1}\left( iy\right) =o\left( \exp \left( 2\left\vert \text{Im}\sqrt{iy}\right\vert d\right) \right) \end{equation*} as $y$ $\left( \text{real}\right) \rightarrow \infty .$ In this case, the assumption $\left( \ref{hypo8}\right) $ can be replaced by \begin{equation*} N_{W}(t)+N_{W^{\infty }}(t)\geq AN_{\sigma \left( B\right) }(t)+\left( \frac{ d}{\pi }-A\right) N_{\sigma \left( B^{\infty }\right) }(t)-\frac{A}{2} +\epsilon . \end{equation*} $(2)$ If $\beta \ $and $\gamma $ are known a priori$,$ by $\left( \ref{ass} \right) $ $\left( \text{for }m=-1\right) $ and $\left( \ref{F1111}\right) $ one has \begin{equation*} \left\vert F\left( iy\right) \right\vert =o\left( \exp \left( 2\left\vert \text{Im}\sqrt{iy}\right\vert d\right) \right) \text{ and }F_{1}\left( iy\right) =o\left( \exp \left( 2\left\vert \text{Im}\sqrt{iy}\right\vert d\right) \right) \end{equation*} as $y$ $\left( \text{real}\right) \rightarrow \infty .$ In this case, the assumption $\left( \ref{hypo8}\right) $ can be replaced by \begin{equation*} N_{W}(t)+N_{W^{\infty }}(t)\geq AN_{\sigma \left( B\right) }(t)+\left( \frac{ d}{\pi }-A\right) N_{\sigma \left( B^{\infty }\right) }(t)-\frac{A}{2}. \end{equation*} \end{remark}
\subsection{Case III: $q$ is known on $\left[ b,\protect\pi \right] ,$ where $b\in \left( 0,d\right) $}
\subsubsection{Pairs of Eigenvalues and Normalizing Constants}
\begin{theorem} \label{theorem copy(4)}Assume Hypothesis \ref{hypo1} and suppose that $q,$ $ \widetilde{q}\in $ $C^{m}$ near $b\in \left( 0,d\right) ,$ $q=\widetilde{q}$ a.e. on $\left[ b,\pi \right] ,$ $\beta =\widetilde{\beta },$ $\gamma = \widetilde{\gamma }$ and \begin{eqnarray} &&N_{W}(t)+N_{W_{1}}(t)+N_{W^{\infty }}(t)+N_{W_{1}^{\infty }}(t) \label{hypo9} \\ &\geq &AN_{\sigma \left( B\right) }(t)+\left( \frac{2b}{\pi }-A\right) N_{\sigma \left( B^{\infty }\right) }(t)-\frac{A}{2}-\frac{m+1}{2} \notag \end{eqnarray} for sufficiently large $t\in
\mathbb{R}
.$ Then $h=\widetilde{h}$ and $q=\widetilde{q}$ a.e. on $\left[ 0,\pi \right] .$ \end{theorem}
\begin{proof} Denote \begin{equation*} H\left( \lambda \right) :=\frac{F\left( \lambda \right) }{G_{\Xi }\left( \lambda \right) }, \end{equation*} where $G_{\Xi }\left( \lambda \right) $ is similarly defined as in $\left( \ref{G3}\right) $ and $F\left( \lambda \right) $ is defined by $\left( \ref {FFF}\right) .$ Then it follows from Lemma \ref{Fqh} that if $\beta = \widetilde{\beta },$ $\gamma =\widetilde{\gamma },$ \begin{equation} F\left( \lambda \right) =\left. \left\langle \varphi \left( x,\lambda \right) ,\widetilde{\varphi }\left( x,\lambda \right) \right\rangle \right\vert _{x=b}.\text{ } \label{FLAMUDA3} \end{equation} In addition, if $q=\tilde{q}$ a.e. on $\left[ b,\pi \right] ,$ and $q,$ $ \tilde{q}\in C^{m}$ near $b\in \left( 0,d\right) ,$ one observes from $ \left( \ref{faifai}\right) $ and Proposition \ref{ooo copy(1)} that \begin{equation} \left\vert F\left( iy\right) \right\vert =o\left( \left\vert y\right\vert ^{- \frac{m+1}{2}}\exp \left( 2\left\vert \text{Im}\sqrt{iy}\right\vert b\right) \right) \text{ as }y\text{ }\left( \text{real}\right) \rightarrow \infty \text{.} \label{FIY3} \end{equation} By Lemma \ref{number} and $\left( \ref{hypo9}\right) ,$ we have \begin{equation*} \left\vert G_{\Xi }\left( iy\right) \right\vert \geq M\left\vert y\right\vert ^{-\frac{m+1}{2}}\exp \left( 2\left\vert \text{Im}\sqrt{iy} \right\vert b\right) . \end{equation*} Therefore, \begin{equation*} \left\vert H\left( iy\right) \right\vert \leq \left\vert \frac{o\left( \left\vert y\right\vert ^{-\frac{m+1}{2}}\exp \left( 2\left\vert \text{Im} \sqrt{iy}\right\vert b\right) \right) }{M\left\vert y\right\vert ^{-\frac{m+1 }{2}}\exp \left( 2\left\vert \text{Im}\sqrt{iy}\right\vert b\right) } \right\vert =o\left( 1\right) \text{ as }y\text{ }\left( \text{real}\right) \rightarrow \infty . \end{equation*} This implies that $H\left( \lambda \right) \equiv 0\ $and thus $F\left( \lambda \right) \equiv 0$ for all $\lambda \in
\mathbb{C}
$ by the argument of the proof of Theorem \ref{theorem}$.$ Then we conclude the statement of this theorem from Lemma \ref{unique by F}. \end{proof}
\subsubsection{Pairs of Eigenvalues and Ratios}
\begin{theorem} \label{theorem copy(5)}Assume Hypothesis \ref{hypo2} and suppose that $q= \widetilde{q}$ a.e. on $\left[ d,\pi \right] ,$ $\beta =\widetilde{\beta },$ $\gamma =\widetilde{\gamma }$ and \begin{equation} N_{W}(t)+N_{W^{\infty }}(t)\geq AN_{\sigma \left( B\right) }(t)+\left( \frac{ b}{\pi }-A\right) N_{\sigma \left( B^{\infty }\right) }(t)-\frac{A}{2} \label{hypo10} \end{equation} for sufficiently large $t\in
\mathbb{R}
,$ where $\epsilon $ is an arbitrary positive constant.$\ $Then $h= \widetilde{h}$ and $q=\widetilde{q}$ a.e. on $\left[ 0,\pi \right] .$ \end{theorem}
\begin{proof} Denote \begin{equation} H_{1}\left( \lambda \right) :=\frac{F_{1}\left( \lambda \right) }{G_{\Theta }\left( \lambda \right) },\text{ }H_{2}\left( \lambda \right) =\frac{ F_{2}\left( \lambda \right) }{G_{\Theta }\left( \lambda \right) }, \label{H2} \end{equation} where $G_{\Theta }\left( \lambda \right) :=G_{W}\left( \lambda \right) G_{W^{\infty }}\left( \lambda \right) ,$ \begin{equation*} G_{W}\left( \lambda \right) :=\prod\limits_{\lambda _{n}\in \widehat{W},n\in S_{B}}\left( 1-\frac{\lambda }{\lambda _{n}}\right) ^{k_{n}},\text{ } G_{W^{\infty }}\left( \lambda \right) :=\prod\limits_{\lambda _{n}^{\infty }\in \widehat{W^{\infty }},n\in S_{B^{\infty }}}\left( 1-\frac{\lambda }{ \lambda _{n}^{\infty }}\right) ^{k_{n}^{\infty }} \end{equation*} and \begin{equation*} F_{1}\left( \lambda \right) :=\varphi \left( b,\lambda \right) -\widetilde{ \varphi }\left( b,\lambda \right) ,\text{ }F_{2}\left( \lambda \right) :=\varphi ^{\prime }\left( b,\lambda \right) -\widetilde{\varphi }^{\prime }\left( b,\lambda \right) . \end{equation*} By a similar method to that of Theorem \ref{theorem copy(1)}, one can easily deduce that $H_{1}\left( \lambda \right) $ and $H_{2}\left( \lambda \right) $ are entire functions of order less than $\frac{1}{2}$ from the facts $\left( \ref{ratios}\right) ,$ $\left( \ref{7}\right) ,$ $\left( \ref{3f}\right) ,$ $ \left( \ref{hhypo 3}\right) $, $H=\widetilde{H}$, $\beta =\widetilde{\beta } , $ $\gamma =\widetilde{\gamma }$ and $q=\widetilde{q}$ a.e. on $\left[ b,\pi \right] .$
In view of Lemma \ref{number} and $\left( \ref{hypo10}\right) ,$ one has \begin{equation} \left\vert G_{\Theta }\left( iy\right) \right\vert \geq M\exp \left( \left\vert \text{Im}\sqrt{iy}\right\vert b\right) . \label{GTHETE} \end{equation} By $\left( \ref{1}\right) $ we also infer that \begin{equation*} \left\vert F_{1}\left( iy\right) \right\vert =O\left( \left\vert y\right\vert ^{-\frac{1}{2}}\exp \left( \left\vert \text{Im}\sqrt{iy} \right\vert b\right) \right) \text{ as }y\text{ }\left( \text{real}\right) \rightarrow \infty . \end{equation*} Therefore, \begin{equation*} \left\vert H_{1}\left( iy\right) \right\vert \leq \left\vert \frac{O\left( \left\vert y\right\vert ^{-\frac{1}{2}}\exp \left( \left\vert \text{Im}\sqrt{ iy}\right\vert b\right) \right) }{M\exp \left( \left\vert \text{Im}\sqrt{iy} \right\vert b\right) }\right\vert =O\left( y^{-\frac{1}{2}}\right) \text{ } \end{equation*} as $y$ $\left( \text{real}\right) \rightarrow \infty .$ Now by Lemma \ref {proposition}$,$ we can obtain that $H_{1}\left( \lambda \right) \equiv 0,$ i.e., $\varphi \left( b,\lambda \right) \equiv \widetilde{\varphi }\left( b,\lambda \right) $ for all $\lambda \in
\mathbb{C}
$. Then it follows from $\left( \ref{FLAMUDA3}\right) $ and $\left( \ref{H2} \right) $ that \begin{equation*} H_{2}\left( \lambda \right) =\frac{\left[ \varphi ^{\prime }\left( b,\lambda \right) -\widetilde{\varphi }^{\prime }\left( b,\lambda \right) \right] \varphi \left( b,\lambda \right) }{G_{\Theta }\left( \lambda \right) \varphi \left( b,\lambda \right) }=\frac{-F\left( \lambda \right) }{G_{\Theta }\left( \lambda \right) \varphi \left( b,\lambda \right) }. \end{equation*} Thus by $\left( \ref{faib}\right) ,$ $\left( \ref{FIY3}\right) $ $\left( \text{for }m=-1\right) $ and $\left( \ref{GTHETE}\right) ,$ we have \begin{equation*} \left\vert H_{2}\left( iy\right) \right\vert \leq \left\vert \frac{o\left( \exp \left( 2\left\vert \text{Im}\sqrt{iy}\right\vert b\right) \right) }{ M\exp \left( \left\vert \text{Im}\sqrt{iy}\right\vert b\right) \frac{1}{2} \exp \left( \left\vert \text{Im}\sqrt{iy}\right\vert b\right) \left( 1+o\left( 1\right) \right) }\right\vert =o\left( 1\right) . \end{equation*} Then by Lemma \ref{proposition} we infer that $H_{2}\left( \lambda \right) \equiv 0$ and then $F\left( \lambda \right) \equiv 0$ for all $\lambda \in
\mathbb{C}
.$ Now we can conclude from Lemma \ref{unique by F} that $h=\widetilde{h}$ and $q=\widetilde{q}$ a.e. on $\left[ 0,\pi \right] .$ The proof is thus completed. \end{proof}
\section*{Appendix}
For the self-adjoint classical Sturm-Liouville operators, an interesting uniqueness result is to assume that the potential $q$ satisfies a local smoothness condition so that some eigenvalues and norming constants can be missing. While in \cite{ges2,wei,zhaoying} the key technique relies on the high-energy asymptotic expansion of the Weyl $m$-function \cite{weyl}, in our non-self-adjoint setting, the key to prove the uniqueness problems (Theorem \ref{theorem}, Theorem \ref{theorem copy(4)}, Remark $\ref{theorem copy(6)},$ Corollary \ref{corollary copy(1)}--\ref{corollary copy(3)}) will be Proposition \ref{ooo copy(1)}, to be established below.
\begin{definition} \label{yyy}For $i=1,2,$ let $y_{i,r}(x,\lambda )$ and $\widetilde{y} _{i,r}(x,\lambda )$ be solutions of $\left( \ref{ly}\right) $ corresponding to the potential $q$ and $\widetilde{q},$ respectively, where $ y_{i,r}(x,\lambda )\ $and $\widetilde{y}_{i,r}(x,\lambda )$ satisfy the initial conditions \begin{eqnarray*} y_{1,r}(r,\lambda ) &=&y_{2,r}^{\prime }(r,\lambda )=1,\text{ } y_{2,r}(r,\lambda )=y_{1,r}^{\prime }(r,\lambda )=0,\ \\ \widetilde{y}_{1,r}(r,\lambda ) &=&\widetilde{y}_{2,r}^{\prime }(r,\lambda )=1,\text{ }\widetilde{y}_{2,r}(r,\lambda )=\widetilde{y}_{1,r}^{\prime }(r,\lambda )=0,\ r\in \left[ 0,\pi \right) . \end{eqnarray*} For simplicity, denote $y_{1}(x,\lambda ):=y_{1,0}(x,\lambda ),$ $ y_{2}(x,\lambda ):=y_{2,0}(x,\lambda ),$ $\widetilde{y}_{1}(x,\lambda ):= \widetilde{y}_{1,0}(x,\lambda ),$ $\widetilde{y}_{2}(x,\lambda ):=\widetilde{ y}_{2,0}(x,\lambda ).$ \end{definition}
\begin{proposition} \label{ooo copy(1)}Let $x_{0}\in \left( r,\pi \right] \ $where $r\in \left[ 0,\pi \right) $ and assume that $q,$ $\widetilde{q}\in $ $C^{m}\left[ x_{0}-\delta ,x_{0}\right] \ $for some sufficiently small $\delta >0$ and some $m\in
\mathbb{N}
_{0}.$ If $q_{-}^{(j)}(x_{0})=\widetilde{q}_{-}^{(j)}(x_{0})$ for $ j=0,1,\ldots ,m,$ then \begin{eqnarray} && \label{1111} \\ y_{1,r}(x_{0},\lambda )\widetilde{y}_{1,r}^{\prime }(x_{0},\lambda )-y_{1,r}^{\prime }(x_{0},\lambda )\widetilde{y}_{1,r}(x_{0},\lambda ) &=&o\left( \frac{\exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert \left( x_{0}-r\right) \right) }{\left\vert \sqrt{\lambda }\right\vert ^{m+1}} \right) , \notag \\ && \label{1112} \\ y_{1,r}(x_{0},\lambda )\widetilde{y}_{2,r}^{\prime }(x_{0},\lambda )-y_{1,r}^{\prime }(x_{0},\lambda )\widetilde{y}_{2,r}(x_{0},\lambda ) &=&o\left( \frac{\exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert \left( x_{0}-r\right) \right) }{\left\vert \sqrt{\lambda }\right\vert ^{m+2}} \right) , \notag \\ && \label{1113} \\ \widetilde{y}_{1,r}^{\prime }(x_{0},\lambda )y_{2,r}(x_{0},\lambda )- \widetilde{y}_{1,r}(x_{0},\lambda )y_{2,r}^{\prime }(x_{0},\lambda ) &=&o\left( \frac{\exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert \left( x_{0}-r\right) \right) }{\left\vert \sqrt{\lambda }\right\vert ^{m+2}} \right) , \notag \\ && \label{1114} \\ y_{2,r}(x_{0},\lambda )\widetilde{y}_{2,r}^{\prime }(x_{0},\lambda )-y_{2,r}^{\prime }(x_{0},\lambda )\widetilde{y}_{2,r}(x_{0},\lambda ) &=&o\left( \frac{\exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert \left( x_{0}-r\right) \right) }{\left\vert \sqrt{\lambda }\right\vert ^{m+3}} \right) \notag \end{eqnarray} as $\left\vert \lambda \right\vert \rightarrow \infty $ in $\Lambda _{\zeta }:=\left\{ \lambda \in
\mathbb{C}
:\zeta <\text{Arg}\left( \lambda \right) <\pi -\zeta \text{ for }\zeta >0\right\} .$ \end{proposition}
\begin{remark} For $f\in $ $C^{m}\left[ x_{0}-\delta ,x_{0}\right] ,$ we adopt following notations in this section: $\ $ \begin{eqnarray*} f_{-}^{(0)}(x_{0}) &:&=f(x_{0}),\text{ }f_{-}^{(1)}(x_{0}):=\lim\limits_{x \rightarrow x_{0}^{-}}\frac{f(x)-f(x_{0})}{x-x_{0}}, \\ f_{-}^{(j)}(x_{0}) &:&=\lim\limits_{x\rightarrow x_{0}^{-}}\frac{ f^{(j-1)}(x)-f_{-}^{(j-1)}(x_{0})}{x-x_{0}}\text{ for }j=2,3,\ldots ,m. \end{eqnarray*} In addition, $f\in $ $C^{m}\left[ x_{0}-\delta ,x_{0}\right] \ $implies $ \lim\limits_{x\rightarrow x_{0}^{-}}f^{(j)}(x)=f_{-}^{(j)}(x_{0})$ for $ j=0,1,\ldots ,m.$ \end{remark}
The proof of Proposition \ref{ooo copy(1)} will be given at the end of this appendix after the proof of the following lemma.
\begin{lemma} \label{ooo}Let $x_{0}\in \left( 0,\pi \right] $ and $q,$ $\widetilde{q}$ $ \in $ $C^{m}\left[ 0,x_{0}\right] \ $for some $m\in
\mathbb{N}
_{0}.$ If \begin{equation} q_{-}^{(j)}(x_{0})=\widetilde{q}_{-}^{(j)}(x_{0}) \label{qq} \end{equation} for $j=0,1,\ldots ,m,$ then \begin{eqnarray} y_{2}(x_{0},\lambda )\widetilde{y}_{2}^{\prime }(x_{0},\lambda )-y_{2}^{\prime }(x_{0},\lambda )\widetilde{y}_{2}(x_{0},\lambda ) &=&o\left( \frac{\exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert x_{0}\right) }{\left\vert \sqrt{\lambda }\right\vert ^{m+3}}\right) , \label{4} \\ y_{1}(x_{0},\lambda )\widetilde{y}_{1}^{\prime }(x_{0},\lambda )-y_{1}^{\prime }(x_{0},\lambda )\widetilde{y}_{1}(x_{0},\lambda ) &=&o\left( \frac{\exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert x_{0}\right) }{\left( \sqrt{\lambda }\right) ^{m+1}}\right) , \\ y_{1}(x_{0},\lambda )\widetilde{y}_{2}^{\prime }(x_{0},\lambda )-y_{1}^{\prime }(x_{0},\lambda )\widetilde{y}_{2}(x_{0},\lambda ) &=&o\left( \frac{\exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert x_{0}\right) }{\left\vert \sqrt{\lambda }\right\vert ^{m+2}}\right) , \\ \widetilde{y}_{1}^{\prime }(x_{0},\lambda )y_{2}(x_{0},\lambda )-\widetilde{y }_{1}(x_{0},\lambda )y_{2}^{\prime }(x_{0},\lambda ) &=&o\left( \frac{\exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert x_{0}\right) }{ \left\vert \sqrt{\lambda }\right\vert ^{m+2}}\right) \end{eqnarray} as $\left\vert \lambda \right\vert \rightarrow \infty $ in the sector $ \Lambda _{\zeta }.$ \end{lemma}
We shall prove Lemma \ref{ooo} by analyzing the asymptotic expansion of the fundamental solutions $\left( \text{see Lemma \ref{y2} and Lemma }\ref{yy2} \right) $. Now we first give some preliminary facts and notations.
Recall the solution $y_{2}$ defined by Definition \ref{yyy}, then it follows from \cite{inverse} that \begin{equation} y_{2}(x,\lambda )=\sum\limits_{p=0}^{\infty }S_{p}(x,\lambda ),\text{ } y_{2}^{\prime }(x,\lambda )=\sum\limits_{p=0}^{\infty }C_{p}(x,\lambda ), \label{3.43} \end{equation} where $S_{0}(x,\lambda )=\frac{\sin (\sqrt{\lambda }x)}{\sqrt{\lambda }}$, $ C_{0}(x,\lambda )=\cos \left( \sqrt{\lambda }x\right) ,\ $and for $p\geq 1,$ \begin{eqnarray} S_{p}(x,\lambda ) &=&\int_{0}^{x}\frac{\sin (\sqrt{\lambda }\left( x-t\right) )}{\sqrt{\lambda }}q(t)S_{p-1}(t,\lambda )\mathrm{d}t,\text{ } \label{4.1} \\ C_{p}(x,\lambda ) &=&\int_{0}^{x}\cos \left( \sqrt{\lambda }\left( x-t\right) \right) q(t)S_{p-1}(t,\lambda )\mathrm{d}t\text{.} \label{4.2} \end{eqnarray} In what follows, we adopt the following notations: \begin{equation*} \left( \pm \right) _{j}=\left\{ \begin{array}{c} -1\text{ if }j=4s,4s+1, \\ 1\text{ if }j=4s+2,4s+3, \end{array} \right. \end{equation*} and \begin{equation*} \nu _{2s}(x,\lambda ):=\frac{\sin (\sqrt{\lambda }x)}{\left( 2\sqrt{\lambda } \right) ^{2s}},\text{ }\nu _{2s+1}(x,\lambda ):=\frac{\cos (\sqrt{\lambda }x) }{\left( 2\sqrt{\lambda }\right) ^{2s+1}},\text{ }s\in
\mathbb{N}
_{0}. \end{equation*} Then we have the following statement relating to $S_{p}$ defined by $\left( \ref{4.1}\right) .$
\begin{lemma} \label{y2}Assume that $q$ $\in $ $C^{m}\left[ 0,\delta \right] $ for some $ \delta >0\ $and some $m\in
\mathbb{N}
.$ Denote $\sigma \left( x\right) :=\int_{0}^{x}q(t)\mathrm{d}t.$ Then for $ x\in \left[ 0,\delta \right] ,$ we have \begin{eqnarray} &&S_{1}(x,\lambda )=\sum\limits_{j=1}^{m+1}\frac{\nu _{j}(x,\lambda )}{\sqrt{ \lambda }}f_{1,j}\left( x\right) +\frac{\left( \pm \right) _{m+2}}{\sqrt{ \lambda }}\int_{0}^{x}\nu _{m+1}(x-2t,\lambda )q^{\left( m\right) }(t)dt, \label{4.4} \\ &&S_{2}(x,\lambda )=\sum\limits_{j=1}^{m+2}\frac{\nu _{j}(x,\lambda )}{\sqrt{ \lambda }}f_{2,j}\left( x\right) +B_{2}\left( x,\lambda \right) , \label{4.5} \\ &&S_{p}(x,\lambda )=\sum\limits_{j=1}^{m+2}\frac{\nu _{j}(x,\lambda )}{\sqrt{ \lambda }}f_{p,j}\left( x\right) +B_{p}\left( x,\lambda \right) \text{ for } p=3,\ldots ,m+2 \label{4.6} \end{eqnarray} where \begin{eqnarray*} B_{2}\left( x,\lambda \right) &=&-\frac{\left( \pm \right) _{m+3}}{\sqrt{ \lambda }}\int_{0}^{x}\nu _{m+2}(x-2t,\lambda )\left( \sum\limits_{j=1}^{m+1}\left( \pm \right) _{j}\left( q(t)f_{1,j}\left( t\right) \right) ^{\left( m+1-j\right) }\right) dt \\ &&+\frac{\left( \pm \right) _{m+2}}{\sqrt{\lambda }}\int_{0}^{x}\frac{\sin \sqrt{\lambda }\left( x-t\right) }{\sqrt{\lambda }}q(t)\int_{0}^{t}\nu _{m+1}(t-2s,\lambda )q^{\left( m\right) }(s)dsdt, \\ B_{p}\left( x,\lambda \right) &=&-\frac{\left( \pm \right) _{m+3}}{\sqrt{ \lambda }}\int_{0}^{x}\nu _{m+2}(x-2t,\lambda )\sum\limits_{j=1}^{m+1}\left( \pm \right) _{j}\left( q(t)f_{p-1,j}(t)\right) ^{\left( m+1-j\right) }\left( t\right) dt \\ &&+\int_{0}^{x}\frac{\sin (\sqrt{\lambda }\left( x-t\right) )}{\sqrt{\lambda }}q(t)\left[ \frac{\nu _{m+2}(t,\lambda )}{\sqrt{\lambda }}f_{p-1,m+2}\left( t\right) +B_{p-1}\left( t,\lambda \right) \right] \mathrm{d}t \end{eqnarray*} for $p=3,\ldots m+2,$ and the functions $f_{p,j}\left( x\right) $ are defined by the recurrence relations \begin{eqnarray*} \text{ }f_{1,j}\left( x\right) &=&\left( \pm \right) _{j}\left( \sigma ^{\left( j-1\right) }\left( x\right) -(-1)^{j-1}\sigma ^{\left( j-1\right) }\left( 0\right) \right) ,\text{ } \\ f_{p,p}\left( x\right) &=&(-1)^{p}\int_{0}^{x}q(t)f_{p-1,p-1}\left( t\right) dt\ \text{for }p=2,\ldots ,m+2, \\ f_{p,j}\left( x\right) &=&-\sum\limits_{s=1}^{j-2}\left( \pm \right) _{s}\left( \pm \right) _{j}\left( \left( qf_{p-1,s}\right) ^{\left( j-s-2\right) }\left( x\right) -(-1)^{j-1}\left( qf_{p-1,s}\right) ^{\left( j-s-2\right) }\left( 0\right) \right) \\ &&+(-1)^{j}\int_{0}^{x}q(t)f_{p-1,j-1}\left( t\right) dt\ \text{for }j>p\ \text{and }p=2,\ldots ,m+2, \\ f_{p,j}\left( x\right) &=&0\text{ for }j<p. \end{eqnarray*} Moreover$,$ $f_{p,j}\in C^{m+p-j+1}\left[ 0,\delta \right] .$ \end{lemma}
\begin{proof} In order to prove this lemma, we will follow the technique in \cite[Lemma 4.2 ]{savchuk}. We first note that \begin{equation} \frac{\sin (\sqrt{\lambda }\left( x-t\right) )}{\sqrt{\lambda }}\nu _{j}(t,\lambda )=\left( -1\right) ^{j+1}\nu _{j+1}(x,\lambda )+\nu _{j+1}(x-2t,\lambda ) \label{4.9} \end{equation} and for $f\in C^{1}\left[ 0,x\right] ,$ \begin{eqnarray} &&\int_{0}^{x}\nu _{j}(x-2t,\lambda )f\left( t\right) dt \label{4.10} \\ &=&\nu _{j+1}(x,\lambda )\left( f\left( x\right) -\left( -1\right) ^{j}f\left( 0\right) \right) +\left( -1\right) ^{j+1}\int_{0}^{x}\nu _{j+1}(x-2t,\lambda )f^{\prime }\left( t\right) dt. \notag \end{eqnarray} In view of $\left( \ref{4.9}\right) $ and $\left( \ref{4.10}\right) ,$ one can easily deduce the expression $\left( \ref{4.4}\right) .$ Now we turn to deduce the expressions for the other functions $S_{j}.$ Suppose that $ f_{j}\in C^{m+1-j}\left[ 0,x\right] ,$ then from $\left( \ref{4.9}\right) ,$ we know that for $j=1,\ldots ,m+1,$ \begin{eqnarray} &&\int_{0}^{x}\frac{\sin (\sqrt{\lambda }\left( x-t\right) )}{\sqrt{\lambda } }\nu _{j}(t,\lambda )f_{j}\left( t\right) dt \label{le} \\ &=&\left( -1\right) ^{j+1}\nu _{j+1}(x,\lambda )\int_{0}^{x}f_{j}\left( t\right) dt+\int_{0}^{x}\nu _{j+1}(x-2t,\lambda )f_{j}\left( t\right) dt, \notag \end{eqnarray} Moreover, integrating by parts the second summand on the right-hand side of the above equality $m+1-j$ times and using $\left( \ref{4.9}\right) ,$ it follows that for $j=1,\ldots ,m,$ \begin{eqnarray} &&\int_{0}^{x}\frac{\sin (\sqrt{\lambda }\left( x-t\right) )}{\sqrt{\lambda } }\nu _{j}(t,\lambda )f_{j}\left( t\right) dt \label{le1} \\ &=&\left( -1\right) ^{j+1}\nu _{j+1}(x,\lambda )\int_{0}^{x}f_{j}\left( t\right) dt-\sum_{s=j+2}^{m+2}\nu _{s}(x,\lambda )\left( \pm \right) _{j}\left( \pm \right) _{s}(f_{j}^{\left( s-j-2\right) }\left( x\right) \notag \\ &&-(-1)^{s-1}f_{j}^{\left( s-j-2\right) }\left( 0\right) )-\left( \pm \right) _{j}\left( \pm \right) _{m+3}\int_{0}^{x}\nu _{m+2}(x-2t,\lambda )f_{j}^{\left( m+1-j\right) }\left( t\right) dt. \notag \end{eqnarray} Therefore, by virtue of $\left( \ref{le}\right) $ $($for $j=m+1)$ and $ \left( \ref{le1}\right) ,$ for $x\in \left( 0,\delta \right] $ we have that \begin{eqnarray} &&\sum\limits_{j=1}^{m+1}\int_{0}^{x}\frac{\sin (\sqrt{\lambda }\left( x-t\right) )}{\sqrt{\lambda }}\nu _{j}(t,\lambda )f_{j}\left( t\right) dt \label{4.11} \\ &=&\left( -1\right) ^{2}\nu _{2}(x,\lambda )\int_{0}^{x}f_{1}\left( t\right) dt \notag \\ &&+\sum_{j=3}^{m+2}\nu _{j}(x,\lambda )(-\sum\limits_{s=1}^{j-2}\left( \pm \right) _{s}\left( \pm \right) _{j}(f_{s}^{\left( j-s-2\right) }\left( x\right) -(-1)^{j-1}f_{s}^{\left( j-s-2\right) }\left( 0\right) ) \notag \\ &&+\left( -1\right) ^{j}\int_{0}^{x}f_{j-1}\left( t\right) dt)-\left( \pm \right) _{m+3}\int_{0}^{x}\nu _{m+2}(x-2t,\lambda )\sum\limits_{j=1}^{m+1}\left( \pm \right) _{j}f_{j}^{\left( m+1-j\right) }\left( t\right) dt. \notag \end{eqnarray} Now in view of $\left( \ref{4.1}\right) $ and $\left( \ref{4.4}\right) ,$ we obtain that for $x\in \left( 0,\delta \right] ,$ \begin{eqnarray*} S_{2}(x,\lambda ) &=&\frac{1}{\sqrt{\lambda }}\sum\limits_{j=1}^{m+1} \int_{0}^{x}\frac{\sin (\sqrt{\lambda }\left( x-t\right) )}{\sqrt{\lambda }} \nu _{j}(t,\lambda )q(t)f_{1,j}\left( t\right) \mathrm{d}t \\ &&+\frac{\left( \pm \right) _{m+2}}{\sqrt{\lambda }}\int_{0}^{x}\frac{\sin \sqrt{\lambda }\left( x-t\right) }{\sqrt{\lambda }}q(t)\int_{0}^{t}\nu _{m+1}(t-2s,\lambda )q^{\left( m\right) }(s)dsdt. \end{eqnarray*} Making use of $\left( \ref{4.11}\right) $ with $f_{j}\left( t\right) $ replaced by $q(t)f_{1,j}\left( t\right) $ and in virtue of the fact $ qf_{1,j}\in C^{m+1-j}\left[ 0,\delta \right] $ for $j$ $=1,\ldots ,m+1,$ we obtain $\left( \ref{4.5}\right) .$ Next, from $\left( \ref{4.1}\right) $ and $\left( \ref{4.5}\right) ,$ it follows that for $x\in \left( 0,\delta \right] ,$ \begin{eqnarray*} S_{3}(x,\lambda ) &=&\frac{1}{\sqrt{\lambda }}\sum\limits_{j=1}^{m+1} \int_{0}^{x}\frac{\sin (\sqrt{\lambda }\left( x-t\right) )}{\sqrt{\lambda }} \nu _{j}(t,\lambda )q(t)f_{2,j}\left( t\right) \mathrm{d}t \\ &&+\int_{0}^{x}\frac{\sin (\sqrt{\lambda }\left( x-t\right) )}{\sqrt{\lambda }}q(t)\left[ \frac{\nu _{m+2}(t,\lambda )}{\sqrt{\lambda }}f_{2,m+2}\left( t\right) +B_{2}\left( t,\lambda \right) \right] \mathrm{d}t. \end{eqnarray*} Then the expression $\left( \ref{4.6}\right) $ for $S_{3}$ can be proved by using $\left( \ref{4.11}\right) $ and letting $f_{j}\left( t\right) :=q(t)f_{2,j}\left( t\right) .$ The proof of the relation $\left( \ref{4.6} \right) $ for $p=4,\ldots ,m+2\ $can be carried out in the same way. \end{proof}
As a consequence of Lemma \ref{y2}, we have the following assertion relating to $C_{p}$ defined by $\left( \ref{4.2}\right) .$
\begin{lemma} \label{yy2}Assume that $q$ $\in $ $C^{m}\left[ 0,\delta \right] $ for some $ \delta >0\ $and some $m\in
\mathbb{N}
.$ Denote $\sigma \left( x\right) :=\int_{0}^{x}q(t)\mathrm{d}t.$ Then for $ x\in \left[ 0,\delta \right] ,$ we have \begin{eqnarray*} C_{1}(x,\lambda ) &=&\frac{-f_{1,1}\left( x\right) }{2}\frac{\nu _{0}(x,\lambda )}{\sqrt{\lambda }}+\sum\limits_{j=1}^{m}\frac{\nu _{j}(x,\lambda )}{\sqrt{\lambda }}\left[ f_{1,j}^{\prime }\left( x\right) + \frac{(-1)^{j+1}f_{1,j+1}\left( x\right) }{2}\right] \\ &&+\frac{\left( \pm \right) _{m+2}}{\sqrt{\lambda }}\int_{0}^{x}\frac{d\nu _{m+1}(x-2t,\lambda )}{dx}q^{\left( m\right) }(t)dt, \\ C_{2}(x,\lambda ) &=&\sum\limits_{j=1}^{m+1}\frac{\nu _{j}(x,\lambda )}{ \sqrt{\lambda }}\left[ f_{2,j}^{\prime }\left( x\right) +\frac{ (-1)^{j+1}f_{2,j+1}\left( x\right) }{2}\right] +D_{2}\left( x,\lambda \right) \\ C_{p}(x,\lambda ) &=&\sum\limits_{j=1}^{m+1}\frac{\nu _{j}(x,\lambda )}{ \sqrt{\lambda }}\left[ f_{p,j}^{\prime }\left( x\right) +\frac{ (-1)^{j+1}f_{p,j+1}\left( x\right) }{2}\right] +D_{p}\left( x,\lambda \right) \ \text{for }p=3,\ldots ,m+2 \end{eqnarray*} where $f_{p,j}\left( x\right) $ are the functions defined in Lemma $\ref{y2}$ , and \begin{eqnarray*} D_{2}\left( x,\lambda \right) &=&-\frac{\left( \pm \right) _{m+3}}{\sqrt{ \lambda }}\int_{0}^{x}\frac{d\nu _{m+2}(x-2t,\lambda )}{dx}\left( \sum\limits_{j=1}^{m+1}\left( \pm \right) _{j}\left( q(t)f_{1,j}\left( t\right) \right) ^{\left( m+1-j\right) }\right) dt \\ &&+\frac{\left( \pm \right) _{m+2}}{\sqrt{\lambda }}\int_{0}^{x}\cos \sqrt{ \lambda }\left( x-t\right) q(t)\int_{0}^{t}\nu _{m+1}(t-2s,\lambda )q^{\left( m\right) }(s)dsdt, \\ D_{p}\left( x,\lambda \right) &=&-\frac{\left( \pm \right) _{m+3}}{\sqrt{ \lambda }}\int_{0}^{x}\frac{d\nu _{m+2}(x-2t,\lambda )}{dx} \sum\limits_{j=1}^{m+1}\left( \pm \right) _{j}\left( qf_{p-1,j}\right) ^{\left( m+1-j\right) }\left( t\right) dt \\ &&+\int_{0}^{x}\cos \sqrt{\lambda }\left( x-t\right) q(t)\left[ \frac{\nu _{m+2}(t,\lambda )}{\sqrt{\lambda }}f_{p-1,m+2}\left( t\right) +B_{p-1}\left( t,\lambda \right) \right] \mathrm{d}t \end{eqnarray*} $\ $for $p=3,\ldots ,m+2.$ \end{lemma}
\begin{lemma} Assume that $q$ $\in $ $C^{m}\left[ 0,\delta \right] $ for some $\delta >0\ $ and some $m\in
\mathbb{N}
_{0}.$ Then for $x\in \left[ 0,\delta \right] ,$ $y_{2}(x,\lambda )$ and $ y_{2}^{\prime }(x,\lambda )$ can be rewritten as the following form: \begin{eqnarray} y_{2}(x,\lambda ) &=&\frac{\sin \left( \sqrt{\lambda }x\right) }{\sqrt{ \lambda }}+\sum\limits_{j=1}^{m+2}a_{j}\left( x\right) \frac{\nu _{j}(x,\lambda )}{\sqrt{\lambda }} \label{y} \\ &&+\frac{\left( \pm \right) _{m+2}}{\sqrt{\lambda }}\int_{0}^{x}\nu _{m+1}(x-2t,\lambda )q^{\left( m\right) }(t)dt \notag \\ &&+\sum\limits_{p=2}^{m+2}B_{p}\left( x,\lambda \right) +\sum\limits_{p=m+3}^{\infty }S_{p}\left( x,\lambda \right) , \notag \end{eqnarray} and \begin{eqnarray} y_{2}^{\prime }(x,\lambda ) &=&\cos \left( \sqrt{\lambda }x\right) +\sum\limits_{j=0}^{m+1}b_{j}\left( x\right) \frac{\nu _{j}(x,\lambda )}{ \sqrt{\lambda }} \label{y2'} \\ &&+\frac{\left( \pm \right) _{m+2}}{\sqrt{\lambda }}\int_{0}^{x}\frac{d\nu _{m+1}(x-2t,\lambda )}{dx}q^{\left( m\right) }(t)dt \notag \\ &&+\sum\limits_{p=2}^{m+2}D_{p}\left( x,\lambda \right) +\sum\limits_{p=m+3}^{\infty }C_{p}\left( x,\lambda \right) , \notag \end{eqnarray} where \begin{equation*} a_{j}\left( x\right) =\sum\limits_{p=1}^{m+2}f_{p,j}\left( x\right) \ \text{ for }j=1,\ldots m+1,\text{ }a_{m+2}\left( x\right) =\sum\limits_{p=2}^{m+2}f_{p,m+2}\left( x\right) , \end{equation*} and \begin{eqnarray*} b_{0}\left( x\right) &=&\frac{-f_{1,1}\left( x\right) }{2},\text{ } b_{m+1}\left( x\right) =\sum\limits_{p=2}^{m+2}\left( f_{p,m+1}^{\prime }\left( x\right) +\frac{(-1)^{m+2}f_{p,m+2}\left( x\right) }{2}\right) , \\ b_{j}\left( x\right) &=&\sum\limits_{p=1}^{m+2}\left( f_{p,j}^{\prime }\left( x\right) +\frac{(-1)^{j+1}f_{p,j+1}\left( x\right) }{2}\right) , \text{ }j=1,2,\ldots ,m. \end{eqnarray*} \end{lemma}
\begin{proof} For $m\in
\mathbb{N}
,$ the expressions $\left( \ref{y}\right) $ and $\left( \ref{y2'}\right) $ can be directly obtained from $\left( \ref{3.43}\right) ,$ Lemma \ref{y2} and Lemma \ref{yy2}. For $m=0,$ the proof can be carried out in the same way even simpler. \end{proof}
\begin{remark} \label{yy2 copy(1)}For $g\in L^{1}\left[ 0,x\right] ,$ one notes that the following identities \begin{eqnarray} \int_{0}^{x}\sin \left( \sqrt{\lambda }t\right) g(t)dt &=&o\left( \exp \left( \left\vert \text{Im}\sqrt{\lambda }\right\vert x\right) \right) , \label{by1} \\ \int_{0}^{x}\cos \left( \sqrt{\lambda }t\right) g(t)dt &=&o\left( \exp \left( \left\vert \text{Im}\sqrt{\lambda }\right\vert x\right) \right) \label{by2} \end{eqnarray} hold \cite{inverse}. By virtue of $\left( \ref{by1}\right) $ and $\left( \ref {by2}\right) $, it is easy to deduce that \begin{equation*} B_{p}\left( x,\lambda \right) =o\left( \frac{\exp \left( \left\vert \text{Im} \sqrt{\lambda }\right\vert x\right) }{\left\vert \sqrt{\lambda }\right\vert ^{m+3}}\right) ,\text{ }B_{p}^{\prime }\left( x,\lambda \right) =o\left( \frac{\exp \left( \left\vert \text{Im}\sqrt{\lambda }\right\vert x\right) }{ \left\vert \sqrt{\lambda }\right\vert ^{m+2}}\right) \end{equation*} and \begin{equation*} D_{p}\left( x,\lambda \right) =o\left( \frac{\exp \left( \left\vert \text{Im} \sqrt{\lambda }\right\vert x\right) }{\left\vert \sqrt{\lambda }\right\vert ^{m+2}}\right) ,\text{ }D_{p}^{\prime }\left( x,\lambda \right) =o\left( \frac{\exp \left( \left\vert \text{Im}\sqrt{\lambda }\right\vert x\right) }{ \left\vert \sqrt{\lambda }\right\vert ^{m+1}}\right) \end{equation*} hold for $p=2,3,\ldots ,m+2.$ \end{remark}
\begin{remark} \label{yy2 copy(3)}Note that \begin{eqnarray*} S_{p}(x,\lambda ) &=&\int_{0\leq t_{1}\leq \cdots \leq t_{p+1}:=x}\prod\limits_{i=1}^{p}s_{\lambda }(t_{i+1}-t_{i}\,)s_{\lambda }(t_{1})q(t_{i})\mathrm{d}t_{1}\cdots \,\mathrm{d}t_{p}, \\ C_{p}(x,\lambda ) &=&\int_{0\leq t_{1}\leq \cdots \leq t_{p+1}:=x}c_{\lambda }(t_{p+1}-t_{p})\prod\limits_{i=1}^{p-1}s_{\lambda }(t_{i+1}-t_{i}\,)s_{\lambda }(t_{1})q(t_{i})\mathrm{d}t_{1}\cdots \,\mathrm{ d}t_{p}, \end{eqnarray*} where $s_{\lambda }(x):=\frac{\sin (\sqrt{\lambda }x)}{\sqrt{\lambda }} ,c_{\lambda }(x):=\cos (\sqrt{\lambda }x),$ and \begin{equation*} \left\vert \frac{\sin (\sqrt{\lambda }x)}{\sqrt{\lambda }}\right\vert \leq \frac{\exp \left( \left\vert \text{Im}\sqrt{\lambda }\right\vert x\right) }{ \left\vert \sqrt{\lambda }\right\vert },\text{ }\left\vert \cos (\sqrt{ \lambda }x)\right\vert \leq \exp \left( \left\vert \text{Im}\sqrt{\lambda } \right\vert x\right) . \end{equation*} Thus for $\lambda \in
\mathbb{C}
$ and $\left\vert \lambda \right\vert $ being large enough$,$ one has \begin{eqnarray*} \left\vert S_{p}(x,\lambda )\right\vert &\leq &\frac{\exp \left( \left\vert \text{Im}\sqrt{\lambda }\right\vert x\right) }{\left\vert \sqrt{\lambda } \right\vert ^{m+4}}\frac{\left( \int_{0}^{x}\left\vert q(t)\right\vert \mathrm{d}t\right) ^{p}}{p!},\text{ }p\geq m+3, \\ \left\vert C_{p}(x,\lambda )\right\vert &\leq &\frac{\exp \left( \left\vert \text{Im}\sqrt{\lambda }\right\vert x\right) }{\left\vert \sqrt{\lambda } \right\vert ^{m+3}}\frac{\left( \int_{0}^{x}\left\vert q(t)\right\vert \mathrm{d}t\right) ^{p}}{p!},\text{ }p\geq m+3. \end{eqnarray*} This directly yields that as $\left\vert \lambda \right\vert \rightarrow \infty ,$ \begin{equation*} \sum\limits_{p=m+3}^{\infty }S_{p}(x,\lambda )=O\left( \frac{\exp \left( \left\vert \text{Im}\sqrt{\lambda }\right\vert x\right) }{\left\vert \sqrt{ \lambda }\right\vert ^{m+4}}\right) ,\text{ }\sum\limits_{p=m+3}^{\infty }C_{p}(x,\lambda )=O\left( \frac{\exp \left( \left\vert \text{Im}\sqrt{ \lambda }\right\vert x\right) }{\left\vert \sqrt{\lambda }\right\vert ^{m+3}} \right) . \end{equation*} Similarly, one can also obtain that$\ \sum\limits_{p=m+3}^{\infty }C_{p}^{\prime }(x,\lambda )=O\left( \frac{\exp \left( \left\vert \text{Im} \sqrt{\lambda }\right\vert x\right) }{\left\vert \sqrt{\lambda }\right\vert ^{m+2}}\right) $ as $\left\vert \lambda \right\vert \rightarrow \infty .$ \end{remark}
Now we turn to prove Lemma \ref{ooo}.
\begin{proof}[Proof of Lemma \protect\ref{ooo}] We only aim to prove the relation $\left( \ref{4}\right) ,$ since the other statements can be treated similarly. We first denote \begin{equation*} g\left( x\right) :=\left\{ \begin{array}{l} q\left( x\right) ,\text{ }x\in \left[ 0,x_{0}\right] , \\ s\left( x\right) ,\text{ }x\in \left( x_{0},x_{0}+\delta \right] , \end{array} \right. \text{ }\widetilde{g}\left( x\right) :=\left\{ \begin{array}{l} \widetilde{q}\left( x\right) ,\text{ }x\in \left[ 0,x_{0}\right] , \\ s\left( x\right) ,\text{ }x\in \left( x_{0},x_{0}+\delta \right] , \end{array} \right. \end{equation*} where $s\left( x\right) =\sum\limits_{j=0}^{m}q_{-}^{(j)}(x_{0})\left( x-x_{0}\right) ^{j}$ and $\delta $ is some positive constant$.$ Then by $ \left( \ref{qq}\right) $ it is easy to see that $g,$ $\widetilde{g}$ $\in $ $ C^{m}\left[ 0,x_{0}+\delta \right] \ $and \begin{equation} g^{\left( j\right) }(x_{0})=q_{-}^{(j)}(x_{0})=\widetilde{g}^{\left( j\right) }(x_{0})\text{ for }j=0,1,\ldots ,m. \label{eeeee} \end{equation} For $i=1,2,$ let $w_{2}(x,\lambda )$ and $\widetilde{w}_{2}(x,\lambda )$ be the fundamental solutions of the equations \begin{equation*} -y^{\prime \prime }+g\left( x\right) y=\lambda y\text{\ and}-y^{\prime \prime }+\widetilde{g}\left( x\right) y=\lambda y,\text{ }x\in \left( 0,x_{0}+\delta \right) \end{equation*} respectively, where $w_{2}(x,\lambda )$ and $\widetilde{w}_{2}(x,\lambda )$ are determined by the initial conditions \begin{equation*} w_{2}(0,\lambda )=\text{ }\widetilde{w}_{2}(0,\lambda )=0,\text{ } w_{2}^{\prime }(0,\lambda )=\widetilde{w}_{2}^{\prime }(0,\lambda )=1. \end{equation*} By $\left( \ref{y}\right) ,$ $\left( \ref{y2'}\right) $, Lemma \ref{y2}, Lemma \ref{yy2}, Remark \ref{yy2 copy(1)} and Remark \ref{yy2 copy(3)}$,$ it is easy to see that there exist functions $r_{k},u_{k},z_{k}\in C^{1}\left[ 0,x_{0}+\delta \right] $ such that for $x\in \left[ 0,x_{0}+\delta \right] ,$ \begin{eqnarray} &&w_{2}(x,\lambda )\widetilde{w}_{2}^{\prime }(x,\lambda )-w_{2}^{\prime }(x,\lambda )\widetilde{w}_{2}(x,\lambda ) \notag \\ &=&\sum\limits_{k=0}^{m+3}r_{k}(x)\frac{\sin \left( \sqrt{\lambda }x\right) \cos \left( \sqrt{\lambda }x\right) }{\left( \sqrt{\lambda }\right) ^{k}} +\sum\limits_{k=0}^{m+3}u_{k}(x)\frac{\cos ^{2}\left( \sqrt{\lambda } x\right) }{\left( \sqrt{\lambda }\right) ^{k}} \notag \\ &&+\sum\limits_{k=0}^{m+3}z_{k}(x)\frac{\sin ^{2}\left( \sqrt{\lambda } x\right) }{\left( \sqrt{\lambda }\right) ^{k}}+\frac{\left( \pm \right) _{m+2}}{\sqrt{\lambda }}I_{1}\left( x,\lambda \right) +I_{2}\left( x,\lambda \right) \notag \\ &=&\sum\limits_{k=0}^{m+3}\frac{r_{k}(x)}{2}\frac{\sin \left( 2\sqrt{\lambda }x\right) }{\left( \sqrt{\lambda }\right) ^{k}}+\sum\limits_{k=0}^{m+3}\frac{ u_{k}(x)-z_{k}(x)}{2}\frac{\cos \left( 2\sqrt{\lambda }x\right) }{\left( \sqrt{\lambda }\right) ^{k}} \label{yy'} \\ &&+\sum\limits_{k=0}^{m+3}\frac{u_{k}(x)+z_{k}(x)}{2\left( \sqrt{\lambda } \right) ^{k}}+\frac{\left( \pm \right) _{m+2}}{\sqrt{\lambda }}I_{1}\left( x,\lambda \right) +I_{2}\left( x,\lambda \right) , \notag \end{eqnarray} where \begin{eqnarray} I_{1}\left( x,\lambda \right) &=&\frac{\sin \left( \sqrt{\lambda }x\right) }{ \sqrt{\lambda }}\int_{0}^{x}\frac{d\nu _{m+1}(x-2t,\lambda )}{dx}\left( \widetilde{g}^{\left( m\right) }(t)-g^{\left( m\right) }(t)\right) dt \label{definition of I1} \\ &&+\cos \left( \sqrt{\lambda }x\right) \int_{0}^{x}\nu _{m+1}(x-2t,\lambda )\left( g^{\left( m\right) }(t)-\widetilde{g}^{\left( m\right) }(t)\right) dt \notag \\ &=&\left\{ \begin{array}{c} \int_{0}^{x}\frac{\cos \left( 2\sqrt{\lambda }t\right) }{\left( 2\sqrt{ \lambda }\right) ^{m+1}}\left( g^{\left( m\right) }(t)-\widetilde{g}^{\left( m\right) }(t)\right) dt\text{ if }m\ \text{is even}, \\ \int_{0}^{x}\frac{\sin \left( 2\sqrt{\lambda }t\right) }{\left( 2\sqrt{ \lambda }\right) ^{m+1}}\left( \widetilde{g}^{\left( m\right) }(t)-g^{\left( m\right) }(t)\right) dt\text{ if }m\ \text{is odd}, \end{array} \text{ }\right. \notag \end{eqnarray} and as $\left\vert \lambda \right\vert \rightarrow \infty ,$ \begin{equation*} I_{2}\left( x,\lambda \right) =o\left( \frac{\exp \left( 2\left\vert \text{Im }\sqrt{\lambda }\right\vert x\right) }{\left\vert \sqrt{\lambda }\right\vert ^{m+3}}\right) ,\text{ }I_{2}^{\prime }\left( x,\lambda \right) =o\left( \frac{\exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert x\right) }{ \left\vert \sqrt{\lambda }\right\vert ^{m+2}}\right) . \end{equation*} In view of $\left( \ref{yy'}\right) $ and the fact $g=\widetilde{g}$ on $ \left[ x_{0},x_{0}+\delta \right] ,$ one deduces that for $x\in \left[ x_{0},x_{0}+\delta \right] ,$ \begin{eqnarray*} &&\left( w_{2}(x,\lambda )\widetilde{w}_{2}^{\prime }(x,\lambda )-w_{2}^{\prime }(x,\lambda )\widetilde{w}_{2}(x,\lambda )\right) ^{\prime } \\ &=&r_{0}(x)\sqrt{\lambda }\cos \left( 2\sqrt{\lambda }x\right) -\left( u_{0}(x)-z_{0}(x)\right) \sqrt{\lambda }\sin \left( 2\sqrt{\lambda }x\right) \\ &&+\sum\limits_{k=0}^{m+2}\frac{u_{k}^{\prime }(x)+z_{k}^{\prime }(x)}{ 2\left( \sqrt{\lambda }\right) ^{k}}+\sum\limits_{k=0}^{m+2}\left( r_{k+1}(x)+\frac{u_{k}^{\prime }(x)-z_{k}^{\prime }(x)}{2}\right) \frac{\cos \left( 2\sqrt{\lambda }x\right) }{\left( \sqrt{\lambda }\right) ^{k}} \\ &&-\sum\limits_{k=0}^{m+2}\left( u_{k+1}(x)-z_{k+1}(x)-\frac{r_{k}^{\prime }(x)}{2}\right) \frac{\sin \left( 2\sqrt{\lambda }x\right) }{\left( \sqrt{ \lambda }\right) ^{k}}+o\left( \frac{\exp \left( 2\left\vert \text{Im}\sqrt{ \lambda }\right\vert x\right) }{\left\vert \sqrt{\lambda }\right\vert ^{m+2}} \right) \\ &=&0. \end{eqnarray*} This forces that for $x\in \left[ x_{0},x_{0}+\delta \right] ,$ $ u_{0}(x)-z_{0}(x)=r_{0}(x)\equiv 0,$ \begin{equation*} r_{k+1}(x)+\frac{u_{k}^{\prime }(x)-z_{k}^{\prime }(x)}{2}=0,\text{ } u_{k+1}(x)-z_{k+1}(x)-\frac{r_{k}^{\prime }(x)}{2}=0\ \text{for }k=0,\ldots ,m+2, \end{equation*} and thus for $k=0,1,\ldots ,m+3$ and $x\in \left[ x_{0},x_{0}+\delta \right] ,$ one has \begin{equation} u_{k}(x)-z_{k}(x)=r_{k}(x)\equiv 0. \label{equal} \end{equation} Therefore, by $\left( \ref{yy'}\right) $ and $\left( \ref{equal}\right) $ we infer that \begin{eqnarray} &&w_{2}(x_{0},\lambda )\widetilde{w}_{2}^{\prime }(x_{0},\lambda )-w_{2}^{\prime }(x_{0},\lambda )\widetilde{w}_{2}(x_{0},\lambda ) \label{step1} \\ &=&\sum\limits_{k=0}^{m+3}\frac{u_{k}(x_{0})+z_{k}(x_{0})}{2\left( \sqrt{ \lambda }\right) ^{k}}+\frac{\left( \pm \right) _{m+2}}{\sqrt{\lambda }} I_{1}\left( x_{0},\lambda \right) +o\left( \frac{\exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert x_{0}\right) }{\left\vert \sqrt{\lambda } \right\vert ^{m+3}}\right) . \notag \end{eqnarray}
Next, we aim to show that \begin{equation} I_{1}\left( x_{0},\lambda \right) =o\left( \frac{\exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert x_{0}\right) }{\left\vert \sqrt{\lambda } \right\vert ^{m+2}}\right) \label{I1} \end{equation} as $\left\vert \lambda \right\vert \rightarrow \infty $ in the sector $ \Lambda _{\zeta }.$ Due to the definition $\left( \ref{definition of I1} \right) $ of $I_{1}\left( x,\lambda \right) ,$ it is sufficient to prove \begin{eqnarray} \int_{0}^{x_{0}}\cos \left( 2\sqrt{\lambda }t\right) \left( g^{\left( m\right) }(t)-\widetilde{g}^{\left( m\right) }(t)\right) dt &=&o\left( \frac{ \exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert x_{0}\right) }{ \left\vert \sqrt{\lambda }\right\vert }\right) , \label{cos} \\ \int_{0}^{x_{0}}\sin \left( 2\sqrt{\lambda }t\right) \left( g^{\left( m\right) }(t)-\widetilde{g}^{\left( m\right) }(t)\right) dt &=&o\left( \frac{ \exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert x_{0}\right) }{ \left\vert \sqrt{\lambda }\right\vert }\right) . \label{sin} \end{eqnarray} In fact, by $\left( \ref{eeeee}\right) $ $\left( \text{for }j=m\right) $ and the fact $g,$ $\widetilde{g}$ $\in $ $C^{m}\left[ 0,x_{0}+\delta \right] $ we infer that given any $\epsilon >0,$ there exists a sufficiently small constant $\delta _{0}>0$ such that \begin{equation*} \left\vert g^{\left( m\right) }(t)-\widetilde{g}^{\left( m\right) }(t)\right\vert <\epsilon \text{ on }\left[ x_{0}-\delta _{0},x_{0}\right] , \end{equation*} and thus for $\lambda \in \Lambda _{\zeta }$ and $\left\vert \lambda \right\vert $ being sufficiently large, we obtain \begin{eqnarray*} &&\left\vert \int_{0}^{x_{0}}\cos \left( 2\sqrt{\lambda }t\right) \left( g^{\left( m\right) }(t)-\widetilde{g}^{\left( m\right) }(t)\right) dt\right\vert \\ &\leq &\left\vert \int_{0}^{x_{0}-\delta _{0}}\cos \left( 2\sqrt{\lambda } t\right) \left( g^{\left( m\right) }(t)-\widetilde{g}^{\left( m\right) }(t)\right) dt\right\vert +\left\vert \int_{x_{0}-\delta _{0}}^{x_{0}}\cos \left( 2\sqrt{\lambda }t\right) \left( g^{\left( m\right) }(t)-\widetilde{g} ^{\left( m\right) }(t)\right) dt\right\vert \\ &\leq &\max_{t\in \left[ 0,x_{0}\right] }\left\vert g^{\left( m\right) }(t)- \widetilde{g}^{\left( m\right) }(t)\right\vert \int_{0}^{x_{0}-\delta _{0}}\left\vert \cos \left( 2\sqrt{\lambda }t\right) \right\vert dt+\epsilon \int_{x_{0}-\delta _{0}}^{x_{0}}\left\vert \cos \left( 2\sqrt{\lambda } t\right) \right\vert dt \\ &\leq &\max_{t\in \left[ 0,x_{0}\right] }\left\vert g^{\left( m\right) }(t)- \widetilde{g}^{\left( m\right) }(t)\right\vert \int_{0}^{x_{0}-\delta _{0}}\exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert t\right) dt+\epsilon \int_{x_{0}-\delta _{0}}^{x_{0}}\exp \left( 2\left\vert \text{Im} \sqrt{\lambda }\right\vert t\right) dt \\ &\leq &\max_{t\in \left[ 0,x_{0}\right] }\left\vert g^{\left( m\right) }(t)- \widetilde{g}^{\left( m\right) }(t)\right\vert \frac{\exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert \left( x_{0}-\delta _{0}\right) \right) }{2\left\vert \text{Im}\sqrt{\lambda }\right\vert }+\epsilon \frac{\exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert x_{0}\right) }{ 2\left\vert \text{Im}\sqrt{\lambda }\right\vert } \\ &\leq &\frac{\max\limits_{t\in \left[ 0,x_{0}\right] }\left\vert g^{\left( m\right) }(t)-\widetilde{g}^{\left( m\right) }(t)\right\vert }{2\exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert \delta _{0}\right) }\frac{ \exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert x_{0}\right) }{ \left\vert \text{Im}\sqrt{\lambda }\right\vert }+\frac{\epsilon }{2\sin \frac{\zeta }{2}}\frac{\exp \left( 2\left\vert \text{Im}\sqrt{\lambda } \right\vert x_{0}\right) }{\left\vert \sqrt{\lambda }\right\vert } \\ &\leq &\frac{\epsilon }{\sin \frac{\zeta }{2}}\frac{\exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert x_{0}\right) }{\left\vert \sqrt{\lambda } \right\vert }, \end{eqnarray*} where we have used the inequalities \begin{equation*} \left\vert \cos \left( 2\sqrt{\lambda }t\right) \right\vert \leq \exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert \left\vert t\right\vert \right) \text{ for }\lambda \in
\mathbb{C}
,t\in
\mathbb{R}
\end{equation*} and \begin{equation} \left\vert \text{Im}\sqrt{\lambda }\right\vert \geq \left\vert \sqrt{\lambda }\right\vert \sin \frac{\zeta }{2}\text{ for }\lambda \in \Lambda _{\zeta }. \label{inequaltiy} \end{equation} This proves the equality $\left( \ref{cos}\right) .$ Note that $\left( \ref {sin}\right) $ can be treated similarly, and thus $\left( \ref{I1}\right) $ is proved.
Now by $\left( \ref{step1}\right) $ and $\left( \ref{I1}\right) $ we have that \begin{eqnarray*} &&w_{2}(x_{0},\lambda )\widetilde{w}_{2}^{\prime }(x_{0},\lambda )-w_{2}^{\prime }(x_{0},\lambda )\widetilde{w}_{2}(x_{0},\lambda ) \\ &=&\sum\limits_{k=0}^{m+3}\frac{u_{k}(x_{0})+z_{k}(x_{0})}{2\left( \sqrt{ \lambda }\right) ^{k}}+o\left( \frac{\exp \left( 2\left\vert \text{Im}\sqrt{ \lambda }\right\vert x_{0}\right) }{\left\vert \sqrt{\lambda }\right\vert ^{m+3}}\right) \end{eqnarray*} as $\left\vert \lambda \right\vert \rightarrow \infty $ in the sector $ \Lambda _{\zeta }.$ This together with $\left( \ref{inequaltiy}\right) $ directly yields that \begin{equation*} w_{2}(x_{0},\lambda )\widetilde{w}_{2}^{\prime }(x_{0},\lambda )-w_{2}^{\prime }(x_{0},\lambda )\widetilde{w}_{2}(x_{0},\lambda )=o\left( \frac{\exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert x_{0}\right) }{\left\vert \sqrt{\lambda }\right\vert ^{m+3}}\right) \end{equation*} as $\left\vert \lambda \right\vert \rightarrow \infty $ in the sector $ \Lambda _{\zeta }.$ Now $\left( \ref{4}\right) $ is\ proved, since by the definition of $g\ $and $\widetilde{g}$ we can infer that \begin{eqnarray*} w_{2}(x_{0},\lambda ) &=&y_{2}(x_{0},\lambda ),\text{ }\widetilde{w} _{2}(x_{0},\lambda )=\widetilde{y}_{2}(x_{0},\lambda ), \\ w_{2}^{\prime }(x_{0},\lambda ) &=&y_{2}^{\prime }(x_{0},\lambda ),\text{ } \widetilde{w}_{2}^{\prime }(x_{0},\lambda )=\widetilde{y}_{2}^{\prime }(x_{0},\lambda ). \end{eqnarray*} \end{proof}
Now we are in a position to prove Proposition \ref{ooo copy(1)}.
\begin{proof}[Proof of Proposition \protect\ref{ooo copy(1)}] Note that \begin{eqnarray} &&y_{2,r}(x_{0},\lambda )\widetilde{y}_{2,r}^{\prime }(x_{0},\lambda )-y_{2,r}^{\prime }(x_{0},\lambda )\widetilde{y}_{2,r}(x_{0},\lambda ) \label{yyyy} \\ &=&\left[ y_{2,r}\left( x_{0}-\delta ,\lambda \right) y_{1,x_{0}-\delta }(x_{0},\lambda )+y_{2,r}^{\prime }\left( x_{0}-\delta ,\lambda \right) y_{2,x_{0}-\delta }(x_{0},\lambda )\right] \times \notag \\ &&\left[ \widetilde{y}_{2,r}\left( x_{0}-\delta ,\lambda \right) \widetilde{y }_{1,x_{0}-\delta }^{\prime }(x_{0},\lambda )+\widetilde{y}_{2,r}^{\prime }\left( x_{0}-\delta ,\lambda \right) \widetilde{y}_{2,x_{0}-\delta }^{\prime }(x_{0},\lambda )\right] \notag \\ &&-\left[ y_{2,r}\left( x_{0}-\delta ,\lambda \right) y_{1,x_{0}-\delta }^{\prime }(x_{0},\lambda )+y_{2,r}^{\prime }\left( x_{0}-\delta ,\lambda \right) y_{2,x_{0}-\delta }^{\prime }(x_{0},\lambda )\right] \times \notag \\ &&\left[ \widetilde{y}_{2,r}\left( x_{0}-\delta ,\lambda \right) \widetilde{y }_{1,x_{0}-\delta }(x_{0},\lambda )+\widetilde{y}_{2,r}^{\prime }\left( x_{0}-\delta ,\lambda \right) \widetilde{y}_{2,x_{0}-\delta }(x_{0},\lambda ) \right] \notag \\ &=&B_{1}(\lambda )\left[ y_{1,x_{0}-\delta }(x_{0},\lambda )\widetilde{y} _{1,x_{0}-\delta }^{\prime }(x_{0},\lambda )-y_{1,x_{0}-\delta }^{\prime }(x_{0},\lambda )\widetilde{y}_{1,x_{0}-\delta }(x_{0},\lambda )\right] \notag \\ &&+B_{2}(\lambda )\left[ y_{1,x_{0}-\delta }(x_{0},\lambda )\widetilde{y} _{2,x_{0}-\delta }^{\prime }(x_{0},\lambda )-y_{1,x_{0}-\delta }^{\prime }(x_{0},\lambda )\widetilde{y}_{2,x_{0}-\delta }(x_{0},\lambda )\right] \notag \\ &&+B_{3}(\lambda )\left[ \widetilde{y}_{1,x_{0}-\delta }^{\prime }(x_{0},\lambda )y_{2,x_{0}-\delta }(x_{0},\lambda )-\widetilde{y} _{1,x_{0}-\delta }(x_{0},\lambda )y_{2,x_{0}-\delta }^{\prime }(x_{0},\lambda )\right] \notag \\ &&+B_{4}(\lambda )\left[ y_{2,x_{0}-\delta }(x_{0},\lambda )\widetilde{y} _{2,x_{0}-\delta }^{\prime }(x_{0},\lambda )-y_{2,x_{0}-\delta }^{\prime }(x_{0},\lambda )\widetilde{y}_{2,x_{0}-\delta }(x_{0},\lambda )\right] \notag \end{eqnarray} where \begin{eqnarray*} B_{1}(\lambda ) &=&y_{2,r}\left( x_{0}-\delta ,\lambda \right) \widetilde{y} _{2,r}\left( x_{0}-\delta ,\lambda \right) =O\left( \left\vert \lambda ^{-1}\right\vert \exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert \left( x_{0}-\delta -r\right) \right) \right) , \\ B_{2}(\lambda ) &=&y_{2,r}\left( x_{0}-\delta ,\lambda \right) \widetilde{y} _{2,r}^{\prime }\left( x_{0}-\delta ,\lambda \right) =O\left( \left\vert \sqrt{\lambda }\right\vert ^{-1}\exp \left( 2\left\vert \text{Im}\sqrt{ \lambda }\right\vert \left( x_{0}-\delta -r\right) \right) \right) , \\ B_{3}(\lambda ) &=&\widetilde{y}_{2,r}\left( x_{0}-\delta ,\lambda \right) y_{2,r}^{\prime }\left( x_{0}-\delta ,\lambda \right) =O\left( \left\vert \sqrt{\lambda }\right\vert ^{-1}\exp \left( 2\left\vert \text{Im}\sqrt{ \lambda }\right\vert \left( x_{0}-\delta \right) -r\right) \right) , \\ B_{4}(\lambda ) &=&\widetilde{y}_{2,r}^{\prime }\left( x_{0}-\delta ,\lambda \right) y_{2,r}^{\prime }\left( x_{0}-\delta ,\lambda \right) =O\left( \exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert \left( x_{0}-\delta -r\right) \right) \right) . \end{eqnarray*} The above asymptotics of $B_{1},$ $B_{2},$ $B_{3},$ $B_{4}$ can be obtained from $\left( \ref{y2y2}\right) .$ Therefore, one can easily deduce from Lemma \ref{ooo} that \begin{eqnarray*} y_{1,x_{0}-\delta }(x_{0},\lambda )\widetilde{y}_{1,x_{0}-\delta }^{\prime }(x_{0},\lambda )-y_{1,x_{0}-\delta }^{\prime }(x_{0},\lambda )\widetilde{y} _{1,x_{0}-\delta }(x_{0},\lambda ) &=&o\left( \frac{\exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert \delta \right) }{\left\vert \sqrt{ \lambda }\right\vert ^{m+1}}\right) , \\ y_{1,x_{0}-\delta }(x_{0},\lambda )\widetilde{y}_{2,x_{0}-\delta }^{\prime }(x_{0},\lambda )-y_{1,x_{0}-\delta }^{\prime }(x_{0},\lambda )\widetilde{y} _{2,x_{0}-\delta }(x_{0},\lambda ) &=&o\left( \frac{\exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert \delta \right) }{\left\vert \sqrt{ \lambda }\right\vert ^{m+2}}\right) , \\ \widetilde{y}_{1,x_{0}-\delta }^{\prime }(x_{0},\lambda )y_{2,x_{0}-\delta }(x_{0},\lambda )-\widetilde{y}_{1,x_{0}-\delta }(x_{0},\lambda )y_{2,x_{0}-\delta }^{\prime }(x_{0},\lambda ) &=&o\left( \frac{\exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert \delta \right) }{\left\vert \sqrt{\lambda }\right\vert ^{m+2}}\right) , \\ y_{2,x_{0}-\delta }(x_{0},\lambda )\widetilde{y}_{2,x_{0}-\delta }^{\prime }(x_{0},\lambda )-y_{2,x_{0}-\delta }^{\prime }(x_{0},\lambda )\widetilde{y} _{2,x_{0}-\delta }(x_{0},\lambda ) &=&o\left( \frac{\exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert \delta \right) }{\left\vert \sqrt{ \lambda }\right\vert ^{m+3}}\right) \end{eqnarray*} as $\left\vert \lambda \right\vert \rightarrow \infty $ in $\Lambda _{\zeta }.$ Thus the equality $\left( \ref{1114}\right) $ can be directly obtained from $\left( \ref{yyyy}\right) .$ The statements $\left( \ref{1111}\right) -\left( \ref{1113}\right) $ can be proved similarly. \end{proof}
\begin{remark} \label{ooo copy(4)}If $q\ $and $\widetilde{q}\ $are both assumed to be in $ L_{
\mathbb{C}
}^{1}\left[ 0,\pi \right] ,$ then one can easily find that relations $\left( \ref{1111}\right) -\left( \ref{1114}\right) $ still hold by taking $m=-1$.$\ $In fact, in this case, $y_{2,r}(x,\lambda )$ and $y_{2,r}^{\prime }(x,\lambda )$ have the following asymptotic form \cite{book}: \begin{eqnarray} && \label{y2y2} \\ &&y_{2,r}(x,\lambda ) \notag \\ &=&\frac{\sin (\sqrt{\lambda }\left( x-r\right) )}{\sqrt{\lambda }}-Q\left( x\right) \frac{\cos (\sqrt{\lambda }\left( x-r\right) )}{2\lambda }+o\left( \frac{\exp \left( \left\vert \text{Im}\sqrt{\lambda }\right\vert \left( x-r\right) \right) }{\left\vert \lambda \right\vert }\right) , \notag \\ &&y_{2,r}^{\prime }(x,\lambda ) \notag \\ &=&\cos (\sqrt{\lambda }\left( x-r\right) )+Q\left( x\right) \frac{\sin ( \sqrt{\lambda }\left( x-r\right) )}{2\sqrt{\lambda }}+o\left( \frac{\exp \left( \left\vert \text{Im}\sqrt{\lambda }\right\vert \left( x-r\right) \right) }{\left\vert \sqrt{\lambda }\right\vert }\right) , \notag \end{eqnarray} where $Q\left( x\right) =\int_{r}^{x}q(t)dt.$ Therefore, it is easy to see that \begin{eqnarray*} &&y_{2,r}(x_{0},\lambda )\widetilde{y}_{2,r}^{\prime }(x_{0},\lambda )-y_{2,r}^{\prime }(x_{0},\lambda )\widetilde{y}_{2,r}(x_{0},\lambda ) \\ &=&\frac{\int_{r}^{x_{0}}\left( \widetilde{q}(t)-q(t)\right) dt}{2\lambda } +o\left( \frac{\exp \left( 2\left\vert \text{Im}\sqrt{\lambda }\right\vert \left( x_{0}-r\right) \right) }{\left\vert \lambda \right\vert }\right) . \end{eqnarray*} This directly yields $\left( \ref{1114}\right) .$ $\left( \ref{1111}\right) -\left( \ref{1113}\right) $ can be treated similarly. \end{remark}
\end{document} | arXiv |
\begin{definition}[Definition:Inverse of Elementary Column Operation]
Let $e$ be an elementary column operation which transforms a matrix $\mathbf A$ to another matrix $\mathbf B$.
Let $e'$ be an elementary column operation which transforms $\mathbf B$ back to $\mathbf A$.
Then $e'$ is the '''inverse of the elementary column operation $e$'''.
\end{definition} | ProofWiki |
\begin{definition}[Definition:Stereographic Projection]
Let $\PP$ be a the plane.
Let $\mathbb S$ be a sphere which is tangent to $\PP$ at the origin $\tuple {0, 0}$.
Let the diameter of $\mathbb S$ perpendicular to $\PP$ through $\tuple {0, 0}$ be $NS$ where $S$ is the point $\tuple {0, 0}$.
Let the point $N$ be referred to as the '''north pole''' of $\mathbb S$ and $S$ be referred to as the '''south pole''' of $\mathbb S$.
Let $A$ be a point on $P$.
Let the line $NA$ be constructed.
:900px
Then $NA$ passes through a point of $\mathbb S$.
Thus any point on $P$ can be represented by a point on $\mathbb S$.
With this construction, the point $N$ on $\mathbb S$ maps to no point on $\mathbb S$.
\end{definition} | ProofWiki |
\begin{definition}[Definition:Monomial Matrix]
A '''monomial matrix (of order $n$)''' is an $n \times n$ square matrix with:
:exactly one element in each row and column which is not $0$
:$0$ elsewhere.
\end{definition} | ProofWiki |
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Extracellular matrix
Protein design
Collagens are the most abundant proteins of the extracellular matrix, and the hierarchical folding and supramolecular assembly of collagens into banded fibers is essential for mediating cell-matrix interactions and tissue mechanics. Collagen extracted from animal tissues is a valuable commodity, but suffers from safety and purity issues, limiting its biomaterials applications. Synthetic collagen biomaterials could address these issues, but their construction requires molecular-level control of folding and supramolecular assembly into ordered banded fibers, comparable to those of natural collagens. Here, we show an innovative class of banded fiber-forming synthetic collagens that recapitulate the morphology and some biological properties of natural collagens. The synthetic collagens comprise a functional-driver module that is flanked by adhesive modules that effectively promote their supramolecular assembly. Multiscale simulations support a plausible molecular-level mechanism of supramolecular assembly, allowing precise design of banded fiber morphology. We also experimentally demonstrate that synthetic fibers stimulate osteoblast differentiation at levels comparable to natural collagen. This work thus deepens understanding of collagen biology and disease by providing a ready source of safe, functional biomaterials that bridge the current gap between the simplicity of peptide biophysical models and the complexity of in vivo animal systems.
Collagens are fibrous proteins. They are the most abundant protein in animal tissues, are a valuable commodity as biomaterials, and are actively involved in biomechanical and biological processes, such as dissipating deformation stress of tissues1,2,3, inducing cell polarity during cell–matrix adhesion4, and directing bone mineralization5,6,7. Collagens have an unusual amino acid composition: rich in glycine (~30%) and proline (~15%)8, and thus do not follow globular protein folding rules such as hydrophobicity driven core formation, or presentation of a hydrophilic surface to avoid aggregation9. Collagens do not form common secondary structure units like α-helices or β-strands; instead they fold into triple-helices (Fig. 1a), where three polypeptide chains are tethered by a network of main-chain hydrogen bonds, not a hydrophobic core10,11. Subsequent to folding, thousands to millions of triple-helices undergo self-assembly to form crystalline supramolecular fibers, providing a foundation for the extracellular matrix9. Although collagens are among the most widely applied biomaterials (e.g., cosmetics, foods, sutures in surgery, and cell-culture substrates in tissue culture), their atypical structural features have hindered understanding of their folding and assembly mechanisms.
Fig. 1: Assembly hierarchy of synthetic collagens built with a modular design strategy vs. natural Type I collagen.
a The folded unit of collagen is the triple-helix, consisting of three supercoiled polypeptide chains with repeating Xaa-Yaa-Gly triplets. b In natural Type I collagen, thousands to millions of long thin triple-helices (~1000 residues per chain, diameter ~1 nm, length ~300 nm) are offset and packed in parallel to form fibers9. Type I collagen folding and supramolecular assembly have been extensively reviewed elsewhere60. c Our synthetic collagens were engineered with two adhesives (white) modules flanking a functional-driver (blue) module. Adhesive modules were either hydrophobic, consisting of n Pro-Pro-Gly triplets on both ends of functional-driver modules, or electrostatic, consisting of n Pro-Arg-Gly triplets positioned N-terminal to the functional-driver module, and with the same number of Glu-Pro-Gly triplets at the C-terminal end. Sequences of the functional-driver modules were based on fragments of length m from the Scl2 collagen-like protein of S. pyogenes. Type I collagen fibers have a distinct supramolecular structure, with a pattern of alternating bands of a characteristic periodicity (d); these appear as dark and light bands upon negative staining and visualization with transmission electron microscopy (TEM) (e). The dark band represents the "gap region" caused by a void between consecutive triple-helices; the light band is the overlap region between neighboring triple-helices. One example, P10BP10 (P10 refers to 10 PPG triplets, B is the central 27-triplet fragment from Scl2) forms a banded fiber. Periodic bands of P10BP10 fibers are visible with TEM (f) and atomic force microscopy (AFM) amplitude scans (g), experimentally confirming the intended banding morphology. h Application testing ultimately showed that our synthetic collagens form banded fibers and strongly promote the osteoblastic adhesion and differentiation of osteoblast precursor cells.
Type I collagen exists physiologically as a distinct supramolecular structure of banded fibers with a pattern of alternating dark and light bands that are unmistakable when imaged by an electron microscope9,12. This banding originates from an offset, parallel packing of triple-helices13,14,15,16 (Fig. 1b). The dark band is referred to as the gap region, caused by a void between consecutive triple-helices12. The light band is the overlap region between neighboring triple-helices. Natural Type I collagen forms robust banded fibers. It is sourced from animal tissues such as bovine skin or rat tails and is the major type of collagen used in biomedical research and industry17. However, animal-sourced Type I collagen suffers from drawbacks such as variable quality and purity, and the risk of prion contamination17,18. Synthetic collagens produced under controlled laboratory conditions could address these issues. However, it remains challenging to recapitulate the supramolecular assembly—and thus the banding pattern—of natural collagens19,20,21,22.
Here, we describe a class of synthetic banded fiber collagens, elucidate a detailed mechanism for banded fiber formation, and use these designed materials to experimentally probe the role of banding in promoting osteoblastic differentiation. Individual collagen polypeptide chains are constructed from two components: an adhesive module that promotes supramolecular assembly of folded triple-helices into a banded fiber, and a functional-driver module that can be altered to tailor the material behavior. By varying the lengths and sequences of the two modules, we can precisely specify the widths of the gap and overlap bands. The assembly mechanism is established using both atomistic models and coarse-grained simulations. This class of synthetic collagens advances our physical understanding of collagen supramolecular assembly from the nanometer to micron scale and provides a testbed for studying collagen biology and developing functional biomaterials.
Modular design and fiber production
Adhesive modules were constructed from tripeptide repeats designed for electrostatic or hydrophobic inter-helical interactions (Fig. 1c). Hydrophobic adhesive modules were formed from tandem repeats of a motif: (Pro-Pro-Gly)n sequence, abbreviated as Pn. The electrostatic attachment was promoted using a basic domain composed of (Pro-Arg-Gly)n triplets and an acidic domain composed of (Glu-Pro-Gly)n triplets on either end of the functional driver, respectively abbreviated Rn and En. In the case of electrostatic adhesive modules, the total length was maintained at 15 residues with the remainder of the module composed of Pro-Pro-Gly triplets (Fig. 1c).
The functional-driver module is not responsible for assembly, and by design should be a largely unconstrained repeated Xaa-Yaa-Gly triplet sequence that can be tailored for specific applications. We explored the plasticity of the functional-driver sequence using fragments of Streptococcus pyogenes Scl2, a collagen-like bacterial protein with a triplet insertion23. Three fragments, referred to as A–C, were chosen for their similar lengths but different amino acid compositions and net charges24,25,26. Functional-driver modules constructed from combinations of fragments with varied compositions and lengths were inserted between two adhesive modules (Fig. 1c). Our nomenclature for the synthetic collagens is based on the arrangement of modules: for example, we use "R5BE5" for a variable domain constructed from the Scl2 B fragment flanked by electrostatic attachment modules and use "P10BBP10" for two tandem Scl2 B fragments flanked by ten Pro-Pro-Gly triplets. Complete sequences of all constructs are in Supplementary Table 1.
The folding and assembly protocol for the designed synthetic collagens parallels many aspects of natural collagen assembly. Type I collagen is expressed in the cell in a procollagen form, where the triple-helix is flanked by globular domains that assist in folding; these are subsequently cleaved prior to supramolecular assembly of banded fibers27. Bacterial Scl2 also has a globular domain (V-domain) that facilitates folding and expression24. We included the V-domain prior to the first adhesive module, separated by a trypsin cleavage site (Supplementary Table 1). Synthetic collagens folded into triple-helices within Escherichia coli cells. After expression and purification, the synthetic collagen was digested with trypsin to remove the V-domain28 and incubated at 4 °C for several days, allowing for supramolecular assembly into banded fibers. Details of protein production and validation are described in the "Methods" section and Supporting Information (Supplementary Figs. 1–3).
Controlling fiber morphology
According to the design scheme, the number of amino acids in the adhesive module should specify the length of the overlap region, and the functional-driver module size should determine the length of the gap. The first design studied—P10BP10—formed fibers with periodic light and dark bands, as observed in negative staining electron microscopy images (Fig. 1f). Assembly of banded fibers was robust, and reproducible multiple times under a variety of solution conditions including 50–150 mM NaCl concentrations, pH 3–11, incubation times from 0.5 to 70 days, and temperatures from 4 to 37 °C (Supplementary Figs. 4–6). Although P10BP10 assembles into fibers, micro-rheology experiments show fibers do not form a viscoelastic gel either at a low (1 mg mL−1) or high (5 mg mL−1) concentration (Supplementary Fig. 7).
In a triple-helix, each residue spans 0.29 nm along the super-helical axis, allowing one to estimate the size of a domain based on the length of its sequence. Band widths were ~10 nm (light) and 24.9 nm (dark), matching the expected sizes of the P10 (30 residues) and B (81 residues) sequences, respectively (Fig. 3c). Surface contour maps of P10BP10 banded fibers imaged by atomic force microscopy (AFM) showed the lighter bands were ~1 nm high, matching the diameter of a triple-helix (Fig. 1g, Supplementary Fig. 8). Combined, TEM and AFM support the design approach where the light bands (the overlap region) in P10BP10 are formed by adhesive modules, and the dark bands (the gap region) are formed by the functional-driver modules.
To investigate the three-dimensional morphology of synthetic collagen fibers, we generated P10BP10 and P10BBP10 fibers under the same conditions used for TEM and imaged the samples using a Talos Arctica cryo-electron microscope. Tomograms of P10BP10 fibers showed that collagen fibers form 3D bundles that are 500 – 800 nm in length and taper at the ends. These bundles exhibited a distinct 3D structure and periodic light and dark bands (Fig. 2a-c, Supplementary Mov. 1). Measurements of light bands (23.6 nm) and dark bands (9.5 nm) were consistent with dimensions of gap and overlap bandwidths respectively, but of opposite intensity as compared to TEM. Using Fourier transforms, we calculated the average periodicity to be 32.4 nm (Fig. 2d), consistent with measurements in real space (33.2 nm, Fig. 2b) and in AFM (32.7 nm, Supplementary Fig. 8). To examine the domain organization, we performed volume rendering of the P10BP10 fiber tomograms using UCSF Chimera and observed that collagen fibers within the bundles show a pattern of dense overlap zones and sparse gap zones (Fig. 2e, Supplementary Mov. 1). This observation further supports that the overlap region (dense zone) is formed by tightly stacked adhesive modules and that the gap region (sparse zone) is formed by loosely stacked functional-driver modules. Tomograms of P10BBP10 fibers revealed similar 3D fiber bundles and a characteristic periodic light and dark banding pattern with a 55.6 nm periodicity, consistent with observations in TEM (Supplementary Fig. 23, Supplementary Mov. 2). The light band widths of P10BBP10 were twice of that of P10BP10 fibers. The banded fibers formed by this class of synthetic collagens were also three-dimensional.
Fig. 2: Cryo-electron tomography of P10BP10 collagen fibrils.
a Tilt series of collagen fibers at tilt angles ranging from −60° to +60°. b Slice views of the tomogram of P10BP10 collagen fibers with characteristic banding patterns. Inset in b shows a zoomed-in view with measurement of periodicity. c Cross-section of a collagen bundle as indicated by a dashed yellow line in b. d Fourier transform of the image in b showing banding periodicity. e Volume rendering of collagen fibers showing dense overlap and sparse gap zones.
We tested our design approach by preventing supramolecular banded fiber assembly by several strategies: denaturing the triple-helix structure or altering the order of modules. The P10BP10 triple-helix unfolds with a transition temperature (Tm) of 52 °C (measured by CD—Supplementary Fig. 9j–l), and 55.2 °C measured by differential scanning calorimetry (DSC) (Supplementary Fig. 10c). Heating P10BP10 above the Tm prior to assembly prevented banded fiber formation (Supplementary Fig. 11). Synthetic collagens with alternate module topologies—i.e. containing one adhesive module before, in the middle of, or after the functional-driver B module (P10B, ½BP10½B and BP10)—folded into triple-helices (Supplementary Fig. 9a–c), but did not assemble into banded fibers. Thus, a folded triple-helix and a topology of two adhesive modules flanking a functional-driver are prerequisites for synthetic collagens that form banded fibers.
We show that one can independently target the properties of dark and light bands by varying the functional-driver and adhesive modules, respectively. The Scl2 B fragment was incorporated into PnBPn and R5BE5 designs, producing banded fibers where the dark band consisted of a width of around 24 nm, matching the calculated length of an 81-residue triple-helix. In our design, the composition of the functional-driver module is relatively unconstrained as long as it folds into a triple-helical structure. To test this, we used Scl2 fragments A and C as functional-driver modules that are of similar size to B, but their amino acid compositions are unique, with A having very few charges, and C rich in charged residues. All three designs, P10AP10, P10BP10, and P10CP10 readily assembled into fibers, with similar bandwidths (Fig. 3a–c, e, h and Supplementary Figs. 12, 13). Thus, the functional-driver module sequence can be modified without noticeably perturbing banded fiber morphology.
Fig. 3: Fine-tuning width of dark and light bands by exchanging functional driver and adhesive modules.
a and b The width of the dark bands remained similar when the functional-driver modules had the same length yet had differing amino acid compositions in P10BP10 (a) and P10CP10 (b); the bandwidth distribution is plotted in c. The width of the dark bands increased as the functional-driver modules increased from 27 to 81 triplets in P10AP10 (e), P10BBP10 (f), and P10ABCP10 (g); the width distribution is plotted in h. The width of the light bands increased as the adhesive modules increased from 5 to 12 triplets in R5BE5 (i), P8BP8 (j), and P12BP12 (k); the width distribution of both the light and dark bands are plotted in l. d Observed bandwidths were plotted against those calculated from module length, multiplied by the average helical rise (~2.9 Å) per residue61. The 67 nm Type I collagen D-period assumes a repeat length of 234 residues32. The dashed lines indicated that the light and dark bands of the synthetic collagen, respectively, correspond to the overlap and gap region of Type I collagen. Solutions of the designed collagens at 0.5 mM in 10 mM phosphate buffer (pH 7.0) were incubated at 4 °C for 3 days, then negatively stained, and visualized with Transmission Electron Microscopy (TEM) (bandwidths were measured using ImageJ, n = 200). Images are all at the same magnification.
It should be possible to increase the width of the dark band by lengthening the functional driver. P10BBP10 and P10ABCP10 folded into triple-helices (Supplementary Fig. 9j–l, 10) and assembled into banded fibers. The dark bandwidths of P10BBP10 and P10ABCP10 were twice and three times that of P10BP10, while the light bandwidths were unchanged (Fig. 3e–h and Supplementary Fig. 13). Indeed, the 59 nm dark bandwidth of P10BBP10 is similar that of the Type I collagen 67 nm D-period9,12, showing one can engineer banded fibers with biologically relevant dimensions.
Molecular mechanism of adhesion
We explored two strategies of triple-helix adhesion based on modules composed of amino acids that make hydrophobic or electrostatic interactions. Briefly, we anticipated that the strengths of these interactions would scale with the size of the interface. To assess the size and energetic requirements for hydrophobically-driven assembly, we constructed a series of synthetic collagens of the form PnBPn, of varying lengths n = 5, 8, 10, and 12. All designs folded into triple-helices as measured by CD, with Tm values higher than the parent Scl2 domain28 (Supplementary Fig. 9d–f). Aggregation, but not banding, was observed for the P5 adhesive module in P5BP5 and P5BBP5 (Supplementary Fig. 14). When n ≥ 8, all PnBPn designs formed banded fibers (Fig. 3j–l and Supplementary Fig. 15), where the light bandwidth was directly proportional to n, confirming that the light band is formed by the adhesive modules.
Eight repeats of a Pro-Pro-Gly triplet were sufficient to promote assembly, suggesting an interaction threshold for adhesion-dependent on the size of the interface. To quantify this potential requirement, we computed interactions between pairs of triple-helical adhesive modules using the protCAD software platform29; note that protCAD was developed for structure-guided protein engineering, and enables sampling of intramolecular torsional rotations as well as intermolecular degrees of freedom. Interaction energies were scored using the AMBER ff14sb molecular mechanics force field30. Interhelical packing was optimized by sampling rigid body translational and rotational parameters31. In the optimized conformation, an extensive packing interface was generated between triple-helical ridges formed by the proline sidechains (Fig. 4a). As the module length, n, increased from 5 to 12 triplets, the calculated interaction energy score increased linearly from −30 to −70 kcal mol−1 (Fig. 4d and Supplementary Table 2). Based on these calculations, the interaction threshold for supramolecular assembly would lie somewhere between −30 and −50 kcal mol−1.
Fig. 4: Multi-scale computational simulations to understand the assembly mechanism of the banded fibers.
a–d Atomic-level models of the interactions between adhesive modules were built with the software package ProtCAD29, with conformations optimized to achieve the lowest interaction score based on simulated annealing. Structural models of P8 + P8 (a) and R5 + E5 (b), with the lowest interaction scores shown as space-filling models. Predicted salt bridges in R5 + E5 are shown with dashed lines in c. Predicted interaction scores of Pn + Pn (diamonds) and Rn + En (circles) were plotted against the number, n, of the triplets in the adhesive modules in d. Filled data points represent designs for which banded fibers were observed. e Fiber morphology and dimensions were simulated with a diffusion-limited aggregation simulation35. A triple-helical unit, [(Xaa-Yaa-Gly)1]3, was simplified as a sphere. The triple-helices of P10BP10 were simplified as a rod with adhesive ends (gray) and central functional-driver regions (black). The assembly process was simulated by allowing 1200 rods to diffuse randomly along a hexagonal lattice, until contacting an existing rod through their adhesive ends. In the simulated fiber, the adhesive modules formed regions with a high density of spheres, corresponding to the light bands under TEM as shown in f, while the functional-driver modules formed low-density regions, corresponding to the dark bands. g As the length of the functional-driver modules increased in P10BP10, P10BBP10, and P10ABCP10, the fiber dimensions—including the mean values for thickness (upper panel) and length (lower panel)—were obtained from DLA simulations and were co-plotted with those in the TEM images (analyzed using ImageJ).
Electrostatic adhesion-driven assembly was also investigated. R5BE5 folded into triple-helices and assembled into banded fibers. The light band had a mean width of 5.8 nm, matching the expected dimensions of a 15-residue sequence. The dark bandwidths were consistent with that of Scl2 fragment B, unaltered by switching from hydrophobic to electrostatic adhesive domains (Fig. 3i–l and Supplementary Fig. 15). Designs with shorter electrostatic adhesive modules—R3BE3 and R1BE1—did not assemble into banded fibers. This further supports the plausibility of an energetic interaction threshold for the electrostatic-driven supramolecular assembly.
We again used protCAD to compute the interaction energy scores, this time with atomic models of electrostatic adhesive modules. In addition to inter-helical degrees of freedom, we also sampled sidechain rotameric states of the Arg and Glu amino acids. Electrostatic modules tended to associate at larger inter-helical distances than hydrophobic ones (Fig. 4a, b), likely accommodating the larger Arg and Glu sidechains at the interface. Decreased contributions from van der Waals packing forces due to greater inter-helical separations were compensated by electrostatic interactions. In the R5 + E5 calculations, Arg and Glu sidechains interdigitated, forming multiple strong (~3.3 Å) interhelical salt bridges (Fig. 4b–c). The interaction score was proportional to the number of charged residues in each adhesive module. Given that R3BE3 and R1BE1 did not form banded fibers, an interaction threshold for assembly between −30 and −50 kcal mol−1 apparently exists for electrostatically driven adhesion. This is the same range as for hydrophobic interactions, suggesting a common interaction threshold for supramolecular assembly across all designs.
It was thought that the functional-driver sequences derived from Scl2 might coincidentally contain regions that promote interhelical associations through hydrophobic or electrostatic interactions, which could confound our proposed mechanism for assembly. As atomistic calculations are impractical for evaluating all possible supramolecular interactions at this scale, we applied a discrete, primary sequence-based approach used previously to model natural collagen interactions20,32. Briefly, two sequences are aligned in a series of poses by shifting one amino acid at a time (Supplementary Fig. 16a). Interaction scores are calculated based on the number of hydrophobic and electrostatic contacts, allowing us to rank the different poses. As expected, when this calculation was applied to Type I collagen, the highest interaction scores occur at staggers every 231–236 residues (Supplementary Fig. 17), consistent with the length of D-spacing9,12. For synthetic collagens P10BP10, P10BBP10, and P10ABCP10, top-scoring poses comprised fully overlapping adhesive domains (Supplementary Fig. 16b, c). Therefore, interactions between the proline-rich adhesive modules, rather than incidental interactions between functional-driver modules, likely drive assembly.
Simulating supramolecular assembly
The multiscale structural features of natural collagens, from the atomic to the macroscopic, all contribute to biological activity27,33. When assembled in vitro—as would be done in tissue culture for example—animal collagens will form long banded fibers that are thicker in the middle and tapered at the ends34. This is consistent with an assembly pathway known as diffusion-limited aggregation (DLA)35, where individual triple-helices attach to a growing fiber at any point along its length (Supplementary Fig. 18). Micrographs of our synthetic collagens showed this anticipated morphology, indicating DLA as the likely operating mechanism of supramolecular assembly. It would be useful to predict supramolecular features on the micron scale for synthetic collagen assemblies composed of thousands or more triple-helices. As atomistic methods are computationally expensive at such scales, we used a minimalistic coarse-grained Monte Carlo DLA model, developed previously with Type I collagen35 and synthetic peptides36, to predict supramolecular features on the ten to hundred-micron scale.
DLA simulations were performed using coarse-grained models of synthetic collagens, where individual triple-helices were approximated as rigid rods composed of adhesive and functional driver modules (Fig. 4e, f). DLA simulations produced banded fibers consistent with the ones observed through TEM and cryoET: these were thicker in the middle and tapered at the ends. As expected, the coarse-grained models showed that the gap region dark bands formed by functional-driver modules had a lower packing density than the overlap regions (light bands) formed by the adhesive domains (Supplementary Fig. 19). These spaces would allow a negative TEM stain to infiltrate the gap regions and produce a darker color. Conversely, overlapping attachment domains had a higher density in DLA models, consistent with light bands resisting stain.
Models from DLA simulations were also compared to observed fiber thicknesses and lengths. Simulated fibers were generated for P10BP10 containing 50–3000 individual triple-helices (Supplementary Fig. 20). Simulated dimensions best-matched observations for fibers composed of ~1200 rods (Fig. 4g). The length and thickness of the simulated fibers increased with the number of rods (Supplementary Fig. 20). However, the aspect ratio (length/thickness) decreased only slightly, still close to the values extracted from TEM (Supplementary Fig. 21), consistent with previous diffusion models of collagen37. Simulated dimensions best-matched observations for fibers composed of ~1200 rods (Fig. 4g). These observations held for the longer P10BBP10 constructs. Only for the longest synthetic collagen, P10ABCP10, did the simulations significantly diverge from observations. This may be due to a number of factors, including potential deviations from a rigid rod approximation for long designs that would increase the anisotropy of diffusion38. It would be useful to explore alternate models of collagen that incorporate triple-helical flexibility39. These findings, including the edge case with P10ABCP10, demonstrate how coarse-grained modeling tools enhance our physical understanding of the collagen assembly process. In establishing DLA as the pathway from triple-helices to banded fibers, we can computationally sample module configurations, prior to studying them in the laboratory, showcasing the utility of synthetic collagens as a testbed for developing designer biomaterials.
Banded fibers support osteoblastic differentiation
Type I collagen is the primary extracellular matrix protein in bone, and its assembly into a banded fiber is known to be essential for osteogenesis and matrix mineralization5,6,40. We, therefore, tested whether the cell culture of osteoblast precursor MC3T3-E1 cells on synthetic collagens that form banded fibers would significantly promote cell adhesion and differentiation over related, non-banded protein substrates. Cell behaviors on synthetic banded fiber collagen substrates (P10BP10 and P10BBP10) were compared to related fiber-incompetent substrates (B and P5BP5). Type I collagen and bovine serum albumin (BSA) were included as positive and negative controls, respectively. Our results ultimately showed that cultured osteoblasts preferentially thrived and differentiated on synthetic banded collagen substrates.
Osteoblast viabilities and proliferation rates for all substrates were comparable (Fig. 5a), confirming that our synthetic collagens are biocompatible. Synthetic collagen banded fiber and Type I collagen substrates promoted higher cell adhesion than the banding-incompetent substrates and BSA control (Fig. 5b). Natural collagen could promote the formation of focal adhesion sites and recruitment of cytoplasmic actin-binding proteins (e.g., vinculin and α-actinin)41,42. We measured VCL and ACTN mRNA levels, and found that the presence of the banded fiber substrates promoted the transcription of both genes (Fig. 5c). Cytoskeletal phenotypes in response to cell adhesion were evident when we assayed cell spreading area and performed cytoskeletal staining with phalloidin (Fig. 5d, e). The cells which were adhered to the banded fiber substrates had larger spreading areas with more obvious cytoskeleton extension, and had prominent cell dendrites, indicating accelerated maturation of such cells (Fig. 5d, e).
Fig. 5: Proliferation, adhesion, and osteoblastic differentiation of MC3T3-E1 osteoblast precursor cells induced by the banded fibers of synthetic collagens.
a–e A non-osteoblastic inducing cell culture was seeded on plates coated with several types of synthetic collagen samples as labeled at the top of the figure. Cell viability at various time points after seeding (0, 1, 3, or 5 days) was analyzed with CCK-8 assays (a), In order to confirm cell adhesion at the transcription level, cell adhesion rates were evaluated (b), and the relative mRNA expression of the adhesion-related genes VCL and ACTN were measured using qPCR after seeding for 6 h (c). d The cells were stained with Phalloidin staining and visualized with fluorescence microscopy (scale bar = 100 μm); e stained areas of the adhered cells were calculated using ImageJ. Osteoblastic-differentiation tests were carried out on plates coated with the same collagen samples as in a–e. The experiments were carried out in a 24-well cell-culture plate for 7 days. f, g Representative alkaline phosphatase (ALP) staining images of a whole well (upper panels) and the central region of the well (scale bar 500 μm, lower panels) were taken under a light microscope, and the ALP-positive area of the whole well was calculated. h In order to test the osteoblastic differentiation at the transcription level, the relative mRNA expression of the osteoblastic-differentiation-related gene RUNX2 was measured with qPCR. All of the experiments had three replicates, and the data are represented as the mean ± s.d. (n = 3). The significance of differences among groups was assessed using ordinary one-way ANOVA with multiple comparisons test.
The banded fiber substrates also promoted MC3T3-E1 cell differentiation into mature osteoblasts. Alkaline phosphatase (ALP) is an extracellular bone enzyme that increases the local concentration of inorganic phosphate and facilitates mineralization43, and is a typical biomarker of osteoblastic differentiation and incipient mineralization. ALP-stained areas for P10BP10 and P10BBP10 were comparable to that of Type I collagen and significantly larger than for the B, P5BP5, or BSA sample groups (Fig. 5f, g). The expression of RUNX2, an osteoblastic-specific transcription factor44, was induced by the synthetic banded fiber collagens P10BP10 at levels higher than that of the banding-incompetent substrate, BSA control, and even natural Type I collagen (Fig. 5h).
The similar outcomes of cells on P10BP10 and P10BBP10 suggest that common features of these two designs and that of Type I collagen stimulate differentiation. We hypothesize that the banding of the synthetic collagen fibers aligns the engineered integrin binding sites (GFPGER) in the fragment B45 (see sequences in Supplementary Table 1), resulting in clustering of cell receptors, promoting and downstream signaling (Supplementary Fig. 24)46.
Synthetic collagens that form banded fibers enhanced cell compatibility of MC3T3-E1 osteoblast precursor cells, and promoted the expression of osteoblastic differentiation markers ALP and RUNX2 in these cells, consistent with observations that the supramolecular structure of natural collagens is actively involved in osteogenesis and bone mineralization5,6,40. Previous studies have reported that collagen from animal models with osteogenesis-relevant diseases (e.g., osteogenesis imperfecta (OI) and osteopenia) display aberrant Type I collagen banding47,48,49. Reported issues from the industry regarding the purity and quality of animal-extracted collagens also limit both the precision of data and the confidence in results from experiments addressing the pathogenic impacts of aberrant collagen structures. In addition, by engineering disease-causing mutations into collagen modules designed based on hypotheses from clinical or biological insights, synthetic collagens can be further used to investigate the molecular mechanisms of cell–matrix interactions in osteogenesis, as well as pathogenesis-related mechanisms in great detail.
We designed, modeled, and tested an innovative class of synthetic collagens comprising an adhesive module to promote supramolecular assembly into classic collagen banded fibers, and a functional-driver module that can be precisely tailored to obtain desired material behaviors. The robust expression and assembly of these banded fibers for a range of materials with different combinations of modules can provide pure, safe, and functionally designable collagen biomaterials. Our demonstration that P10BP10 and P10BBP10 could promote the differentiation of MC3T3-E1 osteoblast precursor cells at levels comparable to natural collagen confirms that these banded fibers are valuable as biomedical research tools and as a testbed for designer functional biomaterials. Although the designed collagens did not form hydrogels, they still can be potentially used as coating materials on metal implants50. It may be possible to design functional drivers that promote the formation of a supramolecular viscoelastic gel.
Computational models that accurately recapitulate the supramolecular assembly process for these fibrous proteins represent a major conceptual advance to better define the rules governing the supramolecular assembly of natural collagen. Specifically, we have established energetic interaction thresholds between adhesive modules and have set up a coarse-grained DLA model to help understand how protein length and composition dictate the morphology of assembled fibers. Notably, we show that extended proline-rich sequences can be used to modulate the strength of intra-fiber adhesion, and changing the lengths of adhesive and functional-driver modules can vary the period of fiber banding at biologically relevant scales.
Functionally designable synthetic collagens with tunable banding morphologies provide an unprecedented tool for exploring collagen–collagen and collagen–cell interactions in biology and disease. To date, investigations of collagen diseases such as OI have largely depended on animal models, where the specific role of collagen cannot be studied in isolation47,48,49, or on collagen peptides and fragments33, which do not recapitulate the supramolecular nature of natural collagen fibers. These designs provide a supramolecular platform for hosting OI mutations of interest, enabling characterization of their precise impacts in disrupting collagen fiber assembly and during osteogenesis. Our synthetic banded fibers provide a flexible platform that fills the gap between simplified biophysical models of collagen-like peptides and complex in vivo systems of model animals and can foster discoveries about collagen-involved biological and pathological processes.
Plasmid construction, protein expression, and purification
Synthetic genes of the designed collagens (Supplementary Table 1), which contained a 5' (NcoI) and 5'UTR ([GC]) and 3' (BamHI), were optimized for the E. coli expression system. For each designed collagen, its gene was cloned into the pColdIII-Tu vector through NcoI and BamHI. The pColdIII-Tu vector was obtained from pColdIII site-directed mutation of the NdeI to NcoI by primer S1 (CTCGAGGGATCCGAATTCA) and A1 (GAGCTCCATGGGCACTTTG). All verified constructs were transformed into E. coli BL21 (DE3) (Supplementary Fig. 1). The primary seed culture (1 mL) was used to inoculate 100 mL of TB medium containing 100 µg mL−1 ampicillin at 37 °C cultured for 24 h. The bacteria liquid was cooled to 25 °C and induced with 1 mM IPTG. After 10 h incubation, cells were further cooled to 15 °C for 14 h51. Cells were harvested by centrifugation at 4 °C. The cell pellets were resuspended in binding buffer (20 mM NaPO4, 500 mM NaCl, 10 mM imidazole, pH 7.4), lysed with sonication, centrifugated at 10,000×g for 20 min at 4 °C and filtered with the microporous membrane (0.45 μm) to remove the cell debris. The designed collagen with a His6 tag at N-terminus was purified with 5 mL HisTrap HP column using the stepwise increasing concentrations of imidazole (130, 160, and 400 mM). The globular folding domain, V-domain, of the designed collagen was removed by incubating with trypsin at 1:1000 (w:w) ratio at 25 °C overnight, which was dialyzed and freeze-dried. As the trypsin is autolyzed into fragments within several hours52, the fragments were eliminated from the protein solutions with the dialysis. The molecular weights and purity (>95%) of the designed collagens were validated by SDS–PAGE and MALDI-TOF mass spectrometry (Supplementary Figs. 2 and 3).
Circular dichroism (CD) spectroscopy
CD Spectra were obtained via an applied photophysics chirascan with a Peltier temperature controller, using quartz cuvettes with 1 mm path length (Model 110-OS, Hellma USA). Protein solutions (1 mg mL−1 in 10 mM PB buffer) were equilibrated at 4 °C for ~24 h before measurements. Wavelength scans were collected at 4 °C from 190 to 260 nm (step 1 nm, averaging time 5 s). For temperature transition experiments from 4 to 80 °C, the ellipticity was monitored at 220 nm at an average heating rate of 1 °C for 6 min−1 with 8 s equilibration time. The first derivative of the melting curves was used to obtain the melting temperatures of collagen samples.
Differential scanning calorimetry (DSC)
DSC was carried out for the collagen proteins in a NANO DSC instrument (TA Instruments) coupled with a thermal data analysis system, Nano-Analyze software. The collagen solution was at 5 mg/mL in 10 mM phosphate buffer at pH 7 and incubated at 4 °C for 3 days before measurements. The 10 mM phosphate buffer was used for the baseline scan at least three times. Sample solutions were loaded at 20 °C into the cell and heated at a rate of 1 °C/min.
MALDI-TOF mass spectrometry
MALDI-TOF was used to confirm the molecular mass of the designed collagen (Bruker Daltonics UltrafleXtreme, USA). 1 μL of 1 mg mL−1 sample was dropped in a 600 μm AnchorChip target plate (Bruker) and allowed to dry, using 5 mg mL−1 2,5-dihydroxyacetophenone, 25% (v/v) ethanol, 1.5 mg mL−1 diammonium hydrogen citrate as the matrix in linear mode, BSA was used for external calibration. Or using 10 mg mL−1 sinapinic acid in 50% acetonitrile, 0.1% TFA as the matrix. 1 μL of the sample was applied onto a plate with apo-myoglobin as the internal standard. Data was acquired with AB Sciex 4800 at positive linear mid-mass mode.
Solutions of the designed collagens at 0.5 mM in 10 mM phosphate buffer (pH 7.0) were incubated at 4 °C for 3 days. The solution was placed onto a copper grid and was incubated for 45 s, after which the excess solvent was absorbed from the grid using filter paper. The sample was negatively stained with 0.75% phosphotungstic acid for 20 s, and the excess stain was absorbed away and imaged with Hitachi H-7650 (Hitachi, Japan) Electron Microscope. The light- and dark-band widths were measured with ImageJ about 200 times from at least 5 TEM images.
Atomic force microscope (AFM)
Tapping mode was performed on a Cypher ES AFM (Asylum Research, Oxford Instruments) at room temperature. AC240 silicon nitride cantilevers sourced from Oxford Instruments were used for imaging in the air. These cantilevers have a nominal spring constant of kc = 2.0 N m−1, and radii of r = 7 nm. Sample solutions were deposited onto freshly cleaved mica, allowed 5 min for binding, and then washed with 2 mL of water. Samples were then allowed to air dry for 1 h in a laminar flow cabinet before being placed in the AFM for imaging and analyzed using Asylum Research software. Before the Fourier transform was used to analyze the banded fibers in AFM images, EMD denoising protocol was used to remove the baseline drift and noise53. The data processing was carried out with Matlab_R2016a with a source code, which was uploaded to GitHub (https://github.com/JinyuanHu/FFT_AFM.git)
Microrheology
Solutions of the designed collagens and Type I collagen at 1 and 5 mg mL−1 in 10 mM phosphate buffer (pH 7.0) were incubated at 4 °C for 3 days. Then Young's moduli, G' and G", were determined as previously described54. Fluorescent polystyrene beads were added to the solution and record the trajectories of the fluorescent beads were by fluorescence microscopy, then determine the mean square displacement as a function of time and Young's moduli, G' and G", were determined from the Generalized Stokes–Einstein relationship.
Calculation of inter-triple helical interactions
The interaction energy scores of the designed collagen were computed according to Hulmes et al. using a sequence-based model32. Charged pair (K/R, D/E) and hydrophobic pair (V, M, I, L, F, P) were given a score of one. Repulsion pairs were given a score of negative ones. All other pairs were scored zero. The segment length for computed is ±3 for charge interactions, and ±2 for hydrophobic interactions (Supplementary Fig. 16). The interaction score was computed at every one residue shift when the above chain shifts laterally from the N- to C-terminus of the neighboring chain. In order to mimic the natural arrangement of Type I collagen, 0.6 D gap (156 amino acids) was set when the interaction scores were calculated (Supplementary Fig. 17). For the PnXPn collagen, (PPG)10 was overlapped to match the full overlap of (PPG)10 in self-assembly process (Supplementary Fig. 16).
Computational modeling of association of attachment modules
The structural model of Pn was obtained by shortening or elongating a high-resolution (1.3 Å) crystal structure of (PPG)10 (PDB: 1K6F)55. Pn was aligned along the principal z-axis by minimizing the projection on the x–y plane. The relative positions of the other off-centered triple helices in Pn + Pn were described with five geometric parameters, Ω, θ, φ, r, and Δh31 (Supplementary Fig. 22). Ω was a tilting angle between two triple helices. φ was the self-rotation angle of the off-centered triple helices, which rotated around the centered one with the angle θ. r and Δh were the distances along x- and z-axis, respectively. When Ω = 0°or 180°, association energies of Pn + Pn were optimized with a Monte Carlo Simulated Annealing (MCSA) program with 16,000 steps from 40 to 0.001 °C. During the optimization, all atoms within the same triple helices were kept fixed in order to keep the intra-helical energies constant. The van der Waals energy between the two triple helices was calculated using a 12-6 Lenard–Jones potential with a distance cutoff 6.0 Å. The AMBER ff14SB force field parameters were used for nonbonding interactions30.
Rn + En association was modeled with a similar protocol used for Pn. Atomic models were generated by mutating (PPG)10 with protCAD. Parallel and antiparallel associations (Ω = 0°, 180°) were sampled. Together with the five geometric parameters, the side chain rotamers of Arg and Glu were optimized with MCSA.
Simulation of collagen self-assembly
A coarse-grained model of self-assembly simulating the process of diffusion-limited aggregation (DLA) was established with a modified Java script previously reported36. The simulation flowchart was presented in Supplementary Fig. 18. A triple helical fragment, [(PPG)10]3, was simplified as a sphere with 1 nm diameter. In P10BP10, the adhesive module P10 was simplified as 10 spheres. As the helical rise of a triplet with low Pro content is about 0.8–0.9 nm20, the 27-triplet functional-driver module of P10BP10 was modeled as 25 spheres. The spheres formed a rigid rod. A seed rod was placed initially at the center and additional rods were placed at the periphery to randomly diffuse along a 3D hexagonal lattice until contacting an existing rod. Considering viscous resistance, the probability of diffusing axially or radially was adjusted:
$$f=\eta S\frac{{{\rm {d}}v}}{{{\rm {d}}t}}$$
where f is viscous resistance; \(\eta\), viscosity of the fluid; S, cross-sectional area in the direction of movement; \(\frac{{{\rm {d}}v}}{{{\rm {d}}t}}\), fluid velocity gradient.
The probability of a movement P along axial (Pz) and radial (Pxy) is inversely proportional to the resistance.
$${P}_{z}/{P}_{{xy}}={f}_{{xy}}/{f}_{z}=\eta {S}_{{xy}}\frac{{{\rm {d}}v}}{{{\rm {d}}t}}/\eta {S}_{z}\frac{{{\rm {d}}v}}{{{\rm {d}}t}}$$
In this simulation, it can be assumed that \(\frac{{{\rm {d}}v}}{{{\rm {d}}t}}\) and \(\eta\) have the same values in the axial and lateral movements. Collagen can be considered as a cylinder with length L and diameter D.
$${P}_{z}/{P}_{{xy}}={S}_{{xy}}/{S}_{z}={LD}/({{{{{\rm{\pi }}}}}}{D}^{2}/4)=4L/{{{{{\rm{\pi }}}}}}D$$
DLA models of fiber assembly were generated by repeating, 100 times, the simulation of seven sets of rods (50, 100, 300, 600, 1000, 1500, and 3000 rods), where each rod represented the P10BP10, P10BBP10, or P10ABCP10 construct. Fiber length and width were taken to be the weighted average of the total set of simulations per rod type.
A 96-well culture plate (3599, Corning, USA) was pre-coated by 10 μg cm−2 collagens, including B, P5BP5, P10BP10, P10BBP10 and Type I collagen (C7661, SIGMA, USA), at 4 °C overnight. Then the sample solution was removed and the plate was dried in the air for 12 h. After 2 h ultraviolet sterilization, the plate was blocked by 5% BSA (A600332-0100, Sangon Biotech, China) for 2 h. Then 2000 MC-3T3 cells were seeded into the wells and Cell Counting Kit-8 (BA00208, BIOSS, China) was used to detect the cell proliferation according to the manufacturer's instructions. At days 0, 1, 3, and 5, 10 μl Cell Counting Kit-8 solution was added into each well and the absorbance at 450 nm was measured through SYNER GY H1 (BioTek, USA). The culture medium contains 90% α-MEM medium (SH30265.01, Hyclone, USA), 10% fetal bovine serum (FND500, ExCell, China), and 1% Penicillin–Streptomycin (KGY0023, KeyGEN, China). The culture medium was renewed every two days.
Cell adhesion assay
A 96-well culture plate was coated with the same samples as the cell proliferation assay. The 104 MC-3T3 cells were seeded into each well for 6 h. The culture medium was removed and the plate was washed with PBS gently 3 times. A new culture medium was supplemented and 10 μL Cell Counting Kit-8 solution (BA00208, BIOSS, China) was added to each well. The absorbance at 450 nm was measured through SYNER GY H1 (BioTek, USA). The adhesion percentage was calculated by dividing the number of adherent cells by the seeding cells.
Quantitative polymerase chain reaction assay (qPCR)
A 96-well culture plate was coated with the same samples as the cell proliferation assay. After cell seeding for 6 h, the adhesion-related gene expression of MC-3T3 cells was detected with qPCR. MC-3T3 cells were also cultured in the plate under osteoblastic differentiation induction for 7 days, and then osteoblastic gene expression was detected with qPCR. Total RNA was isolated from cells using TRIzol Reagent (15596-018, Invitrogen, USA) and cDNA synthesis was conducted using the Transcriptor First Strand cDNA Synthesis Kit (Roche, Switzerland) according to the manufacturer's instruction. The qPCR experiments were performed with an SYBR Premix Ex Taq II kit (Takara, Japan). The gene expression levels were normalized to the Gapdh expression level. The sequence of the primers used is listed in Supplementary Table 3.
Cytoskeleton staining
A 24-well culture plate was coated with the same samples as those in the cell proliferation assay. About 4000 MC-3T3 cells were cultured in each well for 6 h. Then, the plate was washed through PBS 3 times and the cells were fixed by 4% PFA for 30 min. After incubation in 0.3% Triton X-100 (ST797, Beyotime, China) for 15 min, cells were stained with FITC-Phalloidin (KD0382, Kingmorn, China) for 30 min. Finally, cell nuclei were counterstained with DAPI (D9542, Sigma, USA) for 10 min. The images were taken using a fluorescence microscope. The area occupied by cells was determined by Image J56.
Alkaline phosphatase staining
A 24-well culture plate was coated with the same sample as those in the cell proliferation assay. MC-3T3 cells were cultured in a plate under osteoblastic differentiation induction for 7 days. The plate was washed with PBS 3 times, and 4% PFA was used to fix cells for 30 min. Then the Alkaline phosphatase (ALP) staining was performed using the ALP Staining Kit (P0321M, Beyotime, China) according to the manufacturer's instructions. The cells were incubated with ALP staining solution at 37 °C for 1 h. The images were taken using a light microscope and the ALP staining area was determined by Image J.
Tomography data collection and analysis
EM grids for cryoET structural studies were prepared using a self-assembled collagen solution. P10BP10 and P10BBP10 samples were mixed with 6 nm nanogold particles to facilitate tilt series alignment during image processing. An aliquot of 3.5 μl of collagen fibers was then applied onto glow-discharged Quantifoil holey grids (R2/2, Cu, 200 mesh; Quantifoil) before plunge frozen using a Leica EM GP plunger (Leica Microsystems, Buffalo Grove, IL, USA) in a humidity (95%) and temperature (20 °C) controlled chamber. Images and tilt series of the samples were collected on a Talos Arctica cryo-electron microscope (Thermo Fisher Scientific, Waltham, MA, USA) operated at 200 kV, equipped with a post-column BioQuantum energy filter (the slit was set to 20 eV) and a K2 direct electron detector. Automated data collection was performed using SerialEM57. Tomographic data was acquired using a dose-symmetric tilt scheme with 3° intervals and ×31,000 microscope magnification with a corresponding pixel size of 4.36 Å/pixel. Data collection was performed in counting mode, with spot size 9, 100 μm condenser, and 70 μm objective apertures. Tilt series of both P10BP10 and P10BBP10 were collected at a defocus range of −8 to −11 μm with a cumulative dose of ~160 e−/Å2. The latest EMAN2 tomography workflow was used for all tilt series alignment and tomogram reconstruction58.
Quantitative measurements (bandwidths, periodicity, etc.) were performed using functionalities in EMAN2. Volume rendering and visualization of tomograms were done using UCSF Chimera59. Measurements of periodicity in real space involved calculating the length of the fiber along multiple periodicities, divided by the total number of periods. To evaluate whether measurements of periodicity in real space v. Fourier space were statistically different, we performed a two-sample t-test with equal variance.
All numerical data are expressed as the mean ± s.d. The significance of differences among groups for cell culture were assessed using ordinary one-way ANOVA with multiple comparisons test using Prism 8.2.1 (**** P < 0.0001, ** P < 0.01, * P < 0.05). P < 0.05 was considered statistically significant. Experiments were repeated at least three times independently with similar results in all agarose gel electrophoresis, SDS-PAGE, TEM, Cryo-ET, and AFM experiments. The statistical analyses were performed with OriginPro 2019b software for the TEM at n = 200, and for the DLA models at n = 100.
Electron density maps of P10BP10 and P10BBP10 synthetic collagen fibers have been deposited in the Electron Microscopy Data Bank under the accession codes EMD-27664 and EMD-27665, respectively. Scripts and code are available at https://github.com/JinyuanHu. The data that support the findings of this study are available from the corresponding author.
Code availability
The protCAD modeling platform used in this study is available at https://github.com/JinyuanHu/protcad/releases. The code for AFM data processing is available under https://github.com/JinyuanHu/FFT_AFM/releases. The code for DLA simulation is available at https://github.com/JinyuanHu/dla/releases.
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We wish to thank the core imaging facilities at the School of Biotechnology, Jiangnan University, and Robert Wood Johnson Medical School, Rutgers University for help obtaining TEM images. This work was supported by the National Key R&D Program of China (No. 2018YFA0901600) and the National Natural Science Foundation of China (No. 22078129) to FX, National Institute of Health of US (No. R35- GM136431) to JB, NASA NAI 80NSSC18M0093 (VN), and National Natural Science Foundation of China (No. 82270963, No. 82061130222) and Foundation of SCST (No. 20XD1424000, 201409006400) to YS.
These authors contributed equally: Jinyuan Hu, Junhui Li.
Ministry of Education Key Laboratory of Industrial Biotechnology, School of Biotechnology, Jiangnan University, 214122, Wuxi, China
Jinyuan Hu, Lingling Wang & Fei Xu
Department of Oral Implantology, School of Stomatology, Tongji University, Shanghai Engineering Research Center of Tooth Restoration and Regeneration, Shanghai, China
Junhui Li & Yao Sun
Department of Cell Biology and Neuroscience, Rutgers, The State University of New Jersey, Piscataway, NJ, 08854, USA
Jennifer Jiang & Wei Dai
Institute for Quantitative Biomedicine, Rutgers, The State University of New Jersey, Piscataway, NJ, 08854, USA
Department of Chemistry and Chemical Biology, Rutgers, The State University of New Jersey, Piscataway, NJ, 08854, USA
Jonathan Roth & Jean Baum
Department of Biochemistry and Molecular Biology, Robert Wood Johnson Medical School, Rutgers, The State University of New Jersey, New Brunswick, NJ, 08901, USA
Kenneth N. McGuinness & Vikas Nanda
Center for Advanced Biotechnology and Medicine and the Department of Biochemistry and Molecular Biology, Robert Wood Johnson Medical School, Rutgers, The State University of New Jersey, Piscataway, NJ, 08854, USA
Jinyuan Hu
Junhui Li
Jennifer Jiang
Lingling Wang
Jonathan Roth
Kenneth N. McGuinness
Jean Baum
Wei Dai
Yao Sun
Vikas Nanda
Fei Xu
F. X., J. H., and V. N. designed experiments, synthesized materials, performed experiments, analyzed data and wrote the manuscript. J. L. and Y. S. collected and analyzed the cell culture data and wrote the manuscript. J. J. and W. D. collected and analyzed the cryoET data and wrote the manuscript. L. W. collected the TEM data. K. M. took part in DLA simulation and discussions. J. R., and J. B. collected and analyzed the AFM data. F. X. and V. N. supervised the research and wrote the manuscript. All authors discussed the results and commented on the manuscript.
Correspondence to Yao Sun, Vikas Nanda or Fei Xu.
Patent US20210079064A1—"Preparation of Type I Collagen-Like Fiber and Method for Regulating and Controlling the D-periodic of Fiber Thereof" Fei Xu, Jinyuan Hu, Vikas Nanda, David I. Shreiber, Meng Zhang, Sonal Gahlawat.
Nature Communications thanks Biplab Sarkar and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
Hu, J., Li, J., Jiang, J. et al. Design of synthetic collagens that assemble into supramolecular banded fibers as a functional biomaterial testbed. Nat Commun 13, 6761 (2022). https://doi.org/10.1038/s41467-022-34127-6 | CommonCrawl |
Clinical study design
Clinical study design is the formulation of trials and experiments, as well as observational studies in medical, clinical and other types of research (e.g., epidemiological) involving human beings.[1] The goal of a clinical study is to assess the safety, efficacy, and / or the mechanism of action of an investigational medicinal product (IMP)[2] or procedure, or new drug or device that is in development, but potentially not yet approved by a health authority (e.g. Food and Drug Administration).[3] It can also be to investigate a drug, device or procedure that has already been approved but is still in need of further investigation, typically with respect to long-term effects or cost-effectiveness.[4]
Some of the considerations here are shared under the more general topic of design of experiments but there can be others, in particular related to patient confidentiality and ethics.
Outline of types of designs for clinical studies
Treatment studies
• Randomized controlled trial[5]
• Blind trial[6]
• Non-blind trial[7]
• Adaptive clinical trial[8]
• Platform Trials
• Nonrandomized trial (quasi-experiment)[9]
• Interrupted time series design[10] (measures on a sample or a series of samples from the same population are obtained several times before and after a manipulated event or a naturally occurring event) - considered a type of quasi-experiment
Observational studies
1. Descriptive
• Case report[11]
• Case series[12]
• Population study[13]
2. Analytical
• Cohort study[11]
• Prospective cohort[14]
• Retrospective cohort
• Time series study
• Case-control study
• Nested case-control study
• Cross-sectional study
• Community survey (a type of cross-sectional study)
• Ecological study
Important considerations
When choosing a study design, many factors must be taken into account. Different types of studies are subject to different types of bias. For example, recall bias is likely to occur in cross-sectional or case-control studies where subjects are asked to recall exposure to risk factors. Subjects with the relevant condition (e.g. breast cancer) may be more likely to recall the relevant exposures that they had undergone (e.g. hormone replacement therapy) than subjects who don't have the condition.
The ecological fallacy may occur when conclusions about individuals are drawn from analyses conducted on grouped data. The nature of this type of analysis tends to overestimate the degree of association between variables.
Seasonal studies
Conducting studies in seasonal indications (such as allergies, Seasonal Affective Disorder, influenza, and others) can complicate a trial as patients must be enrolled quickly. Additionally, seasonal variations and weather patterns can affect a seasonal study.[15][16]
Other terms
• The term retrospective study is sometimes used as another term for a case-control study.[17] This use of the term "retrospective study" is misleading, however, and should be avoided because other research designs besides case-control studies are also retrospective in orientation.
• Superiority trials are designed to demonstrate that one treatment is more effective than a given reference treatment. This type of study design is often used to test the effectiveness of a treatment compared to placebo or to the currently best available treatment.
• Non-inferiority trials are designed to demonstrate that a treatment is at least not appreciably less effective than a given reference treatment. This type of study design is often employed when comparing a new treatment to an established medical standard of care, in situations where the new treatment is cheaper, safer or more convenient than the reference treatment and would therefore be preferable if not appreciably less effective.
• Equivalence trials are designed to demonstrate that two treatments are equally effective.
• When using "parallel groups", each patient receives one treatment; in a "crossover study", each patient receives several treatments but in different order.
• A longitudinal study assesses research subjects over two or more points in time; by contrast, a cross-sectional study assesses research subjects at only one point in time (so case-control, cohort, and randomized studies are not cross-sectional).
See also
• Conceptual framework
• Epidemiological methods
• Epidemiology
• Experimental control
• Meta-analysis
• Operationalization
• Academic clinical trials
• Design of experiments
• Research design
References
1. Miquel Porta (2014) "A dictionary of epidemiology", 6th edn, New York: Oxford University Press. ISBN 9780199976737.
2. "Investigational Medicinal Product (IMP) | Noclor". www.noclor.nhs.uk.
3. Ann (April 14, 2006). "Clinical Study Management". ProMedica International. Retrieved June 4, 2019.
4. Nichols, Hannah (May 18, 2018). "How do clinical trials work and who can participate?". Medical News Today. Retrieved June 4, 2019.
5. Shiel, William C. Jr. (December 21, 2018). "Randomized control trial". Medicine Net. Retrieved June 4, 2019.
6. "Are these data real? Statistical methods for the detection of data fabrication in clinical trials". The BMJ. July 28, 2005. Retrieved June 4, 2019.
7. Shiel, William C. Jr. (December 21, 2018). "Nonblinded study". Medicine Net. Retrieved June 4, 2019.
8. Mahajan, Rajiv; Gupta, Kapil (August 2012). "Adaptive design clinical trials: Methodology, challenges and prospect". Indian Journal of Pharmacology. 42 (4): 201–7. doi:10.4103/0253-7613.68417. PMC 2941608. PMID 20927243.
9. "NCI Dictionary of Cancer Terms". National Cancer Institute. 2 February 2011. Retrieved June 4, 2019.
10. Kontopantelis, E.; Doran, T.; Springate, D. A.; Buchan, I.; Reeves, D. (June 9, 2015). "Regression based quasi-experimental approach when randomisation is not an option: interrupted time series analysis". The BMJ. 350: h2750. doi:10.1136/bmj.h2750. PMC 4460815. PMID 26058820.
11. Mathes, Tim; Pieper, Dawid (July 17, 2017). "Clarifying the distinction between case series and cohort studies in systematic reviews of comparative studies: potential impact on body of evidence and workload". BMC Med Res Methodol. 17 (1): 107. doi:10.1186/s12874-017-0391-8. PMC 5513097. PMID 28716005.
12. "NCI Dictionary of Cancer Terms". National Cancer Institute. 2 February 2011. Retrieved June 4, 2019.
13. "NCI Dictionary of Cancer Terms". National Cancer Institute. 2 February 2011. Retrieved June 4, 2019.
14. "NCI Dictionary of Cancer Terms". National Cancer Institute. 2 February 2011. Retrieved June 4, 2019.
15. Yamin Khan; Sarah Tilly. "Flu, Season, Diseases Affect Trials". Applied Clinical Trials Online. Archived from the original on 11 July 2011. Retrieved 26 February 2010.
16. Yamin Khan; Sarah Tilly. "Seasonality: The Clinical Trial Manager's Logistical Challenge" (PDF). published by: Pharm-Olam International (POI). Archived from the original (PDF) on 15 July 2011. Retrieved 26 April 2010.
17. "Prospective vs. Retrospective Studies". Stats Direct. Retrieved May 30, 2019.
External links
• Some aspects of study design Tufts University web site
• Comparison of strength Description of study designs from the National Cancer Institute
Clinical research and experimental design
Overview
• Clinical trial
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(EBM I to II-1)
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• Platform trial
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(EBM II-2 to II-3)
• Cross-sectional study vs. Longitudinal study, Ecological study
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| Wikipedia |
All squares end with square digits (i.e. end with 0, 1, 4 or 9), if n is divisible by both 2 and 3, then n2 ends with 0, if n is not divisible by 2 or 3, then n2 ends with 1, if n is divisible by 2 but not by 3, then n2 ends with 4, if n is not divisible by 2 but by 3, then n2 ends with 9. If the unit digit of n2 is 0, then the dozens digit of n2 is either 0 or 3, if the unit digit of n2 is 1, then the dozens digit of n2 is even, if the unit digit of n2 is 4, then the dozen digit of n2 is 0, 1, 4, 5, 8 or 9, if the unit digit of n2 is 9, then the dozen digit of n2 is either 0 or 6. (More specially, all squares of (primes ≥ 5) end with 1)
The numbers n such that the concatenation of n and the unit (1), i.e. 10n+1 (all squares of primes except 4 and 9 are of this form), is square, are all even numbers, and the half of these n are exactly the generalized pentagonal numbers, and such numbers are important to Euler's theory of partitions, as expressed in his pentagonal number theorem (the pentagonal number theorem, originally due to Euler, relates the product and series representations of the Euler function. It states that
$ \prod_{n=1}^{\infty}\left(1-x^{n}\right)=\sum_{k=-\infty}^{\infty}\left(-1\right)^{k}x^{k\left(3k-1\right)/2}=1+\sum_{k=1}^\infty(-1)^k\left(x^{k(3k+1)/2}+x^{k(3k-1)/2}\right). $
In other words,
$ (1-x)(1-x^2)(1-x^3)(1-x^4) \cdots = 1 - x - x^2 + x^5 + x^7 - x^{10} - x^{13} + x^{1\mathcal{X}} + x^{22} - \cdots. $
The exponents 1, 2, 5, 7, 10, 13, 1X, 22, ... on the right hand side are given by the formula Template:Math for k = 1, −1, 2, −2, 3, −3, 4, −4, ... and are called (generalized) pentagonal numbers. This holds as an identity of convergent power series for $ |x|<1 $, and also as an identity of formal power series.
A striking feature of this formula is the amount of cancellation in the expansion of the product), also, the identity implies a marvelous recurrence for calculating $ p(n) $, the number of partitions of n (p(n)):
$ p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+p(n-10)+p(n-13)-p(n-1\mathcal{X})-p(n-22)+\cdots $
$ p(0)=1 $
or more formally,
$ p(n)=\sum_k (-1)^{k-1}p(n-g_k) $
Also the sum of divisors of n (σ(n)):
$ \sigma(n)=\sigma(n-1)+\sigma(n-2)-\sigma(n-5)-\sigma(n-7)+\sigma(n-10)+\sigma(n-13)-\sigma(n-1\mathcal{X})-\sigma(n-22)+\cdots $
but if the last term is σ(0) (this situation appears if and only if n itself is generalized pentagonal number, i.e. the concatenation of 2n and 1 is square), then we change it to n.
where the summation is over all nonzero integers k (positive and negative) and $ g_k $ is the kth generalized pentagonal number. Since $ p(n)=0 $ for all $ n<0 $, the series will eventually become zeroes, enabling discrete calculation, besides, generalized pentagonal numbers are closely related to centered hexagonal numbers (also called hex numbers, the hex numbers are end with 1, 7, 7, 1, 1, 7, 7, 1, ...). When the array corresponding to a centered hexagonal number is divided between its middle row and an adjacent row, it appears as the sum of two generalized pentagonal numbers, with the larger piece being a pentagonal number proper.
The digital root of a square is 1, 3, 4, 5 or E.
No repdigits with more than one digit are squares, in fact, a square cannot end with three same digits except 000.
No four-digit palindromic numbers are squares. (we can easily to prove it, since all four-digit palindromic number are divisible by 11, and since they are squares, thus they must be divisible by 112 = 121, and the only four-digit palindromic number divisible by 121 are 1331, 2662, 3993, 5225, 6556, 7887, 8EE8, 9119, X44X and E77E, but none of them are squares)
n-digit palindromic squares
square roots
number of n-digit palindromic squares
1, 4, 9 1, 2, 3 3
none none 0
121, 484 11, 22 2
10201, 12321, 14641, 16661, 16E61, 40804, 41414, 44944 101, 111, 121, 12E, 131, 202, 204, 212 8
160061 42E 1
1002001, 102X201, 1093901, 1234321, 148X841, 4008004, 445X544, 49XXX94 1001, 1015, 1047, 1111, 1221, 2002, 2112, 2244 8
100020001, 102030201, 104060401, 1060E0601, 121242121, 123454321, 125686521, 1420E0241, 1444X4441, 1468E8641, 14X797X41, 1621E1261, 163151361, 1XX222XX1, 400080004, 404090404, 410212014, 4414X4144, 4456E6544, 496787694, 963848369 10001, 10101, 10201, 10301, 11011, 11111, 11211, 11E21, 12021, 12121, 1229E, 1292E, 12977, 14685, 20002, 20102, 20304, 21012, 21112, 22344, 31053 19
1642662461 434X5 1
10000200001, 10221412201, 10444X44401, 12102420121, 12345654321, 141E1E1E141, 14404X40441, 16497679461, 40000800004, 40441X14404, 41496869414, 44104X40144, 49635653694 100001, 101101, 102201, 110011, 111111, 11E13E, 120021, 12X391, 200002, 201102, 204204, 210012, 223344 11
It is conjectured that if n is divisible by 4, then there are no n-digit palindromic squares.
Rn2 (where Rn is the repunit with length n) is a palindromic number for n ≤ E, but not for n ≥ 10 (thus, for all odd number n ≤ 19, there is n-digit palindromic square 123...321), besides, 11n (also 1{0}1n, i.e. 101n, 1001n, 10001n, etc.) is a palindromic number for n ≤ 5, but not for n ≥ 6, and it is conjectured that no palindromic numbers are n-th powers if n ≥ 6.
The square numbers using no more than two distinct digits are 0, 1, 4, 9, 14, 21, 30, 41, 54, 69, 84, X1, 100, 121, 144, 344, 400, 441, 484, 554, 900, 3000, 4344, 9944, 10000, 11XX1, 16661, 40000, 41414, 44944, 47744, 66969, 90000, 111101, 114144, 300000, 444404, 454554, 999909, 1000000, 1141144, 3333030, 4000000, 4544554, 9000000, 11110100, 30000000, 41144144, 44440400, 99990900, XXXXXXX1, 100000000, 333303000, 400000000, 900000000, 1111010000, 3000000000, 4444040000, 9999090000, 10000000000, 33330300000, 40000000000, 90000000000, 111101000000, 300000000000, 444404000000, 999909000000, 1000000000000, ...
41414 is the largest undulating square (of the form ababab...), note that the only 2 distinct digits in it (1 and 4) and the only 2 distinct two-digit numbers formed by its digits (14 and 41) are also all squares.
A cube can end with all digits except 2, 6 and X (in fact, no perfect powers end with 2, 6 or X), if n is not congruent to 2 mod 4, then n3 ends with the same digit as n; if n is congruent to 2 mod 4, then n3 ends with the digit (the last digit of n +− 6).
The cube numbers using no more than two distinct digits are 0, 1, 8, 23, 54, X5, 1000, 1331, 8000, 1000000, 8000000, 1000000000, 8000000000, 1000000000000, ...
The digital root of a cube can be any number.
21 and 201 are both squares, and it is conjectured that no other numbers of the form 2000...0001 (i.e. of the form 2×10n+1) are squares.
10814 and 100814 are both squares, and 10854 and 100853 are both cubes.
If k≥2, then nk+2 ends with the same digit as nk, thus, if i≥2, j≥2 and i and j have the same parity, then ni and nj end with the same digit.
If x^2 + y^2 = z^2 (that is, {x, y, z} is a Pythagorean triple, then xy end with 0 (and thus xyz also end with 0).
Squares (and every powers) of 0, 1, 4, 9, 54, 69, 369, 854, 3854, 8369, E3854, 1E3854, X08369, ... end with the same digits as the number itself. (since they are automorphic numbers, from the only four solutions of x2−x=0 in the ring of 10-adic numbers (dozadic numbers), these solutions are 0, 1, ...2E21E61E3854 and ...909X05X08369, since 10 is neither a prime nor a prime power, the ring of the 10-adic numbers is not a field, thus there are solutions other than 0 and 1 for this equation in 10-adic numbers)
The triangular numbers using no more than two distinct digits are 0, 1, 3, 6, X, 13, 19, 24, 30, 39, 47, 56, 66, 77, 89, X0, E4, 191, 303, 446, 550, 633, 66X, 6X6, 1117, 3X3X, 3EE3, 6060, 6161, 6366, 6999, 6EE6, 8989, 9779, 23223, 35553, 50050, 77677, 113113, 303333, 331331, 600600, X33X33, 3030330, 60006000, 333666333, 6000060000, 600000600000, ...
The pronic numbers using no more than two distinct digits are 0, 2, 6, 10, 18, 26, 36, 48, 60, 76, 92, E0, 110, 606, 656, 992, XX0, EE6, 1118, 2232, 7878, EE00, 10100, 33330, 46446, 6XXX6, X00X0, 118118, 226226, 606666, 662662, EEE000, 1001000, 6060660, EEEE0000, 100010000, EEEEE00000, 10000100000, EEEEEE000000, 1000001000000, ...
Except for 6 and 24, all even perfect numbers end with 54. Additionally, except for 6, 24 and 354, all even perfect numbers end with 054 or 854. Besides, if any odd perfect number exists, then it must end with 1, 09, 39, 69 or 99.
The digital root of an even perfect number is 1, 4, 6 or X.
Since 10 is the smallest abundant number, all numbers end with 0 are abundant numbers, besides, all numbers end with 6 except 6 itself are also abundant numbers.
unit digit of nk
Template:Diagonal split header
1 X 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
1 E 1 E 1 E 1 E 1 E 1 E 1 E 1 E 1 E 1 E 1 E 1 E 1
The period of the unit digits of powers of a number must be a divisor of 2 (= λ(10), where λ is the Carmichael function).
possible unit digit of an nth power
1 any number
even number ≥ 2 0, 1, 4, 9 (the square digits)
odd number ≥ 3 0, 1, 3, 4, 5, 7, 8, 9, E (all digits != 2 mod 4)
final two digits of nk
01 02 04 08 14 28 54 X8 94 68 14 28 54 X8 94 68 14 28 54 X8 94 68 14 28 54
01 05 21 X5 41 85 61 65 81 45 X1 25 01 05 21 X5 41 85 61 65 81 45 X1 25 01
01 0X 84 E4 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54
01 0E X1 2E 81 4E 61 6E 41 8E 21 XE 01 0E X1 2E 81 4E 61 6E 41 8E 21 XE 01
01 11 21 31 41 51 61 71 81 91 X1 E1 01 11 21 31 41 51 61 71 81 91 X1 E1 01
01 12 44 08 94 X8 54 28 14 68 94 X8 54 28 14 68 94 X8 54 28 14 68 94 X8 54
01 1E 81 5E 41 9E 01 1E 81 5E 41 9E 01 1E 81 5E 41 9E 01 1E 81 5E 41 9E 01
The period of the final two digits of powers of a number must be a divisor of 10 (= λ(100)).
More generally, for every n≥2, the period of the final n digits of powers of a number must be a divisor of 10n−1 (= λ(10n)).
digital root of nk
1 2 4 8 5 X 9 7 3 6 1 2 4 8 5 X 9 7 3 6 1 2 4 8 5
1 X 1 X 1 X 1 X 1 X 1 X 1 X 1 X 1 X 1 X 1 X 1 X 1
1 E E E E E E E E E E E E E E E E E E E E E E E E
The period of the digital roots of powers of a number must be a divisor of X (= λ(E)).
possible digital root of an nth power
= 1, 3, 7, 9 (mod X) any number
= 2, 4, 6, 8 (mod X) 1, 3, 4, 5, 9, E
= 5 (mod X) 1, X, E
> 0 and divisible by X 1, E
The unit digit of a Fibonacci number can be any digit except 6 (if the unit digit of a Fibonacci number is 0, then the dozens digit of this number must also be 0, thus, all Fibonacci numbers divisible by 6 are also divisible by 100), and the unit digit of a Lucas number cannot be 0 or 9 (thus, no Lucas number is divisible by 10), besides, if a Lucas number ends with 2, then it must end with 0002, i.e., this number is congruent to 2 mod 104.
In the following table, Fn is the n-th Fibonacci number, and Ln is the n-th Lucas number.
digit root of Fn
digit root of Ln
1 1 1 1 1 21 37501 5 81101 E
2 1 1 3 3 22 5X301 8 111103 7
3 2 2 4 4 23 95802 2 192204 7
4 3 3 7 7 24 133E03 X 2X3307 3
5 5 5 E E 25 209705 1 47550E X
6 8 8 16 7 26 341608 E 758816 2
7 11 2 25 7 27 54E111 1 1012125 1
8 19 X 3E 3 28 890719 1 176X93E 3
9 2X 1 64 X 29 121E82X 2 2780X64 4
X 47 E X3 2 2X 1XE0347 3 432E7X3 7
E 75 1 147 1 2E 310EE75 5 6XE0647 E
10 100 1 22X 3 30 5000300 8 E22022X 7
11 175 2 375 4 31 8110275 2 16110875 7
12 275 3 5X3 7 32 11110575 X 25330XX3 3
13 42X 5 958 E 33 1922082X 1 3E441758 X
14 6X3 8 133E 7 34 2X3311X3 E 6477263E 2
15 E11 2 2097 7 35 47551X11 1 X3EE4197 1
16 15E4 X 3416 3 36 75882EE4 1 148766816 3
17 2505 1 54E1 X 37 101214X05 2 23075X9E1 4
18 3XE9 E 8907 2 38 176X979E9 3 379305607 7
19 6402 1 121E8 1 39 2780E0802 5 5X9X643E8 E
1X X2EE 1 1XE03 3 3X 432E885EE 8 967169X03 7
1E 14701 2 310EE 4 3E 6XE079201 2 13550121EE 7
20 22X00 3 50002 7 40 E22045800 X 2100180002 3
(Note that F2X begins with L1X, and F2E begins with L1E)
The period of the digit root of Fibonacci numbers is X.
The period of the unit digit of Fibonacci numbers is 20, the final two digits is also 20, the final three digits is 200, the final four digits is 2000, ..., the final n digits is 2×10n−1 (n ≥ 2). (see Pisano period)
There are only 13 possible values (of the totally 100 values, thus only 13%) of the final two digits of a Fibonacci number (see Template:Oeis).
Except 0 = F0 and 1 = F1 = F2, the only square Fibonacci number is 100 = F10 (100 is the square of 10), thus, 10 is the only base such that 100 is a Fibonacci number (since 100 in a base is just the square of this base, and 0 and 1 cannot be the base of numeral system), and thus we can make the near value of the golden ratio: F11/F10 = 175/100 = 1.75 (since the ratio of two connected Fibonacci numbers is close to the golden ratio, as the numbers get large). Besides, the only cube Fibonacci number is 8 = F6.
1 2 21 E2X20X8 41 5317E5804588X8 61 256906X1X93096E8934X8 81 11X12X743504482569888538X0X8 X1 65933E8691303X448E712227X7E11448X8
2 4 22 1X584194 42 X633XE408E5594 62 4E161183966171E566994 82 238259286X08944E17554X758194 X2 10E667E51626078895E22445393X2289594
3 8 23 38E48368 43 190679X815XXE68 63 9X302347710323XE11768 83 4744E6551815689X32XX992E4368 X3 21E113XX305013556EX4488X76784556E68
4 14 24 75X94714 44 361137942E99E14 64 1786046932206479X23314 84 9289E0XX342E15786599765X8714 X4 43X2279860X026XE1E88955931348XE1E14
5 28 25 12E969228 45 702273685E77X28 65 3350091664410937846628 85 16557X198685X2E350E7730E95228 X5 878453750180519X3E556XE6626959X3X28
6 54 26 25E716454 46 120452714EE33854 66 66X0163108821673491054 86 30XE3837514E85X6X1E3261E6X454 X6 15348X72X0340X3787XXE19E10516E787854
7 X8 27 4EE2308X8 47 2408X5229EX674X8 67 111803062154431269620X8 87 619X7472X29E4E9183X6503E188X8 X7 2X695925806818735399X37X20X31E3534X8
8 194 28 9EX461594 48 48158X457E912994 68 2234061042X886251704194 88 103792925857X9E634790X07X35594 X8 5916E64E41143526X777873841863X6X6994
9 368 29 17E8902E68 49 942E588E3E625768 69 446810208595504X3208368 89 20736564E4E397E06936181386XE68 X9 E631E09X82286X5193335274835079191768
X 714 2X 33E5605E14 4X 1685XE55X7E04E314 6X 891420414E6XX0986414714 8X 41270E09X9X773X116703427519E14 XX 1E063X179445518X36666X52946X136363314
E 1228 2E 67XE00EX28 4E 314E9XXE93X09X628 6E 1562840829E1981750829228 8E 82521X179793278231206852X37X28 XE 3X107833688XX358711118X56918270706628
10 2454 30 1139X01E854 50 629E799E678179054 70 2E05481457X37432X1456454 90 144X4383373665344624114X5873854 E0 78213467155986E52222358E1634521211054
11 48X8 31 2277803E4X8 51 1057E377E1343360X8 71 5X0X9428E3872865828E08X8 91 2898874672710X6890482298E5274X8 E1 1344269122XE751XX44446E5X3068X424220X8
12 9594 32 4533407X994 52 20E3X733X268670194 72 E8196855X752550E455X1594 92 557552912522191560944575XX52994 E2 26885162459E2X39888891XE86115884844194
13 16E68 33 8X668139768 53 41X792678515120368 73 1E43714XE92X4XX1X8XE82E68 93 XE2XX5624X44362E01688E2E98X5768 E3 5154X3048E7X58775555639E5022E549488368
14 31E14 34 159114277314 54 839365134X2X240714 74 3X872299E6589983959E45E14 94 19X598E049888705X03155X5E758E314 E4 X2X986095E38E532XXXE077XX045XX96954714
15 63X28 35 2E6228532628 55 147670X269858481228 75 79524577E0E577476E7X8EX28 95 378E75X09755520E8062XE8EE2E5X628 E5 185975016EX75XX65999X1339808E99716X9228
16 107854 36 5E0454X65054 56 29312185174E4942454 76 136X48E33X1XE32931E395E854 96 735E2E8172XXX41E41059E5EX5XE9054 E6 34E72X031E92E990E77782677415E7723196454
17 2134X8 37 EX08X990X0X8 57 5662434X329X96848X8 77 271895X67839X65663X76EE4X8 97 126EX5E432599883X820E7XEE8E9E60X8 E7 69E258063E65E761E3334513282EE32463708X8
18 426994 38 1E81597618194 58 E104869865797149594 78 52356E91347790E107931EX994 98 251E8EX864E775479441E39EE5E7E0194 E8 117X4E4107E0EE303X6668X26545EX6490721594
19 851768 39 3E42E73034368 59 1X20951750E372296E68 79 X46E1E62693361X213663E9768 99 4X3E5E9509E32X936883X77EXEE3X0368 E9 23389X8213X1EX60791115850X8EE90961242E68
1X 14X3314 3X 7X85E26068714 5X 38416X32X1X724571E14 7X 1891X3E051667038427107E7314 9X 987XEE6X17X659671547933E9EX780714 EX 467579442783E90136222E4X195EE61702485E14
1E 2986628 3E 1394EX50115228 5E 74831865839248E23X28 7E 356387X0X3112074852213E2628 9E 17539EE183390E7122X93667E7E9341228 EE 912E36885347E60270445X9836EEE0320494EX28
20 5751054 40 2769E8X022X454 60 12946350E476495X47854 80 6E075381862241294X4427X5054 X0 32X77EX346761E2245967113E3E6682454 100 1625X7154X693E0052088E97471EEX0640969E854
For all digits 1 ≤ d ≤ X (i.e. all digits other than the largest digit (E)), there exists 0 ≤ n ≤ 20 such that 2n starts with the digit d. (This is not true for the digit E, the smallest power of 2 starts with the digit E is indeed 221 = E2X20X8)
21XE = 59E18922E81631X39875663E89X853X91E595336X6114815X5X6929933X288E774E479575X628 may be the largest power of 2 not contain the digit 0, it has 65 digits.
The number 229 = 2368 (see power of 2#Powers of two whose exponents are powers of two) is very close to googol (10100), since it has EE digits. (thus, the Fermat number F9 (=229+1) is very close to googol)
1001 is the first four-digit palindromic number, and it is also the smallest number expressible as the sum of two cubes in two different ways, i.e. 1001 = 1 + 1000 (=13 + 103) = 509 + 6E4 (=93 + X3) (see taxicab number for other numbers), and it is also the smallest absolute Euler pseudoprime, note that there is no absolute Euler-Jacobi pseudoprime and no absolute strong pseudoprime. Since 1001 = 7×11×17, we can use the divisibility rule of 1001 (i.e. form the alternating sum of blocks of three from right to left) for the divisibility rule of 7, 11 and 17. Besides, if 6k+1, 10k+1 and 16k+1 are all primes, then the product of them must be a Carmichael number (absolute Fermat pseudoprime), the smallest case is indeed 1001 (for k = 1), but 1001 is not the smallest Carmichael number (the smallest Carmichael number is 3X9).
All values of n > 45 for incrementally largest values of minimal x > 1 (or minimal y > 0) satisfying Pell's equation $ x^2-ny^2=1 $ end with 1, and the dozens digit of all such values n > 2X1 are odd. (these values n are 2, 5, X, 11, 25, 3X, 45, 51, 91, 131, 1E1, 291, 2X1, 2E1, 391, 471, 711, 751, 971, X91, E31, ...)
The denominator of every nonzero Bernoulli number (except $ B_0=1 $ and $ B_1=-\frac{1}{2} $) ends with 6.
If n ends with 2 and n/2 is prime (or 1), then the denominator of the Bernoulli number $ B_n $ is 6 (this is also true for some (but not all) n ends with ᘔ and n/2 is prime). (if the denominator of the Bernoulli number $ B_n $ is 6, then n ends with 2 or ᘔ, but n/2 needs not to be prime or 1, the first counterexample is n = 82, the denominator of the Bernoulli number $ B_{82} $ is 6, but 82/2 = 41 = 72 is neither prime nor 1)
$ \sqrt{2} $ is very close to 1.5, since a near-value for $ \sqrt{2} $ is 15/10 (=N4/P4, where Nn is nth NSW number, and Pn is nth Pell number, Nn/Pn is very close to $ \sqrt{2} $ when n is large). Besides, $ \sqrt{5} $ is very close to 2.2X, since a near-value for $ \sqrt{5} $ is 22X/100 (= L10/F10, where Ln is nth Lucas number, and Fn is nth Fibonacci number, Ln/Fn is very close to $ \sqrt{5} $ when n is large).
The recurring dozenal of the reciprocal of n terminates if and only if n is 3-smooth number (or harmonic number[1]) (i.e. n is regular to 10 if and only if n is 3-smooth number (or harmonic number)), since the 3-smooth numbers (or the harmonic numbers) are the numbers that evenly divide powers of 10. The 3-smooth numbers up to 1000 are 1, 2, 3, 4, 6, 8, 9, 10, 14, 16, 20, 23, 28, 30, 40, 46, 54, 60, 69, 80, 90, X8, 100, 116, 140, 160, 183, 194, 200, 230, 280, 300, 346, 368, 400, 460, 509, 540, 600, 690, 714, 800, 900, X16, X80, 1000. They are exactly the numbers k such that $ \phi(6k)=2k $, where $ \phi $ is the Euler's totient function. The sum of the reciprocals of the 3-smooth numbers is equal to 3, i.e. 1/1 + 1/2 + 1/3 + 1/4 + 1/6 + 1/8 + 1/9 + 1/10 + ... = 1 + 0.6 + 0.4 + 0.3 + 0.2 + 0.16 + 0.14 + 0.1 + ... = 3. Brief proof: 1/1 + 1/2 + 1/3 + 1/4 + 1/6 + 1/8 + 1/9 + 1/10 + ... = (Sum_{m>=0} 1/(2^m)) * (Sum_{n>=0} 1/(3^n)) = (1/(1-1/2)) * (1/(1-1/3)) = (2/(2-1)) * (3/(3-1)) = 3.
The 3-smooth numbers (or the numbers n such that the reciprocal of n terminates) ≤ 10 are 1, 2, 3, 4, 6, 8, 9, and 10, all of these numbers except 8 and 9 are divisors of 10 (8 is because it has more prime factors 2 than 10, and 9 is because it has more prime factors 3 than 10) (thus, the numbers of digits of the reciprocal of all these n except 8 and 9 are all 1, while the numbers of digits of the reciprocal of 8 and 9 are 2), and the numbers 8 and 9 are in the Catalan's conjecture (i.e. 8 and 9 are the only case of two consecutive perfect powers), besides, the product of 8 and 9 is 60, which is the smallest Achilles number, besides, the concatenation of 8 and 9 is 89, which is the smallest Ziesel number and the smallest integer such that the factorization of $ x^n-1 $ over Q includes coefficients other than $ \pm 1 $ (i.e. the 89th cyclotomic polynomial, $ \Phi_{89} $, is the first with coefficients other than $ \pm 1 $), besides, the repunit with length k (Rk) (where k = the concatenation of n and the unit (1), i.e. k = 10n+1) is prime for both n = 8 and n = 9, and not for any other n ≤ 1000, besides, the squares of 8 and 9 are the only two 2-digit automorphic numbers, besides, 8 and 9 are the only two natural numbers n such that centered n-gonal numbers (the kth centered n-gonal number is n×Tk+1, where Tk is the kth triangular number) cannot be primes (8 is because all centered 8-gonal numbers are square numbers (4-gonal numbers), 9 is because all centered 9-gonal numbers are triangular numbers (3-gonal numbers) not equal to 3, but all square numbers and all triangular numbers not equal to 3 are not primes, in fact, all polygonal numbers with rank > 2 are not primes, i.e. all primes p cannot be a polygonal number (except the trivial case, i.e. each p is the second p-gonal number)), assuming the Bunyakovsky conjecture is true. (i.e. 8 and 9 are the only two natural number n such that $ \frac{n}{2}x^2+\frac{n}{2}x+1 $ is not irreducible) (Note that for n = 10, the centered 10-gonal numbers are exactly the star numbers)
The smallest n≥1 such that 4×60n−1 is prime (where 4 is the smallest composite number, and 60 is the smallest Achilles number, note that 4×60n is 3-smooth for all n≥1, thus 1/(4×60n) terminates in dozenal, and note that 4×60n is exactly 10000 (104) when n=2) is 460089, which contains the only four composite 3-smooth digits (4, 6, 8, 9) exactly once and from small to large in order (4 —> 6 —> 8 —> 9), and with two 0's (zero digits) inside the middle, this prime proves the generalized Riesel problem base 60 (proves that 205 is the smallest generalized Riesel number in base 60, i.e. 205 is the smallest k such that gcd(k−1, 60−1) = 1 and k×60n−1 is composite for all n≥1, note that 205 = 4×60+4+1 = 4×(60+1)+1).
The 3-smooth numbers (or the numbers n such that the reciprocal of n terminates) ≤ 20 are 1, 2, 3, 4, 6, 8, 9, 10, 14, 16, and 20, all of these numbers are divisors of 100, note that the next two 3-smooth numbers (23 and 28) are not divisors of 100 (23 is because it has more prime factors 3 than 100, and 28 is because it has more prime factors 2 than 100) (thus, the numbers of digits of the reciprocal of all these n are all ≤2, while the numbers of digits of the reciprocal of 23 and 28 are 3).
Regular n-gon is constructible using neusis, or an angle trisector if and only if the reciprocal of $ \varphi(n) $ is terminating number (where $ \varphi $ is Euler's totient function) (i.e. $ \varphi(n) $ is 3-smooth, or $ \varphi(n) $ is regular to 10), thus the n ≤ 1000 such that regular n-gon is constructible using neusis, or an angle trisector are 3, 4, 5, 6, 7, 8, 9, X, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 26, 28, 2X, 2E, 30, 31, 32, 33, 34, 36, 39, 40, 43, 44, 46, 48, 49, 50, 53, 54, 55, 58, 5X, 60, 61, 62, 64, 66, 68, 69, 70, 71, 76, 77, 7E, 80, 81, 86, 88, 89, 90, 91, 93, 94, 96, 99, 9E, X0, X6, X8, XX, E1, E3, E4, E8, 100, 102, 104, 108, 109, 110, 114, 116, 117, 120, 122, 123, 130, 132, 135, 139, 13X, 140, 141, 142, 143, 150, 154, 156, 160, 162, 163, 165, 166, 168, 170, 176, 17X, 180, 183, 187, 190, 193, 194, 195, 197, 198, 1X2, 1X6, 1X8, 1X9, 1E4, 1E9, 200, 203, 204, 208, 214, 216, 220, 223, 228, 22E, 230, 232, 233, 239, 240, 244, 246, 253, 259, 260, 264, 265, 26X, 276, 278, 280, 282, 284, 286, 293, 299, 2X0, 2X8, 2E0, 300, 301, 304, 306, 30X, 310, 314, 31E, 320, 323, 330, 338, 340, 341, 345, 346, 347, 349, 352, 360, 366, 367, 368, 369, 36X, 372, 374, 384, 390, 394, 395, 396, 3X3, 3X8, 3E3, 3E6, 400, 401, 403, 406, 408, 409, 414, 417, 428, 430, 440, 445, 446, 454, 45X, 460, 464, 466, 469, 473, 475, 476, 480, 487, 488, 490, 4X6, 4X7, 4E6, 500, 508, 509, 50X, 518, 519, 530, 534, 537, 539, 540, 541, 543, 544, 548, 549, 550, 566, 576, 57E, 580, 583, 594, 5X0, 5E3, 600, 602, 608, 609, 610, 618, 620, 628, 63X, 640, 646, 660, 669, 671, 674, 680, 682, 685, 689, 68X, 690, 692, 696, 699, 6X4, 6E3, 700, 710, 712, 714, 716, 718, 724, 728, 739, 748, 753, 760, 768, 76X, 770, 773, 781, 786, 794, 7X6, 7E0, 7E1, 800, 801, 802, 806, 810, 814, 816, 828, 832, 839, 853, 854, 860, 86E, 875, 880, 88X, 890, 891, 8X8, 8E1, 8E8, 8EE, 900, 901, 903, 908, 910, 916, 926, 92X, 930, 940, 947, 952, 954, 959, 960, 969, 977, 990, 992, 9X1, 9E0, X00, X03, X13, X14, X16, X17, X18, X19, X23, X34, X36, X60, X68, X72, X76, X79, X80, X82, X83, X86, X88, X8E, X94, X96, XX0, E10, E27, E30, E3X, E40, E43, E46, E55, E68, E69, E80, E99, EX6, 1000.
If and only if n is a divisor of 20, then m2 = 1 mod n for every integer m coprime to n.
If and only if n is a divisor of 20, then the Dirichlet characters mod n are all real.
If and only if n is a divisor of 20, then n is divisible by all numbers less than or equal to the square root of n.
If and only if n is a divisor of 20, then k−1 is prime for all divisors k>2 of n.
If and only if n+1 is a divisor of 20, then $ \tbinom{n}{k}=\tfrac{n!}{k!(n-k)!} $ is squarefree for all 0 ≤ k ≤ n, i.e. all numbers in the nth row of the Pascal's triangle are squarefree (the topmost row (i.e. the row which contains only one 1) of the Pascal's triangle is the 0th row, not the 1st row). (Note that all such n are primes or 1 or 0, and 20 is the largest number m such that if n+1 is a divisor of m, then n is prime or 1 or 0, besides, if and only if m is a divisor of 20, then m satisfies this condition)
If we only have the numbers 1 to 20 (including 1 and 20), then only the primes dividing 10 (i.e. primes ≤3) can be squared, since 5^2 = 21 > 20, and for the numbers such that the reciprocal of n terminates, they can only have at most 2 digits (which is the case of 8, 9, 14, 16 and 20), since the numbers with terminate reciprocal with >2 digits, they must be divisible by either 2^5 = 28 or 3^3 = 23, but both are >20. (these (prime power) numbers are >20: (prime > 3)^(>1), (odd prime)^(>2), 2^(>4))
If xy≤20, then at least one of x and y is a divisor of 10 (this is not true for xy=21, 5×5=21, but 5 is not a divisor of 10).
googol (mod n) = googolplex (mod n) for all 1 ≤ n ≤ 20 (but not for n = 21).
For all numbers n ≤ 100 (but not for n = 101, and not for n = smallest prime > 100 (i.e. 105)), there is k ≤ 6 such that nk−1 or nk+1 (or both) is prime. (note that for n = 101, k = 7, 8 and 9 also not satisfy this condition, the smallest k satisfying this condition for n = 101 is X)
100 is the smallest number whose nth power can be written as the sum of (≥2 and ≤n) positive nth powers for sum n (n=5, the formula is 100^5 = 23^5 + 70^5 + 92^5 + E1^5, only four 5th powers).
100 is the maximum number of steps of numbers < 4X7 (the smallest number that reach a number > 5414 (the record which gotten by the start value 23), namely 100E74) in Collatz sequence (for the numbers 461, 466, 467 and 477).
The exponents on the right hand side of $ (1-x)(1-x^2)(1-x^3) \cdots = 1 - x - x^2 + x^5 + x^7 - x^{10} - x^{13} + x^{1\mathcal{X}} + x^{22} - \cdots. $ are exactly the numbers n such that 20n+1 is square. (note that 10 is one of such numbers)
All negative-Pell solvable numbers (i.e. numbers n such that x2−ny2 = −1 is solvable) end with negative-Pell solvable digits (i.e. end with 1, 2, 5 or X).
By Benford's law, the probability for the leading digit d (d ∈ {1, 2, 3, 4, 5, 6, 7, 8, 9, X, E}) occurs (for some sequences, e.g. powers of 2 (1, 2, 4, 8, 14, 28, 54, X8, 194, 368, 714, 1228, 2454, ...) and Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 11, 19, 2X, 47, 75, 100, ...)) are:
1 34.2% 7 7.9%
2 1E.6% 8 6.X%
4 10.E% X 5.6%
5 X.7% E 5.1%
6 8.E%
(Note: the percentage in the list are also in dozenal, i.e. 20% means 0.2 or $ \frac{20}{100}=\frac{1}{6} $, 36% means 0.36 or $ \frac{36}{100}=\frac{7}{20} $, 58.7% means 0.587 or $ \frac{587}{1000} $)
Star numbers are exactly the numbers obtained as the concatenation of a triangular number followed by 1 (the triangular numbers are 0, 1, 3, 6, X, 13, 19, 24, 30, 39, 47, 56, 66, ..., and the star numbers are 1, 11, 31, 61, X1, 131, 191, 241, 301, 391, 471, 561, 661, ...), thus, all star numbers end with 1. (The star numbers are exactly the centered 10-gonal numbers)
The Hilbert numbers are the numbers end with 1, 5 or 9. (i.e. = 1 mod 4)
The Lagado numbers are the numbers end with 1, 4, 7 or X. (i.e. = 1 mod 3)
The smallest two 4-digit palindromic numbers (1001 and 1111) are both Ziesel numbers, they are also the smallest two palindromic numbers which cannot be prime when read in any base, and the smallest 4-digit palindromic number (1001) is exactly the smallest absolute Euler pseudoprime and the smallest number expressible as the sum of two cubes in two different ways, i.e. 1001 = 1 + 1000 (=13 + 103) = 509 + 6E4 (=93 + X3).
The largest 4-digit number (EEEE) is a member of a betrothed number pair (its betrothed number is 5600 (also a 4-digit number, note that 5600 is E-smooth), and if we calculate EEEE/gcd(EEEE, 5600), we get the 4-digit repunit (1111)).
"the smallest (Fermat) pseudoprime to both base 2 and base 3 that is not Carmichael number" (1691) and "conjectured largest panconsummate number" (1961) are both strobogrammatic numbers (the same upside down). (these two numbers are both composite, however, the strobogrammatic numbers 160091 and 190061 are primes) (of course, the Zeisel numbers 1001 and 1111 are also strobogrammatic numbers)
All prime numbers end with prime digits or 1 (i.e. end with 1, 2, 3, 5, 7 or E), more generally, except for 2 and 3, all prime numbers end with 1, 5, 7 or E (1 and all prime digits that do not divide 10), since all prime numbers other than 2 and 3 are coprime to 10.
The density of primes end with 1 is relatively low, but the density of primes end with 5, 7 and E are nearly equal. (since all prime squares except 4 and 9 end with 1, no prime squares end with 5, 7 or E)
Except (3, 5), all twin primes end with (5, 7) or (E, 1), and the density of these two types of twin primes are nearly equal.
The sum of any pair of twin primes (other than (3, 5)) ends with 0.
If n ≥ 3 and n is not divisible by E, then there are infinitely many primes with digit sum n.
All palindromic primes except 11 has an odd number of digits, since all even-digit palindromic numbers are divisible by 11. The palindromic primes below 1000 are 2, 3, 5, 7, E, 11, 111, 131, 141, 171, 181, 1E1, 535, 545, 565, 575, 585, 5E5, 727, 737, 747, 767, 797, E1E, E2E, E6E.
All lucky numbers end with digit 1, 3, 7 or 9.
Except for 3, all Fermat primes end with 5. (In fact, there are only 5 known Fermat primes (3, 5, 15, 195 and 31E15) and it is conjectured that there are no more Fermat primes, interestingly, all digits of all known Fermat primes are odd)
Except for 3, all Mersenne primes end with 7. (Besides, all Mersenne primes except 3 and 7 end with one of the only two 2-digit Mersenne primes (27 and X7))
Except for 2 and 3, all Sophie Germain primes end with 5 or E.
Except for 5 and 7, all safe primes end with E.
A prime p is Gaussian prime (prime in the ring $ Z[i] $, where $ i=\sqrt{-1} $) if and only if p ends with 7 or E (or p=3). (i.e. p = 3 mod 4)
A prime p is Eisenstein prime (prime in the ring $ Z[\omega] $, where $ \omega=\frac{-1+\sqrt{3}i}{2} $) if and only if p ends with 5 or E (or p=2). (i.e. p = 2 mod 3)
A prime p can be written as x2 + y2 if and only if p ends with 1 or 5 (or p=2). (i.e. p = 1 or 2 mod 4)
A prime p can be written as x2 + 3y2 if and only if p ends with 1 or 7 (or p=3). (i.e. p = 0 or 1 mod 3)
All numbers ≤ 20 coprime to 10 are either primes or 1 (unit). (this is not true for 21, 21 is the smallest composite coprime to 10)
All full reptend primes end with 5 or 7. (in fact, for all primes p ≥ 5, (p-1)/(the period length of 1/p) is odd if and only if p is end with 5 or 7, since 10 is a quadratic nonresidue mod p (i.e. $ \left(\frac{10}{p}\right)=-1 $, where $ \left(\frac{m}{n}\right) $ is the Legendre symbol) if and only if p is end with 5 or 7, by quadratic reciprocity, and if 10 is a quadratic residue mod a prime, then 10 cannot be a primitive root mod this prime) However, the converse is not true, 17 is not a full reptend prime, since the recurring digits of 1/17 is 0.076E45076E45..., which has only period 6. If and only if p is a full reptend prime, then the recurring digits of 1/p is cyclic number, e.g. the recurring digits of 1/5 is the cyclic number 2497 (the cyclic permutations of the digits are this number multiplied by 1 to 4), and the recurring digits of 1/7 is the cyclic number 186X35 (the cyclic permutations of the digits are this number multiplied by 1 to 6). The full reptend primes below 1000 are 5, 7, 15, 27, 35, 37, 45, 57, 85, 87, 95, X7, E5, E7, 105, 107, 117, 125, 145, 167, 195, 1X5, 1E5, 1E7, 205, 225, 255, 267, 277, 285, 295, 315, 325, 365, 377, 397, 3X5, 3E5, 3E7, 415, 427, 435, 437, 447, 455, 465, 497, 4X5, 517, 527, 535, 545, 557, 565, 575, 585, 5E5, 615, 655, 675, 687, 695, 6X7, 705, 735, 737, 745, 767, 775, 785, 797, 817, 825, 835, 855, 865, 8E5, 8E7, 907, 927, 955, 965, 995, 9X7, 9E5, X07, X17, X35, X37, X45, X77, X87, X95, XE7, E25, E37, E45, E95, E97, EX5, EE5, EE7. (Note that for the primes end with 5 or 7 below 30 (5, 7, 15, 17, 25 and 27, all numbers end with 5 or 7 below 30 are primes), 5, 7, 15 and 27 are full reptend primes, and since 5×25 = 101 = $ \Phi_4(10) $, the period of 25 is 4, which is the same as the period of 5, and we can use the test of the divisiblity of 5 to test that of 25 (form the alternating sum of blocks of two from right to left), and since 7×17 = E1 = $ \Phi_6(10) $, the period of 17 is 6, which is the same as the period of 7, and we can use the test of the divisiblity of 7 to test that of 17 (form the alternating sum of blocks of three from right to left), thus, 17 and 25 are not full reptend primes, and they are the only two non-full reptend primes end with 5 or 7 below 30)
By Midy theorem, if p is a prime with even period length (let its period length be n), then if we let $ \frac{a}{p}=0.\overline{a_1a_2a_3...a_n} $, then ai + ai+n/2 = E for every 1 ≤ i ≤ n/2. e.g. 1/5 = 0.249724972497..., and 24 + 97 = EE, and 1/7 = 0.186X35186X35..., and 186 + X35 = EEE, all primes (other than 2 and 3) ≤ 37 except E, 1E and 31 have even period length, thus they can use Midy theorem to get an E-repdigit number, the length of this number is the period length of this prime. (see below for the recurring digits for 1/n for all n ≤ 30)
The unique primes below 1060 are E, 11, 111, E0E1, EE01, 11111, 24727225, E0E0E0E0E1, E00E00EE0EE1, 100EEEXEXEE000101, 1111111111111111111, EEEE0000EEEE0000EEEE0001, 100EEEXEE0000EEEXEE000101, 10EEEXXXE011110EXXXE00011, EEEEEEEE00000000EEEEEEEE00000001, EEE000000EEE000000EEEEEE000EEEEEE001, and the period length of their reciprocals are 1, 2, 3, X, 10, 5, 18, 1X, 19, 50, 17, 48, 70, 5X, 68, 53.
If p is a safe prime other than 5, 7 and E, then the period length of 1/p is (p-1)/2. (this is not true for all primes ends with E (other than E itself), the first counterexample is p = 2EE, where the period length of 1/p is only 37)
There is no full reptend prime ends with 1, since 10 is quadratic residue for all primes end with 1. (if so, then this prime p is a proper prime (i.e. for the reciprocal of such primes (1/p), each digit 0, 1, 2, ..., E appears in the repeating sequence the same number of times as does each other digit (namely, (p−1)/10 times)), see repeating decimal#Fractions with prime denominators) (In fact, not only for base 10 such primes do not exist, for all bases = 0 mod 4 (i.e. bases end with digit 0, 4 or 8), such primes do not exist)
5 and 7 are the only two safe primes which are also full reptend primes, since except 5 and 7, all safe primes end with E, and 10 is quadratic residue for all primes end with E. (if so, then this prime p produces a stream of p−1 pseudo-random digits, see repeating decimal#Fractions with prime denominators) (In fact, not only for base 10 there are only finitely many such primes, of course for square bases (bases of the form k2) only 2 may be full reptend prime (if the base is odd), and all odd primes are not full reptend primes, but since all safe primes are odd primes, for these bases such primes do not exist, besides, for the bases of the form 3k2, only 5 and 7 can be such primes, the proof for these bases is completely the same as that for base 10)
period length of 1/p
2 0 111 3 267 266 41E 20E 591 3X 767 766 927 926 E1E 56E 1107 1106 12E5 12E4 14E1 14E 16X7 16X6
3 0 117 116 271 27 421 63 59E 2XE 76E 395 955 954 E21 570 1115 1114 1301 760 14E5 14E4 16E5 188
5 4 11E 6E 277 276 427 426 5E1 159 771 132 95E 48E E25 E24 1125 1124 1317 506 14EE 85E 16E7 16E6
7 6 125 124 27E 13E 431 109 5E5 5E4 775 774 965 964 E2E 575 112E 675 1337 512 150E 865 1705 398
E 1 12E 75 285 284 435 434 5E7 66 77E 39E 971 172 E31 116 1135 1134 133E 77E 1517 1XX 1711 493
11 2 131 76 291 83 437 436 5EE 2EE 785 784 987 32X E37 E36 114E 685 1345 1344 1521 436 1715 1714
15 14 13E 7E 295 294 447 446 611 163 791 3X6 995 994 E45 E44 1151 115 1351 166 1525 1524 1727 1726
17 6 141 20 2X1 150 455 454 615 614 797 796 9X7 9X6 E61 16 1165 1164 1365 1364 1547 1E2 1735 1734
1E E 145 144 2XE 155 457 15X 617 206 7X1 138 9XE 4E5 E67 3X2 1167 42 1367 1366 1561 89 1745 1744
25 4 147 56 2E1 26 45E 22E 61E 30E 7EE 3EE 9E1 9E E6E 595 1185 1184 136E 795 156E 97 1747 1746
27 26 157 12 2EE 37 465 464 637 212 801 140 9E5 9E4 E71 596 118E 6X5 1377 106 1577 1576 1751 32X
31 9 167 166 301 90 46E 7 63E 31E 80E 405 9EE 4EE E91 2E3 1197 472 138E 7X5 157E 89E 1755 1754
35 34 16E 95 307 102 471 13 647 216 817 816 X07 X06 E95 E94 11X1 6E 1391 3E3 1585 1584 1757 1756
37 36 171 96 30E 165 481 24 655 654 825 824 X0E 505 E97 E96 11X5 11X4 1395 1394 1587 2X 176E 995
3E 1E 175 8 315 314 485 44 661 176 82E 415 X11 56 EX5 EX4 11X7 11X6 13X1 7E0 1591 2E6 1781 4E0
45 44 17E 9E 321 170 48E 245 665 138 835 834 X17 X16 EE5 EE4 11XE 6E5 13X7 536 15XE 8E5 1785 1784
4E 25 181 X0 325 324 497 496 66E 335 841 84 X27 156 EE7 EE6 11E7 11E6 13E1 13E 15EE 8EE 178E 9X5
51 13 18E X5 327 10X 4X5 4X4 675 674 851 14X X35 X34 1005 1004 1201 700 13E5 13E4 1601 160 1797 1796
57 56 195 194 32E 175 4E1 9X 687 686 855 854 X37 X36 1011 73 120E 705 1405 1404 1615 1614 17X1 9E0
5E 2E 19E XE 33E 17E 4EE 25E 68E 345 85E 42E X3E 51E 1017 1016 1211 706 1407 326 1621 910 17X5 17X4
61 30 1X5 1X4 347 46 507 182 695 694 865 864 X41 188 1021 610 121E 70E 1425 1424 1625 1624 17EE 9EE
67 22 1X7 46 34E 2E 511 266 69E 34E 867 2X2 X45 X44 1027 1026 1231 123 142E 815 1635 274 1807 682
6E 35 1E1 E6 357 11X 517 516 6X7 6X6 871 152 X4E 525 1041 620 123E 71E 1431 286 1647 276 1815 1814
75 8 1E5 1E4 35E 18E 51E 45 6E1 6E 881 440 X5E 52E 1047 1046 1245 1244 1437 1436 1655 1654 181E X0E
81 14 1E7 1E6 365 364 527 526 701 360 88E 445 X6E 535 104E 625 1255 114 143E 81E 1657 61X 1825 1824
85 84 205 204 375 34 531 276 705 704 8X5 98 X77 X76 1051 313 1257 49X 1445 1444 165E 92E 1831 509
87 86 217 86 377 376 535 534 70E 365 8X7 2E6 X87 X86 1061 16 125E 72E 1457 1456 1667 622 183E X1E
8E 45 21E 10E 391 1X6 541 54 711 71 8XE 455 X91 283 106E 635 1261 730 1461 38 1671 936 184E X25
91 46 221 66 397 396 545 544 71E 36E 8E5 8E4 X95 X94 107E 63E 126E 735 1465 1464 1677 20X 1861 269
95 94 225 224 3X5 3X4 557 556 721 370 8E7 8E6 X9E 54E 1087 1086 127E 73E 1467 562 167E 93E 1865 1864
X7 X6 237 92 3XE 1E5 565 564 727 24X 901 230 XX7 376 109E 64E 1281 740 1471 419 1681 140 186E X35
XE 55 241 120 3E5 3E4 575 574 735 734 905 198 XXE 555 10E1 329 1295 94 1475 1474 1685 1684 1875 1874
E5 E4 24E 125 3E7 3E6 577 116 737 736 907 906 XE7 XE6 10E7 10E6 1297 1296 147E 83E 168E 945 1877 146
E7 E6 251 73 401 100 585 584 745 744 90E 465 XEE 55E 10EE 65E 12X1 75 148E 845 1697 1696 189E X4E
105 104 255 254 40E 205 587 1XX 747 9X 91E 46E E11 1X2 1101 220 12X5 2E8 1495 1494 169E 94E 18X1 210
107 106 25E 12E 415 414 58E 2X5 751 1X3 921 236 E15 228 1105 1104 12X7 12X6 149E 84E 16X1 950 18XE X55
period length
1 E 11 1E0411, 69X3901
2 11 12 157, 7687
3 111 13 51, 471, 57E1
4 5, 25 14 15, 81, 106X95
5 11111 15 X9X9XE, 126180EE0EE
6 7, 17 16 E61, 1061
7 46E, 2X3E 17 1111111111111111111
8 75, 175 18 24727225
9 31, 3X891 19 E00E00EE0EE1
X E0E1 1X E0E0E0E0E1
E 1E, 754E2E41 1E 3E, 78935EX441, 523074X3XXE
10 EE01 20 141, 8E5281
The period level of a prime p ≥ 5 is (p−1)/(period length of 1/p), e.g., $ \frac{1}{17} $ has period level 3, thus the numbers $ \frac{a}{17} $ with integer 1 ≤ a ≤ 16 from 3 different cycles: 076E45 (for a = 1, 7, 8, E, 10, 16), 131X8X (for a = 2, 3, 5, 12, 14, 15) and 263958 (for a = 4, 6, 9, X, 11, 13). Besides, $ \frac{1}{15} $ has period level 1, thus this number is a cyclic number and 15 is a full-reptend prime, and all of the numbers $ \frac{a}{15} $ with integer 1 ≤ a ≤ 14 from the cycle 08579214E36429X7.
There are only 9 repunit primes below R1000: R2, R3, R5, R17, R81, R91, R225, R255 and R4X5 (Rn is the repunit with length n). If p is a Sophie Germain prime other than 2, 3 and 5, then Rp is divisible by 2p+1, thus Rp is not prime. (The length for the repunit (probable) primes are 2, 3, 5, 17, 81, 91, 225, 255, 4X5, 5777, 879E, 198E1, 23175, 311407, ..., note that 879E is the smallest (and the only known) such number ends with E)
By Fermat's little theorem, if p is a prime other than 2, 3 and E, then p divides the repunit with length p−1. (The converse is not true, the first counterexample is 55, which is composite (equals 5×11) but divides the repunit with length 54, the counterexamples up to 1000 are 55, 77, E1, 101, 187, 275, 4X7, 777, 781, E55, they are exactly the Fermat pseudoprimes for base 10 (composite numbers c such that 10c-1 = 1 mod c) which are not divisible by E, they are called "deceptive primes", if n is deceptive prime, then Rn is also deceptive prime, thus there are infinitely may deceptive primes) Thus, we can prove that every positive integer coprime to 10 has a repunit multiple, and every positive integer has a multiple uses only 0's and 1's.
smallest multiple of n uses only 0's and 1's
1 10 10 10 101 10 1001 100 100 1010 11111111111 10
11 10010 1010 100 10111 100 1001 1010 10010 111111111110 11101 100
110111 110 1000 10010 101 1010 101011 1000 111111111110 101110 101101 100
1001001 10010 110 10100 111001 10010 101001 111111111110 10100 111010 10001111 100
10111101 1101110 101110 110 1100101 1000 1011101111111 100100 10010 1010 101011 1010
10010101 1010110 100100 1000 1111 111111111110 1100101 101110 111010 1011010 1100111 100
10101101 10010010 1101110 10010 1011101111111 110 10101011 10100 10000 1110010 100111001 10010
1101001 1010010 1010 1111111111100 10001 10100 1001 111010 1010110 100011110 101101 1000
111011 101111010 1111111111100 1101110 110001 101110 10100111 1100 1011010 11001010 100111 1000
1010111111 10111011111110 10010010 100100 10101001 10010 110101001 1010 1100 1010110 101100011 10100
X0+
111111111101 100101010 1110010 1010110 11100001 100100 1100001 10000 1010010 11110 1111011111 111111111110
E0+
1001 11001010 101000 1011100 101011 111010 11010111 1011010 100011110 11001110 1111111111111111111111 100
n 1 5 7 E 11 15 17 1E 21 25 27 2E
smallest k such that k×n is a repunit 1 275 1X537 123456789E 1 92X79E43715865 8327 69E63848E 634X159788253X72E1 55 509867481E793XX5X1243628E317 45X3976X7E
the length of the repunit k×n 1 4 6 E 2 14 6 E 18 4 26 10
(this k is usually not prime, in fact, this k is not prime for all numbers n < 100 which are coprime to 10 except n = 55, and for n < 1000 which is coprime to 10, this k is prime only for n = 55, 101, 19E, 275 and 46E, and only 19E and 46E are itself prime, other 3 numbers are 5×11, 5×25 and 11×25, and this k for these n are successively 25, 11 and 5, which makes k×n = R4 = 1111 = 5×11×25, besides, this k for n = 46E is 2X3E, which makes k×n = R7 = 1111111, a repunit semiprime, and this k for n = 19E is a X8-digit prime number, with k×n = RXE, another repunit semiprime)
For every prime p except E, the repunit with length p is congruent to 1 mod p. (The converse is also not true, the counterexamples up to 1000 are 4, 6, 10, 33, 55, 77, E1, 101, 187, 1E0, 275, 444, 4X7, 777, 781, E55, they are called "repunit pseudoprimes" (or weak deceptive primes), all deceptive primes are also repunit pseudoprimes, if n is repunit pseudoprime, then Rn is also repunit pseudoprime, thus there are infinitely may repunit pseudoprimes. No repunit pseudoprimes are divisible by 8, 9 or E. (in fact, the repunit pseudoprimes are exactly the weak pseudoprimes for base 10 (composite numbers c such that 10c = 10 mod c) which are not divisible by E) Besides, the deceptive primes are exactly the repunit pseudoprimes which are coprime to 10)
Smallest multiple of n with digit sum 2 are: (0 if not exist)
2, 2, 20, 20, 101, 20, 1001, 20, 200, 1010, 0, 20, 11, 10010, 1010, 200, 100000001, 200, 1001, 1010, 10010, 0, 0, 20, 10000000001, 110, 2000, 10010, 101, 1010, 1000000000000001, 200, 0, 1000000010, 1000000000001, 200, ..., if and only if n is divisible by some prime p with 1/p odd period length, then such number does not exist.
3, 12, 3, 30, 21, 30, 12, 120, 30, 210, 0, 30, 0, 12, 210, 300, 201, 30, 10101, 210, 120, 0, 1010001, 120, 21, 0, 300, 120, 0, 210, 1010001, 1200, 0, 2010, 200001, 30, ..., such number does not exist for n divisible by E, 11 or 25.
4, 4, 13, 4, 13, 40, 103, 40, 130, 130, 0, 40, 22, 1030, 13, 40, 3001, 130, 2002, 130, 103, 0, 11101, 40, 10012, 22, 1300, 1030, 202, 130, 10003, 400, 0, 30010, 101101, 130, ..., such number is conjectured to exist for all n not divisible by E (of course, if n is divisible by E, then such number does not exist).
Smallest multiple of n with digit sum n are:
1, 2, 3, 4, 5, 6, 7, 8, 9, X, E, 1E0, 20E, 22X, 249, 268, 287, 2X6, 45X, 488, 4E6, 1EX, 8E4, 3EX0, 3EE, 23EX, 1899, XX8, 2E79, 4E96, 1EX9, 4XX8, 2EE9, 3XEX, 799X, 5EE90, ..., such number is conjectured to exist for all n.
45 is the smallest prime that produces prime reciprocal magic square, i.e. write the recurring digits of 1/45 (=0.Template:Overline, which has period 44) to 44/45, we get a 44×44 prime reciprocal magic square (its magic number is 1EX), it is conjectured that there are infinitely many such primes, but 45 is the only such prime below 1000, all such primes are full reptend primes, i.e. the reciprocal of them are cyclic numbers, and 10 is a primitive root modulo these primes.
All numbers of the form 34{1} are composite (proof: 34{1n} = 34×10n+(10n−1)/E = (309×10n−1)/E and it can be factored to ((19×10n/2−1)/E) × (19×10n/2+1) for even n and divisible by 11 for odd n). Besides, 34 was proven to be the smallest n such that all numbers of the form n{1} are composite. However, the smallest prime of the form 23{1} is 23{1E78}, it has E7X digits. The only other two n≤100 such that all numbers of the form n{1} are composite are 89 and 99 (the reason of 89 is the same as 34, and the reason of 99 is 99{1n} is divisible by 5, 11 or 25).
The only known of the form 1{0}1 is 11 (see generalized Fermat prime), these are the primes obtained as the concatenation of a power of 10 followed by a 1. If n = 1 mod 11, then all numbers obtained as the concatenation of a power of n (>1) followed by a 1 are divisible by 11 and thus composite. Except 10, the smallest n not = 1 mod 11 such that all numbers obtained as the concatenation of a power of n (>1) followed by a 1 are composite was proven by EX, since all numbers obtained as the concatenation of a power of EX (>1) followed by a 1 are divisible by either E or 11 and thus composite. However, the smallest prime obtained as the concatenation of a power of 58 (>1) followed by a 1 is 10×582781E5+1, it has 459655 digits.
All numbers of the form 1{5}1 are composite (proof: 1{5n}1 = (14×10n+1−41)/E and it can be factored to (4×10(n+1)/2−7) × ((4×10(n+1)/2−7)/E) for odd n and divisible by 11 for even n).
The emirps below 1000 are 15, 51, 57, 5E, 75, E5, 107, 117, 11E, 12E, 13E, 145, 157, 16E, 17E, 195, 19E, 1X7, 1E5, 507, 51E, 541, 577, 587, 591, 59E, 5E1, 5EE, 701, 705, 711, 751, 76E, 775, 785, 7X1, 7EE, E11, E15, E21, E31, E61, E67, E71, E91, E95, EE5, EE7.
The non-repdigit permutable primes below 1010100 are 15, 57, 5E, 117, 11E, 5EEE (the smallest representative prime of the permutation set).
The non-repdigit circular primes below 1010100 are 15, 57, 5E, 117, 11E, 175, 1E7, 157E, 555E, 115E77 (the smallest representative prime of the cycle).
The first few Smarandache primes are the concatenation of the first 5, 15, 4E, 151, ... positive integers.
The only known Smarandache–Wellin primes are 2 and 2357E11.
There are exactly 15 minimal primes, and they are 2, 3, 5, 7, E, 11, 61, 81, 91, 401, X41, 4441, X0X1, XXXX1, 44XXX1, XXX0001, XX000001.
The smallest weakly prime is 6E8XE77.
The largest left-truncatable prime is 28-digit 471X34X164259EX16E324XE8X32E7817, and the largest right-truncatable prime is X-digit 375EE5E515.
The only two base 10 Wieferich primes up to 1010 are 1685 and 5E685, note that both of the numbers end with 685, and it is conjectured that all base 10 Wieferich primes end with 685. (there is also a note for the only two known base 2 Wieferich primes (771 and 2047) minus 1 written in base 2, 8 (= 23) and 14 (= 24), 770 = 010001000100(2) = 444(14) is a repdigit in base 14, and 2046 = 110110110110(2) = 6666(8) is also a repdigit in base 8, see Wieferich prime#Binary periodicity of p − 1)
For the numbers between 5X0 (the smallest number divisible by all of the numbers 1 to 8) and 630 (the square of 26 = 5#) and end with 1, 5, 7 or E (the digits coprime to 10), all numbers whose dozen digit is odd are primes, and all numbers whose dozen digit is even are composites.
For all odd composites c up to 1000, there exists integer a such that GCD(a, c) = 1 and a(c−1)/2 is congruent to neither 1 nor −1 mod c (i.e. c is not an Euler pseudoprime base a), however, this is not true for c = 1001, 1001 is Euler pseudoprime to all bases coprime to itself, i.e. 1001 is an absolute Euler pseudoprime.
There are 1, 2, 3, 5 and 6-digit (but not 4-digit) narcissistic numbers, there are totally 73 narcissistic numbers, the first few of which are 1, 2, 3, 4, 5, 6, 7, 8, 9, X, E, 25, X5, 577, 668, X83, 14765, 938X4, 369862, X2394X, ..., the largest of which is 43-digit 15079346X6E3E14EE56E395898E96629X8E01515344E4E0714E. (see Template:Oeis)
The only two factorions are 1 and 2.
The only seven happy numbers below 1000 are 1, 10, 100, 222, 488, 848 and 884, almost all natural numbers are unhappy. All unhappy numbers get to one of these four cycles: {5, 21}, {8, 54, 35, 2X, 88, X8, 118, 56, 51, 22}, {18, 55, 42}, {68, 84}, or one of the only two fixed points other than 1: 25 and X5.
If we use the sum of the cubes (instead of squares) of the digits, then every natural numbers get to either 1 or the cycle {8, 368, 52E, X20, 700, 247, 2X7, 947, 7X8, 10X7, 940, 561, 246, 200}. (for the example of the famous Hardy–Ramanujan number 1001 = 93 + X3, we know that this sequence with initial term 9X is 9X, 1001, 2, 8, 368, 52E, X20, 700, 247, 2X7, 947, 7X8, 10X7, 940, 561, 246, 200, 8, 368, 52E, X20, 700, 247, 2X7, 947, 7X8, 10X7, 940, 561, 246, 200, 8, ...)
fixed points and cycled for the sequence for sum of n-th powers of the digits
length of these cycles
{1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}, {X}, {E} 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
{1}, {5, 21}, {8, 54, 35, 2X, 88, X8, 118, 56, 51, 22}, {18, 55, 42}, {25}, {68, 84}, {X5} 1, 2, X, 3, 1, 2, 1
{1}, {8, 368, 52E, X20, 700, 247, 2X7, 947, 7X8, 10X7, 940, 561, 246, 200}, {577}, {668}, {6E5, E74, 100X}, {X83}, {11XX} 1, 12, 1, 1, 3, 1, 1
{1}, {X6X, 103X8, 8256, 35X9, 9EXE, 22643, E69, 1102X, 596X, X842, 8394, 6442, 1080, 2455}, {206X, 6668, 4754}, {3X2E, 12396, 472E, X02X, E700, 9X42, 98X9, 13902} 1, 12, 3, 8
The harshad numbers up to 200 are 1, 2, 3, 4, 5, 6, 7, 8, 9, X, E, 10, 1X, 20, 29, 30, 38, 40, 47, 50, 56, 60, 65, 70, 74, 80, 83, 90, 92, X0, X1, E0, 100, 10X, 110, 115, 119, 120, 122, 128, 130, 134, 137, 146, 150, 153, 155, 164, 172, 173, 182, 191, 1X0, 1E0, 1EX, 200, although the sequence of factorials begins with harshad numbers, not all factorials are harshad numbers, after 7! (=2E00, with digit sum 11 but 11 does not divide 7!), 8X4! is the next that is not (8X4! has digit sum 8275 = E×8E7, thus not divide 8X4!). There are no 21 consecutive integers that are all harshad numbers, but there are infinitely many 20-tuples of consecutive integers that are all harshad numbers.
The Kaprekar numbers up to 10000 are 1, E, 56, 66, EE, 444, 778, EEE, 12XX, 1640, 2046, 2929, 3333, 4973, 5E60, 6060, 7249, 8889, 9293, 9E76, X580, X912, EEEE.
The Kaprekar's routine of any four-digit number which is not repdigit converges to either the cycle {3EE8, 8284, 6376} or the cycle {4198, 8374, 5287, 6196, 7EE4, 7375}, and the Kaprekar map of any three-digit number which is not repdigit converges to the fixed point 5E6, and the Kaprekar map of any two-digit number which is not repdigit converges to the cycle {0E, X1, 83, 47, 29, 65}.
Cycles for Kaprekar's routine for n-digit numbers
Number of these cycles
{0} 1 1
{00}, {0E, X1, 83, 47, 29, 65} 1, 6 2
{000}, {5E6} 1, 1 2
{0000}, {3EE8, 8284, 6376}, {4198, 8374, 5287, 6196, 7EE4, 7375} 1, 3, 6 3
{00000}, {64E66, 6EEE5}, {83E74} 1, 2, 1 3
{000000}, {420X98, X73742, 842874, 642876, 62EE86, 951963, 860X54, X40X72, X82832, 864654}, {65EE56} 1, X, 1 3
{0000000}, {841E974, X53E762, 971E943, X64E652, 960EX53, E73E741, X82E832, 984E633, 863E754}, {962E853} 1, 9, 1 3
{00000000}, {4210XX98, X9737422, 87428744, 64328876, 652EE866, 961EE953, X8428732, 86528654, 6410XX76, X92EE822, 9980X323, X7646542, 8320X984, X7537642, 8430X874, X5428762, 8630X854, X540X762, X830X832, X8546632, 8520X964, X740X742, X8328832, 86546654}, {873EE744}, {X850X632} 1, 20, 1, 1 4
The self numbers up to 600 are 1, 3, 5, 7, 9, E, 20, 31, 42, 53, 64, 75, 86, 97, X8, E9, 10X, 110, 121, 132, 143, 154, 165, 176, 187, 198, 1X9, 1EX, 20E, 211, 222, 233, 244, 255, 266, 277, 288, 299, 2XX, 2EE, 310, 312, 323, 334, 345, 356, 367, 378, 389, 39X, 3XE, 400, 411, 413, 424, 435, 446, 457, 468, 479, 48X, 49E, 4E0, 501, 512, 514, 525, 536, 547, 558, 569, 57X, 58E, 5X0, 5E1.
The Friedman numbers up to 1000 are 121=112, 127=7×21, 135=5×31, 144=4×41, 163=3×61, 368=86−3, 376=6×73, 441=(4+1)4, 445=54+4.
The Keith numbers up to 1000 are 11, 15, 1E, 22, 2X, 31, 33, 44, 49, 55, 62, 66, 77, 88, 93, 99, XX, EE, 125, 215, 24X, 405, 42X, 654, 80X, 8X3, X59.
There are totally 71822 polydivisible numbers, the largest of which is 24-digit 606890346850EX6800E036206464. However, there are no E-digit polydivisible numbers contain the digits 1 to E exactly once each. (hence there are also no 10-digit polydivisible numbers using all the digits 0 to E exactly once, since if a number with digits abcdefghijkl is a 10-digit polydivisible number using all the digits 0 to E exactly once, then {a, b, c, d, e, f, g, h, i, j, k, l} = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, X, E}, and then abcdefghijkl is divisible by 10, thus we have l = 0 (by divisibility rule of 10), and {a, b, c, d, e, f, g, h, i, j, k} = {1, 2, 3, 4, 5, 6, 7, 8, 9, X, E}, thus a number with digits abcdefghijk is an E-digit polydivisible numbers using all the digits 1 to E exactly once). (proof: if a number with digits abcdefghijk is an E-digit polydivisible numbers using all the digits 1 to E exactly once, then {a, b, c, d, e, f, g, h, i, j, k} = {1, 2, 3, 4, 5, 6, 7, 8, 9, X, E}, and we have:
f = 6 (since abcdef is divisible by 6) (by divisibility rule of 6)
{d, h} = {4, 8} (since abcd is divisible by 4 and abcdefgh is divisible by 8 (thus by 4)) (by divisibility rule of 4)
{c, i} = {3, 9} (since abc is divisible by 3 and abcdefghi is divisible by 9 (thus by 3)) (by divisibility rule of 3)
{b, j} = {2, X} (since ab is divisible by 2 and abcdefghij is divisible by X (thus by 2)) (by divisibility rule of 2)
thus, we have {a, e, g, k} = {1, 5, 7, E}
Since abcdefgh is divisible by 8, thus gh is divisible by 8 (by divisibility rule of 8), and since {a, e, g, k} = {1, 5, 7, E}, thus g is odd, and h must be 4 (if h = 8 and g is odd, then gh is not divisible by 8), and since abcdefghi is divisible by 9, thus hi is divisible by 9 (by divisibility rule of 9), however, h = 4 and i is either 3 or 9, but neither 43 nor 49 is divisible by 9.
If we do not require the number formed by its first 8 digits divisible by 8, then there are 2 solutions: 1X98265E347 and 7298X65E341 (neither satisfies that the number formed by its first 8 digits is divisible by 4).
If we do not require the number formed by its first 9 digits divisible by 9, then there are 4 solutions: 1X38E694725, 7X981634E25, 7X98E654321, and EX987634125 (only 7X98E654321 satisfies that the number formed by its first 9 digits is divisible by 3).
The candidate Lychrel numbers up to 1000 are 179, 1E9, 278, 2E8, 377, 3E7, 476, 4E6, 575, 5E5, 674, 6E4, 773, 7E3, 872, 8E2, 971, 9E1, X2E, X3E, X5E, X70, XXE, XE0, E2X, E3X, E5X, EXX. The only suspected Lychrel seed numbers up to 1000 are 179, 1E9, X3E and X5E. However, it is unknown whether any Lychrel number exists. (Lychrel numbers only known to exist in these bases: E, 15, 18, 22 and all powers of 2)
Most numbers that end with 2 are nontotient (in fact, all nontotients < 58 except 2X end with 2), except 2 itself, the first counterexample is 92, which equals φ(X1) = φ(E2) and φ(182) = φ(2×E2), next counterexample is 362, which equals φ(381) = φ(1E2) and φ(742) = φ(2×1E2), there are only 9 such numbers ≤ 10000 (the number 2 itself is not counted), all such numbers (except the number 2 itself) are of the form φ(p2) = p(p−1), where p is a prime ends with E.
There is a known generalized Cullen prime for all bases b ≤ 10 (but not for b = 11). (no matter whether you require n ≥ b−1 or not)
There is a known generalized Woodall prime for all bases b ≤ 100 (but not for b = 101). (no matter whether you require n ≥ b−1 or not)
There is a known generalized Carol prime for all even bases b ≤ (100×2 + 100÷2) (=260) (but not for b = next even number (262)).
There is a known generalized Kynea prime for all even bases b ≤ 200 (but not for b = next even number (202)).
The generalized minimal primes problem has at most one unsolved family for all bases b ≤ 20 (but not for b = 21). (there is one unsolved family for b = 15, 17 and 19, and there are no unsolved families for all other b ≤ 20, but for b = 21, there are 10 unsolved families)
There are no n≤100 which is nontotient, noncototient, and untouchable. (the smallest such n is indeed the smallest even number > 100, i.e. 102)
By sieve of Eratosthenes, we can cross out every composites ≤ 20 by sieve the primes dividing 10 (i.e. the primes ≤3) (i.e. the primes 2 and 3). (however, we cannot cross out the composite 21 by sieve the primes dividing 10 (i.e. the primes ≤3) (i.e. the primes 2 and 3))
By sieve of Eratosthenes, we can cross out every composites ≤ 200 by sieve to the prime 10+1 (=11). (however, we cannot cross out the composite 201 by sieve to the prime 10+1 (=11))
For all odd composites c ≤ 1000, there exists integer b coprime to c such that b(c−1)/2 ≠ ±1 (mod c) (i.e. c is not Euler pseudoprime base b). (this is not true for the composite c = 1001, 1001 is the smallest absolute Euler pseudoprime)
The Wagstaff numbers $ \frac{2^p+1}{3} $ is prime for all odd primes p ≤ 20 (but not for p = next odd prime (25)).
There is a known odd generalized Wieferich prime for all prime bases p ≤ 20 (but not for p = next prime (25)).
The smallest Perrin pseudoprime is a near-repunit 111101, this number only contains five 1's and one 0 (no any digit >1), and this number plus 10 is the repunit with length 6, i.e. 111111.
The "beast number" (666), when add a digit "1" before it and add another digit "1" after it, it become a palindromic square 16661, it is the smallest palindromic square whose square root is not palindrome (12E), it is also the smallest palindromic square depending on base.
If we let the musical notes in an octave be numbers in the cyclic group Z10: C=0, C#=1, D=2, Eb=3, E=4, F=5, F#=6, G=7, Ab=8, A=9, Bb=X, B=E (see pitch class and music scale) (thus, if we let the middle C be 0, then the notes in a piano are -33 to 40), then x and x+3 are minor third, x and x+4 are major third, x and x+7 are perfect fifth (thus, we can use 7x for x = 0 to E to get the five degree cycle), etc. (since an octave is 10 semitones, a minor third is 3 semitones, a major third is 4 semitones, and a perfect fifth is 7 semitones, etc.) (if we let an octave be 1, then a semitone will be 0.1, and we can write all 10 notes on a cycle, the difference of two connected notes is 26 degrees or $ \frac{\pi}{6} $ radians) Besides, the x major chord (x) is {x, x+4, x+7} in Z10, and the x minor chord (xm) is {x, x+3, x+7} in Z10, and the x major 7th chord (xM7) is {x, x+4, x+7, x+E}, and the x minor 7th chord (xm7) is {x, x+3, x+7, x+X}, and the x dominant 7th chord (x7) is {x, x+4, x+7, x+X}, and the x diminished 7th triad (xdim7) is {x, x+3, x+6, x+9}, since the frequency of x and x+6 is not simple integer fraction, they are not harmonic, and this diminished 7th triad is corresponding the beast number 666 (three 6's) (also, x and x+6 are tritone, which is not harmonic). Besides, x major scale uses the notes {x, x+2, x+4, x+5, x+7, x+9, x+E}, and x minor scale uses the notes {x, x+2, x+3, x+5, x+7, x+8, x+X}. Besides, the frequency of x+10 is twice as that of x, the frequency of x+7 is 1.6 (=3/2) times as that of x, and the frequency of x+5 is 1.4 (=4/3) times as that of x, they are all simple integer fractions (ratios of small integers), and they all have at most one digit after the duodecimal point, and we can found that 1.610 = X9.8E5809 is very close to 27 = X8, since 217 = 2134X8 is very close to 310 = 217669, the simple frequency fractions found for the scales are only 0.6, 0.8, 0.9, 1.4, 1.6 and 2, however, since the frequency of x+10 is twice as that of x, thus the frequency of x+1 (i.e. a semitone higher than x) is $ \sqrt[10]{2} $ (=20.1) times as that of x. Let f(x) be the frequency of x, then we have f(2)/f(0) = 9/8 (=1.16), f(4)/f(2) = X/9 (=1.14), and f(5)/f(4) = 14/13 (this number is very close to $ \sqrt[10]{2} $), and thus we have that f(5)/f(0) = (9/8) × (X/9) × (14/13) = 4/3. Also, we can found that 20.5 is very close to 1.4, and 20.7 is very close to 1.6.
All orders of non-cyclic simple group end with 0 (thus, all orders of unsolvable group end with 0), however, we can prove that no groups with order 10, 20, 30 or 40 are simple, thus 50 is the smallest order of non-cyclic simple group (thus, all groups with order < 50 are solvable), (50 is the order of the alternating group A5, which is a non-cyclic simple group, and thus an unsolvable group) next three orders of non-cyclic simple group are 120, 260 and 360. (Edit: I found that this is not completely true (although this is true for all orders ≤ 14000), the smallest counterexample is 14X28, however, all such orders are divisible by 4 and either 3 or 5 (i.e. divisible by either 10 or 18), and all such orders have at least 3 distinct prime factors, by these conditions, the smallest possible such order is indeed 50 = 22 × 3 × 5, next possible such order is 70 = 22 × 3 × 7, however, by Sylow theorems, the number of Sylow 7-subgroups of all groups with order 70 (i.e. the number of subgroups with order 7 of all groups with order 70) is congruent to 1 mod 7 and divides 70, hence must be 1, thus the subgroup with order 7 is a normal subgroup of the group with order 70, thus all groups with order 70 have a nontrivial normal subgroup and cannot be simple groups)
The probability for rolling a 6 on a dice is 0.2 or 20%, and the probability for rolling at least one 6 on a dice in 3 rolls is 0.508 (less than one half or 60%), and the probability for rolling at least one 6 on a dice in 4 rolls is 0.6268 (more than one half or 60%), and the probability for rolling a "double 6" on two dices is 0.04 or 4%, and the probability for rolling at least one "double 6" on two dices in 20 rolls is 0.5X9190... (less than one half or 60%), and the probability for rolling at least one "double 6" on two dices in 21 rolls is 0.609685... (more than one half or 60%).
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, X, E
appear in the repeating digits of 1/5 (exactly the even digits ≤ 5 (except 0 (the smallest digit)) and the odd digits ≥ 6 (except E (the largest digit)))
appear in the repeating digits of 1/7 (exactly the odd digits ≤ 5 and the even digits ≥ 6)
appear in the repeating digits of 1/11 (exactly the smallest digit (0) and the largest digit (E))
(note that the number of digits ≤5 is equal to the number of digits ≥6 (both are 6, which is equal to half of the base we used in this wiki (10)), and all digits are either ≤5 or ≥6, but not both)
(note that all of 5, 7, and 11 are primes, and they are the only three primes ≤ 10+1 (=11) and divides neither 10 nor 10−1 (=E))
Cite error: <ref> tags exist, but no <references/> tag was found
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Henry Crapo (mathematician)
Henry Howland Crapo (KRAY-poh;[1] August 12, 1932 – September 3, 2019) was an American-Canadian mathematician who worked in algebraic combinatorics. Over the course of his career, he held positions at several universities and research institutes in Canada and France. He is noted for his work in matroid theory and lattice theory.
Henry Crapo
Henry Crapo at Oberwolfach in 1987
Pronunciation
• KRAY-poh
Born
Henry Howland Crapo
(1932-08-12)August 12, 1932
Detroit, Michigan, United States
DiedSeptember 3, 2019(2019-09-03) (aged 87)
La Vacquerie-et-Saint-Martin-de-Castries, France
CitizenshipAmerican, Canadian
Alma materMassachusetts Institute of Technology (Ph.D.)
Known forMatroid theory
Scientific career
FieldsMathematics
ThesisOn the Theory of Combinatorial Independence (1964)
Doctoral advisorsGian-Carlo Rota, Kenneth Hoffman
Education and career
Crapo was born in Detroit, Michigan, in 1932.[2][3] He received his Ph.D. in 1964 under the supervision of Gian-Carlo Rota and Kenneth Hoffman.[4] He held academic positions at the University of Waterloo, Université de Montréal, INRIA Rocquencourt, and École des Hautes Études en Sciences Sociales.[5] During his time in Waterloo, Crapo became a Canadian citizen.[3]
Crapo is known for his early work in matroid theory, and for related work in lattice theory. He introduced the beta invariant of a matroid,[6] and published the first paper on the Tutte polynomial[7] (though Tutte had already defined an equivalent polynomial in his thesis). Together with Gian-Carlo Rota, Crapo wrote the first book on matroid theory.[8][9] He is also known for Crapo's Complementation Theorem in poset Möbius Inversion.[10][11] Crapo wrote 65 mathematical publications during his career.[12]
Upon his retirement, Crapo moved to the south of France.[3] He continued some mathematical activity, and hosted several small conferences at his house there.[9] He died on September 3, 2019.[13]
Awards and honors
• A special 1999 issue of the journal Advances in Applied Mathematics was dedicated to Crapo on the occasion of his 67th birthday.[5][14]
Personal life
Crapo was a patron of the arts. At the University of Waterloo he donated a collection of rare books on the history of dance and ballet,[15] as well as a copy of the Porcellino sculpture of Florence; the latter shoulder-high bronze sculpture of a wild boar later became a mascot for the University of Waterloo Faculty of Arts.[16] He also donated The Temptation of St. Anthony by James Ensor to the Royal Museum of Fine Arts in Antwerp.[3]
References
1. "Pronunciation: Crapo". MathOverflow. Retrieved October 25, 2019.
2. "Remembering Henry Crapo". Pure Mathematics. University of Waterloo. October 25, 2019. Retrieved October 31, 2019.
3. "Maths Genius Donates James Ensor Painting". Collection. Royal Museum of Fine Arts Antwerp. Retrieved October 25, 2019.
4. Henry Howland Crapo at the Mathematics Genealogy Project
5. Kung, Joseph P.S. (1999). "Guest Editor's Introduction". Advances in Applied Mathematics. Elsevier BV. 23 (1): 1–2. doi:10.1006/aama.1999.0642. ISSN 0196-8858.
6. Crapo, Henry H. (1967). "A higher invariant for matroids". Journal of Combinatorial Theory. Elsevier BV. 2 (4): 406–417. doi:10.1016/s0021-9800(67)80051-6. ISSN 0021-9800. MR 0215744.
7. Crapo, Henry H. (1969). "The Tutte polynomial". Aequationes Mathematicae. Springer Science and Business Media LLC. 3 (3): 211–229. doi:10.1007/bf01817442. ISSN 0001-9054. MR 0215744. S2CID 119602825.
8. Crapo, Henry; Rota, Gian-Carlo (1970). On the Foundations of Combinatorial Theory: Combinatorial Geometries. Cambridge, Massachusetts: M.I.T. Press. ISBN 978-0-262-53016-3. MR 0290980. OCLC 117282.
9. Oxley, James. "Henry Crapo: A Brief Reminiscence". Matroid Union.
10. Crapo, Henry H. (1966). "The Möbius function of a lattice". Journal of Combinatorial Theory. Elsevier BV. 1 (1): 126–131. doi:10.1016/s0021-9800(66)80009-1. ISSN 0021-9800. MR 0193018.
11. Stanley, Richard (2012). Enumerative combinatorics. New York: Cambridge University Press. ISBN 978-1-107-60262-5. MR 2868112. OCLC 777400915.
12. "Henry H. Crapo author profile". MathSciNet. American Mathematical Society.
13. "Henry Crapo Death Notice" (in French). Retrieved October 25, 2019.
14. Penne, Rudi (1999). "Almost Flat Line Configurations". Advances in Applied Mathematics. Elsevier BV. 23 (1): 54–77. doi:10.1006/aama.1999.0647. ISSN 0196-8858. MR 1692976.
15. "Crapo, Henry H." Special Collections & Archives. University of Waterloo Library. Retrieved October 25, 2019.
16. "The Boar in the News". Special Collections & Archives. University of Waterloo Library. March 6, 2015. Retrieved October 25, 2019.
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| Wikipedia |
\begin{document}
\title[Coulhon type inequalities]{A note on Coulhon type inequalities} \author{Joaquim Mart\'{i}n$^{\ast}$} \address{Department of Mathematics\\ Universitat Aut\`{o}noma de Barcelona} \email{[email protected]} \author{Mario Milman**} \address{Department of Mathematics\\ Florida Atlantic University\\ Boca Raton, Fl. 33431} \email{[email protected]} \urladdr{http://www.math.fau.edu/milman} \thanks{$^{\ast}$Partially supported in part by Grants MTM2010-14946, MTM-2010-16232.} \thanks{**This work was partially supported by a grant from the Simons Foundation (\#207929 to Mario Milman).} \thanks{This paper is in final form and no version of it will be submitted for publication elsewhere.} \subjclass{2000 Mathematics Subject Classification Primary: 46E30, 26D10.} \keywords{Sobolev inequalities, modulus of continuity, symmetrization, isoperimetric inequalities, interpolation.}
\begin{abstract} T. Coulhon introduced an interesting reformulation of the usual Sobolev inequalities. We characterize Coulhon type inequalities in terms of rearrangement inequalities. \end{abstract}\maketitle
\section{Introduction}
Let $(X,d,\mu)$ be a connected Borel metric measure space. The perimeter or Minkowski content of a Borel set $A\subset X,$ is defined by \[ \mu^{+}(A)=\lim\inf_{h\rightarrow0}\frac{\mu\left( A_{h}\right) -\mu\left( A\right) }{h}, \] where $A_{h}=\left\{ x\in\Omega:d(x,A)<h\right\} ,$ and the isoperimetric profile $I=I_{(\Omega,d,\mu)}$ is defined by \[ I_{(\Omega,d,\mu)}(t)=\inf_{A}\{\mu^{+}(A):\mu(A)=t\}. \] We assume throughout that $(X,d,\mu)$ is such that $I_{(\Omega,d,\mu)}$ is concave, continuous with $I(0)=0.$ Moreover, we also assume that $(X,d,\mu)$
is such that for each $c\in R,$ and each $f\in Lip_{0}(X),|\nabla f(x)|=0,a.e.$ in the set $\{x:f(x)=c\}.$ Under these conditions\footnote{In \cite{mamiadv} the result is shown for metric probability spaces such that $I(t)$ is symmetric about $1/2,$ in which case we can replace $Lip_{0}(X)$ by $Lip(X)$ in the statement. With minor modifications one can also show its validity for infinite measure spaces (cf. also \cite{mmp}, \cite{mamicon}).} we showed in \cite{mamiadv} that the Gagliardo-Nirenberg-Ledoux inequality \begin{equation}
\int_{0}^{\infty}I(\mu_{f}(t))dt\leq\left\| \left| \nabla f\right|
\right\| _{L{^{1}(X)}},\text{ for all }f\in Lip_{0}(X) \label{gn} \end{equation} is equivalent to \begin{equation}
f^{\ast\ast}(t)-f^{\ast}(t)\leq\frac{t}{I(t)}\left| \nabla f\right| ^{\ast\ast}(t), \label{c0} \end{equation} where $Lip_{0}(X)$ are the functions in $Lip(X)$ of compact support, \[
\left| \nabla f(x)\right| =\lim\sup_{y\rightarrow x}\frac{\left|
f(x)-f(y)\right| }{d(x,y)}, \]
$\mu_{f}(t)=\mu\{\left| f\right| >t\},$ $f^{\ast}$ is the non increasing rearrangement\footnote{For background we refer to \cite{bs} (on rearrangements), and \cite{leoni}, \cite{Maz} (on Sobolev spaces) .} of $f$ with respect to the measure $\mu$ and $f^{\ast\ast}(t)=\frac{1}{t}\int_{0} ^{t}f^{\ast}(s)ds.$
Conversely, if an inequality of the form (\ref{c0}) holds for some continuous concave function $I_{1}(t)$, it was shown in \cite{mamiadv} that $I_{1}(t)$ satisfies the isoperimetric inequality $I_{1}(\mu(A))\leq\mu^{+}(A) $ for any Borel set\footnote{Therefore, $I_{1}(t)\leq\inf\{\mu^{+}(A):\mu(A)=t\}=I(t),$ and consequently $\frac{t}{I(t)}\leq\frac{t}{I_{1}(t)}.$} $A\subset\subset X$. In particular, for $\mathbb{R}^{n}$ it is well known that (cf. \cite[Chapter 1]{Maz}) $I(t)=c_{n}t^{1-1/n},$ and therefore (\ref{c0}) becomes (cf. \cite{bmr} and the references therein) \begin{equation}
f^{\ast\ast}(t)-f^{\ast}(t)\leq c_{n}^{-1}t^{1/n}\left| \nabla f\right| ^{\ast\ast}(t). \label{berta} \end{equation} It follows that (\ref{gn}) gives \[
c_{n}\int_{0}^{\infty}\mu_{f}(t)^{1-1/n}dt=c_{n}\frac{1}{n^{\prime}}\int _{0}^{\infty}t^{1/n^{\prime}}f^{\ast}(t)\frac{dt}{t}\leq\left\| \left|
\nabla f\right| \right\| _{L{^{1}(\mathbb{R}^{n})}} \] i.e. \[
\left\| f\right\| _{L^{^{\frac{n}{n-1}},1}(\mathbb{R}^{n})}\leq c\left\|
\left| \nabla f\right| \right\| _{L{^{1}(\mathbb{R}^{n})}},\text{ for all }f\in Lip_{0}(\mathbb{R}^{n}). \] In other words, (\ref{gn}) represents a generalization of the sharp form of the Euclidean Gagliardo-Nirenberg inequality that uses Lorentz spaces (cf. \cite{poor} and \cite{mmp} (for Euclidean spaces), \cite{le} (Gaussian spaces), and \cite{boho}, \cite{mamiadv} (for metric spaces); for the corresponding rearrangement inequalities we refer to \cite{bmr}, \cite{mmjfa}, \cite{mamiadv}, as well as the references therein).
The corresponding Sobolev inequalities when $\left| \nabla f\right| \in L^{p},$ $p>1$ are also known to self improve (cf. \cite{Maz}, \cite{bakr}, \cite{mamicon}, and the references therein) but an analogous rearrangement inequality characterization in this case has remained an open problem. On the other hand, Coulhon (cf. \cite{cou1}, \cite{cou2}, \cite{cou}) and Bakry-Coulhon-Ledoux \cite{bakr} introduced and studied a different scale of Sobolev inequalities. For $p\in\lbrack1,\infty],$ and $\phi$ an increasing function on the positive half line, these authors studied the validity of inequalities of the form \[
(S_{\phi}^{p})\;\;\;\;\left\| f\right\| _{p}\leq\phi(\left\| f\right\|
_{0})\left\| \left| \nabla f\right| \right\| _{p},\text{ }f\in Lip_{0}(X), \] where \[
\left\| f\right\| _{0}=\mu\{support(f)\},\text{ }\left\| f\right\|
_{p}=\left\| f\right\| _{L^{p}(X)}. \]
In particular, it was shown by Coulhon et al. that the $(S_{\phi}^{p})$ inequalities encapsulate the classical Sobolev inequalities, as well as the Faber-Krahn inequalities. For $p=1,$ $(S_{\phi}^{1})$ is equivalent to the isoperimetric inequality in the sense that\footnote{See Section \ref{sec1} \ below.} \[ \frac{t}{I(t)}\leq\phi(t). \] Moreover, for $p=\infty,$ the $(S_{\phi}^{\infty})$ conditions are explicitly connected with volume growth. For a detailed discussion of the different geometric interpretations for different $p^{\prime}s$ we refer to \cite{cou1}, \cite{gri}, \cite{Maz}, and the references quoted therein.
It follows from this discussion that, for a suitable class of metric measure spaces, the $(S_{\phi}^{1})$ condition can be characterized by means of the symmetrization inequality (\ref{c0}): \[ (S_{\phi}^{1})\text{ holds }\Leftrightarrow\text{(\ref{c0}) holds.} \] The purpose of this paper is to provide an analogous rearrangement characterization of the $(S_{\phi}^{p})$ conditions, \ for $1\leq p<\infty.$ Our main result extends (\ref{c0}) as follows
\begin{theorem} \label{teo}Let $(X,d,\mu)$ be a connected Borel metric measure space as described above, and let $p\in\lbrack1,\infty).$ The following statements are equivalent
\begin{enumerate} \item $(S_{\phi}^{p})$ holds, i.e. \begin{equation}
\left\| f\right\| _{p}\leq\phi(\left\| f\right\| _{0})\left\| \left|
\nabla f\right| \right\| _{p},\text{ for all }f\in Lip_{0}(X). \label{desigualdad01} \end{equation}
\item Let $k\in\mathbb{N}$ be such that $k<p\leq k+1,$ then for all $f\in Lip_{0}(X)$ \begin{equation} \left( \frac{f_{\left( p\right) }^{\ast\ast}(t)}{\phi_{(p)}(t)}\right) ^{1/p}-\left( \frac{f_{(p)}^{\ast}(t)}{\phi_{(p)}(t)}\right) ^{1/p}
\leq2^{\frac{k+1}{p}-1}\left( \left| \nabla f\right| _{(p)}^{\ast\ast }(t)\right) ^{1/p}, \label{norma} \end{equation} where \[ f_{(p)}^{\ast}(t)=\left( f^{\ast}(t)\right) ^{p},\text{ }f_{(p)}^{\ast\ast }(t)=\frac{1}{t}\int_{0}^{t}f_{(p)}^{\ast}(s)ds,\text{ }\phi_{(p)}(t)=\left( \phi(t)\right) ^{p}. \]
\item Let $k\in\mathbb{N}$ be such that $k<p\leq k+1,$ then for all $f\in Lip_{0}(X),$ $f_{(p)}^{\ast}$ is absolutely continuous (cf. \cite{leoni}) and \begin{equation} -\frac{\partial}{\partial t}\left( f_{(p)}^{\ast\ast}(t)\right) ^{1/p}=-\frac{\partial}{\partial t}\left( \frac{1}{t}\int_{0}^{t} f_{(p)}^{\ast}(s)ds\right) ^{1/p}\leq2^{\frac{k+1}{p}}\frac{\phi(t)}
{t}\left( \left| \nabla f\right| _{(p)}^{\ast\ast}(t)\right) ^{\frac{1} {p}}. \label{norma01} \end{equation} \end{enumerate} \end{theorem}
Note that for $p=1$ the inequality (\ref{norma}) of Theorem \ref{teo} coincides with (\ref{c0}). This new characterization for $p\geq1$ is independent of \cite{mamiadv}, and, in fact, it provides a new approach to (\ref{c0}) as well. On the other hand, as it is well known (cf. \cite{cou1}), the $(S_{\phi}^{p})$ conditions get progressively weaker as $p$ increases. Indeed, below we will also show that (\ref{c0}) implies (\ref{norma}) via an extended form of the chain rule, that is valid for metric spaces.
The note is organized as follows. In section \ref{sec1} we give a somewhat more detailed discussion of the $(S_{\phi}^{p})$ conditions and, in particular, we develop a connection with \cite{mamiadv}. In section \ref{secproof} we provide a proof of Theorem \ref{teo} and, finally, in section \ref{secrem}, we discuss, rather briefly, connections with Nash type inequalities, Sobolev and Faber-Krahn inequalities and interpolation/extrapolation theory.
As usual, the symbol $f\simeq g$ will indicate the existence of a universal constant $c>0$ (independent of all parameters involved) so that $(1/c)f\leq g\leq c\,f$, while the symbol $f\preceq g$ means that $f\leq c\,g$. \ \ \ \
\section{The $(S_{\phi}^{p})$ conditions\label{sec1}}
From now on $(X,d,\mu)$ will be a connected metric measure space with a continuous isoperimetric profile $I$ such that $\frac{t}{I(t)}$ increases and such that $I(0)=0.$ Moreover, we also assume that $(X,d,\mu)$ is such that for each $c\in\mathbb{R},$ and each $f\in Lip_{0}(X),|\nabla f(x)|=0,a.e.$ in the set $\{x:f(x)=c\}.$ The isoperimetric profile $I=I_{(\Omega,d,\mu)}$ is defined by \[ I_{(\Omega,d,\mu)}(t)=\inf_{A}\{\mu^{+}(A):\mu(A)=t\}, \] where $\mu^{+}(A)$ is the perimeter or Minkowski content of the Borel set $A\subset X,$ defined by \[ \mu^{+}(A)=\lim\inf_{h\rightarrow0}\frac{\mu\left( A_{h}\right) -\mu\left( A\right) }{h}, \] where $A_{h}=\left\{ x\in\Omega:d(x,A)<h\right\} .$
\subsection{The $(S_{\phi}^{1})$ condition}
From \cite{mamiadv} (cf. also \cite{mamicon}) we know that \begin{equation}
f^{\ast\ast}(t)-f^{\ast}(t)\leq\frac{t}{I(t)}\left| \nabla f\right| ^{\ast\ast}(t),\text{ }f\in Lip_{0}(X), \label{c2.1} \end{equation} is equivalent to the isoperimetric inequality. If we combine these results with the characterization of $(S_{\phi}^{1})$ given in \cite{cou} we can see the equivalence between (\ref{c2.1}) and the $(S_{\phi}^{1})$ condition. To understand the discussion of the next section it is instructive to provide an elementary direct approach. So we shall now show that (\ref{c2.1}) implies $(S_{\phi}^{1})$ with $\phi(t)=t/I(t),$ and that this choice is in some sense the best possible $(S_{\phi}^{1})$ condition.
Suppose that (\ref{c2.1}) holds. Multiplying both sides of (\ref{c2.1}) by $t>0$ we obtain \[ t\left( f^{\ast\ast}(t)-f^{\ast}(t)\right) \leq\frac{t}{I(t)}\int_{0}
^{t}\left| \nabla f\right| ^{\ast}(s)ds. \] Since formally $f^{\ast}(t)=\mu_{f}^{-1}(t),$ drawing a diagram it is easy to convince oneself that \begin{align*} t\left( f^{\ast\ast}(t)-f^{\ast}(t)\right) & =\int_{0}^{t}f^{\ast }(s)ds-tf^{\ast}(t)\\ & =\int_{f^{\ast}(t)}^{\infty}\mu_{f}(s)ds. \end{align*}
Consequently, if we let $t=\left\| f\right\| _{0},$ we see that $f^{\ast
}(\left\| f\right\| _{0})=0,$ $\int_{f^{\ast}(\left\| f\right\| _{0}
)}^{\infty}\mu_{f}(s)ds=\left\| f\right\| _{1},$ and $\int_{0}^{\left\|
f\right\| _{0}}\left| \nabla f\right| ^{\ast}(s)ds=\left\| \left| \nabla f\right| \right\| _{1}.$ Thus, \[
\left\| f\right\| _{1}\leq\frac{\left\| f\right\| _{0}}{I(\left\|
f\right\| _{0})}\left\| \left| \nabla f\right| \right\| _{1}. \] In other words, the $(S_{\phi}^{1})$ condition holds with $\phi(t)=\frac {t}{I(t)},$ and consequently the $(S_{\tilde{\phi}}^{1})$ condition holds for any $\tilde{\phi}(t)\geq\frac{t}{I(t)}.$ On the other hand, consider an $(S_{\tilde{\phi}}^{1})$ condition for a continuous, increasing but arbitrary function $\tilde{\phi}$. Let $A$ be a Borel set, $A\subset\subset X,$ with $\mu(A)=t.$ Formally inserting $f=\chi_{A}$ in the corresponding $(S_{\tilde{\phi}}^{1})$ inequality (this is done rigorously by approximation), yields \[
\left\| \chi_{A}\right\| _{1}=t=\mu(A)\leq\tilde{\phi}(t)\mu^{+}(A). \] Consequently, \begin{align*} \frac{t}{\tilde{\phi}(t)} & \leq\inf\{\mu^{+}(B):\mu(B)=t\}\\ & =I(t), \end{align*} and therefore \[ \frac{t}{I(t)}\leq\tilde{\phi}(t)\text{. } \]
\subsection{$(S_{\phi}^{1})\Rightarrow(S_{\phi}^{p}),$ $p>1$}
In the Euclidean space $\mathbb{R}^{n}$, $I(t)=d_{n}t^{1-1/n},$ $\phi(t)\simeq t^{1/n}$ and the best possible $(S_{\phi}^{1})$ inequality can be written as \[
\left\| f\right\| _{1}\leq c_{n}\left\| f\right\| _{0}^{1/n}\left\|
\left| \nabla f\right| \right\| _{1}. \]
As was shown in \cite{cou1} the corresponding inequalities for $p>1$ then follow by the (classical) chain rule, the fact that $\left\| \left|
f\right| ^{p}\right\| _{0}=\left\| \left| f\right| \right\|
_{0}=\left\| f\right\| _{0},$ and H\"{o}lder's inequality. In detail, \begin{align*}
\left\| f\right\| _{p}^{p} & =\left\| \left| f\right| ^{p}\right\| _{1}\\
& \leq c_{n}p\left\| f\right\| _{0}^{1/n}\left\| \left| f\right|
^{p-1}\left| \nabla\left| f\right| \right| \right\| _{1}\\
& \leq c_{n}p\left\| f\right\| _{0}^{1/n}\left\| f\right\| _{p}
^{p-1}\left\| \left| \nabla\left| f\right| \right| \right\| _{p}. \end{align*} Consequently, \[
\left\| f\right\| _{p}\leq c_{n}p\left\| f\right\| _{0}^{1/n}\left\|
\left| \nabla f\right| \right\| _{p}, \] and therefore, modulo constants, we have that $(S_{\phi}^{1})\Rightarrow (S_{\phi}^{p}),$ for $p>1.$ More generally, this argument, taken from \cite{cou1}, shows that the $(S_{\phi}^{p})$ conditions become weaker as $p$ increases. In the general setting of metric spaces, the classical chain rule needs to be replaced by an inequality\footnote{The underlying elementary inequality is \[
\left| a^{r}-b^{r}\right| \leq r\left| a^{r-1}+b^{r-1}\right| \left|
a-b\right| . \] }: for $r>1,$ \begin{equation}
\left| \nabla f^{r}(x)\right| \leq2r\left| f^{r-1}(x)\right| \left|
\nabla f(x)\right| . \label{chain} \end{equation}
Next, we use the generalized chain rule to explain the origin of the awkward looking condition (\ref{norma}). Informally, we shall now show that\footnote{With slightly more labor the same method will similarly show that, more generally, $(S_{\phi}^{p})\Rightarrow(S_{\phi}^{q}),$ for $q>p.$} $(S_{\phi}^{1})\Rightarrow(S_{\phi}^{p})$ at the level of rearrangements, i.e. (\ref{c0})$\Rightarrow$(\ref{norma}).
Assume the validity of $(S_{\phi}^{1}).$ Let $f\in Lip_{0}(X);$ we may assume without loss that $f$ is positive. Apply the $(S_{\phi}^{1})$ inequality to $f_{(p)}=f^{p},$ where $p>1$ is fixed. Then, by the chain rule (\ref{chain}) \begin{align*}
f_{(p)}^{\ast\ast}(t)-f_{(p)}^{\ast}(t) & \preceq\phi(t)\left| \nabla f\right| _{(p)}^{\ast\ast}(t)\\
& \preceq\phi(t)(f^{p-1}\left| \nabla f\right| )^{\ast\ast}(t). \end{align*} By a result due to O'Neil (cf. \cite[page 88, Exercise 10]{bs}) and H\"{o}lder's inequality \begin{align*}
(f^{p-1}\left| \nabla f\right| )^{\ast\ast}(t) & \leq\frac{1}{t}\int _{0}^{t}(f^{\ast}(s))^{p-1}\left| \nabla f\right| ^{\ast}(s)ds\\ & \leq\frac{1}{t}\left( \int_{0}^{t}f_{(p)}^{\ast}(s)ds\right)
^{1/p^{\prime}}\left( \int_{0}^{t}\left| \nabla f\right| _{(p)}^{\ast }(s)ds\right) ^{1/p}\\
& =\left( f_{(p)}^{\ast\ast}(t)\right) ^{1-1/p}\left( \left| \nabla f\right| _{(p)}^{\ast\ast}(t)\right) ^{1/p}. \end{align*} Combining inequalities we obtain, \[ f_{(p)}^{\ast\ast}(t)-f_{(p)}^{\ast}(t)\preceq\phi(t)\left( f_{(p)}^{\ast
\ast}(t)\right) ^{1-1/p}\left( \left| \nabla f\right| _{(p)}^{\ast\ast }(t)\right) ^{1/p}. \] Hence, \[ \left( f_{(p)}^{\ast\ast}(t)\right) ^{1/p}-\frac{f_{(p)}^{\ast}(t)}{\left(
\left| f\right| _{(p)}^{\ast\ast}(t)\right) ^{1/p^{\prime}}}\preceq
\phi(t)\left( \left| \nabla f\right| _{(p)}^{\ast\ast}(t)\right) ^{1/p}. \] But, since \[
\left( \left| f\right| _{(p)}^{\ast\ast}(t)\right) ^{1/p^{\prime}}
\geq\left( \left| f\right| _{(p)}^{\ast}(t)\right) ^{1/p^{\prime}}=\left(
\left| f\right| _{(p)}^{\ast}(t)\right) ^{1-1/p}, \] we have \[ \left( f_{(p)}^{\ast}(t)\right) ^{1/p}\geq\frac{f_{(p)}^{\ast}(t)}{\left(
\left| f\right| _{(p)}^{\ast\ast}(t)\right) ^{1/p^{\prime}}}, \] and we conclude that \begin{align*} \left( f_{(p)}^{\ast\ast}(t)\right) ^{1/p}-\left( f_{(p)}^{\ast}(t)\right) ^{1/p} & \preceq\left( f_{(p)}^{\ast\ast}(t)\right) ^{1/p}-\frac
{f_{(p)}^{\ast}(t)}{\left( \left| f\right| _{(p)}^{\ast\ast}(t)\right) ^{1/p^{\prime}}}\\
& \preceq\phi(t)\left( \left| \nabla f\right| _{(p)}^{\ast\ast}(t)\right) ^{1/p}. \end{align*} Therefore, \[ \left( \frac{f_{(p)}^{\ast\ast}(t)}{\phi_{(p)}(t)}\right) ^{1/p}-\left(
\frac{f_{(p)}^{\ast}(t)}{\phi_{(p)}(t)}\right) ^{1/p}\preceq\left( \left|
\nabla f\right| _{(p)}^{\ast\ast}(t)\right) ^{1/p}, \] and (\ref{norma}) holds.
\section{Proof of Theorem \ref{teo}\label{secproof}}
Before going through the proof let us make a few useful remarks. Let $\left[ x\right] _{+}=\max(x,0),$ and let $f\geq0,$ then, for all $\lambda>0$, we have \begin{align} \int_{\{f>\lambda\}}\left( f(s)-\lambda\right) d\mu(s) & =\int [f(s)-\lambda]_{+}\,d\mu(s)=\int_{0}^{\infty}[f^{\ast}(s)-\lambda ]^{+}\,ds\label{cata}\\ & =\int_{0}^{\infty}\mu_{\lbrack f^{\ast}-\lambda]_{+}}(s)\,ds=\int_{\lambda }^{\infty}\mu_{f^{\ast}}(s)\,ds=\int_{\lambda}^{\Vert f\Vert_{\infty}}\mu _{f}(s)\,ds.\nonumber \end{align} Thus, inserting $\lambda=f^{\ast}(t)$ in (\ref{cata}), and taking into account that $f^{\ast}$ is decreasing, we obtain \begin{align*} t(f^{\ast\ast}(t)-f^{\ast}(t)) & =\int_{0}^{t}(f^{\ast}(x)-f^{\ast }(t))\,dx=\int_{0}^{\infty}[f^{\ast}(x)-f^{\ast}(t)]_{+}\,dx\\ & =\int_{\{f>f^{\ast}(t)\}}\left[ f(s)-f^{\ast}(t)\right] _{+}d\mu(s). \end{align*} In order to deal with $L^{p}$ norms, $p>1,$ we need to extended the formulae above. This will be achieved through the following variant of the binomial formula, whose proof will be provided at the end of this section.
\begin{lemma} \label{des}Let $p>1,$ and let $k\in\mathbb{N}$ be such that $k<p\leq k+1$. Then, for\ $a\geq b\geq0,$ \begin{equation} (a-b)^{p}\geq a^{p}-b^{p}-\sum_{j=1}^{k}\left( \begin{array} [c]{c} p\\ j \end{array} \right) b^{p-j}(a-b)^{j}, \label{des1} \end{equation} and \begin{equation} a^{p}+b^{p}+\sum_{j=1}^{k}\left( \begin{array} [c]{c} p\\ j \end{array} \right) b^{p-j}(a-b)^{j}\leq(c(p)a+b)^{p}, \label{des2} \end{equation} where $c(p)=2^{\frac{k+1}{p}-1}.$ \end{lemma}
We are now ready to give the proof of Theorem \ref{teo}.
\begin{proof} $1\rightarrow2.$ Suppose that $(S_{\phi}^{p})$ holds. We may assume without loss that $f$ is positive. Let $t>0;$ we will apply (\ref{desigualdad01}) to $\left[ f-f^{\ast}(t)\right] _{+}$. Observe that \[
\left\| \left[ f-f^{\ast}(t)\right] _{+}\right\| _{0}=\mu\{f>f^{\ast }(t)\}\leq t, \]
and, moreover, since $\int_{\{f=f^{\ast}(t)\}}\left| \nabla\left[
f(x)-f^{\ast}(t)\right] \right| dx=0,$ \[
\left\| \nabla\left[ f-f^{\ast}(t)\right] _{+}\right\| _{L^{p}}^{p}
=\int_{\{f>f^{\ast}(t)\}}\left( \left| \nabla f\right| ^{\ast}(s)\right) ^{p}ds. \] Therefore, \begin{align}
\left\| \left[ f-f^{\ast}(t)\right] _{+}\right\| _{p}^{p} & \leq\left\{
\phi(\left\| \left[ f-f^{\ast}(t)\right] _{+}\right\| _{0})\right\}
^{p}\left\| \nabla\left[ f-f^{\ast}(t)\right] _{+}\right\| _{L^{p}} ^{p}\nonumber\\
& \leq\phi(t)^{p}\int_{\{f>f^{\ast}(t)\}}\left( \left| \nabla f\right| ^{\ast}(s)\right) ^{p}ds\nonumber\\
& \leq t\phi(t)^{p}\left( \frac{1}{t}\int_{0}^{t}\left( \left| \nabla f\right| ^{\ast}(s)\right) ^{p}ds\right) \nonumber\\
& =t\phi(t)^{p}\left| \nabla f\right| _{(p)}^{\ast\ast}(t). \label{Sk} \end{align} Now, \begin{align}
\left\| \left[ f-f^{\ast}(t)\right] _{+}\right\| _{p}^{p} & =\int_{\{f>f^{\ast}(t)\}}\left( f(s)-f^{\ast}(t)\right) ^{p}d\mu (s)\nonumber\\ & \geq\int_{\{f>f^{\ast}(t)\}}\left( f^{p}(s)-f^{\ast}(t)^{p}\right) d\mu(s)\nonumber\\ & -\sum_{j=1}^{k}\left( \begin{array} [c]{c} p\\ j \end{array} \right) f^{\ast}(t)^{p-j}\int_{\{f>f^{\ast}(t)\}}(f(s)-f^{\ast}(t))^{j} d\mu(s)\text{ \ (by (\ref{des1}))}\nonumber\\ & =\int_{\{f_{(p)}>f_{(p)}^{\ast}(t)\}}\left( f_{(p)}(s)-f_{(p)}^{\ast }(t)\right) d\mu(s)\nonumber\\ & -\sum_{j=1}^{k}\left( \begin{array} [c]{c} p\\ j \end{array} \right) f^{\ast}(t)^{p-j}\int_{\{f>f^{\ast}(t)\}}(f(s)-f^{\ast}(t))^{j} d\mu(s)\nonumber\\ & =t\left( f_{(p)}^{\ast\ast}(t)-f_{(p)}^{\ast}(t)\right) -\sum_{j=1} ^{k}\left( \begin{array} [c]{c} p\\ j \end{array} \right) f^{\ast}(t)^{p-j}\int_{\{f>f^{\ast}(t)\}}(f(s)-f^{\ast}(t))^{j} d\mu(s). \label{paso1} \end{align} We estimate each of the integrals in the sum using H\"{o}lder's inequality as follows, \begin{align} \int_{\{f>f^{\ast}(t)\}}(f(s)-f^{\ast}(t))^{j}d\mu(s) & \leq\left( \int_{\{f>f^{\ast}(t)\}}(f(s)-f^{\ast}(t))^{p}d\mu(s)\right) ^{\frac{j}{p} }\left( \int_{\{f>f^{\ast}(t)\}}d\mu(s)\right) ^{\frac{p-j}{p}}\nonumber\\ & =\left( \int_{\{f>f^{\ast}(t)\}}(f(s)-f^{\ast}(t))^{p}d\mu(s)\right) ^{\frac{j}{p}}\left( \mu_{f}(f^{\ast}(t))\right) ^{\frac{p-j}{p}}\nonumber\\ & \leq\left( \int_{\{f>f^{\ast}(t)\}}(f(s)-f^{\ast}(t))^{p}d\mu(s)\right) ^{\frac{j}{p}}t^{\frac{p-j}{p}}\nonumber\\
& =\left\| \left[ f-f^{\ast}(t)\right] _{+}\right\| _{p}^{j}t^{\frac {p-j}{p}}\nonumber\\
& \leq\phi(t)^{j}\left( \left| \nabla f\right| _{(p)}^{\ast\ast }(t)\right) ^{\frac{j}{p}}t^{\frac{j}{p}}t^{\frac{p-j}{p}}\text{ \ \ (by (\ref{Sk}))}\nonumber\\
& =t\phi(t)^{j}\left( \left| \nabla f\right| _{(p)}^{\ast\ast}(t)\right) ^{\frac{j}{p}}. \label{paso2} \end{align} Combining (\ref{paso1}) and (\ref{paso2}) we get \begin{align*}
\left\| \left[ f-f^{\ast}(t)\right] _{+}\right\| _{p}^{p} & =t\left( f_{(p)}^{\ast\ast}(t)-f_{(p)}^{\ast}(t)\right) -\sum_{j=1}^{p-1}\left( \begin{array} [c]{c} p\\ j \end{array} \right) f^{\ast}(t)^{p-j}\int_{\{f>f^{\ast}(t)\}}(f^{\ast}(t)-f(s))^{j} d\mu(s)\\ & \geq t\left( f_{(p)}^{\ast\ast}(t)-f_{(p)}^{\ast}(t)\right) -t\left( \sum_{j=1}^{p-1}\left( \begin{array} [c]{c} p\\ j \end{array}
\right) f^{\ast}(t)^{p-j}\phi(t)^{j}\left( \left| \nabla f\right| _{(p)}^{\ast\ast}(t)\right) ^{\frac{j}{p}}\right) . \end{align*} Therefore, we see that \begin{align*}
t\left( f_{(p)}^{\ast\ast}(t)-f_{(p)}^{\ast}(t)\right) & \leq\left\|
\left[ f-f^{\ast}(t)\right] _{+}\right\| _{p}^{p}+t\left( \sum_{j=1} ^{k}\left( \begin{array} [c]{c} p\\ j \end{array}
\right) f^{\ast}(t)^{p-j}\phi(t)^{j}\left( \left| \nabla f\right| _{(p)}^{\ast\ast}(t)\right) ^{\frac{j}{p}}\right) \\
& \leq t\phi(t)^{p}\left| \nabla f\right| _{(p)}^{\ast\ast}(t)+t\left( \sum_{j=1}^{k}\left( \begin{array} [c]{c} p\\ j \end{array}
\right) f^{\ast}(t)^{p-j}\phi(t)^{j}\left( \left| \nabla f\right| _{(p)}^{\ast\ast}(t)\right) ^{\frac{j}{p}}\right) \text{\ \ \ \ (by (\ref{Sk}))}\\
& =t\phi_{(p)}(t)\left( \left| \nabla f\right| _{(p)}^{\ast\ast} (t)+\sum_{j=1}^{k}\left( \begin{array} [c]{c} p\\ j \end{array} \right) \left( \frac{f_{(p)}^{\ast}(t)}{\phi_{(p)}(t)}\right) ^{\frac
{p-j}{p}}\left( \left| \nabla f\right| _{(p)}^{\ast\ast}(t)\right) ^{\frac{j}{p}}\right) . \end{align*} Consequently, \begin{equation}
\frac{f_{(p)}^{\ast\ast}(t)-f_{(p)}^{\ast}(t)}{\phi_{(p)}(t)}\leq\left|
\nabla f\right| _{(p)}^{\ast\ast}(t)+\sum_{j=1}^{k}\left( \begin{array} [c]{c} p\\ j \end{array} \right) \left( \frac{f_{(p)}^{\ast}(t)}{\phi_{(p)}(t)}\right) ^{\frac
{p-j}{p}}\left( \left| \nabla f\right| _{(p)}^{\ast\ast}(t)\right) ^{\frac{j}{p}}. \label{norma1} \end{equation} We can rewrite (\ref{norma1}) as \begin{align*}
\frac{f_{(p)}^{\ast\ast}(t)-f_{(p)}^{\ast}(t)}{\phi_{(p)}(t)} & \leq\left|
\nabla f\right| _{(p)}^{\ast\ast}(t)+\sum_{j=1}^{k}\left( \begin{array} [c]{c} p\\ j \end{array} \right) \left( \frac{f_{(p)}^{\ast}(t)}{\phi_{(p)}(t)}\right) ^{\frac
{p-j}{p}}\left( \left| \nabla f\right| _{(p)}^{\ast\ast}(t)\right) ^{\frac{j}{p}}+\frac{f_{(p)}^{\ast}(t)}{\phi_{(p)}(t)}-\frac{f_{(p)}^{\ast }(t)}{\phi_{(p)}(t)}\\
& =\left( 2^{\frac{k+1}{p}-1}\left( \left| \nabla f\right| _{(p)} ^{\ast\ast}(t)\right) ^{1/p}+\left( \frac{f_{(p)}^{\ast}(t)}{\phi_{(p)} (t)}\right) ^{1/p}\right) ^{p}-\frac{f_{(p)}^{\ast}(t)}{\phi_{(p)}(t)}\text{ \ \ (by (\ref{des2}))} \end{align*} Hence \[ \frac{f_{\left( p\right) }^{\ast\ast}(t)}{\phi_{(p)}(t)}\leq\left(
2^{\frac{k+1}{p}-1}\left( \left| \nabla f\right| _{(p)}^{\ast\ast }(t)\right) ^{1/p}+\left( \frac{f_{(p)}^{\ast}(t)}{\phi_{(p)}(t)}\right) ^{1/p}\right) ^{p}, \] yielding \[ \left( \frac{f_{\left( p\right) }^{\ast\ast}(t)}{\phi_{(p)}(t)}\right)
^{1/p}\leq2^{\frac{k+1}{p}-1}\left( \left| \nabla f\right| _{(p)}^{\ast \ast}(t)\right) ^{1/p}+\left( \frac{f_{(p)}^{\ast}(t)}{\phi_{(p)} (t)}\right) ^{1/p}. \] Summarizing, we have obtained \[ \left( \frac{f_{\left( p\right) }^{\ast\ast}(t)}{\phi_{(p)}(t)}\right) ^{1/p}-\left( \frac{f_{(p)}^{\ast}(t)}{\phi_{(p)}(t)}\right) ^{1/p}
\leq2^{\frac{k+1}{p}-1}\left( \left| \nabla f\right| _{(p)}^{\ast\ast }(t)\right) ^{1/p}. \] $2\rightarrow3.$ Once again we use the elementary inequality \[ \left( x^{p}-y^{p}\right) \leq p\left( x-y\right) \left( x^{p-1} +y^{p-1}\right) ,\text{ \ \ \ (}x\geq y\geq0), \] with $x=\left( f_{\left( p\right) }^{\ast\ast}(t)\right) ^{1/p}$ and $y=\left( f_{(p)}^{\ast}(t)\right) ^{1/p}$. We obtain, \begin{align*} f_{\left( p\right) }^{\ast\ast}(t)-f_{(p)}^{\ast}(t) & \leq p\left( \left( f_{\left( p\right) }^{\ast\ast}(t)\right) ^{1/p}-\left( f_{(p)}^{\ast}(t)\right) ^{1/p}\right) \left( \left( f_{(p)}^{\ast\ast }(t)\right) ^{\frac{p-1}{p}}+\left( f_{(p)}^{\ast}(t)\right) ^{\frac {p-1}{p}}\right) \\
& \leq p2^{\frac{k+1}{p}-1}\phi(t)\left( \left| \nabla f\right| _{(p)}^{\ast\ast}(t)\right) ^{1/p}\left( \left( f_{(p)}^{\ast\ast }(t)\right) ^{\frac{p-1}{p}}+\left( f_{(p)}^{\ast}(t)\right) ^{\frac {p-1}{p}}\right) \text{ \ by (\ref{norma})}\\
& \leq p2^{\frac{k+1}{p}-1}\phi(t)\left( \left| \nabla f\right| _{(p)}^{\ast\ast}(t)\right) ^{1/p}\left( 2\left( f_{(p)}^{\ast\ast }(t)\right) ^{\frac{p-1}{p}}\right) . \end{align*}
Consequently, \[ \frac{1}{p}\left( f_{(p)}^{\ast\ast}(t)\right) ^{\frac{1}{p}-1}\left(
f_{\left( p\right) }^{\ast\ast}(t)-f_{(p)}^{\ast}(t)\right) \leq p2^{\frac{k+1}{p}}\phi(t)\left( \left| \nabla f\right| _{(p)}^{\ast\ast }(t)\right) ^{\frac{1}{p}}. \] Now observe that \[ \frac{1}{p}\left( f_{(p)}^{\ast\ast}(t)\right) ^{\frac{1}{p}-1}\left( \frac{f_{\left( p\right) }^{\ast\ast}(t)-f_{(p)}^{\ast}(t)}{t}\right) =-\frac{\partial}{\partial t}\left( \frac{1}{t}\int_{0}^{t}\left( f^{\ast }(s)\right) ^{p}ds\right) ^{1/p}. \] $3\rightarrow1.$
Let $\Omega\subset\subset X,$ and let $f\in Lip_{0}(\Omega),$ then, for $t=\mu(\Omega),$ we have \[ f_{\left( p\right) }^{\ast\ast}(t)=\frac{1}{t}\int_{0}^{t}\left( f^{\ast
}(t)\right) ^{p}dt=\frac{1}{t}\left\| f\right\| _{p}^{p} \] and, similarly, \[
\left| \nabla f\right| _{(p)}^{\ast\ast}(t)=\frac{1}{t}\left\| \left|
\nabla f\right| \right\| _{p}^{p}. \] Since \[
f_{(p)}^{\ast}(\mu(\Omega))=\inf_{x\in\Omega}\left| f(x)\right| ^{p}=0, \] the inequality (\ref{norma01}) becomes \[
\frac{1}{t}\left\| f\right\| _{p}^{p}\leq p2^{\frac{k+1}{p}}\phi(\mu
(\Omega))\frac{1}{t}\left\| \left| \nabla f\right| \right\| _{p}\left\|
f\right\| _{p}^{p-1}, \] which is (\ref{desigualdad01}), up to constants. \end{proof}
To complete the proof it remains to prove Lemma \ref{des}.
\begin{proof} (of Lemma \ref{des}) We prove (\ref{des1}). Towards this end let us define \[ f(x)=(x-b)^{p}-x^{p}+b^{p}+\sum_{j=1}^{k}\left( \begin{array} [c]{c} p\\ j \end{array} \right) b^{p-j}(x-b)^{j},\text{ \ \ \ \ (}x\geq b). \] An elementary computation shows that $f(b)=\frac{\partial}{\partial x}f(b)=\frac{\partial^{k-1}}{\partial x}f(b)=0.$ Moreover, since \[ \frac{\partial^{k}}{\partial x}f(x)=p(p-1)\ldots(p-k+1)\left( (x-b)^{p-k} -x^{p-k}+b^{p-k}\right) , \] and $0<p-k\leq1,$ we see that \[ \left( x-b\right) ^{p-k}-x^{p-k}+b^{p-k}\geq0, \] consequently, \[ f(x)\geq f(b)=0. \] To see (\ref{des2}) let us write $a=xb$ $(x\geq1).\ $We would like to show that \[ g(x)=(c(p)x+1)^{p}-x^{p}-1-\sum_{j=1}^{k}\left( \begin{array} [c]{c} p\\ j \end{array} \right) (x-1)^{j}\geq0. \] An easy computation shows that $g(1)\geq0,\frac{\partial}{\partial x} g(1)\geq0,\cdots\frac{\partial^{k-1}}{\partial x}g(1)\geq0,$ and $\frac{\partial^{k}}{\partial x}g(1)\geq0.$ Therefore, it will be enough to prove that $\frac{\partial^{k+1}}{\partial x}g(x)\geq0.$ Again, by computation, we find that \[ \frac{\partial^{k+1}}{\partial x}g(x)=p(p-1)\ldots(p-k+1)(p-k)\left( c(p)^{k+1}\left( c(p)x+1\right) ^{p-k-1}-x^{p-k-1}\right) . \] Therefore the desired result will follow if we show that \[ c(p)^{k+1}\left( c(p)x+1\right) ^{p-k-1}-x^{p-k-1}\geq0. \] Since $p-k-1<0,$ this amounts to show \[ \frac{c(p)^{k+1}}{\left( c(p)x+1\right) ^{k+1-p}}\geq\frac{1}{x^{k+1-p} }\Leftrightarrow\frac{c(p)^{\frac{k+1}{k+1-p}}}{c(p)x+1}\geq\frac{1} {x}\Leftrightarrow xc(p)\left( c(p)^{\frac{k+1}{k+1-p}}-1\right) \geq1. \] But since \[ c(p)^{\frac{k+1}{k+1-p}}-1\geq1\Leftrightarrow c(p)\geq2^{\frac{k+1}{p}-1}, \] the desired result follows. \end{proof}
\section{Final Remarks\label{secrem}}
In this section we show the explicit connection of our rearrangement inequalities with the classical Sobolev inequalities, the Nash and Faber-Krahn inequalities and point out possible directions for future research. In particular, using Coulhon inequalities we will show a direct approach to some self-improving properties of Sobolev inequalities for $p>1$.
\subsection{Nash Inequalities}
We start by giving a rearrangement characterization of the Nash type inequalities. It was shown in \cite{bakr} (cf. also \cite{cou1}), that the $(S_{\phi}^{p})$ conditions are equivalent to Nash type inequalities. As a consequence, the results of this paper give a characterization of Nash inequalities in terms of rearrangements which we shall now describe.
We first observe that, with some trivial changes, one can adapt the proof of Proposition 2.4 in \cite{cou1} (case $p=2)$ to obtain the following equivalence (for Nash inequalities for $p>1)$
\begin{proposition} Let $p>1.$ The following are inequalities are equivalent up to multiplicative constants
(i) $(S_{\phi}^{p})$ holds
(ii) There exist positive constants $c_{1}$ and $c_{2}$ such that \[
\left\| f\right\| _{p}\leq c_{1}\phi\left( c_{2}\left( \frac{\left\|
f\right\| _{1}}{\left\| f\right\| _{p}}\right) ^{\frac{p}{p-1}}\right)
\left\| \left| \nabla f\right| \right\| _{p} \] for all $f\in Lip_{0}(X)$. \end{proposition}
The case $\phi(t)=t^{1/n}$, $p=2,$ corresponds to the classical Nash inequality \[
\left\| f\right\| _{2}^{1+2/n}\leq c\left\| f\right\| _{1}^{2/n}\left\|
\left| \nabla f\right| \right\| _{2}. \] Therefore, by Theorem \ref{teo}, Nash's inequality is equivalent to \[ \left( \frac{f_{\left( 2\right) }^{\ast\ast}(t)}{t^{2/n}}\right) ^{1/2}-\left( \frac{f_{(2)}^{\ast}(t)}{t^{2/n}}\right) ^{1/2}\preceq\left(
\left| \nabla f\right| _{(2)}^{\ast\ast}(t)\right) ^{1/2},\text{ }f\in Lip_{0}(\mathbb{R}^{n}). \]
\subsection{Classical Sobolev Inequalities}
We now consider a new approach, via rearrangement inequalities, of the known (cf. \cite{cou}, \cite{bakr}, \cite{cou1} and the references therein) equivalence between the classical Euclidean Sobolev inequalities and Coulhon inequalities. The case $p=1$ of (\ref{norma}) gives us the inequality (\ref{berta}), whose connection to Sobolev inequalities was discussed extensively elsewhere (cf. \cite{mamiadv}).
Let us consider the case $1\leq p<n,$ $\frac{1}{\bar{p}}=\frac{1}{p}-\frac {1}{n}.$ Let $\phi(t)=t^{1/n}.$ We shall denote the corresponding $(S_{\phi }^{p})$ condition by $(S_{n}^{p}).$ Our aim is to prove that $(S_{n}^{p})$ implies the classical Sobolev inequality \[
\left\| f\right\| _{L(\bar{p},p)}\preceq\left\| \nabla f\right\| _{L^{p} },f\in Lip_{0}(\mathbb{R}^{n}), \] where for $1\leq r<\infty,1\leq q\leq\infty,$ \[
\left\| f\right\| _{L(r,q)}=\left\{ \int_{0}^{\infty}\left( f^{\ast }(t)t^{\frac{1}{r}}\right) ^{q}\frac{dt}{t}\right\} ^{1/q}. \] By a well known result, apparently originally due to Maz'ya, weak type Sobolev inequalities self-improve to strong type Sobolev inequalities (cf. \cite{bakr}, \cite{mmp}, and the references therein). We shall discuss this self-improvement in detail in the next subsection. Taking this fact for granted, it will be enough to show that $(S_{n}^{p})$ implies the weak type Sobolev inequality \begin{equation}
\left\| f\right\| _{L(\bar{p},\infty)}\preceq\left\| \nabla f\right\| _{L^{p}},f\in Lip_{0}(\mathbb{R}^{n}), \label{tio} \end{equation} where \[
\left\| f\right\| _{L(\bar{p},\infty)}=\sup_{t}\{f^{\ast}(t)t^{1/\bar{p} }\}. \] To prove (\ref{tio}) let us first recall that since $\bar{p}>1,$ for $f\in Lip_{0}(\mathbb{R}^{n})$ we have (cf. \cite{bs}, \cite{bmr}), \[
\left\| f\right\| _{L(\bar{p},\infty)}\simeq\sup_{t}\{\left( f^{\ast\ast }(t)-f^{\ast}(t)\right) t^{1/\bar{p}}\}. \] We have shown above that $(S_{n}^{p})$ implies (\ref{norma}); therefore it follows that \begin{align*}
\left( f_{p}^{\ast\ast}(t)\right) ^{1/p}-f^{\ast}(t) & \preceq t^{1/n}\left( \left| \nabla f\right| _{p}^{\ast\ast}(t)\right) ^{1/p}\\
& =t^{1/n-1/p}\left\{ \int_{0}^{t}\left| \nabla f\right| ^{\ast} (s)^{p}ds\right\} ^{1/p}. \end{align*} Combining the last inequality with Jensen's inequality we get \begin{align} f^{\ast\ast}(t)-f^{\ast}(t) & \leq\left( f_{p}^{\ast\ast}(t)\right) ^{1/p}-f^{\ast}(t)\nonumber\\
& \preceq t^{1/n-1/p}\left\{ \int_{0}^{t}\left| \nabla f\right| ^{\ast }(s)^{p}ds\right\} ^{1/p}. \label{abaco} \end{align} Summarizing, for $f\in Lip_{0}(\mathbb{R}^{n}),$ \begin{align*}
\left\| f\right\| _{L(\bar{p},\infty)} & \simeq\sup_{t}\{\left( f^{\ast\ast}(t)-f^{\ast}(t)\right) t^{1/\bar{p}}\}\\
& \preceq\sup_{t}\left\{ \int_{0}^{t}\left| \nabla f\right| ^{\ast} (s)^{p}ds\right\} ^{1/p}\\
& \leq\left\| \left| \nabla f\right| \right\| _{p}, \end{align*} as we wished to show.
In this next section we shall discuss in detail the case $p=n,$ and show the self-improvement of Sobolev-Coulhon inequalities.
\subsection{Self-improvement}
There are several known mechanisms to show the self-improvement of Sobolev inequalities. Here we choose to adapt a variant the method apparently first developed by Maz'ya-Talenti (cf. \cite{mamiadv} for a generalized version) using differential inequalities, focussing on the Euclidean case.
For a domain $\Omega\subset\mathbb{R}^{n},$ we have (cf. \cite{mmp} for the classical Euclidean case or \cite{mamiadv} for the general metric space case) the following formulation of the Polya-Szeg\"{o} principle \begin{equation}
\left( \int_{0}^{\left| \Omega\right| }\left( s^{1-\frac{1}{n}}\left( -f^{\ast}\right) ^{^{\prime}}(s)\right) ^{p}ds\right) ^{1/p}\preceq\left(
\int_{0}^{\left| \Omega\right| }\left( \left| \nabla f\right| ^{\ast }(s)\right) ^{p}ds\right) ^{1/p},\text{ }p\geq1,\text{ }f\in Lip_{0} (\Omega). \label{negada} \end{equation}
To use this powerful inequality we now reformulate (\ref{abaco}) as an elementary differential inequality. For $f\in Lip_{0}(\Omega),$ let $F(t):=\left( f^{\ast\ast}(t)-f^{\ast}(t)\right) ^{p}$ $t^{1-\frac{p}{n} },1\leq p<n.$ Then $F$ is a positive, absolutely continuous function (cf. \cite{leoni}), which by (\ref{abaco}) satisfies \[
F(t)\preceq\int_{0}^{t}\left( \left| \nabla f\right| ^{\ast}(s)\right) ^{p}ds. \] It follows that $F(0)=0,$ and therefore we can write $F(t)=\int_{0} ^{t}F^{^{\prime}}(s)ds,t>0.$ We estimate $F$ through this representation. By direct computation, \begin{align*} F^{^{\prime}}(t) & =(1-\frac{p}{n})t^{-\frac{p}{n}}[f^{\ast\ast}(t)-f^{\ast }(t)]^{p}+t^{1-\frac{p}{n}}p[f^{\ast\ast}(t)-f^{\ast}(t)]^{p-1}\left[ \left( f^{\ast\ast}(t)\right) ^{\prime}-\left( f^{\ast}\right) ^{^{\prime} }(t)\right] \\ & =(1-\frac{p}{n})t^{-\frac{p}{n}}[f^{\ast\ast}(t)-f^{\ast}(t)]^{p} +t^{1-\frac{p}{n}}p[f^{\ast\ast}(t)-f^{\ast}(t)]^{p-1}\left[ (-1)\left( \frac{f^{\ast\ast}(t)-f^{\ast}(t)}{t}\right) -\left( f^{\ast}\right) ^{^{\prime}}(t)\right] \\ & =(1-\frac{p}{n}-p)t^{-\frac{p}{n}}[f^{\ast\ast}(t)-f^{\ast}(t)]^{p} +t^{1-\frac{p}{n}}p[f^{\ast\ast}(t)-f^{\ast}(t)]^{p-1}\left( -f^{\ast }\right) ^{^{\prime}}(t). \end{align*} The previous computation, combined with the fact that $F(t)$ is positive, yields \[
(-1)(1-\frac{p}{n}-p)\int_{0}^{\left| \Omega\right| }[f^{\ast\ast
}(t)-f^{\ast}(t)]^{p}t^{-\frac{p}{n}}dt\leq p\int_{0}^{\left| \Omega\right| }t^{1-\frac{p}{n}}[f^{\ast\ast}(t)-f^{\ast}(t)]^{p-1}\left( -f^{\ast}\right) ^{^{\prime}}(t)dt. \] H\"{o}lder's inequality and (\ref{negada}) yields \begin{align*}
& \int_{0}^{\left| \Omega\right| }\left( [f^{\ast\ast}(t)-f^{\ast }(t)]t^{\frac{1}{\bar{p}}}\right) ^{p}\frac{dt}{t}\\
& =\int_{0}^{\left| \Omega\right| }t^{-\frac{p}{n}}[f^{\ast\ast} (t)-f^{\ast}(t)]^{p}dt\\
& \leq\frac{p}{(\frac{p}{n}+p-1)}\int_{0}^{\left| \Omega\right| } t^{1-\frac{p}{n}}[f^{\ast\ast}(t)-f^{\ast}(t)]^{p-1}\left( -f^{\ast}\right) ^{^{\prime}}(t)dt\\
& =\frac{p}{(p-1+\frac{p}{n})}\int_{0}^{\left| \Omega\right| }([f^{\ast \ast}(t)-f^{\ast}(t)]^{p-1}t^{\frac{1-p}{n}})(t^{1-\frac{1}{n}}\left( f^{\ast}\right) ^{^{\prime}}(t))dt\\
& \preceq\left( \int_{0}^{\left| \Omega\right| }\left( t^{\frac{1-p}{n} }[f^{\ast\ast}(t)-f^{\ast}(t)]^{p-1}\right) ^{\frac{p}{p-1}}dt\right)
^{(p-1)/p}\left( \int_{0}^{\left| \Omega\right| }\left( t^{1-\frac{1}{n} }\left( -f^{\ast}\right) ^{^{\prime}}(t)\right) ^{p}dt\right) ^{1/p}\\
& =c_{n,p}\left( \int_{0}^{\left| \Omega\right| }\left( t^{\frac{1} {\bar{p}}}[f^{\ast\ast}(t)-f^{\ast}(t)]\right) ^{p}\frac{dt}{t}\right)
^{1/p^{\prime}}\left\| \left| \nabla f\right| \right\| _{p}. \end{align*}
Consequently, assuming \textit{apriori} that $\int_{0}^{\left| \Omega\right| }\left( t^{\frac{1}{p}-\frac{1}{n}}[f^{\ast\ast}(t)-f^{\ast}(t)]^{p}\right) \frac{dt}{t}<\infty,$ we see that \[
\left\{ \int_{0}^{\left| \Omega\right| }\left( [f^{\ast\ast}(t)-f^{\ast }(t)]t^{\frac{1}{\bar{p}}}\right) ^{p}\frac{dt}{t}\right\} ^{1/p}
\preceq\left\| \left| \nabla f\right| \right\| _{p}. \] For $f\in Lip_{0}(\Omega)$ all the formal calculations above can be easily justified and we find the sharp Sobolev inequality \[
\left\| f\right\| _{L(\bar{p},p)}\simeq\left\{ \int_{0}^{\left|
\Omega\right| }\left( [f^{\ast\ast}(t)-f^{\ast}(t)]t^{\frac{1}{\bar{p}}
}\right) ^{p}\frac{dt}{t}\right\} ^{1/p}\preceq\left\| \left| \nabla f\right| \right\| _{p}. \]
Let us note that the previous calculation also works for $p=n.$ In this case we should let $\frac{1}{\bar{p}}=0$ and we obtain \[
\left\{ \int_{0}^{\left| \Omega\right| }\left( f^{\ast\ast}(t)-f^{\ast
}(t)\right) ^{n}\frac{dt}{t}\right\} ^{1/n}\preceq\left\| \left| \nabla f\right| \right\| _{n}. \] In this case the left hand side should be re-interpreted as the *norm* of $L(\infty,n)$, the space defined by the condition (cf. \cite{bmr}) \[
\left\{ \int_{0}^{\left| \Omega\right| }\left( f^{\ast\ast}(t)-f^{\ast }(t)\right) ^{n}\frac{dt}{t}\right\} ^{1/n}<\infty. \] It was shown in \cite{bmr} that this condition implies the classical exponential integrability results of Trudinger and Brezis-Wainger.
Note that the self-improvement for general $\phi,$ which we have not discussed here, will involve the $p-$Lorentz $\Lambda_{\phi}$ spaces (for further related discussions we refer to \cite{mamicon}).
\subsection{The Morrey-Sobolev theorem}
The connection between rearrangement inequalities and the Morrey-Sobolev theorem (i.e. the case $p>n$ of the Sobolev embedding theorem$)$ has been treated at great length in our recent article \cite{mamiarxiv}. We consider here the corresponding Coulhon variant, but, once again for the sake of brevity, and to avoid technical complications, we shall only sketch the details for Sobolev spaces $W_{0}^{1}(Q)$ on the cube $Q=(0,1)^{n}$.
In this section we let $p>n,$ then $\frac{1}{\bar{p}}=\frac{1}{p}-\frac{1} {n}<0.$ Using the fact that $(-f^{\ast\ast}(t))^{\prime}=\frac{f^{\ast\ast }(t)-f^{\ast}(t)}{t}$ we can integrate the inequality (\ref{abaco}) to obtain \begin{align*} f^{\ast\ast}(0)-f^{\ast\ast}(1) & =\int_{0}^{1}\frac{f^{\ast\ast} (t)-f^{\ast}(t)}{t}dt\\
& \preceq\int_{0}^{1}t^{-\frac{1}{\bar{p}}}\left( \int_{0}^{t}\left| \nabla f\right| ^{\ast}(s)^{p}ds\right) ^{1/p}\frac{dt}{t}\\
& \leq\left\| \left| \nabla f\right| \right\| _{p}\int_{0}^{1} t^{-\frac{1}{\bar{p}}-1}dt\\
& =c_{p}\left\| \left| \nabla f\right| \right\| _{p}. \end{align*} Extending the inequalities we have obtained in this note through the use of signed rearrangements, and using an extension of a scaling argument that apparently goes back to \cite{garsia} (we must refer to \cite[pag. 3]{mamiarxiv} for more details) we find that given $x,y\in Q,$ \begin{align*}
\left| f(x)-f(y)\right| & \preceq\left\| \left| \nabla f\right|
\right\| _{p}\left| x-y\right| ^{n(\frac{1}{n}-\frac{1}{\bar{p}})}\\
& =\left\| \left| \nabla f\right| \right\| _{p}\left| x-y\right| ^{1-\frac{n}{p}}. \end{align*}
\subsection{Further connections}
In this section we mention some problems and possible projects we find of some interest.
In the literature there are other definitions of the notion of gradient in the metric setting (e.g. \cite{haj} and the references therein) and it remains an open problem to fully explore the connections with our development here\footnote{For partial results (restricted to doubling measures) connecting different notions of the gradient with rearrangement inequalities we refer to \cite{badr}, \cite{kalis} and the references therein.}.
We hope to discuss the connection between isoperimetry, rearrangements and discrete Sobolev inequalities elsewhere.
For aficionados of interpolation theory we should note that, while there are obvious connections between the$\ (S_{\phi}^{p})$ conditions and the $J-$method of interpolation or perhaps, even more appropriately, with the corresponding version of this method for the $E-$method of approximation (cf. \cite{jm}), we could not find a treatment in the literature. Such considerations are somehow implicit in the approach given in \cite{bakr}, and more explicitly in the unpublished manuscript \cite{cm}. Likewise, the $\phi$ inequalities that appear in the formulation of Nash's inequality above\footnote{There is an extensive literature on $\phi$ inequalities (cf. \cite{bcr}, and the references therein).}, appear directly related to the $K/J$ inequalities of the extrapolation theory of \cite{jm}.
Still another direction for future research is to develop in more detail the connection of the results in this paper and the work of Xiao \cite{xiao} on the $p-$Faber-Krahn inequality.
Finally in this section we have discussed only the Euclidean case. It will be of interest to develop a detailed treatment of these applications in the general metric case.
\end{document} | arXiv |
# A note on conservative fields ${ }^{1}$
3 November 2005
## Conservative fields
A smooth vector field $\mathbf{F}$ is conservative iff there exists a smooth scalar field $\phi$ such that
$$
\mathbf{F}=\operatorname{grad} \phi
$$
$\phi$ is often called a potential for $\mathbf{F}$.
Since curl grad $\phi=0$ for any scalar field $\phi, \operatorname{curl} \mathbf{F}=0$ is a neccessary condition for $\mathbf{F}$ to be conservative. It isn't suffiecient, however, this is something we will return to, but, for now, we notice that it makes it easy to spot fields that aren't conservative, for example, if $\mathbf{F}=(-y, x, 0)$ then
$$
\nabla \times \mathbf{F}=\left(\begin{array}{l}
0 \\
0 \\
2
\end{array}\right)
$$
On the other hand, it is easy to see that $\mathbf{F}=\mathbf{r}$ is conservative because
$$
\nabla \frac{r^{2}}{2}=\left(\begin{array}{l}
x \\
y \\
z
\end{array}\right)
$$
and therefore $\phi=r^{2} / 2$ is a potential for $\mathbf{F}$. Of course, curl $\mathbf{r}=0$.
## Path independent fields
A field $\mathbf{F}$ is path independent if, for any two points $p_{1}$ and $p_{2}$ the line integral along any path between those points has the same value.
In other words, a field is path independent if the line integral doesn't depend on the path it is taken along. Obviously, the integral around a closed path should be zero for a path independent field and, in fact we can state a lemma:
Lemma: A smooth vector field $\mathbf{F}$ is path independent iff
$$
\oint_{c} \mathbf{F} \cdot \mathrm{dl}=0
$$
for any closed curve $c .^{2}$
${ }^{1}$ Conor Houghton, [email protected], see also http://www.maths.tcd.ie/ houghton/231
${ }^{2}$ The integral sign with a circle on it is a standard notation for an integral around a closed path.
Proof: For a path independent $\mathbf{F}$ choose any two points on a closed curve $c$ and label the two curves from $p_{1}$ to $p_{2}$ as $c_{1}$ and $c_{2}$ so that $c=c_{1}-c_{2}$. Now, by path independence,
$$
\int_{c_{1}} \mathbf{F} \cdot \mathrm{dl}=\int_{c_{2}} \mathbf{F} \cdot \mathrm{dl}
$$
and so
$$
\oint_{c_{1}-c_{2}} \mathbf{F} \cdot \mathbf{d} \mathbf{l}=\oint_{c} \mathbf{F} \cdot \mathbf{d} \mathbf{l}=0
$$
Conversely, two difference paths between two points $p_{1}$ and $p_{2}$ can be subtacted from each other to give a closed path $c=c_{1}-c_{2}$ so
$$
0=\oint_{c} \mathbf{F} \cdot \mathrm{dl}=\oint_{c_{1}-c_{2}} \mathbf{F} \cdot \mathbf{d} \mathbf{l}
$$
implies
$$
\int_{c_{1}} \mathbf{F} \cdot \mathbf{d l}=\int_{c_{2}} \mathbf{F} \cdot \mathbf{d} \mathbf{l}
$$
## Conservative fields are path independent
Theorem: On a connected domain, a smooth vector field $\mathbf{F}$ is path-independent iff it is conservative.
Proof: First, given a conservative field $\mathbf{F}=\nabla \phi$ for some smooth $\phi$ consider the line integral along some curve $c$ from $p_{1}$ to $p_{2}$. Let $\mathbf{r}(t)$ be a parameterization of the curve with $p_{1}=\mathbf{r}\left(t_{1}\right)$ and $p_{2}=\mathbf{r}\left(t_{2}\right)$. Now,
$$
\int_{c} \mathbf{F} \cdot \mathbf{d} \mathbf{l}=\int_{t_{1}}^{t_{2}} \mathbf{F} \cdot \frac{d \mathbf{r}}{d t} d t
$$
$$
\begin{aligned}
& =\int_{t_{1}}^{t_{2}} \nabla \phi \cdot \frac{d \mathbf{r}}{d t} d t \\
& =\int_{t_{1}}^{t_{2}}\left(\frac{\partial \phi}{d x} \frac{d x}{d t}+\frac{\partial \phi}{d y} \frac{d y}{d t}+\frac{\partial \phi}{d y} \frac{d y}{d t}\right) d t
\end{aligned}
$$
where we have used the fact that $\mathbf{F}$ is conservative and we have written the dot product out explicitely. Now, we use the chain rule to rewrite the dot product
$$
\frac{\partial \phi}{d x} \frac{d x}{d t}+\frac{\partial \phi}{d y} \frac{d y}{d t}+\frac{\partial \phi}{d y} \frac{d y}{d t}=\frac{d \phi}{d t}
$$
where $\phi(x, y, z)$ is a function of $t$ along the curve through the $t$ dependence of $x, y$ and $z$ : $\phi(t)=\phi(x(t), y(t), z(t))$. This gives
$$
\left.\int_{c} \mathbf{F} \cdot \mathbf{d} \mathbf{l}=\int_{t_{1}}^{t_{2}} \frac{d \phi}{d t} d t=\phi(t)\right]_{t_{1}}^{t_{2}}=\phi\left(t_{2}\right)-\phi\left(t_{1}\right)
$$
which depends only on the value of the $\phi$ at the begining and end of the curve and is therefore path independent.
To go the other way, for give path independent $\mathbf{F}$ and some fixed point $p$ let
$$
\phi(\mathbf{r})=\int_{c} \mathbf{F} \cdot \mathrm{dl}
$$
where $c$ is a path from $p$ to $\mathbf{r}$. Since the field is path independent $\phi$ is well defined. It clearly depends on the choice of the reference point $p$ and so $\phi$ isn't unique, however, two choices only differ by an overall constant. We now want to prove that $\mathbf{F}=\nabla \phi$. We begin by proving
$$
F_{1}=\frac{\partial \phi}{\partial x}
$$
where $\mathbf{F}=F_{1} \mathbf{i}+F_{2} \mathbf{j}+F_{3} \mathbf{k}$. If this hold it should hold for all paths, so lets choose a path $c=c_{1}+c_{2}$ where $c_{1}$ goes from $p$ to a point $\left(x^{\prime}, y, z\right)$ and $c_{2}$ runs from $\left(x^{\prime}, y, z\right)$ to $(x, y, z)$.
Now,
$$
\begin{aligned}
\phi(\mathbf{r}) & =\int_{c} \mathbf{F} \cdot \mathbf{d} \mathbf{l}=\int_{c_{1}} \mathbf{F} \cdot \mathbf{d} \mathbf{l}+\int_{c_{2}} \mathbf{F} \cdot \mathbf{d} \mathbf{l} \\
& =\int_{c_{1}} \mathbf{F} \cdot \mathbf{d} \mathbf{l}+\int_{x^{\prime}}^{x} F_{1} d x
\end{aligned}
$$
where in the last line we have used that $d y=d z=0$ along the curve $c_{2}$. Now, noting that the first integral doesn't depend on $x$, we differenciate
$$
\frac{\partial}{\partial x} \phi=\frac{d}{d x} \int_{x^{\prime}}^{x} F_{1} d x=F_{1}
$$
using the fundamental theorem of calculus. Now, a similar arguement could be used for the other component and so the theorem is proved.
## Conservative fields and irrotational fields
All conservative fields are irrotational because
$$
\nabla \times \mathbf{F}=\nabla \times \nabla \phi=0
$$
but, the converse isn't true in general, an irrotational field is not neccessarily conservative. However, Stoke's theorem can be used to prove that it is true locally: if a field is irrotation then for any point there is a neighbourhood of the point for which it is conservative.
A more general theorem applies to simply connected domain. A domain is simply connected if any two paths between the same two points can be deformed into each other smoothly. The plane is simply connected but the plane minus a point, say $\mathbf{R}^{2} \backslash(0,0)^{3}$ is not because the missing point gets in the way. So, in the $\mathbf{R}^{2} \backslash(0,0)$ example the path from $(-1,0)$ to $(1,0)$ along the upper half circle can't be deformed into the path along the lower half circle because the missing point gets in the way. Anyway, without proof, we have a theorem:
Theorem: On a simply connected domain a smooth vector field is irrotational iff it is conservative.
${ }^{3}$ The backslash is a set minus, that is
$$
\mathbf{R}^{2} \backslash(0,0)=\left\{(x, y) \in \mathbf{R}^{2} \mid(x, y) \neq(0,0)\right\}
$$
## A note on the uniqueness of solutions to the Laplace Eqn ${ }^{1}$
## January 2007
Proving uniqueness of solutions to the Laplace equation is typical application of the Gauss theorem. Of course, before proving uniqueness of solutions you need to prove a solution exists, this is actually trickier and not something we will deal with here. The Laplace equation is
$$
\nabla^{2} \phi=0
$$
where $\phi$ is defined on some region $D$ with boundary $\delta D$ and here we will be dealing with the common case where $D$ has compact support, there is a $R$ such that $D \subset\{(x, y) \in$ $\left.\mathbf{R}^{2} \mid x^{2}+y^{2}<R\right\}$ and where there are Dirichlet conditions, so $\phi$ is given on $\delta D$. Now, say there are two solutions to this problem $\phi_{1}$ and $\phi_{2}$, then, by linearity
$$
\tilde{\phi}=\phi_{1}-\phi_{2}
$$
also solves the Laplace equation, but with zero boundary conditions on $D$ :
$$
\left.\tilde{\phi}\right|_{\delta D}=0
$$
Now, consider the energy like integral
$$
E=\int_{D} d V|\nabla \tilde{\phi}|
$$
The usual vector identities give the integration by parts like expression
$$
E=\int_{D} d V \nabla \tilde{\phi} \cdot \nabla \tilde{\phi}=\int_{D} d V \nabla(\tilde{\phi} \nabla \tilde{\phi})-\int_{D} \tilde{\phi} \nabla^{2} \tilde{\phi}
$$
Now, the second term is zero because $\tilde{\phi}$ satisfies the Laplace equation and, by Gauss, the first term is a surface integral
$$
E=\int_{\delta D} \tilde{\phi} \nabla \tilde{\phi} \cdot \mathbf{d} \mathbf{S}=0
$$
because $\tilde{\phi}$ vanishes on the boundary. Now, $|\nabla \tilde{\phi}|^{2}$ is non-negative so is integral can only be zero if it vanishes everywhere. Hence
$$
\nabla \tilde{\phi}=0
$$
hence $\tilde{\phi}$ is a constant and, by its boundary condition, this constant must be zero, hence $\phi_{1}=\phi_{2}$, showing the solution, if it exists, is unique. It should be clear that this arguement can also be used to show uniqueness for van Neumann boundary conditions where $\nabla \phi \cdot \mathbf{n}$ is specified on the boundary.
${ }^{1}$ Conor Houghton, [email protected]. ie, see also http://www .maths.tcd.ie/ houghton/231
## A Fourier series example ${ }^{1}$
17 January 2007
The Fourier series for
$$
f(x)= \begin{cases}-1 & -\pi<x<0 \\ 1 & 0<x<\pi\end{cases}
$$
with $f(x+2 \pi)=f(x)$ is
$$
f(x)=\frac{4}{\pi} \sum_{n \text { odd }} \frac{1}{n} \sin n x
$$
If we write
$$
f_{N}(x)=\frac{4}{\pi} \sum_{n \text { odd, } n \leq N} \frac{1}{n} \sin n x
$$
then
${ }^{1}$ Conor Houghton, [email protected], see also http://www.maths.tcd.ie/ houghton/231
## The Bessel Equation ${ }^{1}$
13 April 2007
The Bessel equation is
$$
y^{\prime \prime}+\frac{1}{x} y^{\prime}+\left(1-\frac{\nu^{2}}{x^{2}}\right) y=0
$$
or, multiplying across by $x^{2}$,
$$
x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-\nu^{2}\right) y=0
$$
It is one of the important equation of applied mathematics and engineering mathematics because it is related to the Laplace operator in cylindrical coördinates. The Bessel equation is solved by series solution methods, in fact, to solve the Bessel equation you need to use the method of Fröbenius. It might be expected that Fr" obenius is needed because of the singularities at $x=0$, however, lets pretend we hadn't noticed and try to use the ordinary series solution method:
$$
y=\sum_{n=0}^{\infty} a_{n} x^{n}
$$
Now, by calculating directly
$$
x^{2} y^{\prime \prime}=\sum_{n=0}^{\infty} n(n-1) a_{n} x^{n}
$$
and
$$
x^{2} y^{\prime}=\sum_{n=0}^{\infty} n a_{n} x^{n}
$$
so the equation becomes
$$
\sum_{n=0}^{\infty} n(n-1) a_{n} x^{n}+\sum_{n=0}^{\infty} n a_{n} x^{n}+\sum_{n=0}^{\infty} a_{n} x^{n+2}-\nu^{2} \sum_{n=0}^{\infty} a_{n} x^{n}=0
$$
Hence, if we want to go up to the highest power we need to increase everything to the form $x$ to the $n+2$. By letting $m+2=n$ we get
$$
\sum_{n=0}^{\infty} n(n-1) a_{n} x^{n}=\sum_{m=0}^{\infty}(m+2)(m+1) a_{m+2} x^{m+2}
$$
${ }^{1}$ Conor Houghton, [email protected]. ie please send me any corrections. and
$$
\sum_{n=0}^{\infty} n a_{n} x^{n}=\sum_{m=0}^{\infty}(m+2) a_{m+2} x^{m+2}
$$
and, finally,
$$
\sum_{n=0}^{\infty} n a_{n} x^{n}=a_{0}+a_{1} x+\sum_{m=0}^{\infty} a_{m+2} x^{m+2}
$$
Putting this all back in to the equation, renaming $m$ to $n$ in the usual way, we get
$$
a+0+a_{1} x \sum_{n=0}^{\infty}\left[(n+2)(n+1) a_{n+2}+(n+2) a_{n+2} x^{n}-\nu^{2} a_{n+2}+a_{n}\right] x^{n+2}=0
$$
which gives recursion relation
$$
a_{n+2}=-\frac{a_{n}}{(n+2)^{2}-\nu^{2}}
$$
along with $a_{0}=a_{1}=0$. Thus, while we get a perfectly good two step recursion relation, the extra conditions, on $a_{0}$ and $a_{1}$ lead to the solution being trivial. Hence, the solution of the series form is trivial and, clearly, to find the actual solution, a more general series ansatz is needed.
Fröbenius means that you look for a solution of the form
$$
y=\sum_{n=0}^{\infty} a_{n} x^{n+r}
$$
Now, in terms of this series we have
$$
\begin{aligned}
x^{2} y^{\prime \prime} & =\sum_{n=0}^{\infty} a_{n}(n+r)(n+r-1) x^{n+r} \\
x y^{\prime} & =\sum_{n=0}^{\infty} a_{n}(n+r) x^{n+r} \\
x^{2} y & =\sum_{n=0}^{\infty} a_{n} x^{n+r+2} \\
\nu^{2} y & =\sum_{n=0}^{\infty} \nu^{2} a_{n} x^{n+r}
\end{aligned}
$$
As usual, we move to the highest power, in this case $n+r+2$, without going through the details, this gives
$$
x^{2} y^{\prime \prime}=r(r-1) a_{0} x^{r}+r(r+1) a_{1} x^{r+1}+\sum_{n=0}^{\infty} a_{n+2}(n+r+2)(n+r+1) x^{n+r+2}
$$
and
$$
x y^{\prime}=r a_{0} t^{r}+r(r+1) a_{1} x^{r+1}+\sum_{n=0}^{\infty} a_{n+2}(n+r+2) x^{n+r+2}
$$
and finally
$$
\nu^{2} y=\nu^{2} a_{0} x^{r}+\nu^{2} a_{1} x^{r+1}+\sum_{n=0}^{\infty} a_{n+2} x^{n+r+2}
$$
Now, if we put this all in one equation and set the $x^{r}$ terms to zero, we have
$$
\left[r(r-1)+r-v^{2}\right] a_{0}=0
$$
or, put another way, either $a_{0}=0$ or $r= \pm \nu$. The $x^{r+1}$ term gives
$$
\left[(r+1)^{2}-\nu^{2}\right] a_{1}=0
$$
so, with $r= \pm n u a_{1}=0$
Now, the recusion relation is
$$
\left[(n+r+2)(n+r+1)+(n+r+2)-\nu^{2}\right] a_{n+2}=-a_{n}
$$
so, with $r= \pm \nu$ we have
$$
a_{n+2}=-\frac{a_{n}}{(n \pm \nu+2)^{2}-\nu^{2}}
$$
and so there are two solutions to the Bessel equation, one corresponding to $r=\nu$ and the other with $r=-\nu$.
## $231{\text { Christmas quiz } 2007^{1}}^{2}$
6 December 2007
1. What is
$$
\int_{0}^{2 \pi} d \phi \int_{0}^{\pi} d \theta \int_{0}^{\infty} d r r^{2} \sin \theta e^{-r^{2}} ?
$$
Well
$$
d \phi d \theta d r r^{2} \sin \theta=d V=d x d y d z
$$
and the limits give all of $\mathbf{R}^{3}$, so
$$
\begin{aligned}
\int_{0}^{2 \pi} d \phi \int_{0}^{\pi} d \theta \int_{0}^{\infty} d r r^{2} \sin \theta e^{-r^{2}} & =\int_{-\infty}^{\infty} d x \int_{-\infty}^{\infty} d y \int_{-\infty}^{\infty} d z e^{-x^{2}-y^{2}-z^{2}} \\
& =\left(\int_{-\infty}^{\infty} e^{-x^{2}}\right)^{3}=\pi^{3 / 2}
\end{aligned}
$$
2. When was the college founded? 1592 .
3. What is
$$
\int_{0}^{2 \pi} \cos ^{6} \theta ?
$$
Well use
$$
\cos \theta=\frac{e^{i \theta}+e^{-i \theta}}{2}
$$
so by the binomial theorem
$$
\cos \theta=\frac{1}{2^{6}}(\ldots+20+\ldots)
$$
where we have left out the terms of the form $\exp i n \theta$ for integer $n$, these will all integrate to zero, hence, we integrating, we have
$$
\int_{0}^{2 \pi} \cos ^{6} \theta=\frac{5 \pi}{8}
$$
4. What first is associated with Kenneth Tynan? It is probably best you look this up on Wikipedia.
5. So say $\mathbf{F}=(x, y, z)$ and $\phi=x y z$, let $\hat{\mathbf{n}}$ be the unit normal to the surface $\phi=1$, what is $\mathbf{F} \cdot \hat{\mathbf{n}}$ at $x=y=z=1$ ? Well the key thing is that the normal to a surface $\phi=$ const is given by the gradient of $\phi$, so here the normal is $(y z, x z, x y)$, at $x=y=z=1$ this is $(1,1,1)$ and normalizing it gives $(1 / \sqrt{3}, 1 / \sqrt{3}, 1 / \sqrt{3})$. Dotting this with $\mathbf{F}$ at $x=y=z=1$ gives $\sqrt{3}$.
${ }^{1}$ Conor Houghton, [email protected], see also http://www.maths.tcd.ie/ houghton/231 6. Who won this years nobel prize in literature? Doris Lessing.
7. What is
$$
\int_{C} \mathbf{F} \cdot \mathbf{d l}
$$
for $C$ the semi-circle of unit radius in the $y>0$ half plane about the origin from $x=1$ to $x=-1$ and
$$
\mathbf{F}=\frac{y}{x^{2}+y^{2}} \mathbf{i}-\frac{x}{x^{2}+y^{2}} \mathbf{j}
$$
Well, as we saw in PS6 $\mathbf{F}=-\nabla \theta$, the polar angle and, we have seen before that for a conservative field
$$
\int_{C} \mathbf{F} \cdot \mathbf{d} \mathbf{l}=\phi(b)-\phi(a)
$$
where the curve goes from $a$ to $b$, here, this gives the answer $-\pi$.
8. In Little Miss Sunshine Steve Carell plays the second most distinguished expert on whom? Proust.
## Vector Potentials
Recall that if curl $\mathbf{F}=0$ in a simply-connected region then $\mathbf{F}$ is conservative meaning there exists a scalar potential $\phi$ such that $\mathbf{F}=\operatorname{grad} \phi$.
There is a similar result for solenoidal vector fields:
Theorem If $\operatorname{div} \mathbf{F}=0$ in a region without inner boundaries there exists a vector field $\mathbf{A}$ such that $\mathbf{F}=$ curl $\mathbf{A}$. $\mathbf{A}$ is called a vector potential for $\mathbf{F}$.
Example $\mathbf{F}=\mathbf{B}$ a constant vector (obviously solenoidal). A vector potential is $\mathbf{A}=\frac{1}{2}(\mathbf{B} \times \mathbf{r})$.
The above theorem will not be proved in full here. Rather a (constructive) proof is given for the special case of a star-shaped region.
[A region $D$ is called star-shaped if it has a point $O$ such that the line-segment joining $O$ and any other point in $D$ lies within $D$.]
If $D$ is star-shaped there is a formula for the vector potential for a solenoidal vector field
$$
\mathbf{A}(\mathbf{r})=\int_{0}^{1} d t \mathbf{F}(t \mathbf{r}) \times t \mathbf{r}
$$
where the point $O$ in $D$ is taken to be the origin $(\mathbf{r}=0)$.
Proof To prove this we have to show that taking the curl of the right hand side reproduces the original vector field $\mathbf{F}$. We have
$$
\operatorname{curl} \mathbf{A}(\mathbf{r})=\int_{0}^{1} d t \operatorname{curl}(\mathbf{F}(t \mathbf{r}) \times t \mathbf{r})
$$
Using the vector identity
$$
\begin{gathered}
\nabla \times(\mathbf{F} \times \mathbf{G})=(\nabla \cdot \mathbf{G}) \mathbf{F}+(\mathbf{G} \cdot \nabla) \mathbf{F}-(\nabla \cdot \mathbf{F}) \mathbf{G}-(\mathbf{F} \cdot \nabla) \mathbf{G} \\
\operatorname{curl}(\mathbf{F}(t \mathbf{r}) \times t \mathbf{r})=3 t \mathbf{F}(t \mathbf{r})+t(\mathbf{r} \cdot \nabla) \mathbf{F}(t \mathbf{r})-0-t(\mathbf{F}(t \mathbf{r}) \cdot \nabla) \mathbf{r},
\end{gathered}
$$
using $\operatorname{div} \mathbf{r}=3$ and $\operatorname{div} \mathbf{F}=0$.
A straightforward calculation gives
$$
(\mathbf{F}(t \mathbf{r}) \cdot \nabla) \mathbf{r}=\mathbf{F}(t \mathbf{r})
$$
We also require
$$
(\mathbf{r} \cdot \nabla) \mathbf{F}(t \mathbf{r})=t \frac{d}{d t} \mathbf{F}(t \mathbf{r})
$$
or
$$
\left(x \frac{\partial}{\partial x}+y \frac{\partial}{\partial y}+z \frac{\partial}{\partial z}\right) \mathbf{F}(t x, t y, t z)=t \frac{d}{d t} \mathbf{F}(t x, t y, t z),
$$
which is a generalisation of
$$
a \frac{d}{d a} g(a t)=t \frac{d}{d t} g(a t)
$$
Inserting these two formulae into the expression for curl $(\mathbf{F}(t \mathbf{r}) \times t \mathbf{r})$ gives
$$
\operatorname{curl} \mathbf{A}=\int_{0}^{1} d t\left[2 t \mathbf{F}(t \mathbf{r})+t^{2} \frac{d}{d t} \mathbf{F}(t \mathbf{r})\right]=\int_{0}^{1} d t \frac{d}{d t}\left(t^{2} \mathbf{F}(t \mathbf{r})\right)=\mathbf{F}(\mathbf{r}),
$$
using the FToC.
Note that the vector potential for a solenoidal vector field is not unique; if $\mathbf{A}$ is a vector potential for $\mathbf{F}$ then so is
$$
\mathbf{A}^{\prime}=\mathbf{A}+\operatorname{grad} \phi
$$
where $\phi$ is any scalar field since
$$
\text { curl } \operatorname{grad} \phi=0 \text {. }
$$
In electromagnetic theory this ambiguity is called gauge freedom and (1) is called a gauge transformation.
We have seen that in a simply-connected region an irrotational vector field can be written as a scalar and in a region without inner boundaries (though the proof was only given for star-shaped regions) a solenoidal vector field can be expressed as a curl. Now if $\mathbf{F}$ is neither solenoidal nor irrotational it can be decomposed into a gradient and a curl. There are a number of versions of this statement and they go under various names (Fundamental theorem of vector analysis, Helmholtz' theorem, Hodge decomposition). Here we give a simple version of the decomposition theorem:
Any vector field $\mathbf{F}$ defined in a region $D$ without inner boundaries can be written
$$
\mathbf{F}=\operatorname{grad} \phi+\operatorname{curl} \mathbf{A}
$$
Approach to Proof: Assume that $\mathbf{F}$ is not solenoidal and consider $\mathbf{F}-\nabla \phi$ where $\phi$ is (for now) any scalar field. We have
$$
\operatorname{div}(\mathbf{F}-\nabla \phi)=\operatorname{div} \mathbf{F}-\nabla^{2} \phi .
$$
Now if there exists a scalar field satisfying the equation
$$
\nabla^{2} \phi=\operatorname{div} \mathbf{F}
$$
then $\mathbf{F}-\nabla \phi$ is solenoidal in $\mathrm{D}$ and so we can write
$$
\mathbf{F}-\operatorname{grad} \phi=\operatorname{curl} \mathbf{A} .
$$
To complete the proof we need to show that $\left(^{*}\right)$ always has a smooth solution $\phi .\left(^{*}\right)$ is actually a form of Poisson's equation. More later......
## PART III ODEs
A differential equation is an equation involving a function (or functions) and its derivatives.
An ordinary differential equation (ODE) is a differential equation involving a function (or functions) of one variable.
If the ODE involves the $n$th (and lower) derivatives it is said to be an $n$th order ODE.
Let $y$ be a function of one variable (which we will always call $x$ )
An equation of the form
$$
h_{1}\left(x, y(x), y^{\prime}(x)\right)=0
$$
is a first order ODE.
$$
h_{2}\left(x, y(x), y^{\prime}(x), y^{\prime \prime}(x)\right)=0
$$
is second order.
A function satisfying $y(x)$ the ODE is called a solution of the ODE.
Linear ODEs (2 types)
i) homogeneous. If $y_{1}$ and $y_{2}$ are solutions so is $A y_{1}+B y_{2}$ where $A$ and $B$ are arbitrary constants.
ii) inhomogeneous. If $y_{1}$ and $y_{2}$ are solutions so is $A y_{1}+B y_{2}$ where $A+B=1$.
The general 1st order linear ODE (for a single function) can be written
$$
a(x) y^{\prime}(x)+b(x) y(x)=f(x)
$$
$a, b$ and $f$ are arbitrary functions. The equation is homogeneous if $f=0$.
It is sometimes written in the form
$$
y^{\prime}(x)+p(x) y(x)=f(x) .
$$
$(p=b / a$ and $f / a$ has been renamed as $f)$
The general 2nd order linear ODE
$$
a(x) y^{\prime \prime}(x)+b(x) y^{\prime}(x)+c(x) y(x)=f(x)
$$
$a, b, c$ and $f$ are arbitrary functions (homogeneous if $f=0$ ). We will sometimes write $(*)$ in the form
$$
y^{\prime \prime}(x)+p(x) y^{\prime}(x)+q(x) y(x)=f(x) .
$$
$(* *)$ looks more economical, but we will use $(*)$ as well.
1st Order Case
All solutions of
$$
y^{\prime}(x)+p(x) y(x)=f(x)
$$
can be written
$$
y(x)=C y_{1}(x)+y_{p}(x) .
$$
where $y_{1}(x)$ is a solution of the homogeneous equation $y^{\prime}(x)+p(x) y(x)=0$ and $y_{p}(x)$ is one solution of the full equation.
Proof by explicit construction.
$$
y^{\prime}(x)+p(x) y(x)=f(x)
$$
can be rewritten
$$
\frac{d}{d x} e^{I(x)} y(x)=e^{I(x)} f(x)
$$
where
$$
I(x)=\int_{a}^{x} d z p(z),
$$
( $a$ is an arbitrary constant) which has the property $I^{\prime}(x)=p(x) . I$ is called an integrating factor. Now integrate from $a$ to $x$
$$
e^{I(x)} y(x)-e^{I(a)} y(a)=\int_{a}^{x} d z e^{I(z)} f(z) .
$$
Note that $e^{I(a)}=1$. This gives
$$
y(x)=C y_{1}(x)+y_{p}(x),
$$
with $y_{1}(x)=e^{-I(x)}, y_{p}(x)=e^{-I(x)} \int_{a}^{x} d z e^{I(z)} f(z)$ and $C=y(a)$.
Example Find all solutions of the ODE
$$
y^{\prime}(x)+\frac{1}{x} y(x)=x^{3}
$$
Here $p(x)=1 / x$ which has a non-integrable singularity at $x=0$ ! Work with $x>0($ or $x<0)$.
$I(x)=\int d x p(x)=\log x+c$. Set $c=0($ or $a=1) . e^{I(x)}=x$ so that the ODE can be written
$$
\frac{d}{d x}(x y)=x \cdot x^{3}=x^{4}
$$
Integrating gives $x y=\frac{1}{5} x^{5}+C$ or $y=\frac{1}{5} x^{4}+C / x$, i.e. $y_{1}(x)=1 / x, y_{p}(x)=$ $\frac{1}{5} x^{4}$.
2nd Order Case
All solutions (or the general solution) of $(*)$ or $(* *)$ can be written
$$
y(x)=C_{1} y_{1}(x)+C_{2} y_{2}(x)+y_{p}(x) .
$$
$y_{1}, y_{2}$ are linearly independent solutions of the homogeneous equation
$a(x) y^{\prime \prime}(x)+b(x) y^{\prime}(x)+c(x) y(x)=0 \quad$ or $\quad y^{\prime \prime}(x)+p(x) y^{\prime}(x)+q(x) y(x)=0$,
and $y_{p}(x)$ is a solution of the full equation. $C_{1}$ and $C_{2}$ are arbitrary constants. Proof Not given. written
$y_{p}(x)$ is called a particular integral. The general solution is sometimes
$$
y(x)=y_{c}(x)+y_{p}(x)
$$
where $y_{c}(x)=C_{1} y_{1}(x)+C_{2} y_{2}(x)$ is called the complementary function. It is the general solution of the homogeneous form of the ODE.
## Constant Coefficients
We now consider $\left(^{*}\right)$ in the special case that $a, b$ and $c$ are constants
$$
a y^{\prime \prime}(x)+b y^{\prime}(x)+c y(x)=f(x) .
$$
This type of equation has a nice interpretation as a damped/driven oscillator (where $x$ is time and $y$ is the displacement from equilibrium). Recall the equation for a simple harmonic oscillator
$$
\frac{d^{2} y(t)}{d t^{2}}=-\omega^{2} y(t)
$$
Now add in a damping force proportional to the velocity $d y / d t$ and a driving force $f(t)$ (which may be periodic or non-periodic)
$$
\frac{d^{2} y(t)}{d t^{2}}=-\omega^{2} y(t)-\gamma \frac{d y(t)}{d t}+f(t)
$$
which is a linear ODE with constant coefficents.
The first step in solving ODEs of this type is to find two solutions of the homogeneous equation
$$
a y^{\prime \prime}(x)+b y^{\prime}(x)+c y(x)=0 .
$$
This equation has simple exponential solutions of the form $y(x)=e^{\lambda x}$. Differentiating $y^{\prime}(x)=\lambda e^{\lambda x}$ and $y^{\prime \prime}(x)=\lambda^{2} e^{\lambda x}$ so that
$$
a y^{\prime \prime}+b y^{\prime}+c y=\left(a \lambda^{2}+b \lambda+c\right) y
$$
which is zero provided
$$
a \lambda^{2}+b \lambda+c=0 .
$$
This is called an auxiliary equation. Thus $y_{1}(x)=e^{\lambda_{1} x}$ and $y_{2}(x)=e^{\lambda_{2} x}$ where $\lambda_{1}$ and $\lambda_{2}$ are roots of the (quadratic) auxiliary equation. The complementary function (if $\lambda_{1} \neq \lambda_{2}$ ) is $y_{c}(x)=C_{1} e^{\lambda_{1} x}+C_{2} e^{\lambda_{2} x}$.
If $\lambda_{1}=\lambda_{2}$ we only have one exponential solution. In this case a second solution of the ODE is $y(x)=x e^{\lambda_{1} x}$ and $y_{c}(x)=C_{1} e^{\lambda_{1} x}+C_{2} x e^{\lambda_{1} x}$ (in the oscillator model this special case corresponds to critical damping).
$\underline{\text { Examples }}$ i) $y^{\prime \prime}+3 y^{\prime}+2 y=0$. Auxiliary equation $\lambda^{2}+3 \lambda+2=0$ roots $\overline{\lambda_{1}=-1}, \lambda_{2}=-2$. General solution $y(x)=C_{1} e^{-x}+C_{2} e^{-2 x}$ (over-damping).
ii) $y^{\prime \prime}+2 y^{\prime}+y=0$. Auxiliary equation $\lambda^{2}+2 \lambda+1=0$ with two equal roots $\lambda=-1$. General solution $y(x)=\left(C_{1}+C_{2} x\right) e^{-x}$.
iii) Auxiliary equation $\lambda^{2}+\lambda+1=0$ with complex roots $\lambda=-\frac{1}{2} \pm \frac{1}{2} \sqrt{3} i$. The general complex solution is
$$
y(x)=C_{1} e^{-\frac{1}{2} x+i \frac{1}{2} \sqrt{3} x}+C_{2} e^{-\frac{1}{2} x-i \frac{1}{2} \sqrt{3} x},
$$
where $C_{1}$ and $C_{2}$ are complex constants. The general real solution can be obtained by imposing the constraint $C_{2}=\bar{C}_{1}$ :
$$
y(x)=e^{-\frac{1}{2} x}\left[C_{1} e^{-\frac{1}{2} x}\left(\cos \frac{1}{2} \sqrt{3} x+i \sin \frac{1}{2} \sqrt{3} x\right)+\text { c.c. }\right] .
$$
Writing $C_{1}=\frac{1}{2}(A-i B)$ where $A$ and $B$ are real constants gives
$$
y(x)=e^{-\frac{1}{2} x}\left(A \cos \frac{1}{2} \sqrt{3} x+B \sin \frac{1}{2} \sqrt{3} x\right) .
$$
(underdamped- still oscillates).
Returning to the inhomogeneous form $a y^{\prime \prime}(x)+b y^{\prime}(x)+c y(x)=f(x)$ we still require a particular integral $y_{p}$, i.e. one solution of the full equation. If $f$ is exponential then it is straightforward - $y_{p}$ is an exponential proportional to $f$. For example consider
$$
y^{\prime \prime}+3 y^{\prime}+2 y=e^{x} .
$$
Trying $y_{p}=C e^{x}$ where $C$ is a constant gives $(1+3+2) C e^{x}=e^{x}$ so $C=\frac{1}{6}$, therefore $y_{p}(x)=\frac{1}{6} e^{x}$ is a PI. To obtain the general solution we require the general solution of the homogeneous equation (see earlier example) $y_{c}=$ $C_{1} e^{-x}+C_{2} e^{-2 x}$. General solution $y(x)=y_{c}+y_{p}=C_{1} e^{-x}+C_{2} e^{-2 x}+\frac{1}{6} e^{x}$.
If $f$ is a solution of the homogeneous ODE this doesn't work, e.g. $y^{\prime \prime}+$ $3 y^{\prime}+2 y=e^{-x}$. Trying $y_{p}=C e^{-x}$ gives $C=\infty$. Much as in the equalroot case for the homogeneous equation try $y_{p}=C x e^{-x}$. Differentiating $y_{p}^{\prime}=C e^{-x}-C x e^{-x}$ and $y_{p}^{\prime \prime}=-2 C e^{-x}+C x e^{-x}$. Inserting these into the ODE gives $C x e^{-x}(1-3+2)+C e^{-x}(-2+3)=e^{-x}$ so that $C=1$, i.e. $y_{p}=x e^{-x}$. The general solution can be written $y=\left(C_{1}+x\right) e^{-x}+C_{2} e^{-2 x}$.
If $f$ is a sum of exponentials, e.g. $\sin x=\left(e^{i x}-e^{-i x}\right) /(2 i)$ just add up the PIs corresponding to each exponential term.
If $f$ is not a finite sum of exponentials decompose $f$ into complex exponentials (Fourier analysis).
Example Obtain the general solution of $y^{\prime \prime}(x)+3 y^{\prime}(x)+2 y(x)=f(x)$ where $f$ is the periodic square wave
$$
f(x)=\left\{\begin{array}{c}
1,0<x<\pi \\
-1, \quad-\pi<x<0
\end{array} \quad f(x+2 \pi)=f(x) .\right.
$$
CF: $y_{c}=C_{1} e^{-x}+C_{2} e^{-2 x}$ (as in earlier example)
PI: Expand $f$ as a Fourier series (done it before)
$$
f(x)=\frac{4}{\pi} \sum_{n \text { odd }>0} \frac{1}{n} \sin n x=\frac{2}{\pi i} \sum_{n \text { odd } \in Z} \frac{1}{n} e^{i n x} .
$$
Find PI for $y^{\prime \prime}+3 y^{\prime}+2 y=e^{i n x}$. Trying $y_{p}=C e^{i n x}$ gives $C\left(-n^{2}+3 i n+\right.$ 2) $e^{i n x}=e^{i n x}$ so that
$$
C=\frac{1}{-n^{2}+3 i n+2} .
$$
PI of full problem
$$
y_{p}(x)=\frac{2}{\pi i} \sum_{n \text { odd } \in Z} \frac{e^{i n x}}{n\left(-n^{2}+3 i n+2\right)} .
$$
If $f$ is not periodic write it as a Fourier integral
Example Find the general solution of the ODE $y^{\prime \prime}(x)+2 y^{\prime}(x)+2 y(x)=f(x)$ $\overline{\text { where } f(x)}=e^{-x^{2}}$.
CF: Auxiliary equation $\lambda^{2}+2 \lambda+2=0$ with two complex roots $\lambda=-1 \pm i$ so that $y_{c}=e^{-x}(A \cos x+B \sin x)$.
PI: Write $f$ as a Fourier integral
$$
\begin{gathered}
f(x)=\int_{-\infty}^{\infty} d k e^{i k x} \tilde{f}(k) \\
\tilde{f}(k)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} d x e^{-i k x} e^{-x^{2}}=\frac{1}{2 \sqrt{\pi}} e^{-\frac{1}{4} k^{2}} \quad \text { (problem 4 sheet 12). }
\end{gathered}
$$
Obtain PI for $y^{\prime \prime}+2 y^{\prime}+2 y=e^{i k x}$. Trying $y=C e^{i k x}$ gives $C\left(-k^{2}+2 i k+\right.$ 2) $e^{i k x}=e^{i k x}$ giving
$$
C=\frac{1}{-k^{2}+2 i k+2} .
$$
PI of full problem
$$
y_{p}(x)=\frac{1}{2 \sqrt{\pi}} \int_{-\infty}^{\infty} d k \frac{e^{i k x} e^{-\frac{1}{4} k^{2}}}{\left(-k^{2}+2 i k+2\right)} .
$$
Returning to the general form of $\left(^{*}\right) a(x) y^{\prime \prime}(x)+b(x) y^{\prime}(x)+c(x) y(x)=$ $f(x)$. If two solutions, $y_{1}$ and $y_{2}$, of the homogeneous equation are known then a particular solution of the full equation, $y_{p}$, can be found.
Approach to proof (not examinable)
Consider the equation
$$
a(x) y^{\prime \prime}(x)+b(x) y^{\prime}(x)+c(x) y(x)=\delta\left(x-x^{\prime}\right) .
$$
A solution of this equation is called a Green's function and denoted $G\left(x \mid x^{\prime}\right)$. A solution of $(*)$ is then
$$
y_{p}(x)=\int_{-\infty}^{\infty} d x^{\prime} f\left(x^{\prime}\right) G\left(x \mid x^{\prime}\right)
$$
There is a formula for $G\left(x \mid x^{\prime}\right)$ (note that $G\left(x \mid x^{\prime}\right)$ is not unique)
$$
G\left(x \mid x^{\prime}\right)=\theta\left(x-x^{\prime}\right) \frac{y_{1}(x) y_{2}\left(x^{\prime}\right)-y_{2}(x) y_{1}\left(x^{\prime}\right)}{y_{1}\left(x^{\prime}\right) y_{2}^{\prime}\left(x^{\prime}\right)-y_{2}\left(x^{\prime}\right) y_{1}^{\prime}\left(x^{\prime}\right)} .
$$
The object in the denominator is called the Wronksian of $y_{1}$ and $y_{2}$.
Returning to the homogeneous form of $\left(^{*}\right)$ $a(x) y^{\prime \prime}(x)+b(x) y^{\prime}(x)+c(x) y(x)=0$ no general solution is known but many specific cases have been solved. Some examples
$$
\alpha x^{2} y^{\prime \prime}+\beta x y^{\prime}+\gamma y=0 \quad \text { (Euler's equation) }
$$
$\alpha, \beta, \gamma$ constants.
$$
x^{2} y^{\prime \prime}+x y^{\prime}+\left(\nu^{2}-x^{2}\right) y=0 \quad \text { (Bessel's equation) }
$$
where $\nu$ is a constant.
$$
\left(1-x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+2 \alpha y=0 \quad \text { (Legendre's equation) }
$$
where $\alpha$ is a constant.
$$
y^{\prime \prime}-2 x y^{\prime}+2 \alpha y=0 \quad \text { (Hermite's equation) }
$$
where $\alpha$ is a constant. Solutions of Euler's, Bessel's and Legendre's equations are required in the solution of Laplace's equation (more in part IV).
## Euler's Equation
This can be transformed to the constant coefficient case through the change of variable $x=e^{z}$. This gives
$$
\begin{gathered}
x \frac{d y}{d x}=e^{z} \frac{d y}{d z} \frac{d z}{d x}=e^{z} \frac{d y}{d z}\left(\frac{d z}{d x}\right)^{-1}=\frac{d y}{d z} \\
x^{2} \frac{d^{2} y}{d x^{2}}=x^{2} \frac{d}{d x} \frac{d y}{d x}=x^{2} \frac{d}{d x} \frac{1}{x} \frac{d y}{d z}=-\frac{d y}{d z}+x \frac{d^{2} y}{d z^{2}} \frac{d z}{d x}=-\frac{d y}{d z}+\frac{d^{2} y}{d z^{2}}
\end{gathered}
$$
so the Euler's equation becomes
$$
\alpha \frac{d^{2} y}{d z^{2}}+(\beta-\alpha) \frac{d y}{d z}+\gamma y=0
$$
(constant coefficients). Auxiliary equation
$$
\alpha \lambda^{2}+(\beta-\alpha) \lambda+\gamma=0
$$
The general solution is
$$
y_{c}=C_{1} e^{\lambda_{1} z}+C_{2} e^{\lambda_{2} z}=C_{1} x^{\lambda_{1}}+C_{2} x^{\lambda_{2}} \quad\left(\lambda_{1} \neq \lambda_{2}\right)
$$
where $\lambda_{1}$ and $\lambda_{2}$ are roots of the auxiliary equation.
Euler case solved through change of variable -in general such a simple change of variable is not available. Other methods required.
## Series Solutions
Idea: write $y$ as a power series about $x=0$ or some other point
$$
y(x)=\sum_{n=0}^{\infty} a_{n} x^{n},
$$
and determine a recursion relation for the $a_{n}$ coefficients. Example Consider Hermite's equation $y^{\prime \prime}-2 x y^{\prime}+2 \alpha y=0$.
$$
\begin{gathered}
y(x)=\sum_{n=0}^{\infty} a_{n} x^{n}, \\
x y^{\prime}(x)=\sum_{n=0}^{\infty} n a_{n} x^{n}, \\
x^{2} y^{\prime}(x)=\sum_{n=0}^{\infty} n(n-1) a_{n} x^{n-2} \quad(n=0 \text { and } n=1 \text { terms zero) } \\
=\sum_{m=0}^{\infty}(m+2)(m+1) a_{m+2} x^{m} \quad(\text { relabel } m=n+2)
\end{gathered}
$$
This gives
$$
y^{\prime \prime}-2 x y^{\prime}+2 \alpha y=\sum_{n=0}^{\infty}\left[(n+2)(n+1) a_{n+2}+2(\alpha-n) a_{n}\right] x^{n}=0 .
$$
This implies that the content of the square bracket is zero for all $n$ leading to the recursion relation
$$
a_{n+2}=\frac{2(n-\alpha) a_{n}}{(n+1)(n+2)} .
$$
From this two independent solutions can be obtained.
An even solution: Set $a_{1}=0$. Recursion relation $\rightarrow a_{3}, a_{5}, a_{7}$ etc. all zero. Fix $a_{0}=1$ and apply recursion relation
$$
\begin{aligned}
& a_{2}=-\frac{2 \alpha a_{0}}{1 \cdot 2}=-\frac{2 \alpha}{1 \cdot 2} \\
& a_{4}=\frac{2(2-\alpha) a_{2}}{3 \cdot 4}=2^{2} \frac{(\alpha-2) \alpha}{1 \cdot 2 \cdot 3 \cdot 4} \\
& a_{6}=\frac{2(4-\alpha) a_{4}}{5 \cdot 6}=-\frac{2^{3}(\alpha-4)(\alpha-2) \alpha}{6 !}
\end{aligned}
$$
pattern clear
$$
a_{2 m}=\frac{(-2)^{m}}{(2 m) !}(\alpha-2 m+2)(\alpha-2 m+4) \ldots \alpha
$$
(define $0 !=1$ ). An even solution of Hermite's equation reads
$$
y_{\text {even }}(x) \sum_{m=0}^{\infty} \frac{(-2)^{m}}{(2 m) !}(\alpha-2 m+2)(\alpha-2 m+4) \ldots \alpha x^{2 m} .
$$
This series is convergent (radius of convergence $=\infty$; use ratio test to prove this). For special values of $\alpha$ (even and positive) the series terminates and the solution is a polynomial of degree $\alpha$. For example, when $\alpha=2 a_{4}, a_{6}, a_{8}$ etc. all zero and $y_{\text {even }}(x)=1-2 x^{2}$ (check that this satisfies Hermite's equation $\left.y^{\prime \prime}-2 x y^{\prime}+4 y=0\right)$.
An odd solution: Set $a_{0}=1, a_{1}=1$. Recursion relation $\rightarrow a_{2}, a_{4}, a_{6}$ et. all zero. Odd coefficients can be worked out via the recursion formula. If $\alpha$ is an odd integer the series will terminate producing a polynomial of degree $\alpha$.
General solution of Hermite's equation $y=C_{1} y_{\text {even }}(x)+C_{2} y_{\text {odd }}(x)$. Generating Function
If $\alpha$ is a positive integer one of the solutions to Hermite's equation is polynomial. Such functions are called Hermite polynomials. Remarkably, all the polynomials can be combined into a single generating function.
Consider $\Phi(x, h)=e^{2 x h-h^{2}}$. Expanding this in powers of $h$ :
$$
\Phi(x, h)=\sum_{n=0}^{\infty} \frac{h^{n}}{n !} H_{n}(x)
$$
$H_{n}(x)$ are polynomial solutions of Hermite's equation (see problem sheet 18).
## Method of Frobenius
Homogeneous form of $\left(^{* *}\right) y^{\prime \prime}(x)+p(x) y^{\prime}(x)+q(x) y(x)=0$. Series method worked in Hermite case where both $p$ and $q$ are smooth. If $p$ and $q$ have singularities the series method sometimes fails, e.g. Euler's equation $\alpha x^{2} y^{\prime \prime}+\beta x y^{\prime}+\gamma y=0$ or $p(x)=\beta /(\alpha x)$ and $q(x)=\gamma /\left(\alpha x^{2}\right)$. The explicit solution $y=C_{1} x^{\lambda_{1}}+C_{2} x^{\lambda_{2}}$ not picked up by power series method (unless both roots $\lambda_{1}$ and $\lambda_{2}$ are positive integers).
Way out: expand about a point other than $x=0$
$$
y(x)=\sum_{n=0}^{\infty} a_{n}(x-c)^{n}
$$
where $p(c)$ and $q(c)$ are finite. However, a singular point can often be the 'most symmetric' point.
Frobenius (or generalised series) method allows one to expand about a regular singularity (more later) of $p$ and $q$. Without loss of generality consider an expansion about $x=0$. Consider a solution of the form
$$
y(x)=\sum_{n=0}^{\infty} a_{n} x^{n+s}
$$
where $s$ is some real number. Unlike in the standard power series method $a_{0}$ is always taken to be non-zero ( the odd solution of Hermite's equation would emerge as an $s=1$ Frobenius series with $a_{0} \neq 0$ ). Start with $s$ arbitrary -consistency leads to a quadratic equation for $s$ called the indicial equation. Example Bessel's equation $x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-\nu^{2}\right) y=0\left(p=1 / x, q=1-\nu^{2} / x^{2}\right)$ where $\nu$ is a constant.
$$
\begin{gathered}
y(x)=\sum_{n=0}^{\infty} a_{n} x^{n+s} \\
y^{\prime}(x)=\sum_{n=0}^{\infty} a_{n}(n+s) x^{n+s-1} \\
y^{\prime \prime}(x)=\sum_{n=0}^{\infty} a_{n}(n+s)(n+s-1) x^{n+s-2}
\end{gathered}
$$
Collect contributions to ODE
$$
x^{2} y(x)=\sum_{n=0}^{\infty} a_{n} x^{n+s+2}
$$
This is the least singular term; its first term $a_{0} x^{s+2}$ has a higher power of $x$ than the first terms of $x^{2} y^{\prime \prime}$ etc. Relabel 'more singular' contributions
$$
\begin{aligned}
x^{2} y^{\prime \prime}(x)= & \sum_{n=0}^{\infty} a_{n}(n+s)(n+s-1) x^{n+s} \\
= & a_{0} s(s-1) x^{s}+a_{1}(1+s) s x^{s+1} \\
& +\sum_{m=0}^{\infty} a_{m+2}(n+s+2)(n+s+1) x^{m+s+2} \quad \text { relabel } n=m+2
\end{aligned}
$$
$$
\begin{aligned}
& x y^{\prime}(x)=\sum_{n=0}^{\infty} a_{n}(n+s) x^{n+s} \\
& =a_{0} s x^{s}+a_{1}(1+s) x^{s+1} \\
& +\sum_{m=0}^{\infty} a_{m+2}(m+s+2) x^{m+s+2} \quad \text { relabel } n=m+2 \\
& y(x)=\sum_{n=0}^{\infty} a_{n} x^{n+s} \\
& =a_{0} x^{s}+a_{1} x^{s+1}+\sum_{m=0}^{\infty} a_{m+2} x^{m+s+2} \quad \text { relabel } n=m+2
\end{aligned}
$$
The ODE can be written
$$
\begin{aligned}
& x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-\nu^{2}\right) y \\
& \quad=a_{0}\left[s(s-1)+s-\nu^{2}\right] x^{s}+a_{1}\left[(1+s) s+(1+s)-\nu^{2}\right] x^{s+1} \\
& +\sum_{m=0}^{\infty}\left[(m+s+2)(m+s+1) a_{m+2}+(m+s+2) a_{m+2}-\nu^{2} a_{m+2}+a_{m}\right] x^{m+s+2} \\
& \quad=0 .
\end{aligned}
$$
Since this is true for all $x$ the coefficients of all powers of $x$ must vanish so that
$$
\begin{gathered}
a_{0}\left(s^{2}-\nu^{2}\right)=0 \\
a_{1}\left((1+s)^{2}-\nu^{2}\right)=0 \\
{\left[(m+s+2)^{2}-\nu^{2}\right] a_{m+2}=-a_{m} .}
\end{gathered}
$$
Asuuming $a_{0} \neq 0$ gives $s^{2}-\nu^{2}=0$ (the indicial equation), $a_{1}=0$ and the recursion relation
$$
\left[(a+s+2)^{2}-\nu^{2}\right] a_{m+2}=-a_{m} .
$$
The roots of the indicial equation are $s= \pm \nu$. When $\nu=0$ the indicial equation is $s^{2}=0$ so the method of Frobenius only delivers one solution (see problem 1 on sheet 18). Here we consider the case $\nu=\frac{1}{2}$ so that $s= \pm \frac{1}{2}$. The recursion relation can be written
$$
a_{m+2}=-\frac{a_{m}}{\left(m+2+\frac{1}{2}\right)^{2}-\frac{1}{4}}=-\frac{a_{m}}{(m+2)(m+3)} .
$$
Since $a_{1}=0$ the recursion relation implies that $a_{3}, a_{5}, a_{7}$ etc. are all zero. Fixing $a_{0}=0$ and applying the recursion relation gives
$$
\begin{aligned}
& a_{2}=-\frac{a_{0}}{2 \cdot 3}=-\frac{1}{2 \cdot 3} \\
& a_{4}=-\frac{a_{2}}{4 \cdot 5}=\frac{1}{2 \cdot 3 \cdot 4 \cdot 5}=\frac{1}{5 !} \\
& a_{6}=-\frac{1}{7 !} \quad \text { etc. }
\end{aligned}
$$
The solution is
$$
\begin{aligned}
y(x) & =x^{\frac{1}{2}}-\frac{x^{\frac{5}{2}}}{3 !}+\frac{x^{\frac{9}{2}}}{5 !}-\ldots \\
& =x^{-\frac{1}{2}}\left(x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\ldots\right) \\
& =x^{-\frac{1}{2}} \sin x .
\end{aligned}
$$
(recall $\left.\sin x=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\ldots\right)$
The other root $s=-\frac{1}{2}$ leads to
$$
y(x)=x^{-\frac{1}{2}} \cos x
$$
and so the general solution of the $\nu=\frac{1}{2}$ problem is
$$
y(x)=x^{-\frac{1}{2}}\left(C_{1} \cos x+C_{2} \sin x\right) .
$$
The method of Frobenius gives a series solution of the form
$$
y(x)=\sum_{n=0}^{\infty} a_{n}(x-c)^{n+s}
$$
where $p$ and/or $q$ are singular at $x=c$. Method does not always give the general solution (e.g. $\nu=0$ case of Bessel's equation).
There is a theorem dealing with the applicability of the Frobenius method in the case of regular singularities.
$x=c$ is a regular singular point if $(x-c) p(x)$ and $(x-c)^{2} q(x)$ can be expanded as a power series about $x=c$ (all the singular ODEs we have met have regular singularities, an example of an ODE with a non-regular singularity $x^{3} y^{\prime \prime}+y=0$ since here $q(x)=1 / x^{3}$ so that $x^{2} q(x)=1 / x$ cannot be expanded about $x=0$ ).
If $p$ and $q$ are non-singular at $x=c, x=c$ is called an ordinary point of the ODE $y^{\prime \prime}(x)+p(x) y^{\prime}(x)+q(x) y(x)=0$.
Fuchs' Theorem
$x=c$ is a regular singular or ordinary point of the ODE $y^{\prime \prime}(x)+p(x) y^{\prime}(x)+q(x) y(x)=0$. if and only if
i) The two solutions are Frobenius series.
or
ii) One solution is a Frobenius series, $S_{1}(x)$ and the other solution is of the form $y(x)=S_{1}(x) \log (x-c)+S_{2}(x)$ where $S_{2}(x)$ is another Frobenius series (it is not a solution on its own).
Proof not given.
Case ii) occurs when the indicial equation has equal roots and sometimes when the roots differ by an integer (e.g. $\nu=1$ case of Bessel's equation).
## Finding a Second Solution
If one solution of $y^{\prime \prime}(x)+p(x) y^{\prime}(x)+q(x) y(x)=0$ can be found another one can be constructed. Let $u(x)$ be a solution then try $y(x)=u(x) v(x)$ then a short calculation gives
$$
y^{\prime \prime}+p y^{\prime}+q y=\left(u^{\prime \prime}+p u^{\prime}+q u\right) v+\left(2 u^{\prime}+p u\right) v^{\prime}+u v^{\prime \prime}=0 .
$$
Now since $u$ is, by assumption, a solution the first term on the right hand side is zero giving
$$
\left(2 u^{\prime}+p u\right) v^{\prime}+u v^{\prime \prime}=0 .
$$
This is a first order linear ODE for $v^{\prime}(x)$ (see problem sheet 17). Eigenfunctions and Eigenfunction Expansions
There is a strong analogy between solving some of the 'named' ODEs and finding the eigenvectors (and eigenvalues) of a matrix.
Hermite's equation $y^{\prime \prime}-2 x y^{\prime}+2 \alpha y=0$ can be written
$$
L y=\lambda y
$$
where $L$ is the differential operator
$$
L=-\frac{d^{2}}{d x^{2}}+2 x
$$
and $\lambda=2 \alpha$.
Legendre's equation can be written in the same way, with
$$
L=-\left(1-x^{2}\right) \frac{d^{2}}{d x^{2}}+2 x \frac{d}{d x}, \quad \lambda=\alpha .
$$
Can think of the differential operator $L$ as a matrix (albeit an infinite dimensional one) and the function it acts on, $y$, as a vector.
To make this more precise it is useful to recall some properties of certain finite, say $n \times n$, matrices.
A symmetric matrix $S$ satisfies
$$
S^{T}=S, \quad T=\text { transpose } \quad\left[A^{T}\right]_{i j}=[A]_{j i} .
$$
A Hermitian matrix is an $n \times n$ matrix with complex entries satisfying
$$
H^{\dagger}=H \quad \dagger=\text { adjoint. }
$$
Adjoint $\dagger$ denotes the complex conjugate of the transpose
$$
\left[A^{\dagger}\right]_{i j}=\overline{[A]_{j i}} .
$$
Clearly a real symmetric matrix is Hermitian.
Let $v$ be an $n$-component column vector with complex entries. $v$ is an eigenvector of $H$ if
$$
H v=\lambda v
$$
for some complex number $\lambda$ (the eigenvalue). Theorem: The eigenvalues of a Hermitian matrix are real. Proof Let $v$ be an eigenvector of $H$ with eigenvalue $\lambda$
$$
v^{\dagger} H v=\lambda v^{\dagger} v .
$$
Since $v^{\dagger} v$ is real it suffices to prove that $v^{\dagger} H v$ is real:
$$
\overline{v^{\dagger} H v}=\overline{\left(v^{\dagger} H v\right)^{T}}
$$
(since $v^{\dagger} H v$ is a $1 \times 1$ matrix)
$$
=\left(v^{\dagger} H v\right)^{\dagger}=v^{\dagger} H^{\dagger} v=v^{\dagger} H v
$$
if $H$ is Hermitian. Note that $(A B)^{\dagger}=B^{\dagger} A^{\dagger}$.
Theorem: Eigenvectors of a Hermitian matrix corresponding to distinct eigenvalues are orthogonal, i.e. if $H v_{1}=\lambda_{1} v_{1} \quad H v_{2}=\lambda_{2} v_{2}$ with $\lambda_{1} \neq \lambda_{2}$ then $v_{1}^{\dagger} v_{2}=0$ or $\left(v_{1}, v_{2}\right)=0$ (inner product notation).
Proof We have
$$
v_{2}^{\dagger} H v_{1}=\lambda_{1} v_{2}^{\dagger} v_{1} \quad(A) \quad v_{1}^{\dagger} H v_{2}=\lambda_{1} v_{1}^{\dagger} v_{2}
$$
Take the complex conjugate of (B)
$$
v_{2}^{\dagger} H v_{1}=\lambda_{2} v_{2}^{\dagger} v_{1} \quad\left(\text { using } H^{\dagger}=H, \bar{\lambda}_{2}=\lambda\right) .
$$
Subtracting (A) gives
$$
0=\left(\lambda_{2}-\lambda_{1}\right) v_{2}^{\dagger} v_{1}
$$
so that $\left(v_{2}, v_{1}\right)=0$ if $\lambda_{1} \neq \lambda_{2}$.
If eigenvalues degenerate can choose eigenvectors to be orthogonal (Gram-Schmidt).
Can choose the $n$ eigenvectors $v_{i}(i=1,2, \ldots, n)$ of an Hermitian matrix $\mathrm{H}$ to be orthonormal
$$
v_{i}^{\dagger} v_{j}=\delta_{i j} \quad\left(v_{i}, v_{j}\right)=\delta_{i j} .
$$
Any ( $n$-component) vector $v$ can be written as a linear combination of the $v_{i} \mathrm{~S}$
$$
v=c_{1} v_{1}+c_{2} v_{2}+\ldots+c_{n} v_{n}
$$
where $c_{1}, c_{2}, \ldots c_{n}$ are complex numbers.
Using the orthonormal property
$$
\left(v_{i}, v\right)=c_{i} \quad \text { (c.f. Fourier analysis) }
$$
Also
$$
|v|^{2}=(v, v)=\left|c_{1}\right|^{2}+\left|c_{2}\right|^{2}+\ldots+\left.! c_{n}\right|^{2},
$$
which can be thought of as an $n$-dimensional version of Pythagoras or a finite dimensional version of Parseval.
Resolution of unity: The identity matrix can be written as
$$
I=\sum_{i=1}^{n} v_{i} v_{i}^{\dagger}
$$
since acting on any vector, $v=c_{1} v_{1}+c_{2} v_{2}+\ldots+c_{n} v_{n}$, will reproduce $v$.
The inverse of an Hermitian matrix can be written as
$$
H^{-1}=\sum_{i=1}^{n} \frac{1}{\lambda_{i}} v_{i} v_{i}^{\dagger},
$$
since (left or right) multiplication by $H$ reproduces the identity matrix.
Back to Differential operators
A differential operator $L$ acts on some vector space of functions. Require functions to be such that
$$
\int d x \bar{u}(x)(L u)(x)<\infty .
$$
It is usual to impose further restrictions on the functions such as
i) square integrability $\int_{-\infty}^{\infty} d x \bar{u}(x) u(x)<\infty$.
or
ii) periodicity
or
iii) vanishing at the end points of an interval $[a, b] \subset R ; u(a)=u(b)=0$.
In these cases an inner product can be defined
i) $(u, v)=\int_{-\infty}^{\infty} d x \bar{u}(x) v(x)$
$$
\begin{aligned}
\text { ii) }(u, v) & =\int_{-\pi}^{\pi} d x \bar{u}(x) u(x) \quad(l=2 \pi) \\
\text { iii) }(u, v) & =\int_{a}^{b} d x \bar{u}(x) v(x) .
\end{aligned}
$$
Analogue of a Hermitian matrix:
Suppose
$$
(u, L v)=(L u, v)
$$
for any $u, v$ in the chosen space of functions (finite dimensional case: $H^{\dagger}=H$ $\equiv(H u)^{\dagger} v=u^{\dagger}(H v)$ for any vectors $\left.u, v\right)$. An operator satisfying (***) is called symmetric. If some further technical requirements are met it is called self-adjoint or Hermitian. We will be sloppy and call an operator satisfying $(* * *)$ Hermitian.
Example An operator satisfying $(* * *)$ is
$$
L=-\frac{d^{2}}{d x^{2}}
$$
for any of the three conditions i) ii) iii). Here $(* * *)$ is just
$$
-\int d x \bar{u}(x) v^{\prime \prime}(x)=-\int d x \bar{u}^{\prime \prime} v(x) .
$$
To establish this integrate by parts twice, e.g. for the periodic case
$$
\int_{-\pi}^{\pi} d x \bar{u}(x) v^{\prime \prime}(x)=\left.\bar{u}(x) v^{\prime}(x)\right|_{-\pi} ^{\pi}-\int_{-\pi}^{\pi} d x \bar{u}^{\prime}(x) v^{\prime}(x) .
$$
$\left.\bar{u}(x) v^{\prime}(x)\right|_{-\pi} ^{\pi}$ is zero since $u$ and $v$ are periodic. Integrating by parts once more gives the result.
$L=-d^{2} / d x^{2}$ can be viewed as an Hermitian matrix acting on the space of periodic functions $(l=2 \pi)$. The eigenvectors (or eigenfunctions) are the functions $v_{n}(x)=e^{i n x}(n \in Z)$ with corresponding eigenvalues $\lambda_{n}=n^{2}$. These are orthogonal (as expected since $L$ is Hermitian)
$$
\left(v_{m}, v_{n}\right)=\int_{-\pi}^{\pi} d x \bar{v}_{m}(x) v_{n}(x)=\int_{-\pi}^{\pi} d x e^{-i m x} e^{i n x}=0 \quad(m \neq n) .
$$
Can make them orthonormal
$$
v_{n}(x)=\frac{1}{\sqrt{2 \pi}} e^{i n x} \quad\left(v_{m}, v_{n}\right)=\delta_{m n} .
$$
A periodic function, $f$, can be thought of as a vector in the space acted on by $L$. Expand $f$ in eigenvectors of $L$ :
$$
f(x)=\sum_{n \in Z} c_{n} v_{n}(x)
$$
aka Fourier analysis
$$
c_{m}=\left(v_{m}, f\right)=\frac{1}{\sqrt{2 \pi}} \int_{-\pi}^{\pi} d x e^{-i m x} f(x) .
$$
Parseval
$$
(f, f)=\sum_{n \in Z}\left|c_{n}\right|^{2}
$$
(absence of $2 \pi$ due to normalisation of $v_{n} \mathrm{~S}$ to unity with the $1 / \sqrt{2 \pi}$ factor).
Fourier analysis $\equiv$ expanding in eigenvectors of the Hermitian operator $L=-d^{2} / d x^{2}$. Can choose a different Hermitian operator - this leads to alternative expansions - Legendre series, Hermite series, etc.
Legendre Series
Based on Legendre's equation
$$
\left(1-x^{2}\right) y^{\prime \prime}(x)-2 x y^{\prime}(x)+\alpha y(x)=0
$$
This ODE has (regular) singularities at $x= \pm 1$. If $\alpha$ is of the form $\alpha=$ $n(n+1)$ where $n$ is a non-negative integer then the ODE has a polynomial solution (see problem sheet 17) which is well defined at $x= \pm 1$ (the other solutions blow up at $x=1$ and/or $x=-1$ ).
Recast ODE as an eigenvalue problem:
$$
L y(x)=\lambda y(x), \quad L=-\left(1-x^{2}\right) \frac{d^{2}}{d x^{2}}+2 x \frac{d}{d x} .
$$
Boundary conditions? $y(x)$ finite at $x= \pm 1$ (this excludes all non-polynomial solutions). The inner product is taken as
$$
(u, v)=\int_{-1}^{1} d x \bar{u}(x) v(x) .
$$
It is straightforward to prove that with these boundary conditions $L$ is Hermitian; it helps to write $L$ in the form
$$
L=-\frac{d}{d x}\left(1-x^{2}\right) \frac{d}{d x} .
$$
The eigenfunctions are the polynomial solutions of Legendre's equation $P_{n}(x) \quad(n=0,1,2, \ldots)$ with corresponding eigenvalues $\lambda_{n}=n(n+1)$.
The first few Legendre polynomials are
$P_{0}(x)=1, \quad P_{1}(x)=x, \quad P_{2}(x)=\frac{1}{2}\left(3 x^{2}-1\right)$
For $n$ even/odd $P_{n}$ is even/odd (c.f. Hermite polynomials). There are various formulae for the Legendre polynomials, e.g. they can be combined into a generating function
$$
\Phi(x, h)=\left(1-2 x h+h^{2}\right)^{-\frac{1}{2}}=\sum_{n=0}^{\infty} h^{n} P_{n}(x) .
$$
Actually there is an explicit formula (Rodrigues' formula)
$$
P_{n}(x)=\frac{1}{2^{n} n !} \frac{d^{n}}{d x^{n}}\left(x^{2}-1\right)^{n} .
$$
Since they are the eigenfunctions of an Hermitian operator they are orthogonal
$$
\left(P_{m}, P_{n}\right)=\int_{-1}^{1} d x P_{m}(x) P_{n}(x)=0 \text { if } m \neq n .
$$
Normalisation: they can be made orthonormal, but the following convention is standard
$$
\int_{-1}^{1} d x\left(P_{n}(x)\right)^{2}=\frac{2}{2 n+1} .
$$
Let $f$ be a function defined on $[-1,1]$. Can expand in Legendre polynomials (i.e. in the eigenfunctions of the Hermitian operator $L$ )
$$
f(x)=\sum_{n=0}^{\infty} c_{n} P_{n}(x) .
$$
Much as in Fourier analysis
$$
\left(P_{m}, f\right)=\sum_{n=0}^{\infty} c_{n}\left(P_{m}, P_{n}\right)=\sum_{n=0}^{\infty} c_{n} \frac{2 \delta_{m n}}{2 m+1}=\frac{2 c_{m}}{2 m+1},
$$
so that
$$
c_{m}=\left(m+\frac{1}{2}\right)\left(P_{m}, f\right)=\left(m+\frac{1}{2}\right) \int_{-1}^{1} d x P_{m}(x) f(x)
$$
Examples of Legendre series
i) $f(x)=x \quad-1<x<1$. By inspection $f(x)=P_{1}(x)$, i.e. $c_{1}=1$ and all other Legendre series coefficients zero.
ii) If $f$ is a polynomial of degree $n$ then the Legendre series terminates, i.e. $c_{m}=0$ if $m>n$ (see problem 3 on sheet 18).
iii) $f(x)=1$ for $-1<x<0$ and $f(x)=0$ for $-1<x<0$ Here
$$
c_{m}=\left(n+\frac{1}{2}\right) \int_{-1}^{1} d x P_{m}(x) f(x)=\left(n+\frac{1}{2}\right) \int_{0}^{1} d x P_{m}(x)
$$
Now $c_{m}$ vanishes for all even $m$ except $m=0$ (can you see why this is the case without computing an integral?) $c_{0}=\frac{1}{2}$. To determine the $c_{m}$ for $m$ odd requires the integral
$$
\int_{0}^{1} d x P_{m}(x)
$$
This can be computed using Rodrigues' formula or the generating function (compute the integral $\int_{0}^{1} d x \Phi(x, h)$ ). The result is
$$
\int_{0}^{1} d x P_{2 s+1}(x)=(-1)^{s} \frac{(2 s-1) ! !}{(2 s+2) ! !}
$$
Here we have used double factorial notation:
$$
(2 s+1) ! !=(2 s+1)(2 s-1)(2 s-3) \ldots 1 \quad(2 s) ! !=(2 s)(2 s-2)(2 s-4) \ldots 2 .
$$
In the even case the double factorial can be rewritten as $(2 s) !=2^{s} s !$.
## Hermite's Equation
Try to do the same for Hermite's equation
$$
y^{\prime \prime}(x)-2 x y^{\prime}(x)+2 \alpha y(x)=0 \quad \text { or } L y(x)=\lambda y(x),
$$
where
$$
L=-\frac{d^{2}}{d x^{2}}+2 x \frac{d}{d x}, \quad \lambda=2 \alpha .
$$
Problem: $L$ is not Hermitian!
$(u, L v) \neq(L u, v)$ regardless of boundary conditions. A quick way to prove Hermiticity of $L$ in the Legendre case is to write $L$ in the form
$$
L=-\frac{d}{d x}\left(1-x^{2}\right) \frac{d}{d x} .
$$
Claim: Any $L$ of the form
$$
L=-\frac{d}{d x} p(x) \frac{d}{d x}
$$
where $p$ is some real function is Hermitian (provided suitable boundary conditions are imposed).
Hermite $L$ is not of this form. But it can be written
$$
L=-e^{x^{2}} \frac{d}{d x} e^{-x^{2}} \frac{d}{d x},
$$
which is not Hermitian due to the $e^{x^{2}}$ prefactor. Now change the definition of the inner product
$$
(u, v)_{n e w}=\int d x e^{-x^{2}} \bar{u}(x) v(x) .
$$
With this definition
$$
\begin{aligned}
(u, L v)_{n e w} & =-\int d x \frac{d}{d x} e^{-x^{2}} \frac{d}{d x} v(x) \\
& =-\int d x\left(\frac{d}{d x} e^{-x^{2}} \frac{d}{d x} \bar{u}(x)\right) v(x) \quad \text { int. by parts twice } \\
& =(L u, v)_{n e w},
\end{aligned}
$$
(here boundary terms were dropped).
An operator of the form
$$
L=-\frac{1}{w(x)}\left(\frac{d}{d x} p(x) \frac{d}{d x}-q(x)\right)
$$
where $w(x)>0, p$ and $q$ are real is Hermitian with respect to the inner product
$$
(u, v)_{w}=\int d x w(x) \bar{u}(x) v(x)
$$
provided suitable boundary conditions are imposed. $w(x)$ is called a weight function.
Check Hermiticity in an interval $[a, b] \subset R$ so that
$$
(u, v)_{w}=\int_{a}^{b} d x w(x) \bar{u}(x) v(x) .
$$
Have to check that $(u, L v)_{w}=(L u, v)_{w}$ Integrating by parts twice (this time retaining the boundary terms)
$$
\begin{aligned}
(u, L v)_{w} & =-\left.\bar{u}(x) p(x) v^{\prime}(x)\right|_{a} ^{b}+\left.\bar{u}^{\prime}(x) p(x) v(x)\right|_{a} ^{b}-\int_{a}^{b}\left(\frac{d}{d x} p(x) \frac{d}{d x} \bar{u}(x)\right) v(x) \\
& =(L u, v)_{w}
\end{aligned}
$$
if the boundary terms vanish. This condition can be written
$$
\left.p(x)\left(\bar{u}(x) v^{\prime}(x)-\bar{u}^{\prime}(x) v(x)\right)\right|_{a} ^{b}=0 .
$$
Any boundary conditions satisfying this condition (Lagrange's identity) give Hermiticity with respect to the weight function $w$. For example impose the boundary conditions $y(a)=y(b)=0$ then Lagrange's identity is trivially satisfied (since $u$ and $v$ should be taken as zero at the end-points). In the Legendre case $p(x)=1-x^{2}$ which vanishes at $x= \pm 1$; this is why the Hermiticity of $L$ only requires $y(1)$ and $y(-1)$ to be finite.
The eigenvalue equation, $L y=\lambda y$, is often written
$$
\left(\frac{d}{d x} p(x) \frac{d}{d x}-q(x)+\lambda w(x)\right) y(x)=0 .
$$
This is called the Sturm-Liouville equation. The eigenfunctions of $L$ are real and eigenfunctions corresponding to distinct eigenvalues are orthogonal (c.f. finite dimensional Hermitian matrix).
## Completeness
A function $f$ satisfying the appropriate boundary conditions can be expanded in the eigenfunctions of $L$
$$
f(x)=\sum_{n} c_{n} v_{n}(x)
$$
Another way of expressing this is via
$$
\sum_{n} v_{n}(x) \bar{v}_{n}\left(x^{\prime}\right)=\frac{1}{w(x)} \delta\left(x-x^{\prime}\right)
$$
(here the $v_{n}$ are assumed to be orthonormal) which can be viewed as the infinite dimensional analogue of
$$
\sum_{i=1}^{n} v_{i} v_{i}^{\dagger}=I .
$$
Example Fourier case $L=-d^{2} / d x^{2}, w(x)=1$. Periodic boundary conditions
$$
\begin{gathered}
v_{n}(x)=\frac{e^{i n x}}{\sqrt{2 \pi}} . \\
\sum_{n \in Z} v_{n}(x) \bar{v}_{n}\left(x^{\prime}\right)=\frac{1}{2 \pi} \sum_{n \in Z} e^{i n\left(x-x^{\prime}\right)}=\delta\left(x-x^{\prime}\right) .
\end{gathered}
$$
Setting $x^{\prime}=0$ gives the standard expression for the periodic delta fumction $(l=2 \pi)$
$$
\delta(x)=\frac{1}{2 \pi} \sum_{n \in Z} e^{i n x} .
$$
## Part III: ODEs
A differential equation is an equation involving a function (or functions) and its derivatives. An ordinary differential equation (ODE) is a differential equation involving a function (or functions) of one variable.
If the ODE involves the $n$th (and lower) derivatives it is said to be an $n$th order ODE. Let $y$ be a function of one variable (which we will always call $x$ ) An equation of the form
$$
h_{1}\left(x, y(x), y^{\prime}(x)\right)=0
$$
is a rst order ODE.
$$
h_{2}\left(x, y(x), y^{\prime}(x), y^{\prime \prime}(x)\right)=0
$$
is second order.
A function satisfying $y(x)$ the ODE is called a solution of the ODE.
## $0.1 \quad$ Linear ODEs (2 types)
i) homogeneous. If $y_{1}$ and $y_{2}$ are solutions so is $A y_{1}+B y_{2}$ where $A$ and $B$ are arbitrary constants.
ii) inhomogeneous. If $y_{1}$ and $y_{2}$ are solutions so is $A y_{1}+B y_{2}$ where $A+B=1$.
The general 1st order linear ODE (for a single function) can be written
$$
a(x) y^{\prime}(x)+b(x) y(x)=f(x)
$$
$a, b$ and $f$ are arbitrary functions. The equation is homogeneous if $f=0$.
It is sometimes written in the form
$$
y^{\prime}(x)+p(x) y(x)=f(x) .
$$
$(p=b / a$ and $f / a$ has been renamed as $f)$
The general 2nd order linear ODE
$$
a(x) y^{\prime \prime}(x)+b(x) y^{\prime}(x)+c(x) y(x)=f(x)
$$
$1 a, b, c$ and $f$ are arbitrary functions (homogeneous if $f=0$ ).
We will sometimes write $(*)$ in the form
$$
y^{\prime \prime}(x)+p(x) y^{\prime}(x)+q(x) y(x)=f(x) .
$$
$(* *)$ looks more economical, but we will use $(*)$ as well.
### 1st Order Case
All solutions of
$$
y^{\prime}(x)+p(x) y(x)=f(x)
$$
can be written
$$
y(x)=C y_{1}(x)+y_{p}(x) .
$$
where $y_{1}(x)$ is a solution of the homogeneous equation $y^{\prime}(x)+p(x) y(x)=0$ and $y_{p}(x)$ is one solution of the full equation.
## Proof
by explicit construction.
$$
y^{\prime}(x)+p(x) y(x)=f(x)
$$
can be rewritten
$$
\frac{d}{d x} e^{I(x)} y(x)=e^{I(x)} f(x)
$$
where
$$
I(x)=\int_{a}^{x} d z p(z)
$$
( $a$ is an arbitrary constant) which has the property $I^{\prime}(x)=p(x) . \quad I$ is called an integrating factor. Now integrate from $a$ to $x$
$$
e^{I(x)} y(x) e^{I(a)} y(a)=\int_{a}^{x} d z e^{I(z)} f(z)
$$
Note that $e^{I(a)}=1$. This gives
$$
y(x)=C y_{1}(x)+y_{p}(x),
$$
with $y_{1}(x)=e^{I(x)}, y_{p}(x)=e^{I(x)} \int_{a}^{x} d z e^{I(z)} f(z)$ and $C=y(a)$.
## Example
Find all solutions of the ODE 1
$$
y^{\prime}(x)+\frac{1}{x} y(x)=x^{3}
$$
Here $p(x)=1 / x$ which has a non-integrable singularity at $x=0$ ! Work with $x>0$ (or $x<0)$.
$I(x)=\int d x p(x)=\log x+c$. Set $c=0($ or $a=1) . e^{I(x)}=x$ so that the ODE can be written
$$
\frac{d}{d x}(x y)=x \cdot x^{3}=x^{4} .
$$
Integrating gives $x y=\frac{1}{5} x^{5}+C$ or $y=\frac{1}{5} x^{4}+C / x$, i.e. $y_{1}(x)=1 / x, y_{p}(x)=\frac{1}{5} x^{4}$.
### 2nd Order Case
All solutions (or the general solution) of $(*)$ or $(* *)$ can be written
$$
y(x)=C_{1} y_{1}(x)+C_{2} y_{2}(x)+y_{p}(x) .
$$
$y_{1}, y_{2}$ are linearly independent solutions of the homogeneous equation
$$
a(x) y^{\prime \prime}(x)+b(x) y^{\prime}(x)+c(x) y(x)=0 \quad \text { or } \quad y^{\prime \prime}(x)+p(x) y^{\prime}(x)+q(x) y(x)=0,
$$
or and $y_{p}(x)$ is a solution of the full equation. $C_{1}$ and $C_{2}$ are arbitrary constants.
## Proof
Not given.
$y_{p}(x)$ is called a particular integral. The general solution is sometimes written
$$
y(x)=y_{c}(x)+y_{p}(x)
$$
where $y_{c}(x)=C_{1} y_{1}(x)+C_{2} y_{2}(x)$ is called the complementary function. It is the general solution of the homogeneous form of the ODE.
### Constant Coeffcients
We now consider $(*)$ in the special case that $a, b$ and $c$ are constants
$$
a y^{\prime \prime}(x)+b y^{\prime}(x)+c y(x)=f(x) .
$$
This type of equation has a nice interpretation as a damped/driven oscillator (where $x$ is time and $y$ is the displacement from equilibrium). Recall the equation for a simple harmonic oscillator
$$
\frac{d^{2} y(t)}{d t^{2}}=-\omega^{2} y(t)
$$
Now add in a damping force proportional to the velocity $d y / d t$ and a driving force $f(t)$ (which may be periodic or non-periodic)
$$
\frac{d^{2} y(t)}{d t^{2}}=-\omega^{2} y(t)-\gamma \frac{d y(t)}{d t}+d(t)
$$
which is a linear ODE with constant coeffcents.
The rst step in solving ODEs of this type is to nd two solutions of the homogeneous equation
$$
a y^{\prime \prime}(x)+b y^{\prime}(x)+c y(x)=0 .
$$
This equation has simple exponential solutions of the form $y(x)=e^{\lambda x}$. Differentiating $y^{\prime}(x)=\lambda e^{\lambda x}$ and $y^{\prime \prime}(x)=\lambda^{2} e^{\lambda x}$ so that
$$
a y^{\prime \prime}(x)+b y^{\prime}+c y=\left(a \lambda^{2}+b \lambda+c\right) y
$$
which is zero provided
$$
a \lambda^{2}+b \lambda+c=0
$$
This is called an auxiliary equation. Thus $y_{1}(x)=e^{\lambda_{1} x}$ and $y_{2}(x)=e^{\lambda_{2} x}$ where $\lambda_{1}$ and $\lambda_{2}$ are roots of the (quadratic) auxiliary equation. The complementary function (if $\lambda_{1}=\lambda_{2}$ ) is $y_{c}(x)=C_{1} e^{\lambda_{1} x}+C_{2} e^{\lambda_{2} x}$.
If $\lambda_{1}=\lambda_{2}$ we only have one exponential solution. In this case a second solution of the ODE is $y(x)=x e^{\lambda_{1} x}$ and $y_{c}(x)=C_{1} e^{\lambda_{1} x}+C_{2} x e^{\lambda_{1} x}$ (in the oscillator model this special case corresponds to critical damping).
## Examples
i) $y^{\prime \prime}+3 y^{\prime}+2 y=0$. Auxiliary equation $\lambda^{2}+3 \lambda+2=0$ roots $\lambda_{1}=1, \lambda_{2}=2$. General solution $y(x)=C_{1} e^{x}+C_{2} e^{2 x}$ (over-damping).
ii) $y^{\prime \prime}+2 y^{\prime}+y=0$. Auxiliary equation $\lambda^{2}+2 \lambda+1=0$ with two equal $\operatorname{roots} \lambda=1$. General solution $y(x)=\left(C_{1}+C_{2} x\right) e^{x}$
iii) Auxiliary equation $\lambda^{2}+\lambda+1=0$ with complex roots $\lambda=\frac{1}{2} \pm \frac{1}{2} \sqrt{3} i$.
The general complex solution is
$$
y(x)=C_{1} e^{-\frac{1}{2} x+i \frac{1}{2} \sqrt{3} x}+C_{2} e^{-\frac{1}{2} x-i \frac{1}{2} \sqrt{3} x}
$$
where $C_{1}$ and $C_{2}$ are complex constants. The general real solution can be obtained by imposing the constraint $C_{2}=\bar{C}_{1}$ :
$$
y(x)=e^{\frac{1}{2} x}\left[C_{1} e^{\frac{1}{2} x}\left(\cos \frac{1}{2} \sqrt{3} x+i \sin \cos \frac{1}{2} \sqrt{3} x\right)+\text { c.c. }\right]
$$
Writing $C_{1}=\frac{1}{2}(A i B)$ where $A$ and $B$ are real constants gives
$$
y(x)=e^{\frac{1}{2}} x\left(A \cos \frac{1}{2} \sqrt{3} x+B \sin \frac{1}{2} \sqrt{3} x\right)
$$
(underdamped- still oscillates).
## Part I
## Part IV Partial Differential Equations
## Some linear PDEs involving a scalar field $\phi$
(aka the Equations of Mathematical Physics)
i)
$$
\nabla^{2} \phi=0 \quad \text { Laplace's equation (elliptic) }
$$
ii)
$$
\nabla^{2} \phi=\rho \quad \text { Poisson's equation (elliptic) }
$$
$\phi$ is some scalar field usually called a source term
iii)
$$
\begin{array}{cc}
\left(\nabla^{2}+k^{2}\right) \phi=0 & \text { Helmholtz equation (elliptic) } \\
\left(\nabla^{2}-k^{2}\right) \phi=0 & \text { 'wrong sign' Helmholtz }
\end{array}
$$
$k$ is a real constant
iv)
$$
\begin{gathered}
\nabla^{2} \phi=D \frac{\partial \phi}{\partial t} \quad \text { Heat/diffusion equation (parabolic) } \\
D \text { a constant, } \phi=\phi(\vec{r}, t) \text { time-dependent }
\end{gathered}
$$
v)
$$
\begin{aligned}
\nabla^{2} \phi-\frac{1}{c^{2}} \frac{\partial^{2} \phi}{\partial t^{2}} & =0 \quad \text { Wave equation (hyperbolic) } \\
c & =\text { speed of sound / light }
\end{aligned}
$$
$$
\text { Laplacian } \nabla^{2}=\nabla \cdot \nabla=\operatorname{div} \operatorname{grad}=\frac{d^{2}}{d x^{2}}+\frac{d^{2}}{d y^{2}}+\frac{d^{2}}{d z^{2}}
$$
## Heat/Diffusion Equation
Image some material where the temperature is not constant but with no sources (or sinks) of heat.
Temperature is a scalar field $\phi(\vec{r}, t)$. Heat current $\vec{j}(\vec{r}, t)$ is a vector field s.t. energy flux across an oriented surface $S$.
INSERT IMAGE OF A VECTOR $\vec{n}$ OUT OF A SURFACE $S$
is the surface integral $\int_{S} \vec{j}(\vec{r}, t) \overrightarrow{d A}$
Let $\mathrm{S}$ be a closed (but static) surface. Heat flux out of $\mathrm{S}$
$$
=\int_{S} \vec{j}(\vec{r}, t) \overrightarrow{d A}
$$
$=-$ rate of change of energy in $\mathrm{D}$
INSERT IMAGE HERE
$$
\begin{aligned}
& =-\alpha \frac{\partial}{\partial t} \int_{D} \phi(\vec{r}, t) d V \\
& =-\alpha \int_{D} \frac{\partial \phi}{\partial t}(\vec{r}, t) d V
\end{aligned}
$$
$\alpha$ constant (heat capacity per unit volume)
Apply Gauss' theorem
$$
\int_{S} \vec{j}(\vec{r}, t) \cdot \overrightarrow{d A}=\int_{D} \operatorname{div} \vec{j} d V
$$
so that
$$
\int_{D}\left(\operatorname{div} \vec{j}+\alpha \frac{\partial \phi}{\partial t}\right) d V=0
$$
where $\mathrm{D}$ is any $3 \mathrm{~d}$ region with a smooth boundary. Thus
$$
\operatorname{div} \vec{j}+\alpha \frac{\partial \phi}{\partial t}=0
$$
Assume $\vec{j}=-\beta \operatorname{grad} \phi$ where $\beta$ is the thermal conductivity constant and the - sign indicates that the heat flows from hot to cold regions
$$
\nabla^{2} \phi=D \frac{\partial \phi}{\partial t} \text { with } D=\frac{\alpha}{\beta}
$$
Laplace's Equation is a special case of Poisson's equation and the heat equation
$$
\begin{gathered}
\text { Poisson } \rho=0 \Rightarrow \text { no source } \Rightarrow \nabla^{2} \phi=0 \\
\text { Heat } \frac{\partial \phi}{\partial t}=0 \Rightarrow \text { steady state } \Rightarrow \nabla^{2} \phi=0
\end{gathered}
$$
## Boundary Value Problems
Often wish to solve a PDE subject to some boundary conditions.
Assume $\phi$ satisfies some PDE (e.g. Laplace's equation) in a 3d region D with boundary $S=\partial D$
INSERT IMAGE
### 3 basic kinds of b.c.s.
i) $\phi$ is given on $S \rightarrow$ Dirichlet boundary conditions
ii) $\partial_{n} \phi=\vec{n} \cdot \nabla \phi$ (directional derivative in direction of unit normal $\vec{n}$ ) is given on $\mathrm{S} \rightarrow$ Neumann boundary conditions
iii) $\phi$ and $\partial_{n} \phi$ are given on $S \rightarrow$ Cauchy boundary conditions
Can also have mixed b.c.s. where on different parts of $\mathrm{S}$ different b.c.s are imposed
This is not exhaustive since there are other types of b.c.s. such as periodic boundary conditions.
## $3.2 \quad$ Elliptic Case
Usually the Cauchy b.c.s. are too strong, i.e. no solutions.
Dirichlet b.c.s. more or less lead to a unique solution.
Neumann b.c.s. lead to a unique solution (up to an arbitrary constant in Laplace/Poisson cases).
### Other cases (parabolic and hyperbolic)
more complicated.
## Laplace's Equation
A solution of Laplace's equation is called a harmonic function. Some simple (singular) examples:
$3 \mathrm{~d} \phi=\frac{1}{r}$ is harmonic but singular at the origin.
2d $\phi=\log r, r=\sqrt{x^{2}+y^{2}}$ harmonic but singular at $r=0$.
$1 \mathrm{~d} \phi=x$ is harmonic but singular at $\pm \infty$.
2d Any holomorphic function ( Complex Analysis ) is harmonic!
Suppose we wish to find a non-singular (e.g. $C^{\infty}$ ) harmonic function is some domain D subject to some boundary conditions on $S=\partial D$.
### Theorem: Uniqueness of solns to DBCs and NBCs
The solution of Laplace's equation under Dirichlet's boundary conditions (DBCs), if it exists, is unique. The solution of the problem under Neumann boundary conditions, if it exists, is unique up to an additive constant.
### 1 $\underline{\text { Proof }}$
postponed; digression on vector analysis required.
#### Green's Identities
(not to be confused with Green's theorem in the plane) Let $\phi$ and $\psi$ be smooth functions, (not necessarily harmonic)
## Green's 1st Identity
$$
\int_{D}\left(\phi \nabla^{2} \psi+\nabla \phi \cdot \nabla \psi\right) d V=\int_{S=\partial D} \phi \nabla \psi \cdot \overrightarrow{d A}
$$
## Green's 2nd Identity
$$
\int_{D}\left(\phi \nabla^{2} \psi-\psi \nabla^{2} \phi\right) d V=\int_{S=\partial D}(\phi \nabla \psi-\psi \nabla \phi) \cdot \overrightarrow{d A}
$$
## $\underline{\text { Proofs }}$
1st identity. Apply Gauss' theorem to the vector field $\vec{F}=\phi \nabla \psi$
$$
\operatorname{div} \vec{F}=\phi \nabla \cdot(\nabla \psi)+\nabla \phi \cdot \nabla \psi
$$
vector identity
$$
=\phi \nabla^{2} \psi+\nabla \phi \cdot \nabla \psi
$$
2nd identity. Interchange $\phi$ and $\psi$ in first identity the subtract from the first identity.
#### Proof of uniqueness Theorem
Let $\phi_{1}$ and $\phi_{2}$ be harmonic in D and subject to the same boundary conditions on $S=\partial D$ (either DBCs or NBCs). Consider $\phi=\phi_{1}-\phi_{2}$. Apply Green's 1st identity taking $\psi=\phi$
$$
\begin{gathered}
\int_{D}\left(\phi \nabla^{2} \psi+\nabla \phi \nabla \psi\right) d V=\int \phi \nabla \phi \cdot \overrightarrow{d A} \\
=0\left(\text { since } \phi=\phi_{1}-\phi_{2} \text { is harmonic }\right)=\int \partial_{n} \phi d A \\
=0 \text { for } \operatorname{DBCs}(\phi=0) \\
=0 \text { for } \operatorname{NBCs}\left(\partial_{n} \phi=0\right)
\end{gathered}
$$
Therefore
$$
\int_{D} \nabla \phi \cdot \nabla \phi d V=0
$$
and $\nabla \phi \cdot \nabla \phi$ is non-negative!
which requires $\nabla \phi=0$ or $\phi=$ constant i.e. $\phi_{1}-\phi_{2}=c$ proving the theorem for NBCs. DBCs: c must be zero since $\phi_{1}$ and $\phi_{2}$ agree on $\mathrm{S}$ by assumption.
END FIRST THREE LECTURES
START SECOND THREE LECTURES
## Gauss' Mean Value Theorem
for Harmonic Functions
INSERT IMAGE
Suppose $\phi$ is harmonic in $D \subset \mathbb{R}^{3}$. The average value of $\phi$ over the surface of a sphere of radius $R$ centred at the point $\vec{r}$ is $\phi(\vec{r})$.
#### Note.
This is true for any point in the interior of $D$. The radius $R$ is any number s.t. the sphere (and every point inside it) is in D.
#### Proof
Without loss of generality consider a sphere centred at the origin. Idea is to show that the average
$$
\overline{\phi_{R}}=\frac{1}{4 \pi R^{2}} \int_{x^{2}+y^{2}+z^{2}=R^{2}} \phi d A
$$
is independent of the radius $R$.
Apply Green's 2nd identity to $\phi$ and $\psi=\frac{1}{r}$ in the regions $R_{1}<r<R_{2}$
INSERT IMAGE
$$
\begin{gathered}
\int_{R_{1}<r<R_{2}}\left(\phi \nabla^{2} \frac{1}{r}-\frac{1}{r} \nabla^{2} \phi\right) d V \\
=\int_{r=R_{2}, \text { out }}\left(\phi \nabla \frac{1}{r}-\frac{1}{r} \nabla \phi\right) \cdot \overrightarrow{d A}-\int_{r=R_{1}, \text { out }}\left(\phi \nabla \frac{1}{r}-\frac{1}{r} \nabla \phi\right) \cdot \overrightarrow{d A}
\end{gathered}
$$
Now,
$$
\int_{r=R_{1}, \text { out }} \frac{1}{r} \nabla \phi \cdot \overrightarrow{d A}=\frac{1}{R_{1}} \int_{r=R_{1}, \text { out }} \nabla \phi \cdot \overrightarrow{d A}
$$
because $\frac{1}{r}$ is constant of sphere $r=R_{1}$
$$
\begin{aligned}
& =\frac{1}{R_{1}} \int_{r<=R_{1}} \operatorname{div} \nabla \phi d V \\
& =\frac{1}{R_{1}} \int_{r<=R_{1}} \nabla^{2} \phi d V=0
\end{aligned}
$$
$$
\text { Similarly } \int_{r=R_{2}, \text { out }} \frac{1}{r} \nabla \phi \cdot \overrightarrow{d A}=0
$$
Thus,
$$
\begin{gathered}
0=\int_{r=R_{2}, \text { out }} \phi \nabla \frac{1}{r} \cdot d \vec{A}-\int_{r=R_{1}, \text { out }} \phi \nabla \frac{1}{r} \cdot d \vec{A} \\
\nabla \frac{1}{r}=-\frac{\hat{\vec{r}}}{r^{2}} \text { in both integrals } \vec{n}=\overrightarrow{\hat{r}} \\
0=-\frac{1}{R_{2}^{2}} \int_{r=R_{2}} \phi d A+\frac{1}{R_{1}^{2}} \int_{r=R_{1}} \phi d A \\
\text { or } \phi_{R_{2}}^{-}=\phi_{R_{2}}
\end{gathered}
$$
Letting $R \rightarrow 0, \bar{\phi}_{R}=\phi(0), R$ being the radius of the outer sphere.
## Maximum (minimum) Principle
## for Harmonic Functions
Let $\phi$ be harmonic in a $3 \mathrm{~d}$ (or $2 \mathrm{~d}$ ) domain $\mathrm{D}$. Then $\phi$ never assumes its maximum (or minimum) value at an interior point of $\mathrm{D}$ unless $\phi$ is constant.
#### Proof
Assume $\phi$ has a maximum at some point $P$ in the interior of $D$. For $\mathrm{R}$ sufficiently small the sphere of radius $\mathrm{R}$ centred at $\mathrm{P}$ is inside $\mathrm{D}$. For every point on the sphere $\phi<\phi(P)$ so $\overline{\phi_{R}}<\phi(P)$ contradicting the MVT. A similar argument holds if $\mathrm{P}$ is a minimum.
If $\phi$ is harmonic in $D$ it assumes its maximum and minimum values at the boundary $S=\partial D$
### Physical Interpretation
Heat Equation $\nabla^{2}=D \frac{\partial \phi}{\partial t}$. If $\phi$ reaches a steady state $\frac{\partial \phi}{\partial t}=0$, then $\nabla^{2} \phi=0$, i.e. the temperature is harmonic. Suppose we have a finite lump of matter and the boundary temperature (not necessarily constant) is fixed, e.g. consider a square slab with three sides fixed to be at 0 degrees and the other at 100 degrees. The steady state temperature inside the slab is harmonic. Steady state temperature can never exceed 100 degrees (or fall below 0 degrees); heat would immediately flow out of (or enter) such a hot spot (or cold spot).
SQUARE HEAT IMAGE
## Liouville's Theorem
If $\phi$ is harmonic and bounded throughout $\mathbb{R}^{3}$ (or $\mathbb{R}^{2}$ ) then it is constant.
#### Proof
Not given.
#### Note
$\phi=\frac{1}{r} 3 \mathrm{~d} \phi=\log r(2 \mathrm{~d})$ are unbounded.
### Liouville's Theorem - Complex Analysis Version
If $f$ is holomorphic throughout $\mathbb{C}$ (sometimes called an entire function) and bounded $|f|<C$ then $f$ is constant.
### Solutions
Uniqueness theorem very powerful; any solution with DBCs, however simple is the only solution.
### Examples
#### Example i
## INSERT IMAGE
Let $\phi$ be a harmonic function which is constant, say $\phi=a$, on the boundary of $D$. $\phi(\vec{r})=a$ is trivially a solution of Laplace's equation with the correct b.c.s.. It must be the unique solution to this boundary value problem.
#### Example ii
Let $\phi$ be harmonic in a $2 \mathrm{~d}$ annulus with $\phi=1$ on the other boundary $\left(r=R_{2}\right) \phi=b$ on the inner boundary $\left(r=R_{1}\right)$
## INSERT IMAGE
$\phi=C \log r+D\left(r=\sqrt{x^{2}+y^{2}}\right)$ harmonic but singular at origin,
C, D constants.
$$
a=\phi\left(r=R_{2}\right)=C \log R_{2}+D
$$
$$
\begin{gathered}
b=\phi\left(r=R_{1}\right)=C \log R_{1}+D \\
a-b=C \log \frac{R_{2}}{R_{1}} \\
D=a-C \log R_{2} \\
\phi=\frac{a-b}{\log \frac{R_{2}}{R_{1}}} \log r R_{2}+a
\end{gathered}
$$
In these 2 examples we have guessed a solution (which we know to be unique).
This is clearly insufficient for most problems, e.g. the square where $\phi$ is zero on three sides, and another value, say 1 , on the remaining side.
Insert Image
## Separation of Variables
(a more systematic approach to solving linear PDEs)
Idea is to reduce PDEs involving 2 or more variables to ODEs in each variable.
$$
\operatorname{Try} \phi(x, y)=X(x) Y(y)
$$
where $\mathrm{X}$ depends on $\mathrm{x}$ only and $\mathrm{Y}$ depends on $\mathrm{y}$ only. Laplace's equation becomes
$$
\nabla^{2} \phi=X^{\prime \prime}(x) Y(y)+X(x) Y^{\prime \prime}(y)=0
$$
divide through by $X(x) Y(y)$
$$
\frac{X^{\prime \prime}(x)}{X(x)}+\frac{Y^{\prime \prime}(y)}{Y(y)}=0
$$
with the first term independent of $\mathrm{y}$ and the second independent of $\mathrm{x}$.
$$
\text { Therefore, } \frac{X^{\prime \prime}(x)}{X(x)}=-\frac{Y^{\prime \prime}(y)}{Y(y)}=\text { constant independent of } \mathrm{x} \text { and } \mathrm{y}
$$
3 possibilities constant i) positive, ii) zero, iii) negative.
i)
$$
\begin{array}{ll}
\frac{X^{\prime \prime}(x)}{X(x)}=-\frac{Y^{\prime \prime}(y)}{Y(y)}=k^{2} 2 \text { ODEs, k constant } \\
X^{\prime \prime}(x)=k^{2} X(x) & Y^{\prime \prime}(y)=-k^{2} Y(y) \\
\text { with solutions } & \text { with solutions } \\
X(x)=e^{k x}, e^{-k x} & Y(y)=\sin k y, \cos k y
\end{array}
$$
For each $k 4$ independent solutions of Laplace's equation $\phi=e^{k x} \sin k y, e^{k x} \cos k y e^{-k x} \sin k y, e^{-k x} \cos k y$
ii) constant $=0$
$$
\begin{array}{ll}
X^{\prime \prime}(x)=0 ; & Y^{\prime \prime}(y)=0 \\
X(x)=A x+B & Y(y)=C y+D
\end{array}
$$
$\Rightarrow 4$ independent solutions of Laplace's equation
iii) constant negative
$$
\begin{gathered}
\frac{X^{\prime \prime}(x)}{X(x)}=-\frac{Y^{\prime \prime}(y)}{Y(y)}=-k^{2} \\
X(x)=\sin k x, \cos k x \quad Y(y)=e^{k y}, e^{-k y}
\end{gathered}
$$
For each k 4 solutions
$$
\phi=\sin k x, e^{k y}, \sin k x, e^{-k y}, \cos k x, e^{k y}, \cos k x, e^{-k y}
$$
Lots of solutions! None of these satisfy the b.c.s. for one square problem, however we can implement left and right b.c.s.
$$
\phi(0, y)=\phi(\pi, y)=0
$$
solutions (from iii) ) $\sin k x e^{k y}$ and $\sin k x e^{-k y}$ satisfy left and right b.c.s. if $k$ is an integer.
Consider a linear combination of these solutions
$$
\phi(x, y)=\sum_{n=1}^{\infty} \sin n x\left(b_{n} e^{n y}+\tilde{b_{n}} e^{-n y}\right)
$$
Now try to find $b_{n}$ and $\tilde{b_{n}}$ such that upper and lower b.c.s. are satisfied.
#### Upper boundary condition
$$
0=\phi(x, \pi)=\sum_{n=1}^{\infty} \sin n x\left(b_{n} e^{n \pi}+b_{n} e^{-n \pi}\right)
$$
This is satisfied if
$$
b_{n} e^{n \pi}+\tilde{b_{n}} e^{-n \pi}=0 \text { for all } \mathrm{n} .
$$
#### Lower boundary condition
$$
\phi(x, 0)=\sum_{n=1}^{\infty} \sin n x\left(b_{n}+\tilde{b_{n}}\right)=?
$$
1 is even while the sines are odd. Recall
$$
f(x)=1 \quad 0<x<\pi-1 \quad-\pi<x<0
$$
has the Fourier Series expansion
$$
f(x)=\frac{4}{\pi} \sum_{n \text { odd }>0} \frac{1}{n} \sin n x
$$
Restricting to $0<x<\pi$ we have
$$
1=\frac{4}{\pi} \sum_{n \text { odd }>0} \frac{1}{n} \sin n x
$$
half-range sine series
$$
\begin{aligned}
& \text { Set } b_{n}+\tilde{b_{n}}=\frac{4}{\pi n} \mathrm{n} \text { odd } 0 \mathrm{n} \text { even } \\
& b_{n}=\frac{4}{\pi n\left(1-e^{2 \pi n}\right.} \mathrm{n} \text { odd } 0 \mathrm{n} \text { even } \\
& \phi(x, y)=\frac{4}{\pi} \sum_{n \text { odd }>0} \frac{\sin n x}{n} \frac{e^{n y}}{e^{2 \pi n}-1}
\end{aligned}
$$
## $9 \quad$ Half-range Sine Series
An even function $\mathrm{f}$ can be expanded in sine waves over a half-period
Suppose $f(x+2 \pi)=f(x), f(-k)=f(k)$
The function $\mathrm{f}$ odd defined as
$$
f_{\text {odd }}(x)=\begin{aligned}
& f(x), \quad 0<x<\pi \\
& -f(x), \quad-\pi<x<0
\end{aligned}
$$
is odd and agrees with $f(x)$ for $o<x<\pi$. The usual Fourier expansion of $f_{\text {odd }}$ contains only sines
$$
f(x)=\sum_{n=1}^{\infty} b_{n} \sin n x \quad 0<x<\pi
$$
$$
b_{n}=\frac{2}{\pi} \int_{0}^{\pi} d x \sin n x f(x)
$$
(true if $f$ odd, even or neither)
This is known as a half-range Fourier sine series. If $f$ is odd it coincides with the usual Fourier series.
In Q2 of problem sheet 19 the cosine must be expanded as a sine series.
Back to our square boundary value problem.
Insert Image
Lines of constant $\phi$ must converge at the two lower corners.
Insert Image
Closed loops of constant $\phi$ do not occur - uniqueness theorem would force $\phi$ to be constant inside any such loop.
Other constant boundary conditions:
Insert Image
$a, b, c, d$ constant
can be dealt with through linear combinations of the basic $a=1, b=c=d=0$ square and similar solutions
Insert Image
interchange $x$, and $y$ in $a=1, b=c=d=0$ solution
Insert Image
$x \rightarrow 1-x$ in previous solution
$\phi=$ constant is also a solution.
### Periodic Strip
Periodic in $x$ direction $\pi(x+2 \pi, y)=\phi(x, y)$ and $y$ in some finite range, say $0 \leq y \leq 1$ with DBCs at $y=0$ and $y=1$.
Insert Image
$\phi(x, y=1)=g(x) \quad g(x+2 \pi)=g(x)$
$\phi(x, y=0)=f(x) \quad f(x+2 \pi)=f(x)$
separation of variables:
$\phi(x, y)=X(x) Y(y)$
Solutions (must be periodic in $x$ )
$X(x)=\cos n x$ or $\sin n x \quad Y(y)=e^{n y}$ or $e^{-n y}$
and $X(x)=A+B x \quad Y(y)=C+D y$
$B=0$ periodicity
(redefine $\left.k_{1}=A C, k_{2}=A D\right)$
$$
\phi(x, y)=k_{1}+k_{2} y+\sum_{n=1}^{\infty}\left(a_{n} e^{n y}+\tilde{a_{n}} e^{-n y}\right) \cos n x+\sum_{n=1}^{\infty}\left(b_{n} e^{n y}+\tilde{b_{n}} e^{-n y}\right) \sin n x
$$
Obtain coefficients $k_{1}, k_{2}, a_{n}, \tilde{a_{n}}, b_{n}, \tilde{b_{n}}$ through boundary conditions at $y=0$, and $y=1$.
Note that separation of variables in Cartesian coordinates is not always possible and even when it is, it is not always useful (if the boundary conditions are not suited).
## Seperation of Variables in other Coordinate Sys- tems
In situations with circular symmetry polar coordinates are advantageous. Separation of variables in polar coordinates
$$
\phi(r, \theta)=R(r) \Theta(\theta)
$$
But to solve, for example, Laplace's equation require Laplacian in these coordinates
$$
\nabla^{2}=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}}=\frac{1}{r} \frac{\partial}{\partial r} r \frac{\partial}{\partial r}+\frac{1}{r^{2}} \frac{\partial^{2}}{\partial \theta^{2}}
$$
For more compliacated coordinate systems, e.g. spherical polars, this becomes messy
### Curvilinear Coordinates
Consider a general change of coordinates
$$
\begin{aligned}
x & =x\left(u_{1}, u_{2}, u_{3}\right) \\
y & =y\left(u_{1}, u_{2}, u_{3}\right) \text { or } \underline{r}=\underline{r}\left(u_{1}, u_{2}, u_{3}\right) \\
z & =z\left(u_{1}, u_{2}, u_{3}\right)
\end{aligned}
$$
e.g. $\left(u_{1}, u_{2}, u_{3}\right)=(r, \theta, \phi)$ spherical polars.
Can rewrite grad, div and curl as derivatives w.r.t. the new coordinates
$$
\begin{aligned}
\operatorname{grad} \phi & =\frac{\partial \phi}{\partial x} \underline{i}+\frac{\partial \phi}{\partial y} \underline{j}+\frac{\partial \phi}{\partial z} \underline{k} \\
& =\left(\frac{\partial \phi}{\partial u_{1}} \frac{\partial u_{1}}{\partial x}+\frac{\partial \phi}{\partial u_{2}} \frac{\partial u_{2}}{\partial x}+\frac{\partial \phi}{\partial u_{3}} \frac{\partial u_{3}}{\partial x}\right) \underline{i}+\text { cyclic perms } \\
& =\frac{\partial \phi}{\partial u_{1}} \nabla u_{1}+\frac{\partial \phi}{\partial u_{2}} \nabla u_{2}+\frac{\partial \phi}{\partial u_{3}} \nabla u_{3}
\end{aligned}
$$
Working with other coordinate systems convenient to replace basis vectors $\underline{i}, \underline{j}, \underline{k}$ with unit vectors $\underline{e_{u_{1}}}, \underline{e_{u_{2}}}, \underline{e_{u_{3}}}$ pointing in the direction of increasing $u_{1}, u_{2}, u_{3}$.
Define
$$
\begin{array}{ll}
\underline{e_{u_{1}}}=\frac{1}{h_{u_{1}}} \frac{\partial \underline{r}}{\partial u_{1}} & h_{u_{1}}=\left|\frac{\partial \underline{r}}{\partial u_{1}}\right| \\
\underline{e_{u_{2}}}=\frac{1}{h_{u_{2}}} \frac{\partial \underline{r}}{\partial u_{2}} & h_{u_{2}}=\left|\frac{\partial \underline{r}}{\partial u_{2}}\right| \\
\underline{e_{u_{3}}}=\frac{1}{h_{u_{3}}} \frac{\partial \underline{r}}{\partial u_{3}} & h_{u_{3}}=\left|\frac{\partial \underline{r}}{\partial u_{3}}\right|
\end{array}
$$
(can abbreviate $\underline{e}_{1}=\underline{e}_{u_{1}}, \underline{e}_{2}=\underline{e}_{u_{2}}$ etc and $h_{1}=h_{u_{1}}, h_{2}=h_{u_{2}}$ etc. )
### Spherical Polar Coordinates
$$
\begin{aligned}
x & =r \sin \theta \cos \phi \\
y & =r \sin \theta \sin \phi \\
z & =r \cos \theta \\
\frac{\partial \underline{r}}{\partial r}=\sin \theta \cos \phi \underline{i} & +\sin \theta \cos \phi \underline{j}+\cos \theta \underline{k}
\end{aligned}
$$
$$
\begin{aligned}
& h_{r}=\left|\frac{\partial \underline{r}}{\partial r}\right|=\sqrt{(\sin \theta \cos \phi)^{2}+(\sin \theta \sin \phi)^{2}+\cos ^{2} \theta}=1 \\
& \underline{e_{r}}=\sin \theta \cos \phi \underline{i}+\sin \theta \sin \phi \underline{j}+\cos \theta \underline{k} \\
& =\frac{\underline{r}}{r} \text { sometimes written } \underline{\underline{r}} \\
& \frac{\partial \underline{r}}{\partial \theta}=r \cos \theta \cos \phi \underline{i}+r \cos \theta \sin \phi \underline{j}-r \sin \theta \underline{k} \\
& h_{\theta}=\left|\frac{\partial \underline{r}}{\partial \theta}\right|=r \sqrt{\cos ^{2} \theta \cos ^{2} \phi+\cos ^{2} \theta \sin ^{2} \phi+\sin ^{2} \theta}=r \\
& \underline{e}_{\theta}=\frac{1}{h_{\theta}} \frac{\partial \underline{r}}{\partial \theta}=\cos \theta \cos \phi \underline{i}+\cos \theta \sin \phi \underline{j}-\sin \theta \underline{k} \\
& \frac{\partial \underline{r}}{\partial \phi}=-r \sin \theta \sin \phi \underline{i}+r \sin \theta \cos \phi \underline{j} \\
& h_{\phi}=\left|\frac{\partial \underline{r}}{\partial \phi}\right|=r \sqrt{\sin ^{2} \theta \sin ^{2} \phi+\sin ^{2} \theta \cos ^{2} \phi}=r \sin \theta \\
& \underline{e}_{\phi}=\frac{1}{h_{\phi}} \frac{\partial \underline{r}}{\partial \phi}=-\sin \phi \underline{i}+\cos \phi \underline{j}
\end{aligned}
$$
Note that the new basis vectors $\underline{e}_{r}, \underline{e}_{\theta}$ and $\underline{e}_{\phi}$ are orthogonal $\underline{e}_{r} \cdot \underline{e}_{\theta}=\underline{e}_{\theta} \cdot \underline{e}_{\phi}=\underline{e}_{\phi} \cdot \underline{e}_{r}=0$ $\underline{\text { Claim }}$ The three (un-normalised) basis vectors
$$
\frac{\partial \underline{r}}{\partial u_{1}}, \frac{\partial \underline{r}}{\partial u_{2}}, \frac{\partial \underline{r}}{\partial u_{3}}
$$
are dual to the three gradients
$$
\nabla u_{1}, \nabla u_{2}, \nabla u_{3}
$$
i.e.
$$
\nabla u_{i} \cdot \frac{\partial \underline{r}}{\partial u_{j}}=\delta_{i j} \quad, \text { where } \delta \text { is the Kronecker delta }
$$
$\underline{\text { Proof }}$
$$
\nabla u_{1} \cdot \frac{\partial \underline{r}}{\partial u_{1}}=\frac{\partial u_{1}}{\partial x} \frac{\partial x}{\partial u_{1}}+\frac{\partial u_{1}}{\partial y} \frac{\partial y}{\partial u_{1}}+\frac{\partial u_{1}}{\partial z} \frac{\partial z}{\partial u_{1}}=\frac{\partial u_{1}}{\partial u_{1}}=1
$$
## Orthogonal Curvilinear Coordinates
Assume basis vectors $\underline{e}_{i}$ are orthogonal
$$
\underline{e}_{i} \underline{e}_{j}=\delta_{i j}
$$
(e.g. spherical polars )
Write
$$
\nabla u_{i} \cdot \frac{\partial \underline{r}}{\partial u_{j}}=\delta_{i j}
$$
as
$$
\nabla u_{i} . h_{j} \underline{e}_{j}=\delta_{i j}
$$
or
$$
h_{j} \nabla u_{i} \cdot \underline{e}_{j}=\delta_{i j}
$$
This implies that $\underline{e}_{i}=h_{i} \nabla u_{i}$ or
$$
\nabla u_{i}=\frac{1}{h_{i}} \underline{e}_{i} \text { ( orthogonal coord. system ) }
$$
Inserting this into gradient formula
$$
\begin{aligned}
\operatorname{grad} \phi & =\frac{\partial \phi}{\partial u_{1}} \nabla u_{1}+\frac{\partial \phi}{\partial u_{2}} \nabla u_{2}+\frac{\partial \phi}{\partial u_{3}} \nabla u_{3} \\
& =\frac{1}{h_{1}} \frac{\partial \phi}{\partial u_{1}} \underline{e}_{1}+\frac{1}{h_{2}} \frac{\partial \phi}{\partial u_{2}} \underline{e}_{2}+\frac{1}{h_{3}} \frac{\partial \phi}{\partial u_{3}} \underline{e}_{3}
\end{aligned}
$$
For example in spherical polars $(r, \theta, \phi) h_{r}=1, h_{\theta}=r, h_{\phi}=r \sin \theta$
$$
\operatorname{grad} \Phi=\frac{\partial \Phi}{\partial r} \underline{e}_{r}+\frac{1}{r} \frac{\partial \Phi}{\partial \theta} \underline{e}_{\theta}+\frac{1}{r \sin \theta} \frac{\partial \Phi}{\partial \phi} \underline{e}_{\phi}
$$
( large $\Phi$ not to be confused with angle $\phi !$ )
A vector field $\underline{F}=F_{x} \underline{i}+F_{y} \underline{y}+F_{z} \underline{k}$ can be written in terms of the 'new' basis vectors $\underline{e_{1}}, \underline{e_{2}}, \underline{e_{3}}$.
$$
\underline{F}=F_{u_{1}} \underline{e_{u_{1}}}+F_{u_{2}} \underline{e_{u_{2}}}+F_{u_{3}} \underline{e_{u_{3}}}
$$
abbreviated to
$$
\underline{F}=F_{1} \underline{e_{1}}+F_{2} \underline{e_{2}}+F_{3} \underline{e_{3}}
$$
For example
$$
\begin{aligned}
\underline{F} & =\frac{1}{r^{3}}(x \underline{i}+y \underline{j}+z \underline{k}) \\
& =\frac{1}{r^{2}} e_{r}
\end{aligned}
$$
or $F_{r}=\frac{1}{r^{2}}, F_{\theta}=F_{\phi}=0$
#### Divergence and curl
Divergence of a vector field $\underline{F}=F_{1} \underline{e_{1}}+F_{2} \underline{e_{2}}+F_{3} \underline{e_{3}}$
$$
\div \underline{F}=\frac{1}{h_{1} h_{2} h_{3}}\left[\frac{\partial}{\partial u_{1}}\left(F_{1} h_{2} h_{3}\right)+\frac{\partial}{\partial u_{2}}\left(F_{2} h_{3} h_{1}\right)+\frac{\partial}{\partial u_{3}}\left(F_{3} h_{1} h_{2}\right)\right]
$$
(coord system orthogonal)
Proof Based on identity
$$
\div \frac{e_{1}}{h_{2} h_{3}}=\div \frac{e_{2}}{h_{3} h_{1}}=\div \frac{e_{3}}{h_{1} h_{2}}=0
$$
Write $\underline{F}=\left(F_{1} h_{2} h_{3}\right) \frac{e_{1}}{h_{2} h_{3}}+\left(F_{2} h_{3} h_{1}\right) \frac{e_{2}}{h_{3} h_{1}}+\left(F_{3} h_{1} h_{2}\right) \frac{e_{3}}{h_{1} h_{2}}$
Use $\div \phi \underline{G}=\phi \div \underline{G}+\nabla \phi \cdot \underline{G}$
$$
\begin{aligned}
& \div \underline{F}=\nabla\left(F_{1} h_{2} h_{3}\right) \cdot \frac{e_{1}}{h_{2} h_{3}}+\nabla\left(F_{2} h_{3} h_{1}\right) \cdot \frac{e_{2}}{h_{3} h_{1}}+\nabla\left(F_{3} h_{1} h_{2}\right) \cdot \frac{e_{3}}{h_{1} h_{2}} \\
& \nabla\left(F_{1} h_{2} h_{3}\right)=\frac{e_{1}}{h_{1}} \frac{\partial}{\partial u_{1}}\left(F_{1} h_{2} h_{3}\right)+\frac{e_{2}}{h_{2}} \frac{\partial}{\partial u_{2}}\left(F_{2} h_{3} h_{1}\right)+\frac{e_{3}}{h_{3}} \frac{\partial}{\partial u_{3}}\left(F_{3} h_{1} h_{2}\right)
\end{aligned}
$$
and similarly for $\nabla\left(F_{2} h_{3} h_{1}\right)$ and $\nabla\left(F_{3} h_{1} h_{2}\right)$.
Using $\underline{e_{i}} \cdot \underline{e_{j}}=\delta_{i j}$ gives result.
To prove that $\div \frac{e_{1}}{h_{2} h_{3}}=0+$ cyclic perms.
write $\underline{e_{1}}=\underline{e_{2}} \times \underline{e_{3}}$ (orthonormality, can be $\underline{e_{1}}=\underline{e_{2}} \times \underline{e_{3}}$ )
$$
\frac{e_{1}}{h_{2} h_{3}}=\frac{e_{2}}{h_{2}} \times \frac{e_{3}}{h_{3}}=\nabla u_{2} \times \nabla u_{3}
$$
Using identity
$$
\begin{gathered}
\nabla \dot{(} \underline{F} \times \underline{G})=(\nabla \times \underline{F}) \cdot \underline{G}-(\nabla \times \underline{G}) \cdot \underline{F} \\
\nabla \cdot \frac{e_{1}}{h_{2} h_{3}}=\left(\nabla \times \nabla u_{2}\right) \cdot \nabla u_{3}-\left(\nabla \times \nabla u_{3}\right) \cdot \nabla u_{2}=0
\end{gathered}
$$
$$
\text { curl grad }=0
$$
Curl in orthogonal curv. coords.
$$
\operatorname{curl} \underline{F}=\frac{1}{h_{1} h_{2} h_{3}}\left|\begin{array}{ccc}
h_{1} e_{1} & h_{2} e_{2} & h_{3} e_{3} \\
\frac{\partial}{\partial u_{1}} & \frac{\partial}{\partial u_{2}} & \frac{\partial}{\partial u_{3}} \\
h_{1} F_{1} & h_{2} F_{2} & h_{3} F_{3}
\end{array}\right|
$$
Proof: not given Laplacian
$$
\begin{aligned}
& \nabla^{2} \phi=\text { div grad } \phi \\
&=\div() \\
&=[] \\
&\left.\Rightarrow \nabla^{2}=\frac{1}{[}\right]
\end{aligned}
$$
Note $\mathbf{I}^{1}{ }^{1} 2$ October 2006
## PART I Vector Analysis
## Scalar fields
Standard calculus concerns functions whose domain is either the real line $\mathbf{R}$ or a subset of the real line.
- Definition: A function is a mapping
$$
\phi: D \rightarrow \mathbf{R}
$$
where $D$ is a subset of $\mathbf{R}$.
Here we consider functions over more than one dimension, defined on
$$
\mathbf{R}^{3}=\{(x, y, z) \mid x, y, z \in \mathbf{R}\} .
$$
A function over such a domain is called a scalar field:
- Definition: A scalar field on $D$ is a mapping
$$
\phi: D \rightarrow \mathbf{R}
$$
In the examples we will look at $D$ will be $\mathbf{R}^{3}$ or $\mathbf{R}^{2}$ or a subset of one of these. In the definition the scalar part refers to the fact that the target space is $\mathbf{R}$.
$$
\phi(x, y, z)=x y+z
$$
is an example of a scalar field. Physical examples would include the pressure or temperature field of a fluid: at every point in the fluid there is a pressure and so pressure is a scalar field.
## Integration
We would like to define the integral of a scalar field. For functions
$$
\int_{a}^{b} d x f(x)
$$
may be defined as a limit of a Riemann sum $\mathcal{R}$. (Picture I.1.1). We want to generalize this to an integral of a scalar field, written
$$
\int_{D} d V \phi
$$
${ }^{1}$ Conor Houghton, [email protected], see also http://www.maths.tcd.ie/ ${ }^{\text {houghton/231 }}$
$$
\int_{D} d^{3} x \phi
$$
Because they are easier to draw, we will first consider two-dimensional integrals. The region is divided up into rectangular cells (Picture I.1.2) and the lower Riemann sum is
$$
\mathcal{R}=\sum_{r \in R} A_{r} \phi_{r}
$$
where $R$ is the set of rectangles, $A_{r}$ is the area of the rectangle $r$ and $\phi_{r}$ is the infimum of $\phi$ over $r$, to remind you of what this means, $\phi_{r}$ is the largest number such that $\phi_{r} \leq \phi(x, y)$ for all $(x, y) \in r$, for nice functions the infimum is the same as the minimum. One technical matter is that a prescription is needed to deal with incomplete rectangles at the boundary, one that works is to define $A_{r}$ as the area of the full rectangle, even if it is at the boundary and define
$$
\phi_{r}= \begin{cases}\inf \phi & r \text { a complete rectangle } \\ \min (\inf \phi, 0) & r \text { on the boundary }\end{cases}
$$
Now, we define
$$
\int_{D} d A \phi=\int_{D} d x d y \phi=\sup \mathcal{R}
$$
where the supremum is taken over all possible grids. In two-dimension it is common to use $d A$ instead of $d V$ as the infinitesimal element.
In practice the integral is calculated as an iterated integral. The basic idea of an iterated integral is that you integrate in the $x$ direction first and then in the $y$ direction, or visa versa. In the underlying Riemann sum picture this corresponds to first summing the rectangles in horizontal slices and then summing the slices, or, in the other order, first summing the rectangles in vertical slices and then summing the slices (Picture I.1.3) and, as sketched in the picture, each of the slices should approach a one-dimensional Riemann integral. Hence, if $y=d(x)$ is the upper boundary of the curve and $y=c(x)$ the lower and $a$ and $b$ are the lowest and highest $x$ values
$$
\int_{D} d x d y \phi=\int_{a}^{b} d x \int_{c(x)}^{d(x)} d y \phi(x, y)
$$
- Example: Consider the half-disk $x^{2}+y^{2} \leq 1$ and $x>0$ (Picture I.1.5). The upper boundary is $d(x)=\sqrt{1-x^{2}}$ and the lower boundary is $c(x)=0$. The area of the region is the integral of one over the region, this is clear from the definition given above in terms of the Riemann sum. Now
$$
\text { Area }=\int_{D} d A 1=\int_{-1}^{1} d x \int_{0}^{\sqrt{1-x^{2}}} d y 1=\int_{-1}^{1} d x \sqrt{1-x^{2}}=\frac{\pi}{2}
$$
The centers of mass are defined as
$$
\bar{x}=\frac{\int_{D} d A \rho(x, y) x}{\int_{D} \rho(x, y) d A}
$$
$$
\bar{y}=\frac{\int_{D} d A \rho(x, y) y}{\int_{D} \rho(x, y) d A}
$$
where $\rho(x, y)$ is the density. For uniform density, $\rho(x, y)=1$ it is easy to see that $\bar{x}=0$ by symmetry, the $x<0$ portion cancels the $x>0$. Now,
$$
\begin{aligned}
\bar{y} & =\frac{2}{\pi} \int_{-1}^{1} d x \int_{0}^{\sqrt{1-x^{2}}} d y y \\
& =\left.\frac{2}{\pi} \int_{-1}^{1} d x \frac{y^{2}}{2}\right|^{\sqrt{1-x^{2}}} \\
& =\frac{2}{\pi} \int_{-1}^{1} d x \frac{1-x^{2}}{2}=\frac{4}{3 \pi}
\end{aligned}
$$
Now suppose that the density is $\rho=\sqrt{x^{2}+y^{2}}$ then the mass $\int_{D} d A \rho$ is
$$
\int_{D} d A \rho=\int_{-1}^{1} d x \int_{0}^{\sqrt{1-x^{2}}} d y \sqrt{x^{2}+y^{2}}
$$
which is a messy integral, probably possible with a trigonometric substitution, but who would want to get into it? What is clear is that the integrated integral is not exploiting the symmetry of the problem. This motivates us to look at changes of variable; as in one-dimension, a change in variables can be used to bring an integral into an easier form.
## Changes of variable
Let us first recall the situation with integration in one-dimension when making a change in variable. Say we have $f(x)$ some function of $x$ and $u(x)$ is a new variable and, inverting, we know $x$ in terms of $u$ as $x(u)$, then
$$
\int_{a}^{b} d x f(x)=\int_{u(a)}^{u(b)} d u f(x(u)) \frac{d x}{d u}
$$
- Example: So $f(x)=x^{4}$ and we are interested in
$$
\int_{1}^{2} d x x^{4}=\left.\frac{x^{5}}{5}\right|_{1} ^{2}=\frac{1}{5}-\frac{32}{5}=-\frac{31}{5}
$$
and a new variable is given by $u=x^{2}$. In the interval $[1,2]$ this relationship is invertible with $x=\sqrt{u}$ so $f(u)=(\sqrt{u})^{4}=u^{2}$ and the extra factor in the integral is
$$
\frac{d x}{d u}=\frac{d}{d u} \sqrt{u}=\frac{1}{2 \sqrt{u}}
$$
Finally $u(1)=1$ and $u(2)=4$ so
$$
\int_{1}^{2} d x x^{4}=\int_{1}^{4} d u \frac{u^{2}}{2 \sqrt{u}}=\frac{1}{2} \int_{1}^{4} u^{3 / 2}=\left.\frac{1}{5} u^{5 / 2}\right|_{1} ^{4}=-\frac{31}{5}
$$
as before.
Now, for a two-dimensional integral, consider the change of variables from $(x, y)$ to $(u, v)$ related by
$$
\begin{aligned}
& x=x(u, v) \\
& y=y(u, v)
\end{aligned}
$$
then
$$
\int_{D} d x d y \phi(x, y)=\int_{D} d u d v \phi(x(u, v), y(u, v)) J
$$
where the Jacobian $J$ is the absolute value of the determinant
$$
J=\left\|\begin{array}{ll}
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\
\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}
\end{array}\right\|
$$
Another notation for the Jacobian that is used is
$$
J=\frac{\partial(x, y)}{\partial(u, v)}
$$
An explanation will be given for the Jacobian, but only after we have considered the example of polar coördinates.
- Example: Polar coördinates are given by
$$
\begin{aligned}
& x=r \cos \theta \\
& y=r \sin \theta
\end{aligned}
$$
so $r$ corresponds to the distance from the origin and $\theta$ is the angle distended with the $x$-axis at the origin (Picture I.1.6).
The Jacobian for this coördinate change is
$$
J=\left\|\begin{array}{ll}
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\
\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}
\end{array}\right\|=\left\|\begin{array}{cc}
\cos \theta & -r \sin \theta \\
\sin \theta & r \cos \theta
\end{array}\right\|=r\left(\cos ^{2} \theta+\sin ^{2} \theta\right)=r
$$
Hence
$$
\int_{D} d x d y \phi(x, y)=\int_{D} d r d \theta r \phi(r, \theta)
$$
It is easy to see in a picture (Picture I.1.7) why $d V=d x d y=r d r d \theta$. It is also clear that for simple domains containing the origin this can be written as an iterated integral
$$
\int_{D} d x d y \phi(x, y)=\int_{D} d r d \theta r \phi(r, \theta) \int_{0}^{2 \pi} d \theta \int_{0}^{h(\theta)} d r r \phi(r, \theta)
$$
where $r=h(\theta)$ is the boundary and by simple we mean that the boundary can be written in this way (Picture I.1.8).
Now, to deal with the example from earlier 14, in polar coordinates we have
$$
\int_{D} d A \rho=\int_{-1}^{1} d x \int_{0}^{\sqrt{1-x^{2}}} d y \sqrt{x^{2}+y^{2}}=\int_{0}^{\pi} \int_{0}^{1} d r r^{2}=\frac{\pi}{3}
$$
To work out $\bar{y}$ we would also need $\int_{D} d A \rho y$ which is
$$
\int_{D} d A \rho y=\int_{0}^{\pi} d \theta \int_{0}^{1} d r r^{3} \sin \theta=\frac{1}{4} \int_{0}^{\pi} \theta \sin \theta=\frac{1}{2}
$$
giving
$$
\bar{y}=\frac{3}{2 \pi}
$$
So, back to the interpretation of the Jacobian. Consider what happens if you vary $u$, using the Taylor expansion
$$
\begin{aligned}
x(u+\delta u, v) & \approx x(u, v)+\frac{\partial x}{\partial u} \delta u \\
y(u+\delta u, v) & \approx y(u, v)+\frac{\partial y}{\partial u} \delta u
\end{aligned}
$$
and if you vary $y$
$$
\begin{aligned}
x(u, v+\delta v) & \approx x(u, v)+\frac{\partial x}{\partial v} \delta v \\
y(u, v+\delta v) & \approx y(u, v)+\frac{\partial y}{\partial v} \delta v
\end{aligned}
$$
so the area element corresponding to $\delta u$ and $\delta v$ is a small parallelogram with edges
$$
\begin{aligned}
& \mathbf{e}_{1}=\delta u\left(\frac{\partial x}{\partial u} \mathbf{i}+\frac{\partial y}{\partial u} \mathbf{j}\right) \\
& \mathbf{e}_{2}=\delta v\left(\frac{\partial x}{\partial v} \mathbf{i}+\frac{\partial y}{\partial v} \mathbf{j}\right)
\end{aligned}
$$
and there is a formula for the area of a parallelogram, it is
$$
J=\left\|\begin{array}{ll}
\frac{\partial x}{\partial u} \delta u & \frac{\partial x}{\partial v} \delta v \\
\frac{\partial y}{\partial u} \delta u & \frac{\partial y}{\partial v} \delta v
\end{array}\right\|=J \delta u \delta v
$$
Hence the small element of area corresponding to small variations in $u$ and $v$ is $J \delta u \delta v$ and, roughly speaking, the change of variable formula is the infinitesimal limit of this. Note I. $2^{1} 4$ October 2006
## Three dimensions
The three-dimensional case is a straight-forward extension of the two-dimensional analysis. The Cartesian iterated integral has the form
$$
\int_{D} d V \phi=\int_{a}^{b} d x \int_{c(x)}^{d(x)} d y \int_{e(x, y)}^{f(x, y)} d z \phi(x, y, z)
$$
where $z=e(x, y)$ and $z=f(x, y)$ describe the upper and lower surfaces bounding the domain $D$ (Picture I.2.1). The general coördinate transform
$$
\begin{aligned}
x & =x(u, v, w) \\
y & =y(u, v, w) \\
z & =z(u, v, w)
\end{aligned}
$$
has
$$
d x d y d z=J d u d v d w
$$
where the Jacobian is
$$
J=\frac{\partial(x, y, z)}{\partial(u, v, w)}=\left\|\begin{array}{lll}
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\
\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\
\frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w}
\end{array}\right\|
$$
Of course this formula is more general than this, it applies for any transformation between coördinate systems.
Two commonly used coördinate systems are spherical polar coördinates and cylindrical polar coördinates. The spherical polar coördinates are $r, \theta$ and $\phi$ where $r$ is the distance from the origin, $\theta$, called the polar angle, is the angle distended with the $z$-axis and $\phi$, called the azimuthal angle, is the angle the projection onto the $x y$-plane makes with the $x$-axis (Picture I.2.2). The spherical polars are related to Cartesians by
$$
\begin{aligned}
& x=r \sin \theta \cos \phi \\
& y=r \sin \theta \sin \phi \\
& z=r \cos \theta
\end{aligned}
$$
and $J=r^{2} \sin \theta$. The polar coördinates are $z, \rho$ and $\phi, z$ is the distance along the $z$-axis, as usual, $\rho$ is the length of the projection on the $x y$-plane and $\phi$ is the angle the projection distends with the $x$-axis (Picture I.2.3), they are related to the Cartesians by
$$
\begin{aligned}
& x=\rho \cos \phi \\
& y=\rho \sin \phi \\
& z=z
\end{aligned}
$$
and the Jacobian is $J=\rho$.
${ }^{1}$ Conor Houghton, [email protected], see also http://www .maths.tcd.ie/ houghton/231 - Example: A ball of radius $a$ has a cylindrical hole of radius $b<a$ drilled though its center, what is its volume? Well try cylindrical polars with the $z$-axis corresponding to the axis of the hole (Picture I.2.4). Now to integrate over the remaining material we need to work out the ranges for the various coördinates. Obviously $0 \leq \phi \leq 2 \pi$, by trigonometry, at $z \rho=\sqrt{a^{2}-z^{2}}$ and the sphere and the cylinder touch when $\rho=b$, which happens when $z= \pm \sqrt{a^{2}-b^{2}}$, hence $b \leq \rho \leq \sqrt{a^{2}-z^{2}}$ and $-\sqrt{a^{2}-b^{2}} \leq z \leq$ $\sqrt{a^{2}-b^{2}}$ and the iterated integral is
$$
V=\int_{V} d V=\int_{0}^{2 \pi} d \phi \int_{-\sqrt{a^{2}-b^{2}}}^{\sqrt{a^{2}-b^{2}}} d z \int_{b}^{\sqrt{a^{2}-z^{2}}} d \rho \rho
$$
where the final $\rho$ is the Jacobian. Now, we can do this integral
$$
\begin{aligned}
\int_{0}^{2 \pi} d \phi \int_{-\sqrt{a^{2}-b^{2}}}^{\sqrt{a^{2}-b^{2}}} d z \int_{b}^{\sqrt{a^{2}-z^{2}}} d \rho \rho & =2 \pi \int_{-\sqrt{a^{2}-b^{2}}}^{\sqrt{a^{2}-b^{2}}} d z \frac{a^{2}-z^{2}-b^{2}}{2} \\
& =\left.\pi\left[\left(a^{2}-b^{2}\right) z-\frac{z^{3}}{3}\right]\right|_{z=-\sqrt{a^{2}-b^{2}}} ^{z=\sqrt{a^{2}-b^{2}}} \\
& =\frac{4 \pi}{3}\left(a^{2}-b^{2}\right)^{3 / 2}
\end{aligned}
$$
## Vector fields
A scalar field, defined already, maps points in $\mathbf{R}^{3}$ to real numbers; now we define
- Definition: A vector field is a mapping
$$
\mathbf{F}: D \rightarrow \mathbf{R}^{3}
$$
where $D$ is a subset of $\mathbf{R}^{3}$
so a vector field maps points in $\mathbf{R}^{3}$ to three-dimensional vectors.
## - Example:
$$
\mathbf{F}=\left(x y, y^{2}, z\right)
$$
also written
$$
\mathbf{F}=x y \mathbf{i}+y^{2} \mathbf{j}+z \mathbf{k}
$$
is a vector field, where we have used the usual basis
$$
\begin{aligned}
\mathbf{i} & =(1,0,0) \\
\mathbf{j} & =(0,1,0) \\
\mathbf{k} & =(0,0,1)
\end{aligned}
$$
Physical examples include the electric and magnetic fields, $\mathbf{E}$ and $\mathbf{B}$ in electromagnetism and the fluid velocity $\mathbf{u}(x, y, z)$ in a fluid.
## Vector calculus
Now, the issue is how to define the derivatives of scalar and vector fields. In practice, there are three differential operators used; these will be defined and, hopefully, by looking at examples it will become clearer as to why these particular operators are the ones that are important for physically and mathematically.
- Definition: The gradient of a scalar field $\phi$ is the vector field
$$
\operatorname{grad} \phi=\frac{\partial \phi}{\partial x} \mathbf{i}+\frac{\partial \phi}{\partial y} \mathbf{j}+\frac{\partial \phi}{\partial z} \mathbf{k}
$$
so gradient is a map
$$
\begin{aligned}
\operatorname{grad} \text { : scalar fields } & \mapsto \text { vector fields } \\
\phi & \rightarrow \operatorname{grad} \phi=\frac{\partial \phi}{\partial x} \mathbf{i}+\frac{\partial \phi}{\partial y} \mathbf{j}+\frac{\partial \phi}{\partial z} \mathbf{k}
\end{aligned}
$$
It is common and useful to also use the symbolic notation
$$
\operatorname{grad} \phi=\nabla \phi
$$
where $\nabla$, called nabla is the vector operator
$$
\nabla=\frac{\partial}{\partial x} \mathbf{i}+\frac{\partial}{\partial y} \mathbf{j}+\frac{\partial}{\partial z} \mathbf{k}
$$
Hence, for example,
- Example: The gradient of the scalar field $\phi(x, y, z)=x y+y \cos z$ is
$$
\operatorname{grad} \phi=y \mathbf{i}+(x+\cos z) \mathbf{j}-y \sin z \mathbf{k}
$$
and physical examples include the force on a particle
$$
\mathbf{F}=-\nabla V
$$
in a potential energy field $V(x, y, z)$.
Probably the easiest way to understand the gradient is to relate it to the directional derivative; it is easy to see that a sensible definition of the derivative of $\phi$ is the direction given by a unit vector $\hat{\mathbf{e}}=\left(e_{1}, e_{2}, e_{3}\right)$ is
$$
D_{\hat{\mathbf{e}}} \phi:=\lim _{h \rightarrow 0} \frac{\phi(\mathbf{x}+h \hat{\mathbf{e}})-\phi(\mathbf{x})}{h}
$$
but, by expanding $\phi(\mathbf{x}+h \hat{\mathbf{e}})=\phi\left(x+h e_{1}, y+h e_{2}, z+h e_{3}\right)$ using the Taylor expansion
$$
D_{\hat{\mathbf{e}}} \phi=\hat{\mathbf{e}} \cdot \nabla \phi
$$
Obviously this is maximum for $\hat{\mathbf{e}}$ in the same direction as $\nabla \phi$ so the direction of gradient gives the direction that $\phi$ has its greatest variation in and the length of the gradient is the directional derivative in that direction. Similarly, the gradient of $\phi$ is perpendicular to the level surfaces of $\phi$, so $\operatorname{grad} \phi$ is perpendicular to the surface $\phi=$ constant. Finally, we define - Definition: The stationary points of a scalar field are points where the gradient of the field is zero.
The remaining two differential operators act on vector fields, the divergence, sends a vector field to a scalar field and, we will see, the curl sends a vector field to another vector field.
- Definition: The divergence of a vector field $\mathbf{F}=F_{1} \mathbf{i}+F_{2} \mathbf{j}+F_{3} \mathbf{k}$ is
$$
\operatorname{div} \mathbf{F}:=\frac{\partial F_{1}}{\partial x}+\frac{\partial F_{2}}{\partial y}+\frac{\partial F_{3}}{\partial z}
$$
or in the symbolic notation
$$
\operatorname{div} \mathbf{F}=\nabla \cdot \mathbf{F}
$$
Hence
$$
\begin{aligned}
\operatorname{div} \text { : vector fields } & \mapsto \text { scalar fields } \\
\mathbf{F} & \rightarrow \operatorname{div} \mathbf{F}=\nabla \cdot \mathbf{F}
\end{aligned}
$$
So for example
- Example: The divergence of the vector field $\mathbf{F}=(x y, \sin z, z)$ is $\operatorname{div} \mathbf{F}=1+y$.
We will see that it is significant when a vector field has no divergence and
- Definition: A vector field is called solenoidal if it has a zero divergence.
In electromagnetism the magnetic field $\mathbf{B}$ is solenoidal by the Maxwell equations and in fluid flow the continuity equation for an incompressible liquid has a solenoidal velocity field. In fact, the continuity equation is a good way of getting a handle on how the divergence works, consider a compressible fluid with density field $\rho(x, y, z ; t)$ and velocity field $\mathbf{u}(x, y, z ; t)$, at a given time $t$ and at a given point $(x, y, z) \rho$ gives the density of the fluid and $\mathbf{u}$ gives its velocity. The field $\rho \mathbf{u}$ is the mass transport and the continuity equation is
$$
\frac{\partial \rho}{\partial t}=-\operatorname{div}(\rho \mathbf{u})
$$
so the amount of fluid at a point changes according to the divergence of the mass transport field, hence, roughly speaking we can think of the divergence as giving the net accumulation of the vector field at the point.
## Note I.3 ${ }^{1} 5$ October
- Definition: The curl of a vector field $\mathbf{F}=F_{1} \mathbf{i}+F_{2} \mathbf{j}+F_{3} \mathbf{k}$ is the vector field
$$
\begin{aligned}
\text { curl : vector fields } & \mapsto \text { vector fields } \\
\mathbf{F} & \rightarrow \text { curl } \mathbf{F}
\end{aligned}
$$
with
$$
\operatorname{curl} \mathbf{F}=\left(\frac{\partial F_{3}}{\partial y}-\frac{\partial F_{2}}{\partial z}\right) \mathbf{i}+\left(\frac{\partial F_{1}}{\partial z}-\frac{\partial F_{3}}{\partial x}\right) \mathbf{j}+\left(\frac{\partial F_{2}}{\partial x}-\frac{\partial F_{1}}{\partial y}\right) \mathbf{k}
$$
and, in the symbolic notation this is
$$
\operatorname{curl} \mathbf{F}=\nabla \times \mathbf{F}
$$
This is the easiest way to remember the formula, using the determinant formula for the cross product
$$
\operatorname{curl}=\left|\begin{array}{ccc}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
F_{1} & F_{2} & F_{3}
\end{array}\right|
$$
- Example: If $\mathbf{F}=x y \mathbf{i}+y^{2} \mathbf{j}+z \mathbf{k}$ then applying the formula above gives
$$
\operatorname{curl} \mathbf{F}=-x \mathbf{k}
$$
Again, it is not easy at first to get a picture of what the curl does. One rough idea is that it measures the rotation at a point of the vectors in a vector field. Certainly, this is what happens when you take the curl of the rotational field. Consider the velocity field
$$
\mathbf{u}=\mathbf{w} \times \mathbf{r}
$$
where $\mathbf{r}=(x, y, z)$ is the position vector and $\mathbf{w}=\left(w_{1}, w_{2}, w_{3}\right)$ is some constant vector. Now, $\mathbf{u}$ is perpendicular to both $\mathbf{r}$ and $\mathbf{w}$ and the length of $\mathbf{u}$ is constant on circles around $\mathbf{w}$, hence the velocity field corresponds to rotation around $\mathbf{w}$. We will take its curl, first
$$
\begin{aligned}
\mathbf{u} & =\left|\begin{array}{ccc}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
w_{1} & w_{2} & w_{3} \\
x & y & z
\end{array}\right| \\
& =\left(w_{2} z-w_{3} x\right) \mathbf{i}+\left(w_{3} x-w_{1} z\right) \mathbf{j}+\left(w_{1} y-w_{2} x\right) \mathbf{k}
\end{aligned}
$$
Now, substituting this into the curl formula we get
$$
\nabla \times \mathbf{u}=2 \mathbf{w}
$$
- Definition: A vector field is called irrotational if it has zero curl.
${ }^{1}$ Conor Houghton, [email protected], see also http://www .maths.tcd.ie/ houghton/231 The gradient of a scalar field is irrotational:
$$
\text { curl grad } \phi=0
$$
or, using the symbolic notation $\nabla \times \nabla \phi=0$. This is proved by calculation, using the subscript to denote component, so $\mathbf{v}=\left(v_{1}, v_{2}, v 3\right)$ for any vector, we have
$$
\begin{aligned}
(\nabla \times \nabla \phi)_{1} & =\partial_{y}(\nabla \phi)_{3}-\partial_{z}(\nabla \phi)_{2} \\
& =\partial_{y} \partial_{z} \phi-\partial_{z} \partial_{y} \phi
\end{aligned}
$$
and the other components follow in the same way. We have used the useful notation where
$$
\partial_{x}=\frac{\partial}{\partial x}
$$
and so on.
## Vector identities
There are a number of usefull identities involving grad, div and curl. These are usually proved by direct calculation, expand out the various terms.
Let $\phi$ and $\psi$ be scalar fields and $\mathbf{F}$ and $\mathbf{G}$ be vector fields, then
1.
$$
\nabla(\phi \psi)=\phi \nabla \psi+\psi \nabla \phi
$$
This is a direct consequence of the product rule.
2.
$$
\nabla(\phi \mathbf{F})=\nabla \phi \cdot \mathbf{F}+\phi \nabla \cdot \mathbf{F}
$$
and again this follows by just expanding it out
$$
\begin{aligned}
\nabla(\phi \mathbf{F}) & =\partial_{x}\left(\phi F_{1}\right)+\partial_{y}\left(\phi F_{2}\right)+\partial_{z}\left(\phi F_{3}\right) \\
& =F_{1} \partial_{x} \phi+F_{2} \partial_{y} \phi+F_{3} \partial_{z} \phi+\phi\left(\partial_{x} F_{1}+\partial_{y} F_{2}+\partial_{z} F_{3}\right) \\
& =\nabla \phi \cdot \mathbf{F}+\phi \nabla \cdot \mathbf{F}
\end{aligned}
$$
3.
$$
\nabla \times(\phi \mathbf{F})=\nabla \phi \times \mathbf{F}+\phi \nabla \times \mathbf{F}
$$
This can easily be proved too, just check, say, the $x$-component by direct calculation.
4.
$$
\nabla \cdot(\mathbf{F} \times \mathbf{G})=(\nabla \times \mathbf{F}) \cdot \mathbf{G}-\mathbf{F} \cdot \nabla \times \mathbf{G}
$$
The proof of this is handwritten as (Picture I.3.1) 5.
$$
\nabla \times(\mathbf{F} \times \mathbf{G})=(\nabla \cdot \mathbf{G}) \mathbf{F}+(\mathbf{G} \cdot \nabla) \mathbf{F}-(\nabla \cdot \mathbf{F}) \mathbf{G}-(\mathbf{F} \cdot \nabla) \mathbf{G}
$$
This is one of the harder ones to prove since it involves the unusual operator
$$
(\mathbf{G} \cdot \nabla)=G_{1} \partial_{x}+G_{2} \partial_{y}+G_{3} \partial_{z}
$$
and the proof is given as an exercise on the problem sheets.
6.
$$
\nabla(\mathbf{F} \cdot \mathbf{G})=\mathbf{F} \times(\nabla \times \mathbf{G})+\mathbf{G} \times(\nabla \times \mathbf{F})+(\mathbf{F} \cdot \nabla) \mathbf{G})+(\mathbf{G} \cdot \nabla) \mathbf{F})
$$
This is also given as an exercise.
7.
$$
\nabla \cdot(\nabla \times \mathbf{F})=0
$$
or, the curl of a vector field is solenoidal. This is one of the important vector identities which hints at some of the beautiful constructions in differential geometry. It is easy enough to prove by direct calculation.
8.
$$
\nabla \times \nabla \phi=0
$$
was proved above.
9.
$$
\nabla \times(\nabla \times \mathbf{F})=\nabla(\nabla \cdot \mathbf{F})-\triangle \mathbf{F}
$$
where
$$
\triangle \phi=\nabla \cdot \nabla \phi=\left(\partial_{x}^{2}+\partial_{y}^{2}+\partial_{z}^{2}\right) \phi
$$
is the Laplacian, an operator which occurs frequently in physically significant equations. Obviously
$$
\triangle \mathbf{F}=\left(\triangle F_{3}, \triangle F_{2}, \triangle F_{3}\right)
$$
Note I. $4^{1} 4$ October
## Line and surface integrals
For a vector field there are natural ways of integrating over one and two-dimensional subspaces of $\mathbf{R}^{3}$ to get a number, rather than a vector. These are line and surface integrals.
## Line integrals
Consider two points $P_{1}$ and $P_{2}$ joined by a smooth or piecewise smooth curve $C$ (Picture I.4.1). A small segment of $C$ can be represented by a vector $\delta \mathbf{l}$, meaning that for two proximate points on the curve at $\mathbf{x}_{1}$ and $\mathbf{x}_{2}$ with $\delta \mathbf{l}=\mathbf{x}_{2}-\mathbf{x}_{1}$ then all the points on the curve between $\mathbf{x}_{1}$ and $\mathbf{x}_{2}$ are close to the straight line $\mathbf{x}_{1}+t \delta \mathbf{l}$ where $0 \leq t \leq 1$. Anyway, the idea of the line integral is that it is the limit of the sum
$$
\mathcal{L}=\sum_{k=0 \cdots N-1} \mathbf{F}\left(\mathbf{x}_{k}\right) \cdot \delta \mathbf{l}_{k}
$$
where $\mathbf{x}_{0}=P_{1}, \mathbf{x}_{N}=P_{2}$, the other $x_{k}$ are intermediate points on the curve and $\nabla \mathbf{l}_{k}=$ $\mathbf{x}_{k+1}-\mathbf{x}_{k}$ where the limit is the infinitesimal limit where $N$ becomes infinite and all the lengths of the $\nabla \mathbf{l}$ go to zero. With a bit of effort and a lot of fiddling, this can be made into a rigorous definition, but the important idea is that the line integral
$$
\int_{C} \mathbf{F} \cdot \mathbf{d l}
$$
is the integral along the curve of the projection of $\mathbf{F}$ onto the tangent. Note that this definition orients $C$, reversing the orientation reverses the sign of the integral.
The obvious physical example is work against a force: the work done moving a particle from $P_{1}$ to $P_{2}$ along the curve $C$ against a position dependent force $\mathbf{F}(x, y, z)$ is the line integral $\int_{c} \mathbf{F} \cdot \mathbf{d l}$.
In practise the line integral is usually calculated using a parametric form of the formula. Suppose the points on $C$ are given by $\mathbf{x}(u)$ where $u$ is a parameter, a real number, and it runs from $a$ to $b$ so $\mathbf{x}(a)=P_{1}$ and $\mathbf{x}(b)=P_{2}$. In other words there is a map
$$
\begin{aligned}
{[a, b] } & \hookrightarrow \mathbf{R}^{3} \\
u & \rightarrow \mathbf{x}(u)
\end{aligned}
$$
Now, by Taylor,
$$
\mathbf{x}(u+\delta u) \approx \mathbf{x}(u)+\frac{d \mathbf{x}}{d u} \delta u
$$
so we can identify
$$
\delta \mathbf{l} \leftrightarrow \frac{d \mathbf{x}}{d u} \delta u
$$
${ }^{1}$ Conor Houghton, [email protected], see also http://www .maths.tcd.ie/ houghton/231 and can conclude that
$$
\int_{C} \mathbf{F} \cdot \mathbf{d} \mathbf{l}=\int_{a}^{b} d u \mathbf{F}(\mathbf{x}(u)) \cdot \frac{d \mathbf{x}(u)}{d u}
$$
- Example Integrate the vector field
$$
\mathbf{F}=\frac{1}{2} y \mathbf{i}-\frac{1}{2} x \mathbf{j}
$$
over the semi-circular arc of unit radius in the $z=0$ plane. (Picture I.4.2). So, to get a parameterization of the curve take
$$
\begin{aligned}
& x(u)=\cos u \\
& y(u)=\sin u \\
& z(u)=0
\end{aligned}
$$
with $0 \leq u \leq \pi$. Now,
$$
\frac{d \mathbf{x}(u)}{d u}=-\sin u \mathbf{i}+\cos u \mathbf{j}
$$
and substituting for $x$ and $y$ in the formula for $\mathbf{F}$ we get
$$
\mathbf{F}=\frac{1}{2} \sin u \mathbf{i}-\frac{1}{2} \cos u \mathbf{j}
$$
so that
$$
\mathbf{F} \cdot \frac{d \mathbf{x}(u)}{d u}=-\frac{1}{2} \sin ^{2} u-\frac{1}{2} \cos ^{2} u=-\frac{1}{2}
$$
SO
$$
\int_{C} \mathbf{F} \cdot \mathbf{d} \mathbf{l}=-\frac{1}{2} \int_{0}^{\pi} d u=-\frac{\pi}{2}
$$
## Conservative vector fields and path independence
- Definition: A vector field is called conservative if it is the gradient of a scalar field, so $\mathbf{F}$ is conservative if $\mathbf{F}=\nabla \phi$ for some $\phi$.
If curlF $\neq 0$ then $\mathbf{F}$ cannot be conservative, however, the converse need not hold.
- Definition: A vector field is called path independent if the line integral between any two points is the same for any path.
Any conservative field is path-independent: choose any smooth curve joining points $P_{1}$ and $P_{2}$ parameterized by $u \in[a, b]$, then
$$
\mathbf{F} \cdot \frac{d \mathbf{x}}{d u}=\frac{\partial \phi}{\partial x} \frac{d x}{d u}+\frac{\partial \phi}{\partial y} \frac{d y}{d u}+\frac{\partial \phi}{\partial z} \frac{d z}{d u}=\frac{d \phi(\mathbf{x}(u)}{d u}
$$
so by the Fundamental Theorem of Calculus
$$
\int_{C} \mathbf{F} \cdot \mathbf{d} \mathbf{l}=\phi(\mathbf{x}(b))-\phi(\mathbf{x}(a))
$$
and this answer does not depend on the path.
Now, for a conservative field, let $C_{a}$ and $C_{b}$ be two curves with the same endpoints $P_{1}$ and $P_{2}$ (Picture I.4.3). Since a conservative field is path independent,
$$
\int_{C_{a}} \mathbf{F} \cdot \mathrm{dl}=\int_{C_{b}} \mathbf{F} \cdot \mathrm{dl}
$$
Now consider the closed curve $C=C_{a}-C_{b}$ where the minus in $C_{b}$ means we have reversed the orientation,
$$
\oint_{C} \mathbf{F} \cdot \mathbf{d} \mathbf{l}=\int_{C_{a}} \mathbf{F} \cdot \mathbf{d} \mathbf{l}-\int_{C_{b}} \mathbf{F} \cdot \mathbf{d} \mathbf{l}=0
$$
and for any closed curve $C$
$$
\oint_{C} \mathbf{F} \cdot \mathbf{d} \mathbf{l}=0
$$
- Example: Back to the previous example of the semicircle. It is easy to extend the calculation to the full closed circle to show
$$
\oint_{C} \mathbf{F} \cdot \mathbf{d l}=-\frac{1}{2} \int_{0}^{2 \pi} d u=-\pi
$$
So
$$
\mathbf{F}=\frac{1}{2} y \mathbf{i}-\frac{1}{2} x \mathbf{j}
$$
cannot be conservative. This is consistent with $\operatorname{curl} \mathbf{F}=-\mathbf{k} \neq 0$.
In fact, for a continuous vector field $\mathbf{F}$ in an open and connected domain $D$, the following are equivalent
1. $\mathbf{F}$ is conservative.
2. $\oint_{C} \mathbf{F} \cdot \mathbf{d l}=0$ for all closed paths in $D$
3. $\mathbf{F}$ is path independent.
We have already seen that (2) and (3) are equivalent and that (1) implies (3), to finish, then, we need only prove (3) implies (1). Let $P$ be any point in $D$ and let
$$
\phi(\mathbf{x})=\int_{C(P, \mathbf{x})} \mathbf{F} \cdot \mathbf{d} \mathbf{l}
$$
where $C(P, \mathbf{x})$ is any curve joining $P$ and $\mathbf{x}$. Since the line integral is path independent, $\phi$ is uniquely defined. Now, we want to show that $\mathbf{F}=\nabla \phi$. Again, the result is path independent, so, to prove
$$
F_{1}=\partial_{x} \phi
$$
we use a path that goes from $P$ to $P^{\prime}=\left(x^{\prime}, y, z\right)$ where $P^{\prime}$ is chosen so that the straight line segment from $P^{\prime}$ to $(x, y, z)$ is in $D$ (Picture I.4.4). Now
$$
\phi(\mathbf{x})=\int_{C\left(P, P^{\prime}\right)} \mathbf{F} \cdot \mathbf{d} \mathbf{l}+\int_{x_{1}}^{x} \mathbf{F} \cdot \mathbf{d} \mathbf{l}
$$
So
$$
F_{1}=\partial_{x} \phi
$$
The other components follow by a similar trick.
If $D$ is simply connected all loops are contractile (Picture I.4.5). In this case curlF $=$ 0 is sufficient for $\mathbf{F}$ to be conservative, that is, on simply connected domains, irrotational implies conservative. This will be proved later using the Stokes theorem.
## Note I.5 4 October
## Surface integrals
Consider a two-dimensional surface $S$ embedded in a three-dimensional space (Picture I.5.1) with $\mathbf{F}$ a vector field defined in the domain which contains $S$. Now, consider approximating the surface with small flat pieces, for each piece we construct a vector $\delta \mathbf{A}$ which is normal to the surface and whose magnitude is the area of the piece. Now a scalar can be formed by adding
$$
\sum \mathbf{F} \cdot \delta \mathbf{A}
$$
where the sum is taken over all the small pieces $\delta \mathbf{A}$ and in the sum the field is evaluated at the center of the piece. Roughly speaking, the surface integral
$$
\int_{S} \mathbf{F} \cdot \mathbf{d A}
$$
is the infinitesimal limit of this sum, it is the integral over the surface of the projection of $\mathbf{F}$ onto the normal. As with the line integral, with a bit of care, this rough description can be turned into a definition, but we don't do that here. Physically surface integrals measure net flow of a fluid or the net electric or magnetic flux through a surface.
It is important to note that we have oriented the surface by choosing a direction for the normal; at any point in a surface there are two possible normals and a surface is orientable if it is possible to smoothly choose one of these two possible normals over all the surface. It is not easy here to define what we mean by smoothly choose, but it is easy to to explain, we mean that the normal doesn't hop from one side to the next going from one point to a nearby point. The usual example of an unorientable surface is the Moebius strip and this is illustrated in Picture I.5.2.
To compute surface integrals convert they are usually converted into a standard twodimensional integral using a parametric representation for the surface
$$
\begin{aligned}
& x=x(u, v) \\
& y=y(u, v) \\
& z=z(u, v)
\end{aligned}
$$
or $\mathbf{x}=\mathbf{x}(u, v)$ where $(u, v)$ belongs to some domain $D$ in $\mathbf{R}^{2}$. Using the same approach as for the line integrals we will compute the element of area $\delta \mathbf{A}$ corresponding to small variations in $u$ and $v$. By expanding $\mathbf{x}(u+\delta u, v)$ and $\mathbf{x}(u, v+\delta v)$ using the Taylor expansion we find that to leading order the area element is a parallelogram with sides
$$
\begin{aligned}
& \mathbf{a}=\frac{\partial \mathbf{x}}{\partial u} \delta u \\
& \mathbf{b}=\frac{\partial \mathbf{x}}{\partial v} \delta v
\end{aligned}
$$
${ }^{1}$ Conor Houghton, [email protected], see also http://www.maths.tcd.ie/ houghton/231 Now the area of the parallelogram is $|\mathbf{a} \times \mathbf{b}|$ and the vector $\mathbf{a} \times \mathbf{b}$ is perpendicular to $\mathbf{a}$ and $\mathbf{b}$ and hence to the area element so
$$
\delta \mathbf{A}=\mathbf{a} \times \mathbf{b}=\frac{\partial \mathbf{x}}{\partial u} \times \frac{\partial \mathbf{x}}{\partial v} \delta u \delta v
$$
From this we conclude that the surface integral has parametric form
$$
\int_{S} \mathbf{F} \cdot d A=\int_{D} d u d v \mathbf{F}(\mathbf{x}) \cdot \frac{\partial \mathbf{x}}{\partial u} \times \frac{\partial \mathbf{x}}{\partial v}
$$
where the integrand depends on $U$ and $v$ through the parameterization $\mathbf{x}=\mathbf{x}(u, v)$.
- Example: Consider the flux of $\mathbf{F}=x y \mathbf{k}+z \mathbf{i}$ through the triangle with vertices $(0,0,0),(1,0,0)$ and $(0,2,0)$ (Picture I.5.4) and with orientation upwards, in the $z$-direction. The surface is parameterized by $x(u, v)=u$ with $0 \leq u \leq 1, y(u, v)=v$ with $0 \leq v \leq 2(1-u)$ and $z=0$. Hence
$$
\begin{aligned}
\mathbf{x} & =u \mathbf{i}+v \mathbf{j} \\
\frac{\partial \mathbf{x}}{\partial u} & =\mathbf{i} \\
\frac{\partial \mathbf{x}}{\partial v} & =\mathbf{j}
\end{aligned}
$$
and so
$$
\frac{\partial \mathbf{x}}{\partial u} \times \frac{\partial \mathbf{x}}{\partial v}=\mathbf{k}
$$
and
so
$$
\mathbf{F}(\mathbf{x}) \cdot \frac{\partial \mathbf{x}}{\partial u} \times \frac{\partial \mathbf{x}}{\partial v}=x y=u v
$$
$$
\int_{S} \mathbf{F} \cdot \mathbf{d} \mathbf{A}=\int_{0}^{1} \int_{0}^{2(1-u)} d v u v=\frac{v}{1} 2 \int_{0}^{1} d u u[2(1-u)]^{2}=\frac{1}{6}
$$
A vector field can also be integrated over a closed surface. On a closed surface the choice of orientation is a choice between an inwards or outward pointing normal vector.
- Example: Compute the flux of $\mathbf{F}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}$ out of the sphere $x^{2}+y^{2}+z^{2}=$ $a^{2}$. So, we can parameterize the sphere with the polar and azimuthal angles $(\theta, \phi)$. However, rather than calculating the normal to the parameterized surface using the cross product rule above, it is easier just to note that the normal to a sphere is a radial line and so $\mathbf{F} \cdot \hat{\mathbf{n}}=a$ and $\mathbf{d} \mathbf{A}=\hat{\mathbf{n}} d A$, hence
$$
\int_{S} \mathbf{F} \cdot \mathbf{d A}=a \times \text { surface area }=4 \pi a^{3}
$$
## The integral theorems
There are a number of important theorems relating multi-dimensional integration and the operators of vector calculus. These are all basically consequences of the Fundamental Theorem of Calculus and can be thought of as higher-dimensional prescriptions for integration by parts. Their proofs, though not difficult, tend to quite involved and they will only be sketched here.
## The Stokes Theorem
Let $S$ be a piecewise smooth orientable surface bounded by a piecewise smooth curve $C$ (Picture I.5.5). The orientations of $S$ and $C$ are chosen such that at the edge of the surface $\mathbf{n} \times \delta \mathbf{l}$ points into the surface. (Picture I.5.6). Let $\mathbf{F}$ be a continuously differentiable vector field defined in some domain containing $S$ then
$$
\int_{S} \operatorname{curl} \mathbf{F} \cdot \mathbf{d} \mathbf{A}=\oint_{C} \mathbf{F} \cdot \mathbf{d} \mathbf{l}
$$
Sometimes the notation $\partial S$ is used for the correctly oriented boundary of $S$. Before discussing the proof of the Stokes Theorem, we will first look at Green's Theorem. Green's Theorem can be viewed as Stokes' Theorem for flat surfaces and Green's Theorem is used to prove Stokes' Theorem.
## Green's Theorem in the Plane
Let $D$ be a region in the $x y$-plane bounded by a piecewise smooth curve $C$. If $f(x, y)$ and $g(x, y)$ have continuous first derivatives
$$
\int_{D} d A\left(\frac{\partial}{\partial x} g(x, y)-\frac{\partial}{\partial y} f(x, y)\right)=\int_{C}(f(x, y) d x+g(x, y) d y)
$$
with $C$ oriented anti-clockwise. It is easy to check that Stokes' Theorem for a flat surface, taken without loss of generality to lie in the $x y$-plane and oriented upwards, reduces to Green's Theorem with $F_{1}(x, y, 0)$ identified with $f(x, y)$ and $F_{2}(x, y, 0)$ identified with $g(x, y), F_{3}$ does not enter since $\mathbf{d l}$ is perpendicular to $\mathbf{k}$, the normal.
To prove Green's Theorem we first consider a simple region, $D$ where the integral over $D$ can be written as an iterated Cartesian integral in any order, (Picture I.5.8), so the integral of a scalar field $\phi$ can be written as
$$
\int_{D} d A \phi=\int_{a}^{b} d x \int_{c(x)}^{d(x)} d y \phi=\int_{c}^{d} d y \int_{a(y)}^{b(y)} d x \phi
$$
So, now compute
$$
\int_{D} \frac{\partial f}{\partial y} d A=\int_{a}^{b} d x \int_{c(x)}^{d(x)} d y \frac{\partial f}{\partial y}
$$
$$
\begin{aligned}
& =\int_{a}^{b} d x[f(x, d(x))-f(x, c(x))] \\
& =-\oint_{C} d x f(x, y)
\end{aligned}
$$
where we have used to the Fundamental Theorem of Calculus to get the second equals and in the last line we have put the upper, $f(x, d(x))$, and lower, $f(x, c(x))$ together into a anti-clockwise closed contour integral. Using the opposite order of integration we get
$$
\int_{D} d A \frac{\partial g}{\partial x}=\oint d y g(x, y)
$$
and the theorem follows as the difference of these two.
A regular region is a non-simple region that can be split into simple parts. As illustrated in Picture I.5.9 the boundary contribution from shared boundaries cancels and so the formula for a regular region is the sum of the formula for simple regions. It is also clear that the integral for an internal closed boundary curve needs to be taken clockwise.
## Proving the Stokes Theorem
Like Green's Theorem, Stokes' Theorem is proved by building the general case out of a particular special case where the theorem reduces to something already known, in Green's Theorem this was the Fundamental Theorem of Calculus and here, it will be Green's Theorem. Consider a vector field of the form $\mathbf{F}=F_{3}(x, y, z) \mathbf{k}$, so $F_{1}=F_{2}=0$. Assume $S$ is of the form $z=h(x, y)$ with $(x, y)$ in some domain $D$ in the $x y$-plane (Picture I.5.10). Now compute $\int_{S} \operatorname{curl} \mathbf{F} \cdot \mathbf{d A}$ with the upwards orientation.
$$
\operatorname{curl} \mathbf{F}=\partial_{y} F_{3} \mathbf{i}-\partial_{x} F_{3} \mathbf{j}
$$
The surface is parameterized by
$$
\mathbf{x}=x \mathbf{i}+y \mathbf{j}+h(x, y) \mathbf{k}
$$
and so
$$
\begin{aligned}
& \frac{\partial \mathbf{x}}{\partial x}=\mathbf{i}+\frac{\partial h}{\partial x} \mathbf{k} \\
& \frac{\partial \mathbf{x}}{\partial y}=\mathbf{j}+\frac{\partial h}{\partial y} \mathbf{k}
\end{aligned}
$$
and hence
$$
\frac{\partial \mathbf{x}}{\partial x} \times \frac{\partial \mathbf{x}}{\partial y}=\mathbf{k}-\frac{\partial h}{\partial x} \mathbf{i}-\frac{\partial h}{\partial y} \mathbf{j}
$$
giving
$$
\operatorname{curl} \mathbf{F} \cdot \frac{\partial \mathbf{x}}{\partial x} \times \frac{\partial \mathbf{x}}{\partial y}=-\left(\partial_{y} F_{3}\right) \frac{\partial h}{\partial x}+\left(\partial_{x} F_{3}\right) \frac{\partial h}{\partial y}=-\frac{\partial}{\partial y}\left(F_{3} \frac{\partial h}{\partial x}\right)+\frac{\partial}{\partial x}\left(F_{3} \frac{\partial h}{\partial y}\right)
$$
where, for the last line you need the cross terms to cancel, taking care to account for the two ways $F$ depends on $x$ : explicitly and through the dependence of $h$, so that
$$
\frac{\partial}{\partial x}\left(F_{3} \frac{\partial h}{\partial y}\right)=\partial_{x} F_{3} \frac{\partial h}{\partial y}+\partial_{z} F_{3} \frac{\partial h}{\partial x} \frac{\partial h}{\partial y}+F_{3} \frac{\partial^{2} h}{\partial x \partial y}
$$
Now
$$
\int_{S} \operatorname{curl} \mathbf{F} \cdot \mathbf{d} \mathbf{A}=\int_{D} d A\left(\frac{\partial g}{\partial x}-\frac{\partial f}{\partial y}\right)
$$
where
$$
\begin{aligned}
& f(x, y)=F_{3}(x, y, h(x, y)) \frac{\partial h(x, y)}{\partial x} \\
& g(x, y)=F_{3}(x, y, h(x, y)) \frac{\partial h(x, y)}{\partial y}
\end{aligned}
$$
Thus, we can apply Green's Theorem
$$
\begin{aligned}
\int_{S} \operatorname{curl} \mathbf{F} \cdot \mathbf{d A} & =\int_{C^{\prime}=\partial D}\left(F_{3}(x, y, h(x, y)) \frac{\partial h(x, y)}{\partial x} d x+F_{3}(x, y, h(x, y)) \frac{\partial h(x, y)}{\partial y} d y\right) \\
& =\int_{C=\partial S} F_{3} d z
\end{aligned}
$$
since
$$
d z=\frac{\partial h}{\partial x} d x+\frac{\partial h}{\partial y} d y
$$
Finally $F_{3} d z=\mathbf{F} \cdot \mathbf{d}$ because $\mathbf{F}=\left(0,0, F_{3}\right)$. This shows the theorem holds for $\mathbf{F}=F_{3} \mathbf{k}$ and $S$ of the form $z=h(x, y)$. For more general $S$, split $S$ up into a finite number of surfaces $\left\{S_{1}, S_{2}, \ldots, S_{n}\right\}$ that have the simple form (Picture I.5.11). Now if $\mathbf{F}=F_{1} \mathbf{i}$ or $\mathbf{F}=F_{2} \mathbf{j}$ a similar proof works using $y$ and $z$ or, respectively, $x$ and $z$ to parameterize the surface. Adding up these three results gives the result for a general vector field.
## Applications of Stokes' Theorem
1. Scalar potential In a simply connect region curl $\mathbf{F}=0$ implies that $\mathbf{F}$ is conservative. To see this take any close curve $C$ in the region. Since the region $D$ is simply connected their is a surface $S$ in $D$ whose boundary is $C$. Since curl $\mathbf{F}=0$ on $S \subset D$ Stokes' Theorem implies that
$$
\oint_{C} \mathbf{F} \cdot \mathrm{d} \mathbf{l}=0
$$
In a connected domain this is equivalent to $\mathbf{F}$ being conservative.
2. Area of a plane region. Let $D$ be a region in the $x y$-plane with $C=\partial D$. Apply Green's Theorem to the functions $f=y$ and $g=0$ to get
$$
\text { Area }=\int_{D} d A=-\oint_{C} y d x
$$
In a similar way
$$
\text { Area }=\oint_{C} x d y
$$
Centroid integrals can also be written as line integrals, these will be given on a problem sheet.
3. Cauchy's Theorem, an important theorem in complex analysis is a consequence of Green's Theorem.
## Note I. $6^{1} 4$ October
## The Gauss Theorem
The Gauss, or divergence, theorem states that, if $D$ is a connected three-dimensional region in $\mathbf{R}^{3}$ whose boundary is a closed, piece-wise connected surface $S$ and $\mathbf{F}$ is a vector field with continuous first derivatives in a domain containing $D$ then
$$
\int_{D} d V \operatorname{div} \mathbf{F}=\int_{S} \mathbf{F} \cdot \mathbf{d} \mathbf{A}
$$
where $S$ is oriented with the normal pointing outward (Picture I.6.1). $S$ can be disconnected, if $D$ has one or more inner boundaries the normal points inwards on the inner boundard. In otherwords, the normal always points away from $D$ (Picture I.6.2). The Gauss theorem is proved in a similar way to Green's theorem on the plane, Stoke's theorem is different from the other two because it deals with a curved surface.
First, $D$ is assumed to be simple, again, as before, this means that the integral over $D$ can be written as an iterated Cartesian integral in any order. Next, we take $\mathbf{F}=F_{3} \mathbf{k}$ and integrate using the Fundamental Theorem of Calculus, so,
$$
\begin{aligned}
\int_{D} d V \operatorname{div} \mathbf{F} & =\int_{D} d V \frac{\partial F_{3}}{\partial z} \\
& =\int_{D_{2}} d x d y \int_{e(x, y)}^{f(x, y)} d z \frac{\partial F_{3}}{\partial z} \\
& =\int_{D_{2}} d x d y\left[F_{3}(x, y, f(x, y))-F_{3}(x, y, e(x, y))\right]=\int_{S} \mathbf{F} \cdot \mathbf{d A}
\end{aligned}
$$
where we have used the shorthand $D_{2}$ to denote the parameter region in $(x, y)$, that is, the projection of $D$ onto the $x y$-plane (Picture I.6.3) and, in converting to the surface integral, we are using for the top
$$
\mathbf{r}=x \mathbf{i}+y \mathbf{j}+f(x, y) \mathbf{k}
$$
so that
$$
\mathbf{F} \cdot\left(\frac{\partial \mathbf{r}}{\partial x} \times \frac{\partial \mathbf{r}}{\partial y}\right) d x d y=F_{3} d x d y
$$
with a similar expression, differing by a sign, for the bottom.
Next, for vector fields of the form $\mathbf{F}=F_{1} \mathbf{i}$ and $\mathbf{F}=F_{2} \mathbf{j}$ the proof is the same, but with the $x$ integral done first in the first case and the $y$ integral in the second. Because the result is linear, this proves it for general F. Finally, as with the Green's theorem, it is possible to generalize to beyond simple regions by cutting up more complicated surfaces and noting the formula cancels at the joins.
${ }^{1}$ Conor Houghton, [email protected], see also http://www .maths.tcd.ie/ houghton/231
## Examples and Applications of the Gauss Theorem
- One important consequence of the Gauss theorem is that, if the flux of a smooth vector field out of any closed surface is zero it can be concluded that $\mathbf{F}$ is solenoidal. This applies, for example, to incompressible fluid flows and to magnetic flux.
- The Gauss theorem is used in deriving properties of partial differential equation, the uniqueness of solutions to the Laplace equation with Dirichlet or van Neumann boundary conditions for example.
- In actual calculations, it is sometimes worthwhile to use Gauss's theorem to convert a two-d integral into a three-d one, or visa versa. Similarily, it allow the volume of a region to be calculated as a surface integral,
$$
V=\int_{D} d V=\frac{1}{3} \int_{\delta D} \mathbf{r} \cdot \mathbf{d} \mathbf{A}
$$
since $\operatorname{div} \mathbf{r}=3$.
It is also interesting to note that the Fundemental Theorem of Calculus, Green's Theorem, Stoke's Theorem and Gauss's Theorem, are all special cases of a more general result, valid for flat and curved regions in an arbitrary number of dimensions, called the generalized Stoke's Theorem. Using differential forms this result can be written as
$$
\int_{D} d \mathbf{F}=\int_{\delta D} F
$$
## Vector potentials
Recall that if curl $\mathbf{F}=0$ in a simply-connected region then $\mathbf{F}$ is conservative meaning there exists a scalar potential $\phi$ such that $\mathbf{F}=\operatorname{grad} \phi$. There is a similar result for solenoidal vector fields: if $\operatorname{div} \mathbf{F}=0$ in a region without inner boundaries there exists a vector field $\mathbf{A}$ such that $\mathbf{F}=$ curl $\mathbf{A}$. $\mathbf{A}$ is called a vector potential for $\mathbf{F}$. For example if $\mathbf{F}=\mathbf{B}$ a constant vector then
$$
\mathbf{A}=\frac{1}{2}(\mathbf{B} \times \mathbf{r})
$$
is a vector potential for $\mathbf{F}$.
It is easy to see that $\mathbf{F}=$ curl $\mathbf{A}$ has zero divergence, the converse is trickier and will not be proven here. Rather a, constructive, proof is given for the special case of a star-shaped region. A region $D$ is called star-shaped if it has a point $O$ such that the line-segment joining $O$ and any other point in $D$ lies within $D$ (Picture I.6.4).
If $D$ is star-shaped there is a formula for the vector potential for a solenoidal vector field given by
$$
\mathbf{A}(\mathbf{r})=\int_{0}^{1} d t \mathbf{F}(t \mathbf{r}) \times t \mathbf{r}
$$
where the point $O$ in $D$ is taken to be the origin $(\mathbf{r}=0)$. To prove this is a vector potential for $\mathbf{F}$ we have to show that taking the curl of the right hand side reproduces the original vector field $\mathbf{F}$. We have
$$
\operatorname{curl} \mathbf{A}(\mathbf{r})=\int_{0}^{1} d t \operatorname{curl}(\mathbf{F}(t \mathbf{r}) \times t \mathbf{r})
$$
Using the vector identity
$$
\nabla \times(\mathbf{F} \times \mathbf{G})=(\nabla \cdot \mathbf{G}) \mathbf{F}+(\mathbf{G} \cdot \nabla) \mathbf{F}-(\nabla \cdot \mathbf{F}) \mathbf{G}-(\mathbf{F} \cdot \nabla) \mathbf{G}
$$
we have
$$
\operatorname{curl}(\mathbf{F}(t \mathbf{r}) \times t \mathbf{r})=3 t \mathbf{F}(t \mathbf{r})+t(\mathbf{r} \cdot \nabla) \mathbf{F}(t \mathbf{r})-0-t(\mathbf{F}(t \mathbf{r}) \cdot \nabla) \mathbf{r},
$$
using $\operatorname{div} \mathbf{r}=3$ and $\operatorname{div} \mathbf{F}=0$.
A straightforward calculation gives
$$
(\mathbf{F}(t \mathbf{r}) \cdot \nabla) \mathbf{r}=\mathbf{F}(t \mathbf{r})
$$
We also require
$$
(\mathbf{r} \cdot \nabla) \mathbf{F}(t \mathbf{r})=t \frac{d}{d t} \mathbf{F}(t \mathbf{r})
$$
or
$$
\left(x \frac{\partial}{\partial x}+y \frac{\partial}{\partial y}+z \frac{\partial}{\partial z}\right) \mathbf{F}(t x, t y, t z)=t \frac{d}{d t} \mathbf{F}(t x, t y, t z),
$$
which is a generalisation of
$$
a \frac{d}{d a} g(a t)=t \frac{d}{d t} g(a t)
$$
Inserting these two formulae into the expression for curl $(\mathbf{F}(t \mathbf{r}) \times t \mathbf{r})$ gives
$$
\operatorname{curl} \mathbf{A}=\int_{0}^{1} d t\left[2 t \mathbf{F}(t \mathbf{r})+t^{2} \frac{d}{d t} \mathbf{F}(t \mathbf{r})\right]=\int_{0}^{1} d t \frac{d}{d t}\left(t^{2} \mathbf{F}(t \mathbf{r})\right)=\mathbf{F}(\mathbf{r}),
$$
using the Fundamental Theorem of Calculus.
Note that the vector potential for a solenoidal vector field is not unique; if $\mathbf{A}$ is a vector potential for $\mathbf{F}$ then so is
$$
\mathbf{A}^{\prime}=\mathbf{A}+\operatorname{grad} \phi
$$
where $\phi$ is any scalar field since
$$
\text { curl grad } \phi=0 \text {. }
$$
In electromagnetic theory this ambiguity is called gauge freedom and transforming A to $\mathbf{A}^{\prime}$ is called a gauge transformation.
We have seen that in a simply-connected region an irrotational vector field can be written as a scalar and in a region without inner boundaries, though the proof was only given for star-shaped regions, a solenoidal vector field can be expressed as a curl. Now if $\mathbf{F}$ is neither solenoidal nor irrotational it can be decomposed into a gradient and a curl. There are a number of versions of this statement and they go under various names such as the Fundamental theorem of vector analysis, Helmholtz' theorem and the Hodge decomposition.
Here we give a simple version of the decomposition theorem: any vector field $\mathbf{F}$ defined in a region $D$ without inner boundaries can be written
$$
\mathbf{F}=\operatorname{grad} \phi+\operatorname{curl} \mathbf{A}
$$
To see how a proof works, assume that $\mathbf{F}$ is not solenoidal and consider $\mathbf{F}-\nabla \phi$ where $\phi$ is, for now, any scalar field. We have
$$
\operatorname{div}(\mathbf{F}-\nabla \phi)=\operatorname{div} \mathbf{F}-\nabla^{2} \phi
$$
Now if there exists a scalar field satisfying the equation
$$
\nabla^{2} \phi=\operatorname{div} \mathbf{F}
$$
then $\mathbf{F}-\nabla \phi$ is solenoidal in $\mathrm{D}$ and so we can write
$$
\mathbf{F}-\operatorname{grad} \phi=\operatorname{curl} \mathbf{A} .
$$
To complete the proof we need to show that 22 always has a smooth solution $\phi$. This equation is actually a form of Poisson's equation and will be discuss later in the course.
## Integrating scalars in two and three dimensions
We have defined line and surface integrals of vector fields; it is also possible to define line and surface integrals of scalars, these haven't the same impressive integral theorem results as the integrals of vector fields, but are sometimes useful.
We write
$$
\begin{aligned}
\int_{C} d l \phi & =\int_{C}|\mathbf{d} \mathbf{l}| \phi \\
\int_{S} d S \phi & =\int_{S}|\mathbf{d} \mathbf{S}| \phi
\end{aligned}
$$
where we can define these quantities as some sort of infinitessimal limit of a sum, as we did for the line and surface integrals of vector fields (Picture I.6.5). We can also rewrite these integrals in parametric form, this is easy since we have already seen how to express $\mathbf{d l}$ and $\mathbf{d S}$ in parameteric form. Hence, if a curve $C$ is parameterized by $\mathbf{r}(t)$
$$
\int_{C} d l \phi=\int_{t_{1}}^{t_{2}} d t \phi(\mathbf{r}(t))\left|\frac{\partial \mathbf{r}}{\partial t}\right|
$$
where $\mathbf{r}\left(t_{1}\right)$ and $\mathbf{r}\left(t_{2}\right)$ are the end points of the curve. Similarily, for a surface parameterized by $\mathbf{r}(s, t)$
$$
\int_{S} d S \phi=\int d s d t \phi(\mathbf{r}(s, t))\left|\frac{\partial \mathbf{r}}{\partial s} \times \frac{\partial \mathbf{r}}{\partial t}\right|
$$
As an example, lets work out the circumference of an ellipse, working out the length by integrating $\phi=1$
$$
L=\int_{C} d l
$$
where $C$ is the ellipse
$$
\begin{aligned}
& x=a \cos t \\
& y=b \sin t
\end{aligned}
$$
so $\mathbf{r}=a \cos t \mathbf{i}+b \sin t \mathbf{j}$ and
$$
\frac{\partial \mathbf{r}}{\partial t}=-a \sin t \mathbf{i}+b \cos t \mathbf{j}
$$
giving
$$
\left|\frac{\partial \mathbf{r}}{\partial t}\right|=\sqrt{a^{2} \sin ^{2} t+b^{2} \cos ^{2} t}=a \sqrt{1-\frac{a^{2}-b^{2}}{a^{2}} \cos ^{2} t}
$$
and, assuming without loss of generality, that $a>b$ we define the eccentricity
$$
e^{2}=\frac{a^{2}-b^{2}}{a^{2}}
$$
which gives a measure of how round the ellipse is, for $e=0$ it is a circle, for $e \rightarrow \infty$ it approaches a line, and get
$$
L=a \int_{0}^{2 \pi} \sqrt{1-e^{2} \cos ^{2} t}
$$
This integral is the elliptic integral of the second kind:
$$
L=4 a E(e)
$$
In otherwords, we define a new function by this integral. It is one of a class of integrals and functions used to solve elliptic problems, such as the motion of a pendulum or of a ball rolling on the inside of a sphere. Note II.1 ${ }^{1} 30$ January 2007
## PART II Fourier Analysis
## The Fourier series
First some terminology: a function $f(x)$ is periodic if $f(x+l)=f(x)$ for all $x$ for some $l$, if $l$ is the smallest such number, it is called the period of $f(x)$. It is even if $f(-x)=f(x)$, for all $x$ and odd if $f(-x)=-f(x)$, again, for all $x \cdot \sin x, \cos x, \sin 2 x, \sin 3 x$ and so on are examples of periodic functions: $\sin n x$ has period $2 \pi / n$
Now, consider $\sin ^{3} x$, this is clearly periodic with periodic with period $2 \pi: \sin ^{3}(x+2 \pi)=$ $\sin ^{3} x$. Using the usual trigonometric identities, or otherwise, it can be shown that
$$
\sin ^{3} x=-\frac{1}{4} \sin 3 x+\frac{3}{4} \sin x
$$
In short, $\sin ^{3} x$ can be re-expressed in terms of sines. In fact, this is a much more common property than you might expect, the theory of Fourier series tells us that if $f(x)$ is odd and periodic with period $2 \pi$ then there are $b_{n}$ s such that
$$
f(x)=\sum_{n=1}^{\infty} b_{n} \sin n x
$$
If it is even it has a cosine series instead
$$
f(x)=\frac{1}{2} a_{0}+\sum_{n=1}^{\infty} a_{n} \cos n x
$$
where the half before $a_{0}$ is a standard convention, we will see soon why it is convenient. Anyway, these are Fourier series.
More generally, a periodic function $f(x)$ with period $l$ has Fourier series
$$
f(x)=\frac{1}{2} a_{0}+\sum_{n=1}^{\infty} a_{n} \cos \frac{2 \pi n x}{l}+\sum_{n=1}^{\infty} b_{n} \sin \frac{2 \pi n x}{l}
$$
Leaving aside, for now, issues of convergence, it is easy to calculate what values the $a_{n}$ and $b_{n}$ must have. First, integrating both sides gives
$$
\int_{-l / 2}^{l / 2} d x f(x)=\frac{l}{2} a_{0}+\sum_{n=1}^{\infty} a_{n} \int_{-l / 2}^{l / 2} d x \cos \frac{2 \pi n x}{l}+\sum_{n=1}^{\infty} b_{n} \int_{-l / 2}^{l / 2} d x \sin \frac{2 \pi n x}{l}=\frac{l}{2} a_{0}
$$
${ }^{1}$ Conor Houghton, [email protected], see also http://www.maths.tcd.ie/ houghton/231 where I have assumed I can bring the integrals into the sum signs, the sines and cosines both integrate to zero: sine and cosine integrate to zero if integrated over a whole number of periods and $\cos 2 n \pi / l$ and $\sin 2 n \pi / l$ have period $l / n$. This means that
$$
a_{0}=\frac{2}{l} \int_{-l / 2}^{l / 2} f(x) d x
$$
In fact, the method for calculating the other coefficients is not too different; we multiply across by a sine or cosine and then integrate using the formulae
$$
\begin{aligned}
\int_{-l / 2}^{l / 2} d x \sin \frac{2 \pi m x}{l} \sin \frac{2 \pi n x}{l} & =\frac{l}{2} \delta_{m n} \\
\int_{-l / 2}^{l / 2} d x \cos \frac{2 \pi m x}{l} \cos \frac{2 \pi n x}{l} & =\frac{l}{2} \delta_{m n} \\
\int_{-l / 2}^{l / 2} d x \sin \frac{2 \pi m x}{l} \cos \frac{2 \pi n x}{l} & =0
\end{aligned}
$$
which can be proved, for example, by writing the trigonometric functions in terms of complex exponentials. Hence, multiplying across by $\cos 2 \pi m x / l$ and integrating, we get
$$
\begin{aligned}
\int_{-l / 2}^{l / 2} d x f(x) \cos \frac{2 \pi m x}{l}= & \frac{1}{2} \int_{-l / 2}^{l / 2} d x a_{0} \cos \frac{2 \pi m x}{l} \\
& +\sum_{n=1}^{\infty} a_{n} \int_{-l / 2}^{l / 2} d x \cos \frac{2 \pi n x}{l} \cos \frac{2 \pi m x}{l} \\
& +\sum_{n=1}^{\infty} b_{n} \int_{-l / 2}^{l / 2} d x \sin \frac{2 \pi n x}{l} \cos \frac{2 \pi m x}{l} \\
= & \frac{l}{2} a_{m}
\end{aligned}
$$
so, using this and a similar calculation for sine, we get
$$
\begin{aligned}
a_{n} & =\frac{2}{l} \int_{-l / 2}^{l / 2} d x f(x) \cos \frac{2 \pi m x}{l} \\
b_{n} & =\frac{2}{l} \int_{-l / 2}^{l / 2} d x f(x) \sin \frac{2 \pi m x}{l}
\end{aligned}
$$
where the first equation holds for $n \geq 0$ and the second for $n>0$. It is to have all the $a_{n}$ obey the same general expression that there is the convention to put the half is put in front of the $a_{0}$. As a point of terminology, the $a_{n}$ and $b_{n}$ are called Fourier coefficients and the sines and cosines, or sometimes the sines and cosine along with their coefficient, are called Fourier modes. - Example: Consider the block wave with period $l=2 \pi$ (Picture II.1.1)
$$
f(x)= \begin{cases}1 & 0<x<\pi \\ -1 & -\pi<x<0\end{cases}
$$
with $f(x+2 \pi)=f(x)$.
So
$$
a_{n}=\frac{1}{\pi} \int_{-l / 2}^{l / 2} d x f(x) \cos n x=0
$$
because the integrand is odd, and
$$
b_{n}=\frac{1}{\pi} \int_{-l / 2}^{l / 2} d x f(x) \sin n x=\frac{2}{\pi} \int_{0}^{\pi} d x \sin n x=-\left.\frac{2 \cos n x}{n \pi}\right|_{0} ^{\pi}=\frac{2}{\pi n}\left[1-(-1)^{n}\right]
$$
where we have used $\cos n \pi=(-1)^{n}$. Hence
$$
f(x)=\frac{4}{\pi} \sum_{n \text { odd }} \frac{1}{n} \sin n x
$$
This series is not obviously convergent; the point of Fourier series is that there are theorems to tell us it is. However, there are particular values of $x$ where we can see that the answer is correct, for example, at $x=\pi / 2$, we have $\sin (2 m+1) \pi / 2=(-1)^{m}$ where $m$ is an integer so $2 m-1$ is odd. Putting this back into the series gives
$$
1=\frac{4}{\pi}\left(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9} \ldots\right)
$$
and the right hand side can be derived by Taylor expanding $\tan ^{-1} x$. It is interesting to note that the series as written, up to $1 / 9$ gives $1 \approx 1.06$; the Fourier series gives workable but not efficient approximations and its importance is not in its ability to approximate functions with high numerical accuracy, rather, it quickly captures features of the function, preserving its periodicity and encoding its behaviour at lengths scales bigger than $l / n$, where $n$ is where the series is truncated. Another interesting thing to look at is the behaviour at $x=0$ where the function is discontinuous. Since all the sines are zero, the Fourier series gives zero at $x=0$. This interpolates the discontinuity. This is a feature of the Fourier series, the series does not see what happens at individual points and interpolates over any finite discontinuities. A graph of the Fourier series is given in Note 3.
There are lots of versions of the theorem which tells us the Fourier series exists, different versions impose different conditions on the function and have convergence properties for the series; the version we quote is actually quite vague about the convergence and pretty restrictive on the function and we will call it Dirichlet's Theorem: If $\mathrm{f}$ is periodic and has, in any period, a finite number of maxima and minima and a finite number of discontinuities and $\int_{-l / 2}^{l / 2}|f(x)|^{2} d x$ is finite then the Fourier series converges and converges to $f(x)$ at all points where $f(x)$ is continuous. At a point $a$ where $f(x)$ is discontinuous it converges to
$$
\frac{1}{2}\left[\lim _{x \rightarrow a+} f(x)+\lim _{x \rightarrow a-} f(x)\right]
$$
One annoying thing about Dirichlet's theorem, as quoted, is that it appears to exclude the block wave used in the example, the block wave doesn't have a finite number of maxima and minima, obviously this isn't the sort of function the statement is trying to exclude, it is aimed at functions that oscillate infinitely fast. To fix it you could extend Dirichlet's theorem to functions $f(x)$ such that there is a function $g(x)$, satisfying the properties described by the theorem, such that $g(f(x))$ has the properties required by the theorem.
## Complex Fourier series
As often happens, apart from the slight inconvenience of being complex, complex Fourier series are more straightforward than real ones, there is only once type of Fourier coefficient, $c_{n}$, instead of three, $a_{0}, a_{n}$ and $b_{n}$ for the real series. It is easy to see the existence of a complex exponential series follows from the existence of the sine and cosine series, just replace
$$
\begin{aligned}
\cos x & =\frac{e^{i x}+e^{-i x}}{2} \\
\sin x & =\frac{e^{-i x}-e^{-i x}}{2 i}
\end{aligned}
$$
to get a series of the form
$$
f(x)=\sum_{n=-\infty}^{\infty} c_{n} e^{2 \pi i n x / l} .
$$
Rather than try to work out the formula for the $c_{n}$ from the formulas for the $a_{n}$ and $b_{n}$, we can just take this as a series for $f(x)$ and calculate the $c_{n}$ by a similar trick to the one we used before, we multiply across by $\exp (-2 \pi i m x / l)$ and integrate
$$
\int_{-l / 2}^{l / 2} d x e^{-2 \pi i m x / l} f(x)=\sum_{n=-\infty}^{\infty} c_{n} \int_{-l / 2}^{l / 2} e^{2 \pi i(n-m) x / l}
$$
and use
$$
\int_{-l / 2}^{l / 2} d x e^{2 \pi i(n-m) x / l}=l \delta_{n m}
$$
which is clear if you note the integrand is one for $n=m$ and otherwise, it is easy to see from
$$
e^{i \theta}=\cos \theta+i \sin \theta
$$
that it integrates to zero. This means that
$$
c_{n}=\frac{1}{l} \int_{-l / 2}^{l / 2} d x f(x) e^{-2 \pi i n x / l}
$$
It is interesting to ask what the consequence of $f(x)$ being real is on the $c_{n}$, using a star to mean the complex conjugate lets take the complex conjugate of this equation, using $f *(x)=f(x)$ :
$$
c_{n}^{*}=\frac{1}{l} \int_{-l / 2}^{l / 2} d x f(x) e^{2 \pi i n x / l}=\frac{1}{l} \int_{-l / 2}^{l / 2} d x f(x) e^{-2 \pi i(-n) x / l}=c_{-n}
$$
- Example: It is easy to redo the last by integrating; since we have already done the integrations when working out the $b_{n}$ 's, we will use the previous real series to work out the Fourier coefficients for the complex series, so,
$$
f(x)=\frac{4}{\pi} \sum_{n>0 \text { and odd }} \frac{1}{n} \sin n x=\frac{2}{\pi} \sum_{n>0 \text { and odd }} \frac{1}{n}\left(e^{i n x}-e^{-i n x}\right)=\frac{2}{\pi} \sum_{n \text { odd }} \frac{1}{n} e^{i n x}=
$$
so
$$
c_{n}= \begin{cases}2 /(\text { Tin }) & n \text { odd } \\ 0 & \text { otherwise }\end{cases}
$$
- Example: Consider $f(x)=e^{x}$ for $-\pi<x<\pi$ and $f(x+2 \pi)=f(x)$. So,
$$
\begin{aligned}
c_{n} & =\frac{1}{2 \pi} \int_{-\pi}^{\pi} d x e^{-i n x} e^{x}=\frac{1}{2 \pi} \int_{-\pi}^{\pi} d x e^{(1-i n) x} \\
& =\frac{e^{\pi} e^{-i n \pi}-e^{-\pi} e^{i n \pi}}{2 \pi(1-i n)}=(-1)^{n} \frac{e^{\pi}-e^{-\pi}}{2 \pi(1-i n)} \\
& =\frac{\sinh \pi}{\pi} \frac{(-1)^{n}}{1-i n}
\end{aligned}
$$
and so
$$
f(x)=\frac{\sinh \pi}{\pi} \sum_{n=-\infty}^{\infty} \frac{(-1)^{n}}{1-i n} e^{i n x}
$$
At $x=0$ this gives the amusing formula
$$
1=\frac{\sinh \pi}{\pi}\left(1+\sum_{n=1}^{\infty} \frac{(-1)^{n}}{1-i n}+\sum_{n=1}^{\infty} \frac{(-1)^{n}}{1+i n}\right)=\frac{\sinh \pi}{\pi} \sum_{n=2}^{\infty} \frac{(-1)^{n}}{1+n^{2}}
$$
where the $n=1$ terms cancel the one.
## Parseval's Theorem
Parseval's theorem is a relation between the $L^{2}$ size of $f(x)$ and the Fourier coefficients:
$$
\frac{1}{l} \int_{-l / 2}^{l / 2}|f(x)|^{2} d x=\frac{1}{4} a_{0}^{2}+\frac{1}{2} \sum_{n=1}^{\infty}\left(a_{n}^{2}+b_{n}^{2}\right)
$$
or for the complex series
$$
\frac{1}{l} \int_{-l / 2}^{l / 2}|f(x)|^{2} d x=\sum_{n=-\infty}^{\infty}\left|c_{n}\right|^{2}
$$
This theorem is very impressive, it relates a natural measure for the size of the function on the space of periodic functions to the natural measure for the size of an infinite vector on the space of coefficients. It is easy to prove and convenient too for the complex series
$$
\int_{-l / 2}^{l / 2} d x f(x) f^{*}(x)=\sum_{m, n} c_{n} c_{m}^{*} \int_{-l / 2}^{l / 2} d x e^{2 \pi i(n-m) x / l}=\sum_{m, n} c_{n} c_{m}^{*} \delta_{n m}=l \sum_{n}\left|c_{n}\right|^{2} .
$$
- Example: So, going back to the block wave example, it is easy to check that
$$
\frac{1}{2 \pi} \int_{-l / 2}^{l / 2} d x|f(x)|^{2}=1
$$
SO
$$
1=\frac{8}{\pi^{2}}\left(1+\frac{1}{9}+\frac{1}{25}+\frac{1}{49}+\ldots\right)
$$
Note II. ${ }^{1} 26$ February 2007
## PART I Fourier Analysis
## The Fourier integral
So far we have discussed the expansion of periodic functions; here we extend this discussion to functions which are not periodic. We will see that the expansion will go from a sum over a countably infinite set of basis function, to an integral over a continuum of basis function. It is possible to see why this will happen, the periodic function, with period $l$ say, is expanded into basis functions which repeat after $l$, this does not require the basis functions to have period $l$, but, it does require that their period divides a whole number of times into $l$, reducing the set of basis functions to a countably infinite set.
## The Fourier integral: definition
The expansion is an integral
$$
f(x)=\int_{-\infty}^{\infty} d k f(k) e^{i k x}
$$
where
$$
\tilde{f(k)}=\frac{1}{2 \pi} \int_{-\infty}^{\infty} d k f(x) e^{-i k x}
$$
Here, the basis functions are the so called plane waves, $\exp (i k x)$ and the Fourier coefficients are now functions, $\tilde{f(k)}$. The sum over a discrete index $n$ has been replaced by an integral of a continuous index $k$. No proof is given here, we will revisit these formulas again after we have done the Dirac delta function, and, in the meantime, here is a formal arguement relating the Fourier integral to the Fourier transform. For a function $f(c x)$ with period $l$
$$
f(x)=\sum_{n=-\infty}^{\infty} c_{n} e^{2 \pi i n x / l}
$$
where
$$
c_{n}=\frac{1}{l} \int_{-l / 2}^{l / 2} d y f(y) e^{-2 \pi i n y / l}
$$
or, putting this together
$$
f(x)=\sum_{n=-i n f t y}^{\infty} \frac{2 \pi}{l} e^{2 \pi i n x / l} \frac{1}{\pi} \int_{-l / 2}^{l / 2} d y f(y) e^{-2 \pi i n y / l}
$$
${ }^{1}$ Conor Houghton, [email protected], see also http://www.maths.tcd.ie/ houghton/231 Now for $l$ very large and $k=2 \pi / l$, the sum term above is a Riemann sum, giving
$$
f(x)=\int_{-\infty}^{\infty} d k e^{i k x} \frac{1}{\pi} \int_{-l / 2}^{l / 2} d y f(y) e^{-i k y}
$$
It should be noted that there are a number of different conventions for defining the Fourier integral; these vary in where the $2 \pi$ goes. Some people share it between the $f(x)$ and $\tilde{f}(k)$ formula, giving factors of $1 / \sqrt{2 \pi}$ in front of both itegrals. Others send $k$ to $2 \pi k$ and $f \tilde{(k)}$ to $f \tilde{(k)} / 2 \pi$ so there are no constants multiplying either integral but the exponentials have $2 \pi \mathrm{s}$ in them.
The Fourier integral is not defined for all functions, the integrals defining $f \tilde{f}(k)$ may be divergent. $L^{1}$ is often used, if
$$
\int_{-\infty}^{\infty} d x|f(x)|<\infty
$$
then the Fourier integral exists. However, the Fourier coefficient $f \tilde{(k)}$ may not itself have a Fourier integral: that is, the Fourier transform, the map from a function to its Fourier coefficient, is not closed on $L^{1}$. A space that works in this respect is the Schwatz space S, the space of rapidly decreasin functions, if $f \in S$ then $\tilde{f} \in S$. $f$ is rapidly decreasing if it is infinitely differentiable and
$$
\sup \left|x^{r} \frac{d^{s} f}{d x^{2}}\right|<\infty
$$
for all positive $r$ and $s$ and with the sup taken over all $x$.
- Example: Consider the square wave packet
$$
f(x)= \begin{cases}1 & -1<x<1 \\ 0 & |x|>1\end{cases}
$$
This is an example of an $L^{1}$ function which is not in $S$. Now,
$$
f(k)=\frac{1}{2 \pi} \int_{-1}^{1} d x e^{-i k x}=\left.\frac{e^{-i k x}}{-2 \pi i k}\right|_{-1} ^{1}=\frac{\sin k}{\pi k}
$$
and so
$$
f(x)=\int_{-\infty}^{\infty} d k \frac{\sin k}{\pi k} e^{i k x}
$$
Hence, for example, putting $x=0$ gives
$$
\pi=\int_{-\infty}^{\infty} d k \frac{\sin k}{k}
$$
This is Dirichlet's integral and can also be calculated using contour integration. At $x= \pm 1, f$ has a jump discontinuity, the Fourier integral averages at a jump, just like the Fourier series, we can see this in this case, at $x=1$ :
$$
\int_{-\infty}^{\infty} d k \frac{\sin k}{\pi k} e^{i k}=\int_{-\infty}^{\infty} d k \frac{\sin k}{\pi k}(\cos k+i \sin k)
$$
Now dropping the odd integral over $\sin ^{2} k / k$ we get
$$
\int_{-\infty}^{\infty} d k \frac{\sin k}{\pi k} e^{i k}=\int_{-\infty}^{\infty} d k \frac{\sin k \cos k}{\pi k}=\int_{-\infty}^{\infty} d k \frac{\sin 2 k}{2 \pi k}=\frac{1}{2}
$$
since it is in the Dirichlet integral form.
## Parseval's Theorem
If
$$
\int_{-\infty}^{\infty} d x|f(x)|^{2} d x<\infty
$$
then
$$
\frac{1}{2 \pi} \int_{-\infty}^{\infty} d x|f(x)|^{2}=\int_{-\infty}^{\infty} d k|f \tilde{f}(k)|^{2}
$$
If $f \in S$ this result is sometimes called the Plancherel formula. We will prove this formula later, at the moment we have no formula analogous to
$$
\int_{-\pi}^{\pi} d x e^{i(n-m) x}=2 \pi \delta_{n m}
$$
We will see that the Dirac delta function, which we look at next, provides such a formula.
- Example: Applying this formula for the square wave packet above, we get
$$
\int_{-\infty}^{\infty} d k \frac{\sin ^{2} k}{k^{2}}=\pi
$$
Note II.3 ${ }^{1} 26$ February 2007
## PART I Fourier Analysis
## Distributions
The Heaviside function $\theta(x)$ is defined as
$$
\theta(x)= \begin{cases}1 & x>0 \\ 0 & x<0\end{cases}
$$
$\theta$ is related to the sign fxn
$$
\epsilon(x)= \begin{cases}1 & x>0 \\ -1 & x<0\end{cases}
$$
The question is, what is the integral
$$
I=\int_{-\infty}^{\infty} d x \frac{d}{d x} \epsilon(x)
$$
The first answer is to argue that the sign function is constant everywhere except at $x=0$, therefore its derivative is zero except at one point, this is a set of measure zero and hence $I$ is zero. Another arguement is to use the fundamental theorem of calculus:
$$
I=\int_{-\infty}^{\infty} d x \frac{d}{d x} \epsilon(x)=\epsilon(\infty)-\epsilon(-\infty)=1-(-1)=2
$$
We will develop a formalism, the theory of distributions, where the latter is the correct answer. We will do this in the traditional applied mathematics root, but, in fact, there is a beautiful mathematics theory of distributions which starts by trying to dualize the space of functions and, in this formulation, the fundamental theorem of calculus arguement above is very natural.
Consider the family of smooth functions
$$
\epsilon_{n}(x)=\frac{2}{\pi} \tan ^{-1} n x
$$
These are sigmoid shaped function with
$$
\int_{-\infty}^{\infty} d x \frac{d}{d x} e_{n} x=2
$$
for all $n$ and $\epsilon(x)$ can be viewed as the $n \rightarrow \infty$ limit of $\epsilon_{n}$.
Include a plot here
${ }^{1}$ Conor Houghton, [email protected], see also http://www.maths.tcd.ie/ ${ }^{\text {houghton/231 }}$ Now, we define
$$
\delta_{n}(x)=\frac{1}{2} \frac{d}{d x} \epsilon_{n}(x)=\frac{1}{\pi} \frac{n}{1+n^{2} x^{2}}
$$
Then, we write
$$
\delta(x)=\frac{1}{2} \frac{d}{d x} \epsilon(x)=\frac{d}{d x} \theta(x)
$$
where $\delta(x)$ is the Dirac delta function. In this applied mathematics approach, it is the $n$ goes to infinity limir of $\delta_{n}$ and is zero everywhere except $x=0$ but has integral one from minus infinity to infinity.
In fact, the defining properties of the delta function are
$$
\int_{-\infty}^{\infty} d x \delta(x) f(x)=f(0)
$$
where $f(x)$ is any Schwartz function and $\delta(x)=0$ for $x \neq 0$. Again, one approach to this, from an applied mathematics point of view, is to regard $\delta(x)$ as the $\epsilon$ goes to zero limit of
$$
\delta_{\epsilon}(x)= \begin{cases}1 / \epsilon & 0<x<\epsilon \\ 0 & \text { otherwise }\end{cases}
$$
Now, for sufficiently small $\epsilon f(x) \approx f(0)$ for $0<x<\epsilon$, so
$$
\int_{-\infty}^{\infty} d x f(x) \delta_{\epsilon}(x) \approx f(0) \frac{1}{\epsilon} \int_{0}^{\epsilon} d x=f(0)
$$
A practical approach to the delta function is to calculate its properties relative to its action under an integral. So,
$$
\int_{-\infty}^{\infty} d x \delta(x-a) f(x)=f(a)
$$
by a change of variable, $x^{\prime}=x-a$ inside the integral. Similarily, using integration by parts and $\delta(x)=0$ for $x \neq 0$
$$
\left.\int_{-\infty}^{\infty} \delta^{\prime}(x) f(x)=\delta(x) f(x)\right]_{-\infty}^{\infty}-\int_{-\infty}^{\infty} \delta(x) f^{\prime}(x)=-f^{\prime}(0)
$$
and, using $y=a x$
$$
\int_{-\infty}^{\infty} \delta(a x) f(x)=\frac{1}{a} \int_{-\infty}^{\infty} d y \delta(y) f(y / a)=\frac{1}{a} f(0)
$$
where $a$ is a positive constant, however, if $a$ is negative then the change of variables changes the signs of the limits: if $a<0 x=\infty$ implies $y=-\infty$ and so on. Hence
$$
\int_{-\infty}^{\infty} \delta(a x) f(x)=-\frac{1}{a} f(0)
$$
or, putting it all together
$$
\int_{-\infty}^{\infty} \delta(a x) f(x)=\frac{1}{|a|} f(0)
$$
Note that for $a=-1$, this formula gives $\delta(x)=\delta(-x)$, so the delta function is formally even.
Now, consider integral of the form
$$
\int_{-\infty}^{\infty} d x \delta(h(x)) f(x)
$$
where $h(x)$ is some smooth function. If $h$ has no zeros then integral is zero, so, for example
$$
\int_{-\infty}^{\infty} d x \delta\left(1+x^{2}\right) f(x)=0
$$
Assume that $h$ has one zero at $x=x_{1}$, so $h\left(x_{1}\right)=0$ and assume further that $h^{\prime}\left(x_{1}\right)>0$, this means that there is an interval $(c, d)$ containing $x_{1}$ such that $h(x)$ is increasing on $(c, d)$. Now, $\delta(x)$ is zero for $x \neq 0$ means that
$$
\int_{-\infty}^{i} n f \operatorname{ty} \delta(h(x)) f(x)=\int_{c}^{d} d x \delta(h(x)) f(x)
$$
and, since $h(x)$ is strictly increasing on $(c, d)$ it is invertible on this interval, so we can do a change of variables to $y=h(x)$
$$
\int_{c}^{d} d x \delta(h(x)) f(x)=\int_{h(c)}^{h(d)} d y \frac{d x}{d y} \delta(y) f(x(y))
$$
so $f$ is a function of $y$ through $x=h^{-1}(y)$. Now, using the delta function to do the integral, and rewritting the Jacobian $d x / d y=1 / h^{\prime}$ we get
$$
\int_{-\infty}^{\infty} \delta(h(x)) f(x)=\frac{f\left(x_{1}\right)}{h^{\prime}\left(x_{1}\right)}
$$
Now, if $h^{\prime}\left(x_{1}\right)$ had been negative everything would have been the same except there would have been an extra factor of minus one coming from changing around the limits, hence
$$
\int_{-\infty}^{\infty} \delta(h(x)) f(x)=\frac{f\left(x_{1}\right)}{\left|h^{\prime}\left(x_{1}\right)\right|}
$$
and finally, if there are $n$ isolated zeros with non-zero derivative $x_{i}$ where $i=1, \ldots, n$
$$
\int_{-\infty}^{\infty} \delta(h(x)) f(x)=\sum_{i=1}^{n} \frac{f\left(x_{i}\right)}{\left|h^{\prime}\left(x_{i}\right)\right|}
$$
or, put another way
$$
\delta(h(x))=\sum_{i=1}^{n} \frac{\delta\left(x-x_{i}\right)}{\left|h^{\prime}\left(x_{i}\right)\right|}
$$
- Example:
$$
\delta\left(x^{2}-a^{2}\right)=\frac{1}{2|a|}[\delta(x-a)+\delta(x+a)]
$$
so here $h(x)=x^{2}-a^{2}$ with zeros at $\pm a$ and $h^{\prime}(x)=2 x$
Finally, care needs to be taken with products, for example $\delta(x)^{2}$ and $\delta(x) \theta(x)$ are meaningless and $\delta(x-a) \delta(x-b)$ is zero for all $a \neq b$. The product rule do not work for discontinuities either, for example $\theta(x)^{2}=\theta(x)$ so
$$
\frac{d}{d x} \theta(x)^{2}=\delta(x)
$$
which is not what the product rule predicts. Similarly, for the sign function, $\epsilon(x)^{2}=1$ so
$$
\frac{d}{d x} \epsilon(x)^{2}=0
$$
You might worry that $\epsilon(x)^{2}=1$ does not hold at $x=0$, but no matter, this has no effect under an integral, so, as a distribution, $\epsilon(x)^{2}=1$. Note II.4 ${ }^{1} 5$ March 2007
## PART II Fourier Analysis
## Distributions and the Fourier transform.
First, lets work out the Fourier representation of the delta function, that is, lets write it as a Fourier integral:
$$
\delta(x)=\int_{-\infty}^{\infty} d k \widetilde{\delta(k)} e^{i k x}
$$
where
$$
\widetilde{\delta(k)}=\frac{1}{2 \pi} \int_{-\infty}^{\infty} d k \delta(x) e^{-i k x}
$$
but we can evaluate this integral easily because of the delta function, it gives $\widetilde{\delta(k)}=1 / 2 \pi$ and, hence,
$$
\delta(x)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} d k e^{i k x}
$$
This is an interesting and useful result, substituting $x-x^{\prime}$ for $x$ it gives
$$
\delta\left(x-x^{\prime}\right)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} d k e^{i k\left(x-x^{\prime}\right)}
$$
which is an orthonormality result analogeous to
$$
\delta_{n m}=\frac{1}{2 \pi} \int_{-\pi}^{\pi} d x e^{i x(n-m)}
$$
for the periodic functions.
We can use this result to work out the Fourier integral of the constant function $f(x)=1$ :
$$
\tilde{1}=\frac{1}{2 \pi} \int_{-\infty}^{\infty} d k e^{-i k x}=\delta(-k)=\delta(k)
$$
so, together
$$
\begin{aligned}
\widetilde{\delta(k)} & =\frac{1}{2 \pi} \\
\tilde{1} & =\delta(x)
\end{aligned}
$$
This is similar to what we saw for the Gauss function, the wider and flatter the function the narrower and pointier the transform and visa versa.
${ }^{1}$ Conor Houghton, [email protected], see also http://www.maths.tcd.ie/ houghton/231
## The Fourier integral formulas and the Plancherel formula
Now, we have used the Fourier integral formulas to write down the Fourier representation of the Dirac delta function, there are other ways to do this and they way we have done it, we certainly aren't in a position to use this formula to derive any part of the Fourier integral; however, it is interesting to see how the ideas hang together. Consider some function $f(x)$ :
$$
f(x)=\int_{-\infty}^{\infty} d x^{\prime} f\left(x^{\prime}\right) \delta\left(x-x^{\prime}\right)
$$
Now, substititute for the delta, this is a maneuver know as substituting a complete set:
$$
f(x)=\int_{-\infty}^{\infty} d x^{\prime} f\left(x^{\prime}\right) \frac{1}{2 \pi} \int_{-\infty}^{\infty} d k e^{i k\left(x-x^{\prime}\right)}
$$
and now move around the integrals a bit:
$$
f(x)=\int_{-\infty}^{\infty} d k e^{i k x} \frac{1}{2 \pi} \int_{-\infty}^{\infty} d x^{\prime} f\left(x^{\prime}\right) e^{-i k x^{\prime}}
$$
which is precisely the Fourier integral formula.
Similarily, we can use this for the Plancherel formula:
$$
\int_{\infty}^{\infty} d x|f(x)|^{2}=\int_{\infty}^{\infty} d x f(x)^{*} f(x)=\int_{-\infty}^{\infty} d x \int_{-\infty}^{\infty} d x^{\prime} f(x)^{*} f\left(x^{\prime}\right) \delta\left(x-x^{\prime}\right)
$$
and, again, substitute for the delta:
$$
\begin{aligned}
\int_{\infty}^{\infty} d x|f(x)|^{2} & =\int_{-\infty}^{\infty} d x \int_{-\infty}^{\infty} d x^{\prime} \frac{1}{2 \pi} \int_{-\infty}^{\infty} d k f(x)^{*} f\left(x^{\prime}\right) e^{i k\left(x-x^{\prime}\right)} \\
& =2 \pi \int_{-\infty}^{\infty} d k \frac{1}{2 \pi} \int_{-\infty}^{\infty} d x f(x)^{*} e^{i k x} \frac{1}{2 \pi} \int_{-\infty}^{\infty} d x^{\prime} f\left(x^{\prime}\right) e^{-i k x^{\prime}} \\
& =2 \pi \int_{-\infty}^{\infty} d k \widetilde{f(k)} \widetilde{f(k)}
\end{aligned}
$$
Note III.1 $^{12} 19$ February 2007
## Part III: ODEs
A differential equation is an equation involving derivatives. An ordinary differential equation (ODE) is a differential equation involving a function, or functions, of only one variable. If the ODE involves the $n$th (and lower) derivatives it is said to be an $n$th order ODE. Let $y$ be a function of one variable $x$, for neatness, we will try to always use $x$ as the dependent variable and prime for derivative. An equation of the form
$$
h_{1}\left(x, y(x), y^{\prime}(x)\right)=0
$$
is a first order ODE.
$$
h_{2}\left(x, y(x), y^{\prime}(x), y^{\prime \prime}(x)\right)=0
$$
is second order. A function satisfying the ODE is called a solution of the ODE.
## Linear ODEs ( 2 types)
There are two types of linear ODEs
1. Homogeneous: If $y_{1}$ and $y_{2}$ are solutions so is $A y_{1}+B y_{2}$ where $A$ and $B$ are arbitrary constants.
2. Inhomogeneous: If $y_{1}$ and $y_{2}$ are solutions so is $A y_{1}+B y_{2}$ where $A+B=1$.
where, obviously, the point is in a homogeneous equation, all the terms are $y$ terms, whereas the inhomogeneous equation has an extra forcing term.
- Homogeneous example: The equation
$$
y^{\prime \prime}+p(x) y^{\prime}+q(x) y=0
$$
is homogeneous, where $p(x)$ and $q(x)$ are some, given, functions of $x$. Now substituting $A y_{1}+B y_{2}$ gives
$$
\left(A y_{1}+B y_{2}\right)^{\prime \prime}+p\left(A y_{1}+B y_{2}\right)^{\prime}+q\left(A y_{1}+B y_{2}\right)=A\left(y_{1}^{\prime \prime}+p y_{1}^{\prime}+q y_{1}\right)+B\left(y_{2}^{\prime \prime}+p y_{2}^{\prime}+q y_{2}\right)=0
$$
when $y_{1}$ and $y_{2}$ are solutions.
- Inhomogeneous example: The equation
$$
y^{\prime \prime}+p(x) y^{\prime}+q(x) y=f(x)
$$
${ }^{1}$ Conor Houghton, [email protected], see also http://www.maths.tcd.ie/ houghton/231
${ }^{2}$ Based partly on lecture notes taken by John Kearney is homogeneous, where $p(x), q(x)$ and $f(x)$ are some, given, functions of $x$. Now substituting $A y_{1}+B y_{2}$ gives
$$
\left(A y_{1}+B y_{2}\right)^{\prime \prime}+p\left(A y_{1}+B y_{2}\right)^{\prime}+q\left(A y_{1}+B y_{2}\right)=A\left(y_{1}^{\prime \prime}+p y_{1}^{\prime}+q y_{1}\right)+B\left(y_{2}^{\prime \prime}+p y_{2}^{\prime}+q y_{2}\right)=(A+B) f
$$
when $y_{1}$ and $y_{2}$ are solutions. Hence $A y_{1}+B y_{2}$ is a solution is $A+B=1$.
The general first order linear ODE, for a single function, can be written
$$
a(x) y^{\prime}(x)+b(x) y(x)=f(x)
$$
where $a, b$ and $f(x)$ are arbitrary functions. The equation is homogeneous if $f=0$. A common standard form is write the equation as
$$
y^{\prime}(x)+p(x) y(x)=f(x)
$$
where $p=b / a$ and $f / a$ has been renamed back to $f$.
The general 2 nd order linear ODE is
$$
a(x) y^{\prime \prime}(x)+b(x) y^{\prime}(x)+c(x) y(x)=f(x)
$$
where $a, b, c$ and $f$ are arbitrary functions and the equation is homogeneous if $f=0$. Again, another standard form is
$$
y^{\prime \prime}(x)+p(x) y^{\prime}(x)+q(x) y(x)=f(x)
$$
## First order linear differential equations.
All solutions of
$$
y^{\prime}(x)+p(x) y(x)=f(x)
$$
can be written
$$
y(x)=C y_{1}(x)+y_{p}(x)
$$
where $y_{1}(x)$ is a solution of the corresponding homogeneous equation $y^{\prime}(x)+p(x) y(x)=0$ and $y_{p}(x)$ is one solution of the full equation. This can be demonstrated by explicit construction.
$$
y^{\prime}(x)+p(x) y(x)=f(x)
$$
can be rewritten
$$
\frac{d}{d x} e^{I(x)} y(x)=e^{I(x)} f(x)
$$
where
$$
I(x)=\int_{a}^{x} d z p(z)
$$
and, here, $a$ is an arbitrary constant. Now, $I^{\prime}(x)=p(x)$ and $I$ is called an integrating factor. Integrate from $a$ to $x$
$$
e^{I(x)} y(x)-e^{I(a)} y(a)=\int_{a}^{x} d z e^{I(z)} f(z)
$$
with $e^{I(a)}=1$. This gives
$$
y(x)=C y_{1}(x)+y_{p}(x),
$$
with $y_{1}(x)=e^{-I(x)}, y_{p}(x)=e^{-I(x)} \int_{a}^{x} d z e^{I(z)} f(z)$ and $C=y(a)$. In practise, this method will always find a solution, but, often, it is quicker just to stare at the equation and then guess a solution and check it works.
- Example Find all solutions of the ODE 1
$$
y^{\prime}(x)+\frac{1}{x} y(x)=x^{3} .
$$
Here $p(x)=1 / x$ which has a non-integrable singularity at $x=0$ ! Work with $x>0$, or $x<0$. First, the integrating factor $I(x)=\int d x p(x)=\log x+c$. Set $c=0$, or $a=1$. $e^{I(x)}=x$ so that the ODE can be written
$$
\frac{d}{d x}(x y)=x^{4} .
$$
Integrating gives $x y=\frac{1}{5} x^{5}+C$ or $y=\frac{1}{5} x^{4}+C / x$, that is $y_{1}(x)=1 / x, y_{p}(x)=\frac{1}{5} x^{4}$.
## Second order case
All solutions, or the general solution of
$$
y^{\prime \prime}(x)+p(x) y^{\prime}(x)+q(x) y(x)=f(x)
$$
are given by
$$
y(x)=C_{1} y_{1}(x)+C_{2} y_{2}(x)+y_{p}(x)
$$
where $y_{1}, y_{2}$ are linearly independent solutions of the corresponding homogeneous equation
$$
y^{\prime \prime}(x)+p(x) y^{\prime}(x)+q(x) y(x)=0
$$
and $y_{p}(x)$ is a solution of the full equation. $C_{1}$ and $C_{2}$ are arbitrary constants. This isn't proved here, but it is easy to understand why it would be the case: this is a second order equation so it nears to arbitrary constant, in the initial value problem, one matches $y(0)$ and the other $y^{\prime}(0)$. Now, if you have a solution, adding a solution of the corresponding homogeneous problem gives you another solution and the homogeneous problem also has a two-dimensional space of solutions, so it all mathes up. $y_{p}(x)$ is called a particular integral. The general solution is sometimes written
$$
y(x)=y_{c}(x)+y_{p}(x)
$$
where $y_{c}(x)=C_{1} y_{1}(x)+C_{2} y_{2}(x)$ is called the complementary function. It is the general solution of the homogeneous form of the ODE.
## Constant Coeffcients
We now consider the special case where the coefficients $a, b$ and $c$ are constants
$$
a y^{\prime \prime}(x)+b y^{\prime}(x)+c y(x)=f(x) .
$$
This type of equation has a nice interpretation as a damped/driven oscillator where we will use $t$ instead of $x$ as the variable, since it is time. $y$ is the displacement from equilibrium. Recall the equation for a simple harmonic oscillator
$$
\frac{d^{2} y(t)}{d t^{2}}=-\omega^{2} y(t)
$$
Now add in a damping force proportional to the velocity $d y / d t$ and a driving force $f(t)$, which may be periodic or non-periodic,
$$
\frac{d^{2} y(t)}{d t^{2}}=-\omega^{2} y(t)-\gamma \frac{d y(t)}{d t}+d(t)
$$
which is a linear ODE with constant coeffcents.
So, back to the general constant coefficient form with $x$ as the variable, the first step in solving ODEs of this type is to find two solutions of the homogeneous equation
$$
a y^{\prime \prime}(x)+b y^{\prime}(x)+c y(x)=0 .
$$
This equation has simple exponential solutions of the form $y(x)=e^{\lambda x}$. Differentiating $y^{\prime}(x)=\lambda e^{\lambda x}$ and $y^{\prime \prime}(x)=\lambda^{2} e^{\lambda x}$ so that
$$
a y^{\prime \prime}(x)+b y^{\prime}+c y=\left(a \lambda^{2}+b \lambda+c\right) y
$$
which is zero provided
$$
a \lambda^{2}+b \lambda+c=0
$$
This is called an auxiliary equation. Thus $y_{1}(x)=e^{\lambda_{1} x}$ and $y_{2}(x)=e^{\lambda_{2} x}$ where $\lambda_{1}$ and $\lambda_{2}$ are roots of the quadratic auxiliary equation. The complementary function, if $\lambda_{1} \neq \lambda_{2}$, is $y_{c}(x)=C_{1} e^{\lambda_{1} x}+C_{2} e^{\lambda_{2} x}$.
If $\lambda_{1}=\lambda_{2}$ we only have one exponential solution. In this case a second solution of the ODE is $y(x)=x e^{\lambda_{1} x}$ and $y_{c}(x)=C_{1} e^{\lambda_{1} x}+C_{2} x e^{\lambda_{1} x}$. In the oscillator model this special case corresponds to critical damping. This trick is justified by the fact it works; there are ways to derived it, for example, by converting the equation into two first order equations using $y_{1}=y$ and $y_{2}=y^{\prime}$ and then diagonalizing the corresponding matrix equation and solving using an integrating factor. In practise, the easiest thing is to keep adding powers of $x$ until you have two solutions.
- Example: $y^{\prime \prime}+3 y^{\prime}+2 y=0$ has auxiliary equation $\lambda^{2}+3 \lambda+2=0$ with roots $\lambda_{1}=1$, $\lambda_{2}=2$ so the general solution is
$$
y(x)=C_{1} e^{x}+C_{2} e^{2 x}
$$
This corresponds to over damping. - Example: $y^{\prime \prime}+2 y^{\prime}+y=0$ has auxiliary equation $\lambda^{2}+2 \lambda+1=0$ with two equal roots $\lambda=1$ and so the general solution is
$$
y(x)=\left(C_{1}+C_{2} x\right) e^{x}
$$
- Example: If the auxiliary equation $\lambda^{2}+\lambda+1=0$ with complex roots $\lambda=\frac{1}{2} \pm \frac{1}{2} \sqrt{3} i$ the general complex solution is
$$
y(x)=C_{1} e^{-\frac{1}{2} x+i \frac{1}{2} \sqrt{3} x}+C_{2} e^{-\frac{1}{2} x-i \frac{1}{2} \sqrt{3} x}
$$
where $C_{1}$ and $C_{2}$ are complex constants. The general real solution can be obtained by imposing the constraint $C_{2}=\bar{C}_{1}$ :
$$
y(x)=e^{-\frac{1}{2} x}\left[C_{1}\left(\cos \frac{1}{2} \sqrt{3} x+i \sin \cos \frac{1}{2} \sqrt{3} x\right)+\text { c.c. }\right]
$$
Writing $C_{1}=\frac{1}{2}(A-i B)$ where $A$ and $B$ are real constants gives
$$
y(x)=e^{-\frac{1}{2} x}\left(A \cos \frac{1}{2} \sqrt{3} x+B \sin \frac{1}{2} \sqrt{3} x\right)
$$
this is the underdamped case, it still oscillates. Note III.2 ${ }^{12} 16$ February 2008
## Inhomogeneous 2nd order
Returning to the inhomogeneous form $a y^{\prime \prime}(x)+b y^{\prime}(x)+c y(x)=f(x)$ we still require a particular integral $y_{p}$, that is one solution of the full equation. If $f$ is exponential then it is straightforward: $y_{p}$ is an exponential proportional to $f$.
- For example: consider
$$
y^{\prime \prime}+3 y^{\prime}+2 y=e^{x}
$$
Trying $y_{p}=C e^{x}$ where $C$ is a constant gives
$$
(1+3+2) C e^{x}=e^{x}
$$
so
$$
C=\frac{1}{6}
$$
and, therefore,
$$
y_{p}(x)=\frac{1}{6} e^{x}
$$
is a particular integral. To obtain the general solution we require the general solution of the homogeneous equation, see earlier example, $y_{c}=C_{1} e^{-x}+C_{2} e^{-2 x}$. Thus, the general solution is
$$
y(x)=y_{c}+y_{p}=C_{1} e^{-x}+C_{2} e^{-2 x}+\frac{1}{6} e^{x}
$$
If $f$ is a solution of the homogeneous ODE this doesn't work.
- For example: consider $y^{\prime \prime}+3 y^{\prime}+2 y=e^{-x}$. Trying $y_{p}=C e^{-x}$ gives $C=\infty$. Much as in the equal-root case for the homogeneous equation try $y_{p}=C x e^{-x}$. Differentiating
$$
\begin{aligned}
& y_{p}^{\prime}=C e^{-x}-C x e^{-x} \\
& y_{p}^{\prime \prime}=-2 C e^{-x}+C x e^{-x}
\end{aligned}
$$
Inserting these into the ODE gives
$$
C x e^{-x}(1-3+2)+C e^{-x}(-2+3)=e^{-x}
$$
so that $C=1$ or $y_{p}=x e^{-x}$. The general solution can be written
$$
y=\left(C_{1}+x\right) e^{-x}+C_{2} e^{-2 x}
$$
${ }^{1}$ Conor Houghton, [email protected], see also http://www.maths.tcd.ie/ houghton/231
${ }^{2}$ Based on notes I got from Chris Ford Now, for more general cases, we exploit the linearity. If $f$ is a sum of exponentials, as, for example, $\sin x=\left(e^{i x}-e^{-i x}\right) /(2 i)$, just add up the corresponding particular integrals corresponding to each exponential term; so, for example, if
$$
y^{\prime \prime}+3 y^{\prime}+2 y=\sin x .
$$
We solve
$$
y^{\prime \prime}+3 y^{\prime}+2 y=\frac{1}{2 i} e^{i x}
$$
and
$$
y^{\prime \prime}+3 y^{\prime}+2 y=-\frac{1}{2 i} e^{-i x}
$$
and add the two solutions.
If $f$ is not a finite sum of exponentials decompose $f$ into complex exponentials: we use Fourier analysis.
- Example with Fourier analysis: Obtain the general solution of
$$
y^{\prime \prime}(x)+3 y^{\prime}(x)+2 y(x)=f(x)
$$
where $f$ is the periodic square wave
$$
f(x)= \begin{cases}1 & 0<x<\pi \\ -1 & -\pi<x<0\end{cases}
$$
with $f(x+2 \pi)=f(x)$. The complementary function is $y_{c}=C_{1} e^{-x}+C_{2} e^{-2 x}$, as in earlier example. For the particular integral we expand $f$ as a Fourier series, this we have done before;
$$
f(x)=\frac{4}{\pi} \sum_{n \text { odd }>0} \frac{1}{n} \sin n x=\frac{2}{\pi i} \sum_{n \text { odd } \in Z} \frac{1}{n} e^{i n x}
$$
Now, we find the particular integral for
$$
y^{\prime \prime}+3 y^{\prime}+2 y=e^{i n x}
$$
Trying $y_{p}=C e^{i n x}$ gives
$$
C\left(-n^{2}+3 i n+2\right) e^{i n x}=e^{i n x}
$$
so that
$$
C=\frac{1}{-n^{2}+3 i n+2}
$$
Now, adding the individual particular integrals together to get the particular integral of full problem
$$
y_{p}(x)=\frac{2}{\pi i} \sum_{n \text { odd } \in Z} \frac{e^{i n x}}{n\left(-n^{2}+3 i n+2\right)}
$$
If $f$ is not periodic write it as a Fourier integral
- Fourier integral example: Find the general solution of the ODE
$$
y^{\prime \prime}(x)+2 y^{\prime}(x)+2 y(x)=f(x)
$$
where $f(x)=e^{-x^{2}}$. The auxiliary equation is $\lambda^{2}+2 \lambda+2=0$ with two complex roots $\lambda=-1 \pm i$ so that
$$
y_{c}=e^{-x}(A \cos x+B \sin x)
$$
Write $f$ as a Fourier integral
$$
f(x)=\int_{-\infty}^{\infty} d k e^{i k x} \tilde{f}(k)
$$
where, using the standard formula for the Fourier integral of a Gaußian,
$$
\tilde{f}(k)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} d x e^{-i k x} e^{-x^{2}}=\frac{1}{2 \sqrt{\pi}} e^{-\frac{1}{4} k^{2}}
$$
Now, we obtain PI for $y^{\prime \prime}+2 y^{\prime}+2 y=e^{i k x}$. Trying $y=C e^{i k x}$ gives
$$
C\left(-k^{2}+2 i k+2\right) e^{i k x}=e^{i k x}
$$
giving
$$
C=\frac{1}{-k^{2}+2 i k+2} .
$$
The particular integral of the whole problem is obtained by integrating over the particular integrals for the individual problems labelled by $k$ :
$$
y_{p}(x)=\frac{1}{2 \sqrt{\pi}} \int_{-\infty}^{\infty} d k \frac{e^{i k x} e^{-k^{2} / 4}\left(-k^{2}+2 i k+2\right)}{.}
$$
Note III.3 ${ }^{12} 19$ February 2008
## Green's functions
Returning to the general second order linear inhomogeneous equation
$$
a(x) y^{\prime \prime}(x)+b(x) y^{\prime}(x)+c(x) y(x)=f(x)
$$
If two solutions, $y_{1}$ and $y_{2}$, of the homogeneous equation are known then a particular solution of the full equation, $y_{p}$, can be found by the Green's function method; this is outlined here. Consider the equation
$$
a(x) y^{\prime \prime}(x)+b(x) y^{\prime}(x)+c(x) y(x)=\delta\left(x-x^{\prime}\right) .
$$
A solution of this equation is called a Green's function and denoted $G\left(x \mid x^{\prime}\right)$. A solution of the inhomogeneous equation is then
$$
y_{p}(x)=\int_{-\infty}^{\infty} d x^{\prime} f\left(x^{\prime}\right) G\left(x \mid x^{\prime}\right)
$$
The Green's function is not unique, but one formula for $G\left(x \mid x^{\prime}\right)$ is
$$
G\left(x \mid x^{\prime}\right)=\theta\left(x-x^{\prime}\right) \frac{y_{1}(x) y_{2}\left(x^{\prime}\right)-y_{2}(x) y_{1}\left(x^{\prime}\right)}{y_{1}\left(x^{\prime}\right) y_{2}^{\prime}\left(x^{\prime}\right)-y_{2}\left(x^{\prime}\right) y_{1}^{\prime}\left(x^{\prime}\right)} .
$$
The object in the denominator is called the Wronksian of $y_{1}$ and $y_{2}$. Of course, this still leaves the problem of solving the homogeneous equation
$$
a(x) y^{\prime \prime}(x)+b(x) y^{\prime}(x)+c(x) y(x)=0
$$
## The Euler equation
For $a(x) y^{\prime \prime}(x)+b(x) y^{\prime}(x)+c(x) y(x)=0$ no general solution when the coefficients aren't constants. One important case that can be solved is Euler's equation.
$$
\alpha x^{2} y^{\prime \prime}+\beta x y^{\prime}+\gamma y=0
$$
where $\alpha, \beta, \gamma$ constants. This equation arises when studying Laplace's equation, the most important partial differential equation. Euler's equation is solved by transforming it into the constant coefficient case using a change of variable:
$$
x=e^{z}
$$
${ }^{1}$ Conor Houghton, [email protected], see also http://www.maths.tcd.ie/ ${ }^{\sim}$ houghton/231
${ }^{2}$ Based on notes I got from Chris Ford Using
$$
1=\frac{d x}{d x}=\frac{d}{d x} e^{z}=\frac{d z}{d x} e^{z}
$$
so $d z / d x=e^{-z}$ this gives
$$
x \frac{d y}{d x}=e^{z} \frac{d y}{d z} \frac{d z}{d x}=\frac{d y}{d z}
$$
and
$$
\begin{aligned}
x^{2} \frac{d^{2} y}{d x^{2}} & =x^{2} \frac{d}{d x} \frac{d y}{d x}=x^{2} \frac{d}{d x} \frac{1}{x} \frac{d y}{d z} \\
& =-\frac{d y}{d z}+x \frac{d^{2} y}{d z^{2}} \frac{d z}{d x}=-\frac{d y}{d z}+\frac{d^{2} y}{d z^{2}}
\end{aligned}
$$
so the Euler's equation becomes
$$
\alpha \frac{d^{2} y}{d z^{2}}+(\beta-\alpha) \frac{d y}{d z}+\gamma y=0
$$
which has constant coefficients. The auxiliary equation is
$$
\alpha \lambda^{2}+(\beta-\alpha) \lambda+\gamma=0
$$
with general solution is
$$
y_{c}=C_{1} e^{\lambda_{1} z}+C_{2} e^{\lambda_{2} z}=C_{1} x^{\lambda_{1}}+C_{2} x^{\lambda_{2}}
$$
where $\lambda_{1}$ and $\lambda_{2}$ are roots of the auxiliary equation. If $\lambda_{1}=\lambda_{2}$ then
$$
y_{c}=C_{1} e^{\lambda_{1} z}+C_{2} z e^{\lambda_{1} z}=C_{1} x^{\lambda_{1}}+C_{2} \log x x^{\lambda_{1}}
$$
for $x \geq 0$. Note III.4 ${ }^{12} 31$ March 2008
## Series Solutions
The idea behind series solutions is to write $y$ as a power series about $x=0$ or some other point
$$
y(x)=\sum_{n=0}^{\infty} a_{n} x^{n},
$$
and determine a recursion relation for the $a_{n}$ coefficients.
Here is a simple example where we know what the answer should be
$$
y^{\prime}-y=0
$$
Now, assuming convergence,
$$
y^{\prime}(x)=\sum_{n=0}^{\infty} n a_{n} x^{n-1},
$$
and so substituting into the equation gives
$$
\sum_{n=0}^{\infty} n a_{n} x^{n-1}-\sum_{n=0}^{\infty} a_{n} x^{n}=0
$$
The way series solutions works is we try to express this in the form
$$
\sum_{n} \operatorname{stuff}_{n} x^{n}=0
$$
and the $\operatorname{stuff}_{n}$ gives the recursion relation. The problem here is the different powers of $x$ in the sums, we fix this by doing a change of index $m=n-1$ so
$$
\sum_{n=0}^{\infty} n a_{n} x^{n-1}=\sum_{m=-1}^{\infty}(m+1) a_{m+1} x^{m}
$$
and now the sum limits are different to the limits of the other term in the equation, we fix this by doing the first term seperately, in this case the $m=-1$ term is zero, so
$$
\sum_{m=-1}^{\infty}(m+1) a_{m+1} x^{m}=\sum_{m=0}^{\infty}(m+1) a_{m+1} x^{m}
$$
Now, we $m$ is just a dummy index, we can name it back to $n$ and the equation becomes
$$
y^{\prime}-y=\sum_{n=0}^{\infty}(n+1) a_{n+1} x^{n}-\sum_{n=0}^{\infty} a_{n} x^{n}
$$
${ }^{1}$ Conor Houghton, [email protected], see also http://www.maths.tcd.ie/ ${ }^{\text {houghton/231 }}$
${ }^{2}$ Based on notes I got from Chris Ford
$$
=\sum_{n=0}^{\infty}\left[(n+1) a_{n+1}-a_{n}\right] x^{n}=0
$$
and, since the $x^{n}$ are independent, for this to be true for all $x$ every coefficient of every power of $x$ must vanish, giving
$$
a_{n+1}=\frac{a_{n}}{n+1}
$$
This is the recursion relation, it allows us to calculate higher $a_{n}$ 's from lower ones, for $n=0$ it gives $a_{1}=a_{0}$, for $n=1$ it gives $a_{2}=a_{1} / 2=a_{0} / 2$ and so on. The $a_{0}$ is arbitrary, it is the arbitrary constant you would expect for a first order differential equation. Of course, in this case it is easy to see that $a_{n}=a_{0} / n$ !, as you would expect: we know this equation is solved by $y=A \exp x$ where $A$ is an arbitrary constant.
Here is another example, consider
$$
y^{\prime \prime}-x y=0
$$
so assuming we can find a solution of the form
$$
y(x)=\sum_{n=0}^{\infty} a_{n} x^{n}
$$
we have
$$
x y=\sum_{n=0}^{\infty} a_{n} x^{n+1},
$$
and
$$
y^{\prime \prime}=\sum_{n=0}^{\infty} a_{n} n(n-1) x^{n-2},
$$
Substituting into the equation we have
$$
\sum_{n=0}^{\infty} a_{n} n(n-1) x^{n-2}-\sum_{n=0}^{\infty} a_{n} x^{n+1}=0
$$
and so we take the first term and shift its index up three so that the exponent of $x$ has the same form as in the second term
$$
\begin{aligned}
\sum_{n=0}^{\infty} a_{n} n(n-1) x^{n-2} & =\sum_{m=-3}^{\infty} a_{m+3}(m+3)(m+2) x^{m+1} \\
& =2 a_{2}+\sum_{m=0}^{\infty} a_{m+3}(m+3)(m+2) x^{m+1}
\end{aligned}
$$
where we have removed the $m=-3,-2$ and -1 terms from the sum, but only the -1 term is non-zero. Renaming $m$ to $n$ we have
$$
2 a_{2}+\sum_{n=0}^{\infty} a_{n+3}(n+3)(n+2) x^{n+1}-\sum_{n=0}^{\infty} a_{n} x^{n+1}=0
$$
and, again, substituting every coefficient of every power of $x$ to zero we get recursion relations
$$
a_{2}=0
$$
and
$$
a_{n+3}=\frac{a_{n}}{(n+3)(n+2)}
$$
Now, we have a three-step recursion relation, but with $a_{2}=0$; so there are two arbitrary constants
$$
\begin{aligned}
& a_{0}=y(0) \\
& a_{1}=y^{\prime}(0)
\end{aligned}
$$
where the constants are related to the initial conditions by substituting $x=0$ into the series for $y$ and $y^{\prime}$.
Now, Consider Hermite's equation
$$
y^{\prime \prime}-2 x y^{\prime}+2 \alpha y=0
$$
So, we let
$$
y(x)=\sum_{n=0}^{\infty} a_{n} x^{n}
$$
SO
$$
x y^{\prime}(x)=\sum_{n=0}^{\infty} n a_{n} x^{n}
$$
and
$$
y^{\prime \prime}(x)=\sum_{n=0}^{\infty} n(n-1) a_{n} x^{n-2}=\sum_{m=0}^{\infty}(m+2)(m+1) a_{m+2} x^{m}
$$
where we have changed index to $m=n+2$ and used that the first two terms in the sum are zero. This gives
$$
y^{\prime \prime}-2 x y^{\prime}+2 \alpha y=\sum_{n=0}^{\infty}\left[(n+2)(n+1) a_{n+2}+2(\alpha-n) a_{n}\right] x^{n}=0 .
$$
This implies that the content of the square bracket is zero for all $n$ leading to the recursion relation
$$
a_{n+2}=\frac{2(n-\alpha) a_{n}}{(n+1)(n+2)} .
$$
From this two independent solutions can be obtained.
- An even solution: Set $a_{1}=0$. Recursion relation $\rightarrow a_{3}, a_{5}, a_{7}$ etc. all zero. Fix $a_{0}=1$ and apply recursion relation
$$
a_{2}=-\frac{2 \alpha a_{0}}{1 \cdot 2}=-\frac{2 \alpha}{1 \cdot 2}
$$
$$
\begin{aligned}
& a_{4}=\frac{2(2-\alpha) a_{2}}{3 \cdot 4}=2^{2} \frac{(\alpha-2) \alpha}{1 \cdot 2 \cdot 3 \cdot 4} \\
& a_{6}=\frac{2(4-\alpha) a_{4}}{5 \cdot 6}=-\frac{2^{3}(\alpha-4)(\alpha-2) \alpha}{6 !}
\end{aligned}
$$
so the pattern clear
$$
a_{2 m}=\frac{(-2)^{m}}{(2 m) !}(\alpha-2 m+2)(\alpha-2 m+4) \ldots \alpha
$$
where we define $0 !=1$. An even solution of Hermite's equation reads
$$
y_{e}(x) \sum_{m=0}^{\infty} \frac{(-2)^{m}}{(2 m) !}(\alpha-2 m+2)(\alpha-2 m+4) \ldots \alpha x^{2 m} \text {. }
$$
This series is convergent with radius of convergence $=\infty$; the ratio test can be used to prove this, we don't persue issues of convergence here. For special values of $\alpha$, even and positive, the series terminates and the solution is a polynomial of degree $\alpha$. For example, when $\alpha=2 a_{4}, a_{6}, a_{8}$ and so on, are all zero and
$$
y_{\text {even }}(x)=1-2 x^{2}
$$
You can check that this satisfies Hermite's equation.
- An odd solution: Set $a_{0}=1, a_{1}=1$. Recursion relation $\rightarrow a_{2}, a_{4}, a_{6}$ et. all zero. Odd coefficients can be worked out via the recursion formula. If $\alpha$ is an odd integer the series will terminate producing a polynomial of degree $\alpha$.
The general solution of Hermite's equation is therefore
$$
y=C_{1} y_{e}(x)+C_{2} y_{o}(x)
$$
where $y_{e}$ and $y_{o}$ are the even and odd solutions.
## Generating function
If $\alpha$ is a positive integer one of the solutions to Hermite's equation is polynomial. Such functions are called Hermite polynomials. Remarkably, all the polynomials can be combined into a single generating function. Consider $\Phi(x, h)=e^{2 x h-h^{2}}$. Expanding this in powers of $h$ :
$$
\Phi(x, h)=\sum_{n=0}^{\infty} \frac{h^{n}}{n !} H_{n}(x)
$$
$H_{n}(x)$ are polynomial solutions of Hermite's equation. Note III.5 ${ }^{12} 12$ April 2008
## The method of Fröbenius
For the general homogeneous ordinary differential equation
$$
y^{\prime \prime}(x)+p(x) y^{\prime}(x)+q(x) y(x)=0
$$
the series method works, as in the Hermite case, where both $p$ and $q$ are smooth. If $p$ and $q$ have singularities the series method sometimes fails, and example is Euler's equation $\alpha x^{2} y^{\prime \prime}+\beta x y^{\prime}+\gamma y=0$ or $p(x)=\beta /(\alpha x)$ and $q(x)=\gamma /\left(\alpha x^{2}\right)$. The explicit solution $y=C_{1} x^{\lambda_{1}}+C_{2} x^{\lambda_{2}}$ picked up by power series method, unless both roots $\lambda_{1}$ and $\lambda_{2}$ are positive integers, because $x^{\lambda}$ is not of the form of the ansatz
$$
y=\sum_{n=0}^{\infty} a_{n} x^{n}
$$
for any $a_{n} \mathrm{~s}$, unless $\lambda$ is itself a natural number.
One way out is to expand about a point other than $x=0$ :
$$
y(x)=\sum_{n=0}^{\infty} a_{n}(x-c)^{n}
$$
where $p(c)$ and $q(c)$ are finite. However, a singular point can often be the 'most symmetric' point and, in many cases, exactly the point we are interested in.
Frobenius (or generalised series) method allows one to expand about a regular singularity, described later, of $p$ and $q$. Without loss of generality consider an expansion about $x=0$. Consider a solution of the form
$$
y(x)=\sum_{n=0}^{\infty} a_{n} x^{n+s}
$$
where $s$ is some real number. Unlike in the standard power series method $a_{0}$ is always taken to be non-zero; the odd solution of Hermite's equation would emerge as an $s=1$ Frobenius series with $a_{0} \neq 0$. Starting with $s$ arbitrary consistency will lead to a quadratic equation for $s$ called the indicial equation.
## The Bessel Equation
The Bessel equation is
$$
y^{\prime \prime}+\frac{1}{x} y^{\prime}+\left(1-\frac{\nu^{2}}{x^{2}}\right) y=0
$$
${ }^{1}$ Conor Houghton, [email protected], see also http://www.maths.tcd.ie/ houghton/231
${ }^{2}$ Based on notes I got from Chris Ford or, multiplying across by $x^{2}$,
$$
x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-\nu^{2}\right) y=0
$$
It is one of the important equation of applied mathematics and engineering mathematics because it is related to the Laplace operator in cylindrical coördinates. The Bessel equation is solved by series solution methods, in fact, to solve the Bessel equation you need to use the method of Fröbenius. It might be expected that Fröbenius is needed because of the singularities at $x=0$, however, lets pretend we hadn't noticed and try to use the ordinary series solution method:
$$
y=\sum_{n=0}^{\infty} a_{n} x^{n}
$$
Now, by calculating directly
$$
x^{2} y^{\prime \prime}=\sum_{n=0}^{\infty} n(n-1) a_{n} x^{n}
$$
and
$$
x^{2} y^{\prime}=\sum_{n=0}^{\infty} n a_{n} x^{n}
$$
so the equation becomes
$$
\sum_{n=0}^{\infty} n(n-1) a_{n} x^{n}+\sum_{n=0}^{\infty} n a_{n} x^{n}+\sum_{n=0}^{\infty} a_{n} x^{n+2}-\nu^{2} \sum_{n=0}^{\infty} a_{n} x^{n}=0
$$
Hence, if we want to go up to the highest power we need to increase everything to the form $x$ to the $n+2$. By letting $m+2=n$ we get
$$
\sum_{n=0}^{\infty} n(n-1) a_{n} x^{n}=\sum_{m=0}^{\infty}(m+2)(m+1) a_{m+2} x^{m+2}
$$
and
$$
\sum_{n=0}^{\infty} n a_{n} x^{n}=\sum_{m=0}^{\infty}(m+2) a_{m+2} x^{m+2}
$$
and, finally,
$$
\sum_{n=0}^{\infty} n a_{n} x^{n}=a_{0}+a_{1} x+\sum_{m=0}^{\infty} a_{m+2} x^{m+2}
$$
Putting this all back in to the equation, renaming $m$ to $n$ in the usual way, we get
$$
a+0+a_{1} x \sum_{n=0}^{\infty}\left[(n+2)(n+1) a_{n+2}+(n+2) a_{n+2} x^{n}-\nu^{2} a_{n+2}+a_{n}\right] x^{n+2}=0
$$
which gives recursion relation
$$
a_{n+2}=-\frac{a_{n}}{(n+2)^{2}-\nu^{2}}
$$
along with $a_{0}=a_{1}=0$. Thus, while we get a perfectly good two step recursion relation, the extra conditions, on $a_{0}$ and $a_{1}$ lead to the solution being trivial. Hence, the solution of the series form is trivial and, clearly, to find the actual solution, a more general series ansatz is needed.
Fröbenius means that you look for a solution of the form
$$
y=\sum_{n=0}^{\infty} a_{n} x^{n+r}
$$
Now, in terms of this series we have
$$
\begin{aligned}
x^{2} y^{\prime \prime} & =\sum_{n=0}^{\infty} a_{n}(n+r)(n+r-1) x^{n+r} \\
x y^{\prime} & =\sum_{n=0}^{\infty} a_{n}(n+r) x^{n+r} \\
x^{2} y & =\sum_{n=0}^{\infty} a_{n} x^{n+r+2} \\
\nu^{2} y & =\sum_{n=0}^{\infty} \nu^{2} a_{n} x^{n+r}
\end{aligned}
$$
As usual, we move to the highest power, in this case $n+r+2$, without going through the details, this gives
$$
x^{2} y^{\prime \prime}=r(r-1) a_{0} x^{r}+r(r+1) a_{1} x^{r+1}+\sum_{n=0}^{\infty} a_{n+2}(n+r+2)(n+r+1) x^{n+r+2}
$$
and
$$
x y^{\prime}=r a_{0} t^{r}+r(r+1) a_{1} x^{r+1}+\sum_{n=0}^{\infty} a_{n+2}(n+r+2) x^{n+r+2}
$$
and finally
$$
\nu^{2} y=\nu^{2} a_{0} x^{r}+\nu^{2} a_{1} x^{r+1}+\sum_{n=0}^{\infty} a_{n+2} x^{n+r+2}
$$
Now, if we put this all in one equation and set the $x^{r}$ terms to zero, we have
$$
\left[r(r-1)+r-v^{2}\right] a_{0}=0
$$
or, put another way, either $a_{0}=0$ or $r= \pm \nu$. The $x^{r+1}$ term gives
$$
\left[(r+1)^{2}-\nu^{2}\right] a_{1}=0
$$
so, with $r= \pm n u a_{1}=0$
Now, the recusion relation is
$$
\left[(n+r+2)(n+r+1)+(n+r+2)-\nu^{2}\right] a_{n+2}=-a_{n}
$$
so, with $r= \pm \nu$ we have
$$
a_{n+2}=-\frac{a_{n}}{(n \pm \nu+2)^{2}-\nu^{2}}
$$
and so there are two solutions to the Bessel equation, one corresponding to $r=\nu$ and the other with $r=-\nu$. If $\nu=0$ the situation is more complicated, this example is dealt with in a problem sheet.
Here we consider the case $\nu=\frac{1}{2}$ so that $s= \pm \frac{1}{2}$. The recursion relation can be written
$$
a_{m+2}=-\frac{a_{m}}{\left(m+2+\frac{1}{2}\right)^{2}-\frac{1}{4}}=-\frac{a_{m}}{(m+2)(m+3)} .
$$
Since $a_{1}=0$ the recursion relation implies that $a_{3}, a_{5}$, $a_{7}$ etc. are all zero. Fixing $a_{0}=0$ and applying the recursion relation gives
$$
\begin{aligned}
a_{2} & =-\frac{a_{0}}{2 \cdot 3}=-\frac{1}{2 \cdot 3} \\
a_{4}= & -\frac{a_{2}}{4 \cdot 5}=\frac{1}{2 \cdot 3 \cdot 4 \cdot 5}=\frac{1}{5 !} \\
a_{6}= & -\frac{1}{7 !}
\end{aligned}
$$
and so on. Thus, the solution is
$$
\begin{aligned}
y(x) & =x^{\frac{1}{2}}-\frac{x^{\frac{5}{2}}}{3 !}+\frac{x^{\frac{9}{2}}}{5 !}-\ldots \\
& =x^{-\frac{1}{2}}\left(x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\ldots\right) \\
& =x^{-\frac{1}{2}} \sin x .
\end{aligned}
$$
where we have used $\left.\sin x=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\ldots\right)$
The other root $s=-\frac{1}{2}$ leads to
$$
y(x)=x^{-\frac{1}{2}} \cos x
$$
and so the general solution of the $\nu=\frac{1}{2}$ problem is
$$
y(x)=x^{-\frac{1}{2}}\left(C_{1} \cos x+C_{2} \sin x\right) .
$$
## Fuch's theorem
The method of Frobenius gives a series solution of the form
$$
y(x)=\sum_{n=0}^{\infty} a_{n}(x-c)^{n+s}
$$
where $p$ or $q$ are singular at $x=c$. Method does not always give the general solution, the $\nu=0$ case of Bessel's equation is an example where it doesn't. There is a theorem dealing with the applicability of the Frobenius method in the case of regular singularities.
$x=c$ is a regular singular point if $(x-c) p(x)$ and $(x-c)^{2} q(x)$ can be expanded as a power series about $x=c$. All the singular ODEs we have met have regular singularities, an example of an ODE with a non-regular singularity $x^{3} y^{\prime \prime}+y=0$ since here $q(x)=1 / x^{3}$ so that $x^{2} q(x)=1 / x$ cannot be expanded about $x=0$.
If $p$ and $q$ are non-singular at $x=c, x=c$ is called an ordinary point of the ODE $y^{\prime \prime}(x)+p(x) y^{\prime}(x)+q(x) y(x)=0$.
Fuchs' Theorem states that if $x=c$ is a regular singular or ordinary point of the ODE
$$
y^{\prime \prime}(x)+p(x) y^{\prime}(x)+q(x) y(x)=0
$$
if and only if the two solutions are Frobenius series or one solution is a Frobenius series, $S_{1}(x)$ and the other solution is of the form $y(x)=S_{1}(x) \log (x-c)+S_{2}(x)$ where $S_{2}(x)$ is another Frobenius series, it is not a solution on its own. The proof of this isn't given.
The second case occurs when the indicial equation has equal roots and sometimes when the roots differ by an integer, an example is the $\nu=1$ case of Bessel's equation).
## Finding a Second Solution
If one solution of $y^{\prime \prime}(x)+p(x) y^{\prime}(x)+q(x) y(x)=0$ can be found another one can be constructed. Let $u(x)$ be a solution then try $y(x)=u(x) v(x)$ then a short calculation gives
$$
y^{\prime \prime}+p y^{\prime}+q y=\left(u^{\prime \prime}+p u^{\prime}+q u\right) v+\left(2 u^{\prime}+p u\right) v^{\prime}+u v^{\prime \prime}=0 .
$$
Now since $u$ is, by assumption, a solution the first term on the right hand side is zero giving
$$
\left(2 u^{\prime}+p u\right) v^{\prime}+u v^{\prime \prime}=0 .
$$
This is a first order linear ODE for $v^{\prime}(x)$. Note III.6 ${ }^{12} 22$ April 2008
## Eigenfunctions and Eigenfunction Expansions
There is a strong analogy between solving some of the named ODEs and finding the eigenvectors and eigenvalues of a matrix.
Hermite's equation
$$
y^{\prime \prime}-2 x y^{\prime}+2 \alpha y=0
$$
can be written
$$
L y=\lambda y
$$
where $L$ is the differential operator
$$
L=-\frac{d^{2}}{d x^{2}}+2 x
$$
and $\lambda=2 \alpha$.
Legendre's equation can be written in the same way, with
$$
L=-\left(1-x^{2}\right) \frac{d^{2}}{d x^{2}}+2 x \frac{d}{d x}
$$
with $\lambda=\alpha$.
We can think of the differential operator $L$ as a matrix, albeit an infinite dimensional one, and the function it acts on, $y$, as a vector.
To make this more precise it is useful to recall some properties of certain finite, say $n \times n$, matrices.
- A symmetric matrix $S$ satisfies
$$
S^{T}=S
$$
where the superscript $T$ denotes the transpose $\left[A^{T}\right]_{i j}=[A]_{j i}$.
- A Hermitian matrix is an $n \times n$ matrix with complex entries satisfying
$$
H^{\dagger}=H
$$
where the superscript dagger is the adjoint, the complex conjugate of the transpose
$$
\left[A^{\dagger}\right]_{i j}=\overline{[A]_{j i}} .
$$
${ }^{1}$ Conor Houghton, [email protected], see also http://www.maths.tcd.ie/ ${ }^{\text {houghton/231 }}$
${ }^{2}$ Based on notes I got from Chris Ford Clearly a real symmetric matrix is Hermitian.
Let $v$ be an $n$-component column vector with complex entries. $v$ is an eigenvector of $H$ if
$$
H v=\lambda v
$$
for some complex number $\lambda$, the eigenvalue.
Now, an important theorem is that the eigenvalues of a Hermitian matrix are real. To prove this, let $v$ be an eigenvector of $H$ with eigenvalue $\lambda$
$$
v^{\dagger} H v=\lambda v^{\dagger} v
$$
Since $v^{\dagger} v$ is real it suffices to prove that $v^{\dagger} H v$ is real:
$$
\overline{v^{\dagger} H v}=\overline{\left(v^{\dagger} H v\right)^{T}}
$$
since $v^{\dagger} H v$ is a $1 \times 1$ matrix
$$
\overline{\left(v^{\dagger} H v\right)^{T}}=\left(v^{\dagger} H v\right)^{\dagger}=v^{\dagger} H^{\dagger} v=v^{\dagger} H v
$$
if $H$ is Hermitian. Note that $(A B)^{\dagger}=B^{\dagger} A^{\dagger}$.
Another important property is that the eigenvectors of a Hermitian matrix corresponding to distinct eigenvalues are orthogonal, that is if $H v_{1}=\lambda_{1} v_{1} \quad H v_{2}=\lambda_{2} v_{2}$ with $\lambda_{1} \neq \lambda_{2}$ then $v_{1}^{\dagger} v_{2}=0$ or, in the inner product notation, $\left(v_{1}, v_{2}\right)=0$. To prove this we take the eigenvector equations and multiply them on the left to give
$$
\begin{aligned}
v_{2}^{\dagger} H v_{1} & =\lambda_{1} v_{2}^{\dagger} v_{1} \\
v_{1}^{\dagger} H v_{2} & =\lambda_{1} v_{1}^{\dagger} v_{2}
\end{aligned}
$$
Take the complex conjugate of the second
$$
v_{2}^{\dagger} H v_{1}=\lambda_{2} v_{2}^{\dagger} v_{1}
$$
using $H^{\dagger}=H$ and $\bar{\lambda}_{2}=\lambda_{2}$ Subtracting the first gives
$$
0=\left(\lambda_{2}-\lambda_{1}\right) v_{2}^{\dagger} v_{1}
$$
so that $\left(v_{2}, v_{1}\right)=0$ if $\lambda_{1} \neq \lambda_{2}$. If the eigenvalues are degenerate we can choose eigenvectors to be orthogonal using Gram-Schmidt orthogonalization.
Thus, we can choose the $n$ eigenvectors $v_{i}, i=1,2, \ldots, n$, of an Hermitian matrix $\mathrm{H}$ to be orthonormal
$$
v_{i}^{\dagger} v_{j}=\delta_{i j}
$$
or
$$
\left(v_{i}, v_{j}\right)=\delta_{i j}
$$
Any $n$-component vector $v$ can be written as a linear combination of the $v_{i} \mathrm{~S}$
$$
v=c_{1} v_{1}+c_{2} v_{2}+\ldots+c_{n} v_{n}
$$
where $c_{1}, c_{2}, \ldots c_{n}$ are complex numbers. Using the orthonormal property
$$
\left(v_{i}, v\right)=c_{i}
$$
Also
$$
|v|^{2}=(v, v)=\left|c_{1}\right|^{2}+\left|c_{2}\right|^{2}+\ldots+\left|c_{n}\right|^{2}
$$
which can be thought of as an $n$-dimensional version of Pythagoras or a finite dimensional version of Parseval.
Furthermore, the eigenvector basis can be used to rewrite the matrix
$$
H=\sum_{i=1}^{n} \lambda_{i} v_{i} v_{i}^{\dagger},
$$
since we know by definition
$$
H v=H\left(c_{1} v_{1}+c_{2} v_{2}+\ldots+c_{n} v_{n}\right)=c_{1} \lambda_{1} v_{1}+c_{2} \lambda_{2} v_{2}+\ldots+c_{n} \lambda_{n} v_{n}
$$
and
$$
\sum_{i=1}^{n} \lambda_{i} v_{i} v_{i}^{\dagger} v_{j}=\lambda_{j} v_{j}
$$
so this will have the same effect on $v$. The resolution of unity is a similar formula for the identity matrix, it can be written as
$$
I=\sum_{i=1}^{n} v_{i} v_{i}^{\dagger}
$$
since acting on any vector, $v=c_{1} v_{1}+c_{2} v_{2}+\ldots+c_{n} v_{n}$, will reproduce $v$. The inverse of an Hermitian matrix can be written as
$$
H^{-1}=\sum_{i=1}^{n} \frac{1}{\lambda_{i}} v_{i} v_{i}^{\dagger},
$$
since left or right multiplication by $H$ reproduces the identity matrix.
Now, back to differential operators. A differential operator $L$ acts on some vector space of functions. It is normal to require functions to be such that
$$
\int d x \bar{u}(x)(L u)(x)<\infty
$$
and, in practice, it is usual to impose further restrictions on the functions such as
1. square integrability $\int_{-\infty}^{\infty} d x \bar{u}(x) u(x)<\infty$.
2. periodicity 3. vanishing at the end points of an interval $[a, b] \subset R ; u(a)=u(b)=0$.
In each of these cases an inner product can be defined
1. $(u, v)=\int_{-\infty}^{\infty} d x \bar{u}(x) v(x)$
2. $(u, v)=\int_{-\pi}^{\pi} d x \bar{u}(x) u(x) \quad(l=2 \pi)$
3. $(u, v)=\int_{a}^{b} d x \bar{u}(x) v(x)$.
Once an inner product is defined, we can look for an analogue of a Hermitian matrix. Suppose
$$
(u, L v)=(L u, v)
$$
for any $u, v$ in the chosen space of functions then $L$ is called symmetric. If some further technical requirements are met it is called self-adjoint or Hermitian. We will be sloppy and call any symmetric operator Hermitian. An example is
$$
L=-\frac{d^{2}}{d x^{2}}
$$
for any of the three conditions above. Here the symmetric condition is just
$$
-\int d x \bar{u}(x) v^{\prime \prime}(x)=-\int d x \bar{u}^{\prime \prime} v(x)
$$
To establish this integrate by parts twice, for example, for the periodic case
$$
\int_{-\pi}^{\pi} d x \bar{u}(x) v^{\prime \prime}(x)=\left.\bar{u}(x) v^{\prime}(x)\right|_{-\pi} ^{\pi}-\int_{-\pi}^{\pi} d x \bar{u}^{\prime}(x) v^{\prime}(x)
$$
$\left.\bar{u}(x) v^{\prime}(x)\right|_{-\pi} ^{\pi}$ is zero since $u$ and $v$ are periodic. Integrating by parts once more gives the result.
$L=-d^{2} / d x^{2}$ can be viewed as an Hermitian matrix acting on the space of periodic functions $(l=2 \pi)$. The eigenvectors, or eigenfunctions, are the functions
$$
v_{n}(x)=e^{i n x}
$$
with $n \in \mathbf{Z}$ ) with corresponding eigenvalues
$$
\lambda_{n}=n^{2}
$$
These are orthogonal, as expected since $L$ is Hermitian,
$$
\left(v_{m}, v_{n}\right)=\int_{-\pi}^{\pi} d x \bar{v}_{m}(x) v_{n}(x)=\int_{-\pi}^{\pi} d x e^{-i m x} e^{i n x}=0
$$
for $n \neq m$. Can make them orthonormal
$$
v_{n}(x)=\frac{1}{\sqrt{2 \pi}} e^{i n x}
$$
gives
$$
\left(v_{m}, v_{n}\right)=\delta_{m n}
$$
A periodic function, $f$, can be thought of as a vector in the space acted on by $L$. Expanding $f$ in eigenvectors of $L$
$$
f(x)=\sum_{n \in Z} c_{n} v_{n}(x)
$$
just gives Fourier analysis
$$
c_{m}=\left(v_{m}, f\right)=\frac{1}{\sqrt{2 \pi}} \int_{-\pi}^{\pi} d x e^{-i m x} f(x) .
$$
Fourier analysis is thus equivalent to expanding in eigenvectors of the Hermitian operator $L=-d^{2} / d x^{2}$. We can choose a different Hermitian operator and this leads to alternative expansions: we expect to able to get Legendre series, Hermite series and so on, instead of the Fourier series.
## Legendre Series
Consider Legendre's equation
$$
\left(1-x^{2}\right) y^{\prime \prime}(x)-2 x y^{\prime}(x)+\alpha y(x)=0
$$
This ODE has, regular, singularities at $x= \pm 1$. If $\alpha$ is of the form $\alpha=n(n+1)$ where $n$ is a non-negative integer then the ODE has a polynomial solution, see problem sheet 17, which is well defined at $x= \pm 1$, the other solutions all blow up at $x=1$ or $x=-1$.
We can rewrite the the ODE as an eigenvalue problem:
$$
\begin{aligned}
L y(x) & =\lambda y(x), \\
L & =-\left(1-x^{2}\right) \frac{d^{2}}{d x^{2}}+2 x \frac{d}{d x} .
\end{aligned}
$$
Let us consider in $L$ in the space of functions which are finite throughout $[-1,1]$; this will include the polynomial solutions, but will exclude the other solutions since they blow up at the end points. The inner product is taken as
$$
(u, v)=\int_{-1}^{1} d x \bar{u}(x) v(x) .
$$
It is straightforward to prove that with these boundary conditions $L$ is Hermitian; it helps to write $L$ in the form
$$
L=-\frac{d}{d x}\left(1-x^{2}\right) \frac{d}{d x}
$$
The eigenfunctions are the polynomial solutions of Legendre's equation $P_{n}(x)$ with $n=$ $0,1,2, \ldots$ with corresponding eigenvalues $\lambda_{n}=n(n+1)$. The first few Legendre polynomials are $P_{0}(x)=1, P_{1}(x)=x$ and $P_{2}(x)=\frac{1}{2}\left(3 x^{2}-1\right)$. For $n$ even or odd $P_{n}$ is even or odd, like the Hermite polynomials. There are various formulae for the Legendre polynomials, for example they can be combined into a generating function
$$
\Phi(x, h)=\left(1-2 x h+h^{2}\right)^{-\frac{1}{2}}=\sum_{n=0}^{\infty} h^{n} P_{n}(x) .
$$
The polynomials are orthogonal, they can be made orthonormal, but the following convention is standard
$$
\int_{-1}^{1} d x\left(P_{n}(x)\right)^{2}=\frac{2}{2 n+1} .
$$
Let $f$ be a function defined on $[-1,1]$. Can expand in Legendre polynomials (i.e. in the eigenfunctions of the Hermitian operator $L$ )
$$
f(x)=\sum_{n=0}^{\infty} c_{n} P_{n}(x)
$$
Much as in Fourier analysis
$$
\left(P_{m}, f\right)=\sum_{n=0}^{\infty} c_{n}\left(P_{m}, P_{n}\right)=\sum_{n=0}^{\infty} c_{n} \frac{2 \delta_{m n}}{2 m+1}=\frac{2 c_{m}}{2 m+1},
$$
so that
$$
c_{m}=\left(m+\frac{1}{2}\right)\left(P_{m}, f\right)=\left(m+\frac{1}{2}\right) \int_{-1}^{1} d x P_{m}(x) f(x) .
$$
Hermite's equation
$$
y^{\prime \prime}(x)-2 x y^{\prime}(x)+2 \alpha y(x)=0
$$
can be written as $L y(x)=\lambda y(x)$, where
$$
L=-\frac{d^{2}}{d x^{2}}+2 x \frac{d}{d x}
$$
with $\lambda=2 \alpha$. the problem is $L$ is not Hermitian! $(u, L v) \neq(L u, v)$ regardless of boundary conditions. However if we change the definition of the inner product
$$
(u, v)_{n e w}=\int d x e^{-x^{2}} \bar{u}(x) v(x) .
$$
the operator is Hermitian and similar results as to the Legendre case can be derived using the new inner product. Note IV.1 ${ }^{12} 13$ May 2007
## Part IV: Partial Differential Equations
A partial differential equation is a differential equation involving derivatives of more than one independent variable.
## Some linear PDEs involving a scalar field $\phi$
These equations are often known as the equations of mathematical physics since they are all important in physics and where all studied first because of their physical importance. However, they are now equally important in mathematics.
- Laplace's equation:
$$
\nabla^{2} \phi=0
$$
- Poisson's equation:
$$
\nabla^{2} \phi=\rho
$$
where $\rho$ is some scalar field usually called a source term
- The Helmholtz equation:
$$
\left(\nabla^{2}+k^{2}\right) \phi=0
$$
and the 'wrong sign' Helmholtz equation:
$$
\left(\nabla^{2}-k^{2}\right) \phi=0
$$
where $k$ is a real constant
- The Heat or diffusion equation:
$$
\nabla^{2} \phi=D \frac{\partial \phi}{\partial t}
$$
where $D$ a constant and $\phi=\phi(\mathbf{r}, t)$ is time-dependent.
- The wave equation:
$$
\nabla^{2} \phi-\frac{1}{c^{2}} \frac{\partial^{2} \phi}{\partial t^{2}}=0
$$
where $c$ is the speed of wave propagation.
All of these equation involve the Laplacian:
$$
\triangle=\nabla^{2}=\nabla \cdot \nabla=\operatorname{div} \operatorname{grad}=\frac{d^{2}}{d x^{2}}+\frac{d^{2}}{d y^{2}}+\frac{d^{2}}{d z^{2}}
$$
${ }^{1}$ Conor Houghton, [email protected], see also http://www.maths.tcd.ie/ houghton/231
${ }^{2}$ Substantially based on lecture notes taken by John Kearney
## Heat/Diffusion Equation
Imagine some material where the temperature is not constant but with no sources, or sinks, of heat. Temperature is a scalar field $\phi(\mathbf{r}, t)$. Heat current $\mathbf{j}(\mathbf{r}, t)$ is a vector field such that energy flux across an oriented surface $S$ (Picture IV.1.1) is the surface integral $\int_{S} \mathbf{j}(\mathbf{r}, t) \mathbf{d A}$
Let $\mathrm{S}$ be a closed, static, surface. Heat flux out, $\Phi$, of $\mathrm{S}$ is given by
$$
\begin{aligned}
\Phi=\int_{S} \mathbf{j}(\mathbf{r}, t) \mathbf{d} \mathbf{A}=- \text { rate of change of energy in D } & =-\alpha \frac{\partial}{\partial t} \int_{D} \phi(\mathbf{r}, t) d V \\
& =-\alpha \int_{D} \frac{\partial \phi}{\partial t}(\mathbf{r}, t) d V
\end{aligned}
$$
where $\alpha$ is a constant, the heat capacity per unit volume.
Now, apply Gauß' theorem
$$
\int_{S} \mathbf{j}(\mathbf{r}, t) \cdot \mathbf{d A}=\int_{D} \operatorname{div} \mathbf{j} d V
$$
so that
$$
\int_{D}\left(\operatorname{div} \mathbf{j}+\alpha \frac{\partial \phi}{\partial t}\right) d V=0
$$
where $\mathrm{D}$ is any $3 \mathrm{~d}$ region with a smooth boundary. Thus
$$
\operatorname{div} \mathbf{j}+\alpha \frac{\partial \phi}{\partial t}=0
$$
Assume $\mathbf{j}=-\beta \operatorname{grad} \phi$ where $\beta$ is the thermal conductivity constant and the - sign indicates that the heat flows from hot to cold regions
$$
\nabla^{2} \phi=D \frac{\partial \phi}{\partial t} \text { with } D=\frac{\alpha}{\beta}
$$
## Boundary Value Problems
Typically, we wish to solve a PDE subject to some boundary conditions. Assume $\phi$ satisfies some PDE (e.g. Laplace's equation) in a 3d region D with boundary $S=\delta D$ (Picture IV.1.2). There are three common, basic, types of boundary conditions
## 1. $\phi$ is given on S: Dirichlet boundary conditions.
2. $\partial_{\mathbf{n}} \phi=\mathbf{n} \cdot \nabla \phi$, directional derivative in direction of unit normal $\mathbf{n}$, is given on $\mathrm{S}$ : Neumann boundary conditions.
3. $\phi$ and $\partial_{n} \phi$ are given on S: Cauchy boundary conditions.
4. Can also have mixed b.c.s. where on different parts of S different b.c.s are imposed
5. This is not exhaustive since there are other types of b.c.s. such as periodic boundary conditions. We will concentrate on the first two cases, usually the third case, Cauchy conditions, is too strong and there will be no solution. Typically, Dirichlet boundary conditions lead to a unique solution and Neumann, to a solution which is unique up to an overall additive constant. A useful distinction is often made between the simpler, elliptic equations and parabolic and hyperbolic equations. These terms refer to the relative signs of the derivative terms, something which has implications for the existence and stability of solutions. This won't be discussed here.
## Laplace's Equation
A solution of Laplace's equation is called a harmonic function. Some simple (singular) examples:
- $3 \mathrm{~d} \phi=\frac{1}{r}$ is harmonic but singular at the origin.
- $2 \mathrm{~d} \phi=\log r, r=\sqrt{x^{2}+y^{2}}$ harmonic but singular at $r=0$.
- $1 \mathrm{~d} \phi=x$ is harmonic but singular at $\pm \infty$.
- 2d Any holomorphic function ( Complex Analysis ) is harmonic!
Suppose we wish to find a non-singular (e.g. $C^{\infty}$ ) harmonic function is some domain D subject to some boundary conditions on $S=\partial D$. There are important theorems relating to the uniqueness of the solutions and to the value of the solutions, these theorems are very useful and are also typical of similar theorems for other similar equations. There are also important existence theorems, but we do not deal with these here.
## Theorem: Uniqueness of the Laplace equation with Dirichlet and Neumann boundary conditions.
The solution of Laplace's equation under Dirichlet's boundary conditions, if it exists, is unique. The solution of the problem under Neumann boundary conditions, if it exists, is unique up to an additive constant. This will be proved after considering the Green's identities.
## Green's Identities
These should not be confused with Green's theorem in the plane. Let $\phi$ and $\psi$ be smooth functions, not necessarily harmonic, then
- Green's First Identity
$$
\int_{D}\left(\phi \nabla^{2} \psi+\nabla \phi \cdot \nabla \psi\right) d V=\int_{S=\partial D} \phi \nabla \psi \cdot \mathbf{d} \mathbf{A}
$$
## - Green's second Identity
$$
\int_{D}\left(\phi \nabla^{2} \psi-\psi \nabla^{2} \phi\right) d V=\int_{S=\partial D}(\phi \nabla \psi-\psi \nabla \phi) \cdot \mathbf{d} \mathbf{A}
$$
To prove these, well, for the first identity, pply Gauss' theorem to the vector field $\mathbf{F}=\phi \nabla \psi$
$$
\operatorname{div} \mathbf{F}=\phi \nabla \cdot(\nabla \psi)+\nabla \phi \cdot \nabla \psi=\phi \nabla^{2} \psi+\nabla \phi \cdot \nabla \psi
$$
So, for the second identity, interchange $\phi$ and $\psi$ in first identity the subtract from the first identity.
Now, to prove the uniquenss theorem, let $\phi_{1}$ and $\phi_{2}$ be harmonic in $\mathrm{D}$ and subject to the same boundary conditions on $S=\partial D$ (either DBCs or NBCs). Consider $\phi=\phi_{1}-\phi_{2}$. Apply Green's 1st identity taking $\psi=\phi$
$$
\int_{D}(\phi \triangle \phi+\nabla \phi \cdot \nabla \phi) d V=\int \phi \nabla \phi \cdot \mathbf{d A}
$$
Now, the first term on the left is zero since $\phi=\phi_{1}-\phi_{2}$ is harmonic. The right hand side is also zero because either $\phi$ is zero for Dirichlet boundary conditions or $\partial_{\mathbf{n}} \phi$ is zero for Neumann. Therefore
$$
\int_{D} \nabla \phi \cdot \nabla \phi d V=0
$$
and $\nabla \phi \cdot \nabla \phi$ is non-negative! This requires $\nabla \phi=0$ or $\phi=$ constant, that is, $\phi_{1}-\phi_{2}=c$ proving the theorem for Neumann conditions. For Dirichlet $c$ must be zero since $\phi_{1}$ and $\phi_{2}$ agree on $\mathrm{S}$ by assumption.
#### Gauss' Mean Value Theorem
The Gauss mean value theorem for harmonic functions: suppose $\phi$ is harmonic in $D \subset \mathbf{R}^{3}$. The average value of $\phi$ over the surface of a sphere of radius $R$ centred at the point $\mathbf{r}$ is $\phi(\mathbf{r})$. This is true for any point in the interior of $D$. The radius $R$ is any number such that the sphere, and every point inside it, is in D. To prove it, we assume without loss of generality consider a sphere centred at the origin. Idea is to show that the average
$$
\overline{\phi_{R}}=\frac{1}{4 \pi R^{2}} \int_{x^{2}+y^{2}+z^{2}=R^{2}} \phi d A
$$
is independent of the radius R. Apply Green's 2nd identity to $\phi$ and $\psi=\frac{1}{r}$ in the regions $R_{1}<r<R_{2}$ (Picture 231.IV.1.3).
$$
\int_{R_{1}<r<R_{2}}\left(\phi \nabla^{2} \frac{1}{r}-\frac{1}{r} \nabla^{2} \phi\right) d V
$$
and
$$
=\int_{r=R_{2}, \text { out }}\left(\phi \nabla \frac{1}{r}-\frac{1}{r} \nabla \phi\right) \cdot \mathbf{d} \mathbf{A}-\int_{r=R_{1}, \text { out }}\left(\phi \nabla \frac{1}{r}-\frac{1}{r} \nabla \phi\right) \cdot \mathbf{d} \mathbf{A}
$$
Now,
$$
\int_{r=R_{1}, \text { out }} \frac{1}{r} \nabla \phi \cdot \mathbf{d} \mathbf{A}=\frac{1}{R_{1}} \int_{r=R_{1}, \text { out }} \nabla \phi \cdot \mathbf{d} \mathbf{A}
$$
because $\frac{1}{r}$ is constant of sphere $r=R_{1}$, hence
$$
\int_{r=R_{1}, \text { out }} \frac{1}{r} \nabla \phi \cdot \mathbf{d} \mathbf{A}=\frac{1}{R_{1}} \int_{r<=R_{1}} \operatorname{div} \nabla \phi d V=\frac{1}{R_{1}} \int_{r<=R_{1}} \nabla^{2} \phi d V=0
$$
Similarily
$$
\int_{r=R_{2}, \text { out }} \frac{1}{r} \nabla \phi \cdot \mathbf{d} \mathbf{A}=0
$$
Thus,
$$
0=\int_{r=R_{2}, \text { out }} \phi \nabla \frac{1}{r} \cdot \mathbf{d} \mathbf{A}-\int_{r=R_{1}, \text { out }} \phi \nabla \frac{1}{r} \cdot \mathbf{d A}
$$
SO
$$
\nabla \frac{1}{r}=-\frac{\hat{\mathbf{r}}}{r^{2}} \text { in both integralsn }=\hat{\mathbf{r}}
$$
and
$$
0=-\frac{1}{R_{2}^{2}} \int_{r=R_{2}} \phi d A+\frac{1}{R_{1}^{2}} \int_{r=R_{1}} \phi d A
$$
or
$$
\overline{\phi_{R_{2}}}=\overline{\phi_{R_{2}}}
$$
Letting $R \rightarrow 0, \bar{\phi}_{R}=\phi(0), R$ being the radius of the outer sphere, completing the proof.
This leads to the Maximum (minimum) Principle for Harmonic Functions. Let $\phi$ be harmonic in a 3d (or $2 \mathrm{~d}$ ) domain D. Then $\phi$ never assumes its maximum (or minimum) value at an interior point of $\mathrm{D}$ unless $\phi$ is constant. To prove this, assume $\phi$ has a maximum at some point $P$ in the interior of $D$. For $\mathrm{R}$ sufficiently small the sphere of radius $\mathrm{R}$ centred at $\mathrm{P}$ is inside $\mathrm{D}$. For every point on the sphere $\phi<\phi(P)$ so $\bar{\phi}_{R}<\phi(P)$ contradicting the MVT. A similar argument holds if $\mathrm{P}$ is a minimum. If $\phi$ is harmonic in $D$ it assumes its maximum and minimum values at the boundary $S=\partial D$
This has a physical interpretation. Heat satisfies the heat equation $\nabla^{2}=D \frac{\partial \phi}{\partial t}$. If $\phi$ reaches a steady state $\frac{\partial \phi}{\partial t}=0$, then $\nabla^{2} \phi=0$, that is, the temperature is harmonic. Suppose we have a finite lump of matter and the boundary temperature (not necessarily constant) is fixed, for example, consider a square slab with three sides fixed to be at 0 degrees and the other at 100 degrees (Picture IV.1.4). The steady state temperature inside the slab is harmonic. Steady state temperature can never exceed 100 degrees, or fall below 0 degrees; heat would immediately flow out of, or enter, such a hot or cold spot.
A similar result for the whole of space is known as Liouville's Theorem: if $\phi$ is harmonic and bounded throughout $\mathbb{R}^{3}$ (or $\mathbb{R}^{2}$ ) then it is constant. The proof isn't given here. $\phi=\frac{1}{r}$ in three dimensions and $\phi=\log r$ in two dimensions are unbounded.
## Some solutions
Uniqueness theorem very powerful; any solution with DBCs, however simple is the only solution.
- Let $\phi$ be a harmonic function which is constant, say $\phi=a$, on the boundary of $D$. $\phi(\mathbf{r})=a$ is trivially a solution of Laplace's equation with the correct b.c.s.. It must be the unique solution to this boundary value problem.
- Let $\phi$ be harmonic in a $2 \mathrm{~d}$ annulus with $\phi=1$ on the other boundary $\left(r=R_{2}\right) \phi=b$ on the inner boundary $\left(r=R_{1}\right)$. Now $\phi=C \log r+D\left(r=\sqrt{x^{2}+y^{2}}\right)$ harmonic but singular at origin, with $C$ and $D$ constants.
$$
\begin{aligned}
a & =\phi\left(r=R_{2}\right)=C \log R_{2}+D \\
b & =\phi\left(r=R_{1}\right)=C \log R_{1}+D \\
a-b & =C \log \frac{R_{2}}{R_{1}} \\
D & =a-C \log R_{2}
\end{aligned}
$$
and
$$
\phi=\frac{a-b}{\log \frac{R_{2}}{R_{1}}} \log r R_{2}+a
$$
In these two examples we have guessed a solution (which we know to be unique). This is clearly insufficient for most problems. Note IV.1 ${ }^{12} 13$ May 2007
## Separation of variables
The seperation of variables is the main, systematic, way to solve PDEs. Idea is to reduce PDEs involving two or more variables to ODEs in each variable by assuming, for example, that $\phi(x, y)=X(x) Y(y)$ where $\mathrm{X}$ depends on $\mathrm{x}$ only and $\mathrm{Y}$ depends on $\mathrm{y}$ only. While this is a strong restiction, not many functions in $x$ and $y$ are in this form, we will find that, in practice, we will get a countably infinite number of solutions and these can be added, by linearity.
## Separation of variables and Laplace's equation
Consider the two-dimensional Laplace equation $\triangle \phi(x, y)=0$ on the square region $0 \leq$ $x \leq \pi$ and $0 \leq y \leq \pi$ with zero Dirichlet boundary conditions on three sides: $\phi(0, y)=$ $\phi(\pi, y)=\phi(x, \pi)$ and, on the third side, a constant Dirichlet condition $\phi(x, 0)=1$ (Picture 231.IV.2.1).
Now, under the assumption
$$
\phi(x, y)=X(x) Y(y)
$$
Laplace's equation becomes
$$
\nabla^{2} \phi=X^{\prime \prime}(x) Y(y)+X(x) Y^{\prime \prime}(y)=0
$$
where prime is used to denote differenciation with respect to the arguement, so $X^{\prime}=d X / d x$ and $Y^{\prime}=d Y / d y$. Now divide through by $X(x) Y(y)$
$$
\frac{X^{\prime \prime}(x)}{X(x)}=-\frac{Y^{\prime \prime}(y)}{Y(y)}
$$
with the left hand side independent of $\mathrm{y}$ and the right hand side independent of $\mathrm{x}$. Since $x$ and $y$ are independent variables, this equation can only hold if each side is equal to the same constant
$$
\frac{X^{\prime \prime}(x)}{X(x)}=-\frac{Y^{\prime \prime}(y)}{Y(y)}=\text { constant independent of } \mathrm{x} \text { and } \mathrm{y}=E
$$
Now, we have two seperate equations
$$
\begin{aligned}
X^{\prime \prime} & =E X \\
Y^{\prime \prime} & =-E Y
\end{aligned}
$$
There are three possibilities: $E$ can be zero, strictly positive or strictly negative.
${ }^{1}$ Conor Houghton, [email protected], see also http://www . maths.tcd.ie/ houghton/231
${ }^{2}$ Substantially based on lecture notes taken by John Kearney - Zero: $E=0$ so $X^{\prime \prime}=Y^{\prime \prime}=0$ and
$$
\begin{aligned}
& X(x)=A x+B \\
& Y(y)=C y+D
\end{aligned}
$$
- Positive: $E=k^{2}, k$ real and non-zero. Hence
$$
\begin{aligned}
X^{\prime \prime} & =k^{2} X \\
Y^{\prime \prime} & =-k^{2} Y
\end{aligned}
$$
with solutions
$$
\begin{aligned}
& X(x)=A e^{k x}+B e^{-k x} \\
& Y(y)=C \sin k y+D \cos k y
\end{aligned}
$$
- Negative: $E=-k^{2}, k$ real and non-zero. Hence
$$
\begin{aligned}
X^{\prime \prime} & =-k^{2} X \\
Y^{\prime \prime} & =k^{2} Y
\end{aligned}
$$
with solutions
$$
\begin{aligned}
& X(x)=A \sin k x+B \cos k x \\
& Y(y)=C e^{k y}+D e^{-k y}
\end{aligned}
$$
So, there are lots of solutions, lots of arbitrary constants and, in addition, this new sort of constant, $E$ or $k$ that arrived as part of the seperation of variables assumption. However, we haven't implimented the boundary conditions yet. In fact, none of these solutions satisfies all the boundary conditions on its own.
Lets start with the left and right conditions
$$
\phi(0, y)=\phi(\pi, y)=0
$$
Since $\phi=X Y$ this means that $X(0)=X(\pi)=0$. For the $E=0$ solution $X=A x+B$, $X(0)=0$ gives $B=0$ and, the $X(\pi)=0$ gives $A=0$, so these solutions are trivial. Now, for $E=k^{2}$
$$
X(x)=A e^{k x}+B e^{-k x}
$$
$X(0)=0$ if $A=-B$ but, then
$$
X(\pi)=A\left(e^{k \pi}-e^{-k \pi}\right)
$$
and this can never be zero for non-zero $k$. Finally, for $E=-k^{2}$
$$
X(x)=A \sin k x+B \cos k x
$$
and, so, $X(0)=0$ if $B=0$, then
$$
X(\pi)=A \sin \pi k
$$
and this is zero if $k=n \in \mathbf{N}$, a natural number. Thus, there is an infinite series of solutions satisfying the differential equation and the boundary conditions at $x=0$ and $\pi$.
This is a linear problem and so we can add solutions to give general solution
$$
\phi(x, y)=\sum_{n=1}^{\infty}\left(A_{n} e^{n y}+B_{n} e^{-n y}\right) \sin n x
$$
Now, we need only match the two remaining boundary conditions. First, the zero condition implies
$$
A_{n} e^{n \pi}+B_{n} e^{-n \pi}=0
$$
and so we are left with
$$
\phi(x, 0)=\sum_{n=1}^{\infty} C_{n} \sin n x=1
$$
where $C_{n}=A_{n}+B_{n}$. This is obviously in Fourier series form and so we might hope to choose $C_{n}$ s so that we have a Fourier series for $f(x)=1$, the boundary condition. This is more or less what we do do, the only problem is that $f(x)=1$ is even while the sines are odd. We need to make $f(x)$ into an odd periodic function, so that there is a sine series for it, such that $f(x)=1$ for $0<x<\pi$, since that is boundary condition. Recall
$$
f(x)=1 \quad 0<x<\pi-1 \quad-\pi<x<0
$$
has the Fourier Series expansion
$$
f(x)=\frac{4}{\pi} \sum_{n \text { odd }>0} \frac{1}{n} \sin n x
$$
Restricting to $0<x<\pi$ we have
$$
1=\frac{4}{\pi} \sum_{n \text { odd }>0} \frac{1}{n} \sin n x
$$
Now, solving for the $A_{n}$ and $B_{n}$ we get
$$
\phi(x, y)=\frac{4}{\pi} \sum_{n \text { odd }>0} \frac{\sin n x}{n} \frac{e^{n y}}{e^{2 \pi n}-1}
$$
solving the problem.
If the bottom boundary condition has been replaced by different function the method would not change: extend the function to an odd periodic function and then Fourier analysis it. For general Dirichlet boundary conditions, the easiest thing is to express $\phi$ as the sum of four $\phi$, each with one non-zero boundary condition, one for each side (Picture 231.IV.2.2). Solving the individual problems, each with one non-zero side is easy, they follow the same as for the one above, in fact, one quick way to solve them would be to do a change of variables rotating the above solution. Neumann conditions are equally straight-forward, more complicated boundaries are not; for this whole thing to work, the boundary conditions in individual parts of the boundary need to expressed as conditions on one of the seperated functions, $X$ and $Y$ here. To do this for more complicated boundaries usually requires a change of variables, to, for example, polar coördinates for annular or wedge shaped regions. This is not done here.
## Periodic Strip
Periodic in $x$ direction $\pi(x+2 \pi, y)=\phi(x, y)$ and $y$ in some finite range, say $0 \leq y \leq 1$ with Dirichlet boundary conditions at $y=0$ and $y=1$.
$$
\phi(x, 1)=g(x)
$$
where $g(x+2 \pi)=g(x)$ and
$$
\phi(x, y=0)=f(x)
$$
where $f(x+2 \pi)=f(x)$.
So, separation of variables:
$$
\phi(x, y)=X(x) Y(y)
$$
Solutions must be periodic in $x$,
$$
\begin{aligned}
X(x) & =A \cos n x+B \sin n x \\
Y(y) & =C e^{n y}+D e^{-n y}
\end{aligned}
$$
and $X(x)=A$ with $Y(y)=C+D y$. Now, redefining the constants
$$
\phi(x, y)=k_{1}+k_{2} y+\sum_{n=1}^{\infty}\left(a_{n} e^{n y}+\tilde{a_{n}} e^{-n y}\right) \cos n x+\sum_{n=1}^{\infty}\left(b_{n} e^{n y}+\tilde{b_{n}} e^{-n y}\right) \sin n x
$$
Obtain coefficients $k_{1}, k_{2}, a_{n}, \tilde{a_{n}}, b_{n}, \tilde{b_{n}}$ through boundary conditions at $y=0$, and $y=1$.
## The heat equation
For convenience we will take the 1+1-dimensional heat equation to be
$$
\triangle \phi(x, t)=\frac{\partial \phi(x, t)}{\partial t}
$$
with zero Dirichlet boundary conditions at $x=0$ and $\pi$ and an initial condition
$$
\phi(x, 0)=f(x)
$$
for some known $f(x)$.
We begin, as usual, by seperating
$$
\phi(x, t)=X(x) T(t)
$$
giving
$$
\frac{X^{\prime \prime}}{X}=\frac{T^{\prime}}{T}
$$
As before, this must mean that each side is equal to the same constant, which we will call $E$. There are, again, three cases
- Zero: $E=0$ so $X^{\prime \prime}=T^{\prime}=0$ and
$$
\begin{aligned}
X(x) & =A x+B \\
T(t) & =C
\end{aligned}
$$
- Positive: $E=k^{2}, k$ real and non-zero. Hence
$$
\begin{aligned}
X^{\prime \prime} & =k^{2} X \\
T^{\prime} & =k^{2} T \\
X(x)= & A e^{k x}+B e^{-k x} \\
T(t) & =C e^{k^{2} t}
\end{aligned}
$$
- Negative: $E=-k^{2}, k$ real and non-zero. Hence
$$
\begin{aligned}
X^{\prime \prime} & =-k^{2} X \\
T^{\prime} & =-k^{2} T
\end{aligned}
$$
with solutions
$$
\begin{aligned}
X(x) & =A \sin k x+B \cos k x \\
T(t) & =C e^{-k^{2} t}
\end{aligned}
$$
As before, most of these solutions will not satisfy the $x$ boundary conditons, applying the boundary conditions and adding all solutions gives
$$
\phi(x, t)=\sum_{n=1}^{\infty} A_{n} \sin n x e^{-n^{2} t}
$$
and, so for $t=0$
$$
\phi(x, 0)=\sum_{n=1}^{\infty} A_{n} \sin n x
$$
and so this is used to match the initial condition. If there are Neumann boundary conditions the sine will be replaced by a cosine and the constant solution would have to be included
$$
\phi(x, t)=A_{0}+\sum_{n=1}^{\infty} A_{n} \cos n x e^{-n^{2} t}
$$
Obviously in this case the even periodic extension of $f(x)$ is required.
| Textbooks |
OSA Publishing > Optical Materials Express > Volume 10 > Issue 1 > Page 149
Alexandra Boltasseva, Editor-in-Chief
Feature Issues
InAsSb mole fraction determination using Raman low energy modes
Kacper Grodecki, Krzysztof Murawski, Krystian Michalczewski, Bartłomiej Jankiewicz, and Piotr Martyniuk
Kacper Grodecki,1,* Krzysztof Murawski,1 Krystian Michalczewski,2 Bartłomiej Jankiewicz,3 and Piotr Martyniuk1
1Institute of Applied Physics, Military University of Technology, 2 Kaliskiego Str., 00–908 Warsaw, Poland
2Vigo System S.A., Poznańska 129/133, 05-850 Ozarow Mazowiecki, Poland
3Institute of Optoelectronics, Military University of Technology, 2 Kaliskiego Str., 00–908 Warsaw, Poland
*Corresponding author: [email protected]
Kacper Grodecki https://orcid.org/0000-0001-6433-7321
Bartłomiej Jankiewicz https://orcid.org/0000-0002-1172-8764
K Grodecki
K Murawski
K Michalczewski
B Jankiewicz
P Martyniuk
Issue 1,
•https://doi.org/10.1364/OME.10.000149
Kacper Grodecki, Krzysztof Murawski, Krystian Michalczewski, Bartłomiej Jankiewicz, and Piotr Martyniuk, "InAsSb mole fraction determination using Raman low energy modes," Opt. Mater. Express 10, 149-154 (2020)
Injection of ethanol into supercritical CO2: Determination of mole fraction and phase state using linear Raman scattering (OE)
Unusual optical properties of the Au/Ag alloy at the matching mole fraction (OME)
Mole fraction measurement through a transparent quarl burner using filtered Rayleigh scattering (AO)
Table of Contents Category
Indium arsenide
Physical vapor deposition
Raman microscopy
X ray diffraction
Original Manuscript: July 30, 2019
Revised Manuscript: October 24, 2019
Manuscript Accepted: November 7, 2019
References and links
Equations (3)
The InAs1−xSbx ternary alloy band gap nonlinearly depends on the composition, which provides the opportunity for use of this material in devices operating in a wide range of infrared radiation. We present experimental results for InAs1−xSbx samples for Sb composition from 0.1 to 0.8. The most common way to determine it is by using a high resolution X-ray diffractometer. In a previous article, we showed that energies of folded longitudinal acoustic and folded transverse acoustic Raman peaks are linearly correlated with mole fraction. In this work, we will illustrate how to determine mole fraction using peak energy and calculate the bowing parameter for InAs1−xSbx at 300 K.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Devices for the detection of infrared radiation (motion detectors, smoke and gas detectors) have more and more applications in today [1,2]. Systems based on HgCdTe are well known and widely used. The AIIIBV group technology is more often alternative to HgCdTe-based devices. Advantage of covalent bonding contribution comparing to ionic bonding in HgCdTe makes InAs1−xSbx more stable which the epitaxial layers gives greater mechanical and chemical stability [1,2].The InAs1−xSbx ternary alloy band gap depends on the composition, which gives the opportunity to use this material in devices operating in a wide range of infrared radiation [3].
High Resolution X-Ray Diffraction (HRXRD) is the most commonly used technique to determine mole fraction [4,5]. However this technique is not always optimal, what is discussed in this article in experimental part. Another way to determine mole fraction may be Raman spectroscopy. One can obtain mole fraction using Longitudinal Optical (LO) intensities [6], but only for mole fraction lower than 0.5.
The other idea is to use Raman low energy bands commonly called Disorder Activated Longitudinal Acoustic (DALA) and Disorder Activated Transverse Acoustic DATA [7–11]. In literature in low energies region except DALA there were also Folded Longitudinal Acoustic (FLA) peaks observed as a consequence of long distance ordering [12,13]. In previous article we shown DALA and DATA bands in InAs1−x Sbx may be interpreted as FLA and Folded Transverse Acoustic (FTA) peaks [8,10,11]. We shown the energy of both FLA and FTA peaks decreases with increasing Sb mole fraction [11]. Here we present how to use FLA and FTA energies to determine Sb mole fraction. After determination mole fraction using FLA we may calculate bowing parameter for InAs1−xSbx in 300 K [14].
In this work we present experimental results of InAs1−xSbx samples (Table 1) bulk material grown in Molecular Beam Epitaxy (MBE) VIGO/MUT laboratory on 2 inch (001) GaAs substrate. The sample was obtained using a RIBER Compact 21-DZ solid-source molecular beam epitaxy (MBE) system [15]. Crystallographic properties were measured by High-resolution HRXRD diffractometer of PANalytical X'Pert MRD3. For each sample (Sample no) Sb mole fraction obtained using HRXRD (XSb HRXRD) as well as using FLA (XSb FLA) and FTA (XSb FTA) are presented. We also added energies of FLA (FLA) for each sample, FTA (FTA) peaks for most of samples and bandgap energy obtained using PL (PL). Experiments were performed using Bruker Vertex 70v FT-IR spectrometer, MCT photodetector and lock-in. The whole system was working in Step Scan mode [16]. As a pump beam we used 637 nm line laser chopped mechanically with frequency of 1000 Hz. All samples presented in the article were measured in room temperature. Raman spectra were acquired using Renishaw InVia Raman Microscope equipped with Eclipse filter, x100 objective and 532 nm laser. We used about 50 μW laser power and 30 seconds measurement time to avoid sample heating.
Table 1. The InAs1−xSbx samples parameters.
3. Results and discussion
In Fig. 1 typical Raman spectra for InAs0.05Sb0.96 and InAs are presented. We assigned Longitudinal Acoustic (LO), as well as 2*Transverse Acoustic (2TA) peaks for InAs. For InAs0.05Sb0.96 we marked InSb LO, InAs LO and second order 2LO, 2TO and TO + LO peaks. There are also TO peaks in the spectrum but it is hard to assign them not losing readability of figure. Two low energy peak usually called DALA and DATA are assigned as FLA and FTA as we explained in previous article [11].
Fig. 1. Raman spectra for InAs and InAs0.05Sb0.95 samples.
Download Full Size | PPT Slide | PDF
We are not sure what the phonon branches generating FLA and FTA peaks are. FLA (Fig. 2 a) and FTA (Fig. 2.b) energies for all measures samples are marked as black squares. In literature for Cu-Pt zone folding was observed and L-point becomes Γ point [17]. We assigned theoretical value for LA and TA branches at L point for both InAs and InSb as blue squares [18]. Red line represents linear fit of experimental points while blue line has the same slope as red but is shifted in y-scale by 18.5 cm−1. Obtained blue line almost perfectly connects blue theoretical points. Similar operation we performed for FLA experimental points (Fig. 2.b). In this case blue line in shifted by 7 cm−1 with respect to red line and also connects theoretical blue points.
Fig. 2. a) FLA b) FTA peaks energy with Sb mole fraction (HRXRD).
Our experimental points for FLA are close to obtained in literature assign as DALA [19]. It is hard to interpret why theoretical blue line is shifted with respect to experimental red line, however in other article [13] average LA phonon energy in L point also does not fit to theoretical value.
We decided to use red lines as theoretical curves to calculate mole fraction using FLA and FTA peaks energy. Transforming equations obtained from linear fit (Fig. 2 a and Fig. 2 b) we may calculate mole fraction as:
(1)$${X_{Sb}}FLA = \frac{{FLA - 137}}{{20}}$$
and using FTA
(2)$${X_{Sb}}FTA = \frac{{FTA - 49}}{{10}}$$
Assuming last steps:
1. For each sample we present FLA and FTA energy vs mole fraction obtained using HRXRD (Fig. 2)
2. Linear dependency for both FTA and FTA peaks are plotted (Fig. 2)
3. We treat plot as theoretical curve to obtain mole fraction using FLA and FTA (Eqs. 1 and 2)
The question is why mole fraction obtained using Eqs. (1) and (2) may work better than obtained using HRXRD, which was used to obtain Eqs. (1) and (2). Choosing points with mole fraction below 0.2 (Fig. 2) we can observe that red line behaves like average for group of points. We may not obtain this average if we would have only mole fraction from HRXRD. Average from 12 numbers is one number. However if we have some additional information about points (FLA and FTA energies) we may use additional axis to perform average and that makes Eqs. (1) and (2) possibly better.
Having dependency between these values we can calculate mole fraction for all sample using FLA and FTA energies and plot band gap energy vs mole fraction using all three methods (Fig. 3). To verify if obtained mole fraction values are better than obtained using HRXRD we compare experimental results of bandgap value for all samples versus mole fraction obtained using HRXRD, FLA and FTA. For each points we plot theoretical curve (3) that is commonly accepted in literature [14].
(3)$${E_g}_{InAs1 - xSbx} = {E_g}_{InSb} \times x + {E_g}_{InAs} \times ({1 - x} )- Cx({1 - x} )$$
Where Eg stands for bandgap energy for InAsSb, InAs and InSb, x is Sb mole fraction and C bowing parameter. One can see that for all our figures curves are similar. On the other hand points obtained from Raman method are much better fit to theoretical curve 3 (R2 > 0.9) than using HRHXR (R2 < 0.9). The matching parameter C for HRXRD, FLA and FTA are 0.69, 0.69 and 0.68 respectively. This value is close to obtained by other groups [20–22].
Fig. 3. Band gap versus mole fraction for a) HRXRD, b) FLA and c) FTA.
Disagreement between experimental results and parabolic curve obtained for Fig. 3 may be also possibly explained by differences if ordering for each sample. Ordering has influence on the bandgap energy [14] and differences in ordering may cause errors in fitting in Fig. 3(a). One of the way to measure ordering is to compare bandgap value obtained using PL for different excitation power [23]. In the Fig. 4 we present PL spectra for sample where we observe highest energy shift for different excitation power measured in 300 K (Fig. 4). The difference between peaks' energies is 3 meV. This means any differences in ordering for presented samples does not have strong influence on band gap measurements.
Fig. 4. PL spectra for sample #10139 performed in 300 K with 40 mW (blue curve) and 200 mW (red curve) laser power.
Presented in this article idea to determine mole fraction using Raman low energy peaks (FLA and FTA) may be applied to other III-V alloys. However for each type of analyzing material one have to have enough high sample quality to be able to distinguish FLA or/and FTA peaks. For example in [8,9] quality of Raman spectra is too low to analyze FLA peaks, on the other hand in [12] authors could measure it.
In this article we showed an alternative way to calculate mole fraction value for InAs1−xSbx using FLA and FTA. We calculated and compared bowing parameter using mole fraction determination by HRXRD, FLA and FTA. In all cases obtained parameter is very similar, however fit to data points is much better for FLA and FTA than HRXRD calculated mole fraction. Obtained parameter 0.69 is close to previously reported.
Narodowe Centrum Badań i Rozwoju (TECHMATSTRATEG1/347751/5/NCBR/2017.).
This paper has been completed with the financial support of the - grant no. TECHMATSTRATEG1/347751/5/NCBR/2017.
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2. A. Rogalski, "Infrared detectors: Status and trends," Prog. Quantum Electron. 27(2-3), 59–210 (2003). [CrossRef]
3. J. Piotrowski and A. Rogalski, High-Operating-Temperature Infrared Photodetectors (SPIE, 2007).
4. M. Erkuş and U. Serincan, "Phonon frequency variations in high quality InAs1-xSbx epilayers grown on GaAs," in Applied Surface Science (North-Holland, 2014), 318, pp. 28–31.
5. M. A. Marciniak, R. L. Hengehold, Y. K. Yeo, and G. W. Turner, "Optical characterization of molecular beam epitaxially grown InAsSb nearly lattice matched to GaSb," J. Appl. Phys. 84(1), 480–488 (1998). [CrossRef]
6. K. Murawski, K. Grodecki, D. Benyahia, A. Wysmolek, B. Jankiewicz, and P. Martyniuk, "X-ray and Raman determination of InAsSb mole fraction for x <0.5," J. Cryst. Growth 498, 137–139 (2018). [CrossRef]
7. K. Radhakrishnan, T. H. K. Patrick, H. Q. Zheng, P. H. Zhang, and S. F. Yoon, "InP/InxGa1-xAs (0.53 ≤ x ≤ 0.81) high electron mobility transistor structures grown by solid source molecular beam epitaxy," J. Cryst. Growth 207(1-2), 8–14 (1999). [CrossRef]
8. Z. C. Feng, A. A. Allerman, P. A. Barnes, and S. Perkowitz, "Raman scattering of InGaAs/InP grown by uniform radial flow epitaxy," Appl. Phys. Lett. 60(15), 1848–1850 (1992). [CrossRef]
9. K. Radhakrishnan, T. H. K. Patrick, H. Q. Zheng, P. H. Zhang, and S. F. Yoon, "Study of doping concentration variation in InGaAs/InP high electron mobility transistor layer structures by Raman scattering," J. Vac. Sci. Technol., A 18(2), 713–716 (2000). [CrossRef]
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12. T. Mori, T. Hanada, T. Morimura, K. Shin, H. Makino, and T. Yao, "Surface structure of InGaAs/InP(0 0 1) ordered alloy during and after growth," in Applied Surface Science (North-Holland, 2004), 237(1–4), pp. 230–234.
13. H. M. Cheong, S. P. Ahrenkiel, M. C. Hanna, and A. Mascarenhas, "Phonon signatures of spontaneous CuPt ordering in Ga0.47In0.53As/InP," Appl. Phys. Lett. 73(18), 2648–2650 (1998). [CrossRef]
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15. L. O. Bubulac, A. M. Andrews, E. R. Gertner, and D. T. Cheung, "Backside-illuminated InAsSb/GaSb broadband detectors," Appl. Phys. Lett. 36(9), 734–736 (1980). [CrossRef]
16. M. Motyka, G. Sęk, J. Misiewicz, A. Bauer, M. Dallner, S. Höfling, A. Forchel, G. Sȩk, J. Misiewicz, A. Bauer, M. Dallner, S. Höfling, A. Forchel, G. Sęk, J. Misiewicz, A. Bauer, M. Dallner, S. Höfling, and A. Forchel, "Fourier transformed Photoreflectance and photoluminescence of mid infrared GaSb-based type II quantum wells," Appl. Phys. Express 2(12), 126505 (2009). [CrossRef]
17. Z. Jinghua, T. Xiaohong, and T. Jinghua, "Atomic ordering of AlInP grown by MOVPE at different temperatures in pure ambient N2," CrystEngComm 11(6), 1068–1072 (2009). [CrossRef]
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23. H. Dumont, L. Auvray, and Y. Monteil, "Optical signature of atomic ordering in In0.53Ga 0.47As/InP: Photoluminescence properties and IR response," in Optical Materials (North-Holland, 2003), 24(1–2), pp. 309–314.
Article Order
J. C. Woolley, J. Warner, J. C. Woolley, and J. Warner, "Preparation of InAs-InSb Alloys," J. Electrochem. Soc. 111(10), 1142–1145 (1964).
A. Rogalski, "Infrared detectors: Status and trends," Prog. Quantum Electron. 27(2-3), 59–210 (2003).
J. Piotrowski and A. Rogalski, High-Operating-Temperature Infrared Photodetectors (SPIE, 2007).
M. Erkuş and U. Serincan, "Phonon frequency variations in high quality InAs1-xSbx epilayers grown on GaAs," in Applied Surface Science (North-Holland, 2014), 318, pp. 28–31.
M. A. Marciniak, R. L. Hengehold, Y. K. Yeo, and G. W. Turner, "Optical characterization of molecular beam epitaxially grown InAsSb nearly lattice matched to GaSb," J. Appl. Phys. 84(1), 480–488 (1998).
K. Murawski, K. Grodecki, D. Benyahia, A. Wysmolek, B. Jankiewicz, and P. Martyniuk, "X-ray and Raman determination of InAsSb mole fraction for x <0.5," J. Cryst. Growth 498, 137–139 (2018).
K. Radhakrishnan, T. H. K. Patrick, H. Q. Zheng, P. H. Zhang, and S. F. Yoon, "InP/InxGa1-xAs (0.53 ≤ x ≤ 0.81) high electron mobility transistor structures grown by solid source molecular beam epitaxy," J. Cryst. Growth 207(1-2), 8–14 (1999).
Z. C. Feng, A. A. Allerman, P. A. Barnes, and S. Perkowitz, "Raman scattering of InGaAs/InP grown by uniform radial flow epitaxy," Appl. Phys. Lett. 60(15), 1848–1850 (1992).
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Ahrenkiel, S. P.
Allerman, A. A.
Andrews, A. M.
Auvray, L.
Barnes, P. A.
Bauer, A.
Belenky, G.
Benyahia, D.
Biefeld, R. M.
Broido, D. A.
Bubulac, L. O.
Budner, B.
Cellek, O. O.
Cheong, H. M.
Cherng, Y. T.
Cheung, D. T.
Dallner, M.
Donetsky, D.
Dumont, H.
Erkus, M.
Fastenau, J. M.
Feng, Z. C.
Forchel, A.
Gertner, E. R.
Grodecki, K.
Hanada, T.
Hanna, M. C.
Hengehold, R. L.
Hier, H.
Höfling, S.
Howard, A. J.
Jankiewicz, B.
Jinghua, T.
Jinghua, Z.
Kipshidze, G.
Kurtz, S. R.
Lin, Y.
Lindsay, L.
Liu, A. W. K.
Lubyshev, D.
Ma, K. Y.
Makino, H.
Marciniak, M. A.
Martyniuk, P.
Mascarenhas, A.
Meyer, J. R.
Michalczewski, K.
Misiewicz, J.
Monteil, Y.
Mori, T.
Morimura, T.
Motyka, M.
Murawski, K.
Patrick, T. H. K.
Perkowitz, S.
Piotrowski, J.
Qiu, Y.
Radhakrishnan, K.
Ram-Mohan, L. R.
Reinecke, T. L.
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Sarney, W. L.
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Serincan, U.
Shin, K.
Shterengas, L.
Steenbergen, E. H.
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Svensson, S. P.
Turner, G. W.
Vurgaftman, I.
Warner, J.
Woolley, J. C.
Wysmolek, A.
Xiaohong, T.
Yao, T.
Yeo, Y. K.
Yoon, S. F.
Zhang, P. H.
Zhang, Y.-H.
Zheng, H. Q.
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(1) X S b F L A = F L A − 137 20
(2) X S b F T A = F T A − 49 10
(3) E g I n A s 1 − x S b x = E g I n S b × x + E g I n A s × ( 1 − x ) − C x ( 1 − x )
The InAs1−xSbx samples parameters.
Sample no
Sbx HRXRD
XSb FLA
XSb FTA
FLA[cm−1]
FTA[cm−1]
PL[meV]
10350 0.014 0.18 0.15 134 49 218
10139 0.05 0.06 0.06 137 50 280
10397 0.05 0.14 0.1 135 – 250
10383 0.055 0.05 135.5 – 251
10245 0.059 0.1 0.05 136 49 255
10251 0.084 0.006 0.005 137 – 340
10131 0.27 0.3 0.3 131 45 175
10458 0.495 0.4788 0.5235 126.53 44.97 100
10589 0.495 0.402 0.4275 128.45 47 120
10566 0.5 0.58 0.65 124 43 80
10806 0.55 0.5 0.55 126 43 80
10896 0.62 0.66 0.75 122 42.6 91
10463 0.659 0.468 0.51 126.8 45 91
10515 0.8 0.8 0.88 119 39 110 | CommonCrawl |
Alternating conditional expectations
Alternating conditional expectations (ACE) is an algorithm to find the optimal transformations between the response variable and predictor variables in regression analysis.[1]
Introduction
In statistics, nonlinear transformation of variables is commonly used in practice in regression problems. Alternating conditional expectations (ACE) is one of the methods to find those transformations that produce the best fitting additive model. Knowledge of such transformations aids in the interpretation and understanding of the relationship between the response and predictors.
ACE transform the response variable $Y$ and its predictor variables, $X_{i}$ to minimize the fraction of variance not explained. The transformation is nonlinear and is obtained from data in an iterative way.
Mathematical description
Let $Y,X_{1},\dots ,X_{p}$ be random variables. We use $X_{1},\dots ,X_{p}$ to predict $Y$. Suppose $\theta (Y),\varphi _{1}(X_{1}),\dots ,\varphi _{p}(X_{p})$ are zero-mean functions and with these transformation functions, the fraction of variance of $\theta (Y)$ not explained is
$e^{2}(\theta ,\varphi _{1},\dots ,\varphi _{p})={\frac {\mathbb {E} \left[\theta (Y)-\sum _{i=1}^{p}\varphi _{i}(X_{i})\right]^{2}}{\mathbb {E} [\theta ^{2}(Y)]}}$
Generally, the optimal transformations that minimize the unexplained part are difficult to compute directly. As an alternative, ACE is an iterative method to calculate the optimal transformations. The procedure of ACE has the following steps:
1. Hold $\phi _{1}(X_{1}),\dots ,\phi _{p}(X_{p})$ fixed, minimizing $e^{2}$gives $\theta _{1}(Y)=\mathbb {E} \left[\sum _{i=1}^{p}\varphi _{i}(X_{i}){\Bigg |}Y\right]$
2. Normalize $\theta _{1}(Y)$ to unit variance.
3. For each $k$, fix other $\varphi _{i}(X_{i})$ and $\theta (Y)$, minimizing $e^{2}$ and the solution is:: ${\tilde {\varphi }}_{k}=\mathbb {E} \left[\theta (Y)-\sum _{i\neq k}\varphi _{i}(X_{i}){\Bigg |}X_{k}\right]$
4. Iterate the above three steps until $e^{2}$ is within error tolerance.
Bivariate case
The optimal transformation $\theta ^{*}(Y),\varphi ^{*}(X)$ for $p=1$ satisfies
$\rho ^{*}(X,Y)=\rho ^{*}(\theta ^{*},\varphi ^{*})=\max _{\theta ,\varphi }\rho (\theta (Y),\varphi (X))$
where $\rho $ is Pearson correlation coefficient. $\rho ^{*}(X,Y)$ is known as the maximal correlation between $X$ and $Y$. It can be used as a general measure of dependence.
In the bivariate case, ACE algorithm can also be regarded as a method for estimating the maximal correlation between two variables.
Software implementation
The ACE algorithm was developed in the context of known distributions. In practice, data distributions are seldom known and the conditional expectation should be estimated from data. R language has a package acepack which implements ACE algorithm. The following example shows its usage:
library(acepack)
TWOPI <- 8 * atan(1)
x <- runif(200, 0, TWOPI)
y <- exp(sin(x) + rnorm(200)/2)
a <- ace(x, y)
par(mfrow=c(3,1))
plot(a$y, a$ty) # view the response transformation
plot(a$x, a$tx) # view the carrier transformation
plot(a$tx, a$ty) # examine the linearity of the fitted model
Discussion
The ACE algorithm provides a fully automated method for estimating optimal transformations in multiple regression. It also provides a method for estimating maximal correlation between random variables. Since the process of iteration usually terminates in a limited number of runs, the time complexity of the algorithm is $O(np)$ where $n$ is the number of samples. The algorithm is reasonably computer efficient.
A strong advantage of the ACE procedure is the ability to incorporate variables of quite different type in terms of the set of values they can assume. The transformation functions $\theta (y),\varphi _{i}(x_{i})$ assume values on the real line. Their arguments can, however, assume values on any set. For example, ordered real and unordered categorical variables can be incorporated in the same regression equation. Variables of mixed type are admissible.
As a tool for data analysis, the ACE procedure provides graphical output to indicate a need for transformations as well as to guide in their choice. If a particular plot suggests a familiar functional form for a transformation, then the data can be pre-transformed using this functional form and the ACE algorithm can be rerun.
As with any regression procedure, a high degree of association between predictor variables can sometimes cause the individual transformation estimates to be highly variable, even though the complete model is reasonably stable. When this is suspected, running the algorithm on randomly selected subsets of the data, or on bootstrap samples can assist in assessing the variability.
References
1. Breiman, L. and Friedman, J. H. Estimating optimal transformations for multiple regression and correlation. J. Am. Stat. Assoc., 80(391):580–598, September 1985b. This article incorporates text from this source, which is in the public domain.
• This draft contains quotations from Estimating Optimal Transformations For Multiple Regression And Correlation By Leo Breiman And Jerome Freidman. Technical Report No. 9 July 1982, which is in the public domain.
| Wikipedia |
\begin{document}
\title{Multipartite nonlocality and random measurements} \author{Anna~de~Rosier} \affiliation{Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics and Informatics, University of Gda\'nsk, 80-308 Gda\'nsk, Poland} \author{Jacek~Gruca} \affiliation{Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics and Informatics, University of Gda\'nsk, 80-308 Gda\'nsk, Poland} \author{Fernando~Parisio} \affiliation{Departamento de F\'isica, Federal University of Pernambuco, Recife, PE 50670-901, Brazil} \author{Tam\'as~V\'ertesi} \affiliation{Institute for Nuclear Research, Hungarian Academy of Sciences, H-4001 Debrecen, P.O. Box 51, Hungary} \author{Wies{\l}aw~Laskowski} \affiliation{Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics and Informatics, University of Gda\'nsk, 80-308 Gda\'nsk, Poland}
\begin{abstract} We present an exhaustive numerical analysis of violations of local realism by families of multipartite quantum states. As an indicator of nonclassicality we employ the probability of violation for randomly sampled observables. Surprisingly, it rapidly increases with the number of parties or settings and even for relatively small values local realism is violated for almost all observables. We have observed this effect to be typical in the sense that it emerged for all investigated states including some with randomly drawn coefficients. We also present the probability of violation as a witness of genuine multipartite entanglement. \end{abstract}
\maketitle
\section{Introduction} Quantum multiparticle systems do not provide a mere amplification of the nontrivial effects displayed by two-party systems. Rather, they bring about completely new phenomena and applications. On the fundamental level, multipartite systems, e. g., have been employed to illustrate nonlocality without Bell inequalities \cite{GHZ} and, more recently, to show that finite-speed superluminal causal influences would allow for superluminal signalling between spatially separated parties \cite{gisinNP}. In what concerns applications, one-way quantum computing \cite{oneway} and multipartite secret sharing \cite{liang-gisin} are outstanding examples where complex quantum systems can be employed.
As is the case for multipartite entanglement, the characterization of nonclassical features of multiparticle systems is a hard problem with several open questions \cite{bellNL}. One interesting possibility to analyze the nonclassicality of complex states is to study their correlation properties under random measurements. With this motivation we will be concerned with the following quantity \begin{equation} \mathpzc{P}_V(\rho)=\int f (\Omega)d\Omega, \label{prob} \end{equation} where the integration variables correspond to all parameters that can be varied within a Bell scenario and, $f=1$ only for settings that lead to violations in local realism, and vanishes otherwise. Note that, when properly normalized, $\mathpzc{P}_V$ can be interpreted as a probability of violation of local realism.
The probability $\mathpzc{P}_V$ can be used at different context levels. One can select a particular Bell inequality $I$ and integrate $f_I$ over all possible settings of the corresponding Bell experiment. This was mainly the approach adopted in previous theoretical \cite{PhysRevLett.104.050401,PhysRevA.83.022110} and experimental \cite{Liang} works. This is also the case of ref. \cite{PhysRevA.92.030101}, where the quantity defined in (\ref{prob}) has been considered as a measure of nonlocality and applied in the context of the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality \cite{CGLMP,PhysRevA.65.052325}. This procedure, however, would face increasing difficulties as the number of parties grows. For a relatively modest number of qubits, e. g., the corresponding number of inequivalent Bell inequalities with a fixed (say 2) number of settings is already very large and, thus, addressing one inequality at a time would become prohibitive. On a deeper level we can dispense with the choice of a particular inequality and directly consider the space of behaviors (space of joint probabilities), which local polytopes inhabit. In this case, the integration refers to all possible measurements, the only context information required being the number of measurements per party. This is the approach that we will adopt here, so that we use the probability of violation to evaluate the degree of nonclassicality of several relevant states involving up to five qubits and also bipartite states of qutrits.
This work is presented in the following way. In the next section we provide a brief description of the numeric method to be employed (linear programming). In section III we present our results in the form of several tables and discuss their main consequences. In the last section we give our final remarks and some perspectives.
\section{Description of the method}
In our numerical analysis we consider the most general Bell experiment with $N$ spatially separated observers performing measurements on a given state of $N$ qu$d$its with $d=2$ (qubits) and $d=3$ (qutrits). Each observer can choose among $m_i$ arbitrary observables $\{O^i_{1}, O^i_{2}, ..., O^i_{m_i}\}$ ($i=1,2,...,N$) defined by orthogonal projections $O^i_{j} = \sum_{r_i=0}^{d-1} r^i \proj{v^i_j}$ linked by the general unitary transformations $\ket{v^i_j} = U^i_j |r^i \rangle$. The unitary transformations are parametrized by three angles for qubits: \begin{equation} U^i_{j} (\phi_1^{i,j},\psi_1^{i,j},\chi_1^{i,j})= \left(\begin{array}{c c} \cos\phi_1^{i,j}\text{ e}^{i\psi_1^{i,j}} & \sin\phi_1^{i,j} \text{ e}^{i\chi_1^{i,j}} \\ -\sin\phi_1^{i,j}\text{ e}^{-i\chi_1^{i,j}} & \cos\phi_1^{i,j} \text{ e}^{-i\psi_1^{i,j}} \end{array} \right), \end{equation} and eight angles for qutrits: \begin{eqnarray} U^i_{j}& (\phi_1^{i,j},\psi_1^{i,j},\chi_1^{i,j},\phi_2^{i,j},\psi_2^{i,j},\chi_2^{i,j},\phi_3^{i,j},\psi_3^{i,j})=\nonumber\\ &\left(\begin{array}{c c c}
\cos\phi_1^{i,j}\text{e}^{i\psi_1^{i,j}} & \sin\phi_1^{i,j} e^{i\chi_1^{i,j}} & 0 \\
-\sin\phi_1^{i,j} \text{e}^{-i\chi_1^{i,j}} & \cos\phi_1^{i,j} \text{e}^{-i\psi_1^{i,j}} & 0 \\
0 & 0 & 1 \end{array} \right)\nonumber\\ &\times \left(\begin{array}{ccc}
\cos\phi_2^{i,j}\text{e}^{i\psi_2^{i,j}} & 0 & \sin\phi_2^{i,j} \text{e}^{i\chi_2^{i,j}} \\
0 & 1 & 0\\
-\sin\phi_2^{i,j} \text{e}^{-i\chi_2^{i,j}} & 0 & \cos\phi_2^{i,j} \text{e}^{-i\psi_2^{i,j}} \\ \end{array} \right) \nonumber\\ &\times \left(\begin{array}{ccc}
1 & 0 & 0\\
0 & \cos\phi_3^{i,j}\text{e}^{i\psi_3^{i,j}} & \sin\phi_3^{i,j} \\
0 & -\sin\phi_3^{i,j} & \cos\phi_3^{i,j} \text{e}^{-i\psi_3^{i,j}} \\ \end{array} \right). \end{eqnarray}
A local realistic description of an experiment is equivalent to the existence of a joint probability distribution $p_{\rm lr}(r_1^1, ..., r_{m_1}^1, ..., r_1^N, ..., r_{m_N}^N)$, where $r_{j_i}^i = \{0,1,...,d-1\}$ denotes the result of the measurement of the $i$th observer's $O^i_{j}$ observable. If the model exists, quantum predictions for the probabilities are given by the marginal sums: \begin{eqnarray}
&P(r^1, ..., r^N \left| O^1_{k_1}, ..., O^N_{k_N} \right.) & \nonumber \\ &={\rm Tr}(\rho \proj{v^1_{k_1}} \otimes \cdots \otimes \proj{v^N_{k_N}})& \label{set-eq} \\ &=\sum\limits_{r^1_{j_1}, ..., r^N_{j_N}=0}^{d-1} p_{\rm lr}(r^1_1, ..., r_{m_1}^1, ..., r_1^N, ..., r_{m_N}^N)& \nonumber \end{eqnarray}
where $P(r^1, ..., r^N \left|O^1_{k_1}, ..., O^N_{k_N}\right.)$ denotes the probability of obtaining the result $r^i$ by the $i$th observer while measuring observables $O^i_{k_i}$ and $j_i \neq k_i$ ($i=1,..., N$). It can be shown that for some quantum entangled states the marginal sums cannot be satisfied, which is an expression of Bell's theorem.
Our task is to find, for a given state $\rho$ and a set of observables $O^i_{k_i}$ ($i=1,...,N$; $k_i = 1, ..., m_i)$, whether the local realistic model exists, i.e., all the equations (\ref{set-eq}) can be satisfied. This can be done by means of linear programming (see e.g. \cite{PhysRevLett.85.4418, PhysRevA.82.012118, j.cam.2013.12.003}). It is worth mentioning that the method allows us to reveal nonclassicality even without direct knowledge of Bell inequalities for the given experimental situation.
Finally, we check how many sets of settings (in percents) lead to violation of local realism. We introduce a frequency $\mathpzc{p}_V(\rho)$ which for a sufficiently large statistics converges to the probability of violation $\mathpzc{P}_V(\rho)$. We provided sufficient statistics to not observe changes in results on the third decimal place.
The measurement operators are sampled according to Haar measure \cite{Haar}. The angles $\psi_r$ and $\chi_r$ are taken from uniform distributions on the intervals: $ 0\leq\psi_r <2\pi$ and $0\leq\chi_r<2\pi.$ To generate $\phi_r$ in interval $0\leq\phi_r\leq\frac{\pi}{2}$ it is convenient to use an auxiliary random variable $\xi_r$ distributed uniformly on $0\leq\xi_r<1$ and $\phi_r=\arcsin(\xi_r^{1/2})$ for $r\in\left\{1,2\right\}$ and $\phi_3=\arcsin(\xi_r^{1/4})$. Of course, all variables are generated independently for each observer $i$ and measurement setting $j$.
\section{Results and analysis}
We applied the numerical method to prominent families of quantum states: \begin{itemize} \item[(1)] the generalized $N$ qubit GHZ state \cite{GHZ,GHZ.VS.NG} \begin{equation} \ket{\rm{GHZ}(\alpha)}_N=\sin \alpha \ket{0...0}_N+\cos \alpha \ket{1...1}_N, \nonumber \end{equation} and for $\alpha = \pi/4$, $\ket{\rm{GHZ}(\alpha)}_N \equiv \ket{\rm{GHZ}}_N$; \item[(2)] the four qubit singlet state \cite{singlet4,PhysRevA.68.012304} \begin{eqnarray} \ket{\psi_4^-}&=&\frac{1}{\sqrt{3}}\left(\ket{0011}+\ket{1100}\right)\nonumber \\ &-&\frac{1}{\sqrt{12}}\left(\ket{0101}+\ket{0110}+\ket{1001}+\ket{1010}\right); \nonumber \end{eqnarray} \item[(3)] the $N$ qubit Dicke state with $e$ excitations \cite{Dicke} \begin{equation}
\ket{D_N^e}=\binom{N}{e}^{-1/2}\sum_{\rm permutations}|0...0\underbrace{1...1}_{e}0...0\rangle_N, \nonumber \end{equation} where the special case $e=1$ is referred to as the $N$ qubit W state $\ket{W}_N\equiv\ket{\rm{D}_N^1}$ \cite{W}; \item[(4)] the four qubit cluster state \cite{cluster} \begin{equation} \ket{\rm{Cluster}_4}=\frac{1}{2}\left(\ket{0000}+\ket{0011}+\ket{1100}-\ket{1111} \right); \nonumber \end{equation} \item[(5)] the generalized $N$ qutrit GHZ state \begin{equation} \ket{\rm{GHZ}^{d=3}(\alpha)}_N=\sin\alpha\ket{0...0} +\frac{1}{\sqrt{2}}\cos\alpha(\ket{1...1}+\ket{2...2});\nonumber \end{equation} \item[(6)] the three qutrit singlet state (Aharonov state) \cite{Aharonov} \begin{eqnarray} \ket{A^-}_3&=&\frac{1}{\sqrt{6}}(\ket{012}+\ket{120}+\ket{201} \nonumber\\
&-&\ket{011}-\ket{101}-\ket{110});\nonumber \end{eqnarray} \item[(9)] the three qutrit Dicke states with the sum of excitations equals to $e$ \cite{DickeQt} \begin{eqnarray} \ket{\rm{Q}^1_3}&=&\frac{1}{\sqrt{3}}\sum_{\pi} \pi\left\{\ket{001}\right\}, \nonumber\\ \ket{\rm{Q}^2_3}&=&\frac{1}{\sqrt{15}}\left(2\sum_{\pi}\pi\left\{\ket{011}\right\}+\sum_{\pi}\pi\left\{\ket{002}\right\}\right),\nonumber\\ \ket{\rm{Q}^3_3}&=&\frac{1}{\sqrt{10}}\left(2\ket{111}+\sum_{\pi}\pi\left\{\ket{012}\right\}\right).\nonumber \end{eqnarray} \end{itemize}
We calculated the frequencies $\mathpzc{p}_V(\rho)$ for an increasing number of different settings per site. All results are presented in Tables \ref{tab-qubits}, \ref{tab-random} and \ref{tab-qutrits}. Some states which appear on the tables are not listed above. They will be defined in the appropriate paragraphs. Our results lead to the following observations.
\subsubsection{Comparison with known results}
The probability of violation was previously examined in several contexts. The only analytical result on tight inequalities was obtained in \cite{PhysRevLett.104.050401} for the simplest scenario of two settings and two outcomes, where the probability of violation of different versions of the CHSH inequality \cite{CHSH} has been obtained by the two qubit GHZ state (the Bell state). In this case our numerical method gives the same value (No.~\ref{GHZ2}) as the analytical expression $\mathpzc{p}_V(\rm{GHZ}_2)=\mathpzc{P}_V^{\rm CHSH}(\rm{GHZ}_2) =2(\pi-3) \sim 0.283183$ with accuracy to four decimal places.
For $N>2$, the GHZ state has been studied only numerically. In \cite{PhysRevLett.104.050401} the state was analyzed in the context of WWW{\.Z}B inequality for $N \leq 6$. In \cite{PhysRevA.83.022110} the analysis was extended to $N=15$ qubits (WWW{\.Z}B inequality) and $N=6$ (using a similar linear programming method). In all cases, the results agree with our numerical method.
\subsubsection{Genuine tripartite entanglement criterion}
We note that for any two-qubit state and two measurement settings per party, the probability of violation of local realism cannot be greater than $2(\pi-3)$, i.e., the two-qubit GHZ state gives the highest probability. The analytical proof is deferred to the Appendix.
Then, it is straightforward to prove that for any biproduct state $\ket{\psi_{12}}\otimes\ket{\psi_3}$ the two-qubit quantum probability $P\left(r_1,r_2\left|A_i,B_j\right.\right)$ is described by a local realistic theory if and only if $P\left(r_1,r_2,r_3\left|A_i,B_j,C_k\right.\right)$ does. Hence, in the examined cases of entangled states of $N_E$ particles, multiplied by the product state $\ket{0}^{\otimes N_0}$, the full $(N_E+N_0)$-particle state has, as expected, exactly the same probability of violation as its entangled component alone. The above property comes along with the fact that biseparable states (i.e. convex mixtures of biproduct states) can only lower the probability of violation compared to biproduct states. So we can argue that for any 3-qubit state (including mixed states) with two measurement settings per party, if $\mathpzc{P}_V(\rho)>2(\pi-3)$, this certifies that the three qubit state is genuinely tripartite entangled, that is, it can not be written in any of the forms $\ket{\psi_{12}}\otimes\ket{0}$, $\ket{\psi_{13}}\otimes\ket{0}$ and $\ket{0}\otimes\ket{\psi_{23}}$ and convex combinations of these states. Indeed, data in the table \ref{tab-qubits} indicates that both GHZ3 and W3 states are genuinely tripartite entangled as the respective probabilities: 74.688\% (No.~\ref{GHZ3}) and 54.893\% (No.~\ref{W3}) are much higher than 28.319\%.
One could construct a similar condition for higher number of parties ($N>3$) but in this case one may give only numerical bounds for the critical probability, because analytical results are not known in these cases.
We also considered the probability of violation for the state $\psi_3(\theta)= \cos\theta\ket{111}+\sin\theta\ket{\rm{W}_3}$ (Nos.~\ref{PSI15}-\ref{PSI90}). For all values of angle $\theta>25.975^{\circ}$ one can prove that the state is genuinely three-partite entangled \cite{PhysRevLett.88.170405}, whereas our numerical method reveals the threshold slightly below $30^\circ$. This discrepancy, though small, is due to the fact that our criterion is a necessary but not a sufficient one.
\subsubsection{Non-additivity and multiplicative features of $\mathpzc{P}_V(\rho)$}
The question of additivity seems to be better posed in terms of $\mathpzc{P}_V(\rho)$ than in terms of maximal violations of a Bell inequality. Consider the example of the state $\ket{\rm{GHZ}_2}\otimes\ket{\rm{GHZ}_2}$, for which probability of violation is non-additive, since $\mathpzc{p}_V(\ket{\rm{GHZ}_2}\otimes \ket{\rm{GHZ}_2})\approx 1.7 \mathpzc{p}_V(\ket{\rm{GHZ}_2})$, and is a bit less than half of $\mathpzc{p}_V(\ket{\rm{GHZ}_4})$.
Therefore instead of additivity, we should consider the multiplicative features of $\mathpzc{P}_V(\rho)$. Concerning $\mathpzc{p}_V(\ket{\rm{GHZ}_2})$ and $\mathpzc{p}_V(\ket{\rm{GHZ}_2}\otimes \ket{\rm{GHZ}_2})$, the probabilities that measurement results admit a local realistic description, $\mathpzc{P}_{LR}=1-\mathpzc{P}_V$, should be multiplied. In this particular case, \begin{eqnarray} &&\mathpzc{P}_{LR}(\ket{\rm{GHZ}_2}) = 1 - \mathpzc{P}_V(\ket{\rm{GHZ}_2}) = 1-2(\pi-3), \\ &&\mathpzc{P}_{LR}(\ket{\rm{GHZ}_2} \otimes \ket{\rm{GHZ}_2}) = \mathpzc{P}_{LR}(\ket{\rm{GHZ}_2})^2 = (7-2\pi)^2. \nonumber \end{eqnarray} Hence, $\mathpzc{p}_V(\rho_{\ket{\rm{GHZ}_2}}\otimes \rho_{\ket{\rm{GHZ}_2}}) = 1-(7-2\pi)^2 = 0.486176$ which fits our numerical results up to displayed digits (No.~\ref{GHZ2GHZ2}).
We also examined the product of the two qubit GHZ state with a state that does not violate any two setting Bell inequality, namely the Werner state: $\rho_{\rm{Werner}_2} = 1/\sqrt{2} |{\rm GHZ}\rangle_2 \langle {\rm GHZ}| + (1-1/\sqrt{2})\leavevmode\hbox{\small1 \normalsize \kern-.64em1}/4.$ In this case the probability of violation for the resulting state is the same as for $\ket{\rm{GHZ}_2}$, what can be explained by the above multiplicative feature, since $\mathpzc{P}_{LR}(\rho_{\rm{Werner}_2})=1$.
\subsubsection{Non-maximal probability of violation for GHZ states of more than 3 particles}
We observe a surprising feature, which emerges if the number of qubits is larger than three. It is well known that the $N$-qubit GHZ state maximizes many entanglement conditions and measures \cite{RevModPhys.81.865}. However, already for $N=4$ the probability of violation for the cluster state (No. \ref{cluster2222}) is greater than for the GHZ state (No. \ref{GHZ4}). The situation is even more dramatic for $N=5$, where the probability is greater for any out of 10 randomly sampled pure states (Nos. \ref{rand5a}-\ref{rand5j}).
There is a particular entanglement measure which is in pace with the above observations, namely the generalized Schmidt Rank (SR) \cite{cluster}, corresponding to the minimal number of product states required to represent a given state. The SR of a GHZ state is two for any number of qubits, and it has been shown in \cite{cluster} that the SR behaves as $2^{\left\lfloor N\slash 2\right\rfloor}$ for cluster states of $N$ qubits.
\subsubsection{All typical states of five or more qubits violate local realism for almost all settings}
Even with only two observables per party it becomes almost impossible not to detect non-classicality for states with 5 qubits or more. Any of the studied states (Nos.~\ref{GHZ5}-\ref{R5}) including random 5-qubit states (Nos. \ref{rand5a}-\ref{rand5j}) lead to nearly 100$\%$ probability of violation. In fact, the numbers are so close, that one can not distinguish the states by means of the violation probability. This amounts to an enhancement of the content of Gisin's theorem in the sense that not only all entangled states seem to be nonclassical, but they violate local realism for almost all experimental situations. That is, given an entangled state it is very likely that one can prove its non-classicality on a first try by choosing random observables (note also related recent results in Ref.~\cite{sampling}). This is to be contrasted with the original demonstration \cite{GISIN1991201,GISIN199215}, involving two qubits, where the settings have to be carefully selected. Of course, one can always find some states with a $\mathpzc{p}_V(\rho)$ which is much smaller than 100\% (e.g. Nos. \ref{32}, \ref{23}), but they are strictly less entangled.
\subsubsection{$\mathpzc{p}_V(\rho)$ rapidly increases with the number of settings}
The probability of violation increases significantly also with the number of settings per party. For the two qubit GHZ state, and five measurement settings per site, the corresponding violation probability is almost equal to 1. This means that almost all randomly sampled settings lead to a conflict with local realistic models and to the violation of some Bell inequality.
This rapid growth is more pronounced than it is for robustness against white noise admixture. An increase is also observed in the resistance to noise, but it is usually a much less evident effect and visible particularly in multipartite cases \cite{PhysRevA.82.012118}. For example, due to the recent work~\cite{brierley2016convex}, an increase of $0.58\%$ in the noise resistance of the two-qubit maximally entangled state required 30 settings (see also a previous work~\cite{VERTESI}). It is also conjectured that the above improvement in the noise resistance could not be attained with fewer settings. Note also that one cannot go beyond the increase of $3.682\%$ in noise resistance using an infinite number of projective measurements~\cite{hirsch2016better}.
The dependence of $\mathpzc{p}_V(\rho)$ as a function of the number of settings can be approximated by $1 - a e^{-bx}$, where $a,b$ are constant parameters and $x$ can be either the number of settings referring to one party (with the other number of settings fixed) (Fig. \ref{fig-sett}a) or a product of the number of possible measurement settings (Fig. \ref{fig-sett}b). Of course there are other possible combinations involving the number of settings.
\begin{figure}
\caption{The probability of violation for the two qubit GHZ states vs. (a) the number of measurement settings for the first observer; (b)a product of the number of settings for both observers.}
\label{fig-sett}
\end{figure}
\subsubsection{Nonclassicality of bound entangled states}
A bound entangled state (BES) is entangled but undistillable \cite{HOR}. However, in \cite{PhysRevA.74.010305} it was shown that the 4-qubit bound entangled Smolin state \cite{PhysRevA.63.032306} can maximally violate a 2-setting Bell inequality similar to the standard CHSH inequality. In accordance with this finding, when numerically investigating the possibility of a local realistic description for the Smolin state, even with 2 settings per party, we get small, but nonzero probability of violation, $\mathpzc{p}_V(\rho_{\rm{Smolin}_4})$=0.023\% (No.~\ref{Smolin2222}). Although this value is three orders of magnitude smaller than that for other examined entangled states, it grows very fast (faster than for other entangled states) with the number of settings (Nos.~\ref{Smolin3222}-\ref{Smolin3333}) and the growth seems to be exactly exponential.
In general, if we investigate the $\mathpzc{P}_V$ of a PPT state \cite{peres1996,horodecki1996} and find it to be non-vanishing, then the state must be entangled. Note that this conclusion can be reached even without the knowledge of which Bell inequality is to be violated. This may be particularly useful when the state involves many subsystems. In general, if we investigate the $\mathpzc{P}_V$ of a PPT state \cite{peres1996,horodecki1996} and find it to be non-vanishing, then the state must be entangled. Note that this conclusion can be reached even without the knowledge of which Bell inequality is to be violated. This may be particularly useful when the state involves many subsystems.
The three qubit BES $\rho_{\rm BES}^{2\times 2\times 2}$ introduced in \cite{VB12} also violates some Bell inequality, but this seems to be statistically very rare since we had not observed any violation of local realism for two settings per party. Nevertheless, when the same measurement is applied to every particle, we observed a nonzero probability of violation, $\mathpzc{p}_V(\rho_{\rm BES}^{2\times 2\times 2})$=0.008\%.
The last considered example of bound entangled states is the two qutrit state $\rho_{\rm BES}^{3\times 3}$ that was used to disprove the famous Peres conjecture \cite{peres1999all,ncomms6297}. Despite the fact that this state does not admit a local realistic model, the violation is proved only for judiciously specified observables and inequality, which occur seldom enough, so that we did not find violations in any of the $10^{10}$ randomly chosen settings.
\subsubsection{Two qutrits: coincidence of maximal entanglement and maximal nonclassicality}
Entanglement and nonclassicality are distinct resources. The former corresponds to the purely mathematical concept of state nonseparability while the later amounts to its manifestation in experiments. It is acknowledged that a clear illustration of this point is the unexpected difference between maximally entangled states and states that maximally violate a Bell inequality. In \cite{PhysRevA.92.030101} it is suggested that this anomaly may be an artifact of almost all measures that have been used to quantify nonclassicality. Our numerical results show that, according to the probability of violation, there is no anomaly in the nonclassicality of two qutrit generalized GHZ state. The maximal probability of violation $\mathpzc{p}_V$=24.011\% (No.~\ref{symmGHZqt2}) is attained for the symmetric state $\rm{GHZ}^{d=3}_2(35.26^\circ)$ instead of the asymmetric one: $\rm{GHZ}^{d=3}_2(29.24^\circ)$ [$\mathpzc{p}_V=22.317\%$ (No.~\ref{asymmGHZqt2})], which maximally violates the CGLMP inequality \cite{PhysRevA.65.052325}. A little surprising is the behavior of the probability of violation around $\alpha =0$, where we observe a small local minimum for $\alpha = 6^\circ$ (see Fig. \ref{fig-qutrit}). The minimum remains even if the number of settings per party is increased to three. A possible explanation of this feature could be the fact that there are two relevant Bell inequalities for the considered case -- CHSH and CGLMP inequalities with different functions representing the violation probability. The total probability of violation is a combination of the probabilities for those particular inequalities, what may result in several extremes.
\begin{figure}
\caption{ Probability of violation for 2 qutrit generalized GHZ($\alpha$) state vs. $\alpha$}
\label{fig-qutrit}
\end{figure}
\begin{footnotesize} \begin{longtable}{>{\the\numexpr\value{rowNo}}>{\refstepcounter{rowNo}}cccccc} \caption{\label{tab-qubits} Frequencies of violation of local realism $\mathpzc{p}_V$ observed statistically with random measurements on qubit states} \endfirsthead \endhead \hline \hline \multicolumn{1}{c}{No.} & $N$ & State & Settings & Stat. & $\mathpzc{p}_V$ \\ \hline\hline
\label{GHZ2} & 2 & $\ket{\rm{GHZ}_2}$ & $2\times2$ & $10^{10}$ & 28.318 \\ & 2 & $\ket{\rm{GHZ}_2}$ & $3\times2$ & $10^9$ & 52.401 \\ & 2 & $\ket{\rm{GHZ}_2}$ & $4\times2$ & $10^9$ & 68.654 \\ & 2 & $\ket{\rm{GHZ}_2}$ & $5\times2$ & $10^9$ & 78.947 \\ & 2 & $\ket{\rm{GHZ}_2}$ & $6\times2$ & $10^9$ & 85.391 \\ & 2 & $\ket{\rm{GHZ}_2}$ & $7\times2$ & $10^9$ & 89.482 \\ & 2 & $\ket{\rm{GHZ}_2}$ & $8\times2$ & $10^9$ & 92.150 \\ & 2 & $\ket{\rm{GHZ}_2}$ & $9\times2$ & $10^8$ & 93.945 \\ & 2 & $\ket{\rm{GHZ}_2}$ & $10\times2$& $10^8$ & 95.198 \\ \cline{4-6} & 2 & $\ket{\rm{GHZ}_2}$ & $3\times3$ & $10^9$ & 78.219 \\ & 2 & $\ket{\rm{GHZ}_2}$ & $4\times3$ & $10^9$ & 89.545 \\ & 2 & $\ket{\rm{GHZ}_2}$ & $5\times3$ & $10^9$ & 94.658 \\ & 2 & $\ket{\rm{GHZ}_2}$ & $6\times3$ & $10^9$ & 97.085 \\ & 2 & $\ket{\rm{GHZ}_2}$ & $7\times3$ & $10^9$ & 98.303 \\ & 2 & $\ket{\rm{GHZ}_2}$ & $8\times3$ & $10^8$ & 98.953 \\ \cline{4-6} & 2 & $\ket{\rm{GHZ}_2}$ & $4\times4$ & $10^9$ & 96.169 \\ & 2 & $\ket{\rm{GHZ}_2}$ & $5\times4$ & $10^8$ & 98.460 \\ & 2 & $\ket{\rm{GHZ}_2}$ & $6\times4$ & $10^8$ & 99.321 \\ & 2 & $\ket{\rm{GHZ}_2}$ & $7\times4$ & $10^8$ & 99.672 \\ \cline{4-6} & 2 & $\ket{\rm{GHZ}_2}$ & $5\times5$ & $10^8$ & 99.504 \\ \cline{3-6} & 2 & $\ket{\rm{GHZ}_2(1^\circ)}$ & $2\times2$ & $10^{10}$ & 0.00000025\\ & 2 & $\ket{\rm{GHZ}_2(10^\circ)}$ & $2\times2$ & $10^9$ & 0.093 \\ & 2 & $\ket{\rm{GHZ}_2(20^\circ)}$ & $2\times2$ & $10^9$ & 2.826 \\ & 2 & $\ket{\rm{GHZ}_2(30^\circ)}$ & $2\times2$ & $10^9$ & 14.796 \\ & 2 & $\ket{\rm{GHZ}_2(40^\circ)}$ & $2\times2$ & $10^9$ & 26.599 \\ \cline{2-6} & 3 & $\ket{\rm{GHZ}_2}\otimes \ket{0}$ & $2\times2\times2$ & $10^9$ & 28.317 \\ & 3 & $\ket{\rm{GHZ}_2}\otimes \ket{0}$ & $3\times2\times2$ & $10^9$ & 52.399 \\ \cline{3-6} \label{GHZ3} & 3 & $\ket{\rm{GHZ}_3}$ & $2\times2\times2$ & $10^9$ & 74.688 \\
& 3 & $\ket{\rm{GHZ}_3}$ & $3\times2\times2$ & $10^9$ & 90.132 \\
& 3 & $\ket{\rm{GHZ}_3}$ & $4\times2\times2$ & $10^9$ & 95.357 \\
& 3 & $\ket{\rm{GHZ}_3}$ & $3\times3\times2$ & $10^9$ & 97.245 \\
& 3 & $\ket{\rm{GHZ}_3}$ & $4\times3\times2$ & $10^8$ & 98.926 \\
& 3 & $\ket{\rm{GHZ}_3}$ & $4\times4\times2$ & $10^8$ & 99.590 \\
& 3 & $\ket{\rm{GHZ}_3}$ & $3\times3\times3$ & $10^9$ & 99.542 \\ \cline{3-6}
& 3 & $\ket{\rm{W}_3}$ & $1\times2\times2$ & $10^9$ & 15.244 \\ \label{W3} & 3 & $\ket{\rm{W}_3}$ & $2\times2\times2$ & $10^9$ & 54.893 \\
& 3 & $\ket{\rm{W}_3}$ & $3\times2\times2$ & $10^9$ & 76.788 \\
& 3 & $\ket{\rm{W}_3}$ & $4\times2\times2$ & $10^9$ & 87.287 \\
& 3 & $\ket{\rm{W}_3}$ & $5\times2\times2$ & $10^9$ & 92.465 \\
& 3 & $\ket{\rm{W}_3}$ & $3\times3\times2$ & $10^9$ & 91.366 \\
& 3 & $\ket{\rm{W}_3}$ & $3\times3\times3$ & $10^9$ & 97.797 \\ \cline{3-6} \label{PSI15} & 3 & $\ket{\rm{\psi}_3(15^\circ)}$ & $2\times2\times2$ & $10^9$ & 4.941 \\
& 3 & $\ket{\rm{\psi}_3(20^\circ)}$ & $2\times2\times2$ & $10^9$ & 10.327 \\
& 3 & $\ket{\rm{\psi}_3(25^\circ)}$ & $2\times2\times2$ & $10^8$ & 18.762 \\
& 3 & $\ket{\rm{\psi}_3(25.975^\circ)}$ & $2\times2\times2$ & $10^9$ & 20.786 \\
& 3 & $\ket{\rm{\psi}_3(30^\circ)}$ & $2\times2\times2$ & $10^9$ & 30.323 \\
& 3 & $\ket{\rm{\psi}_3(45^\circ)}$ & $2\times2\times2$ & $10^9$ & 64.382 \\
& 3 & $\ket{\rm{\psi}_3(60^\circ)}$ & $2\times2\times2$ & $10^9$ & 74.689 \\
& 3 & $\ket{\rm{\psi}_3(75^\circ)}$ & $2\times2\times2$ & $10^9$ & 65.377 \\ \label{PSI90} & 3 & $\ket{\rm{\psi}_3(90^\circ)}$ & $2\times2\times2$ & $10^9$ & 54.893 \\ \cline{2-6} & 4 & $\ket{\rm{GHZ}_2}\otimes \ket{00}$ & $2\times2\times2\times2$ & $10^8$ & 28.318 \\ & 4 & $\ket{\rm{GHZ}_2}\otimes \ket{00}$ & $3\times2\times2\times2$ & $10^7$ & 52.407 \\ \cline{3-6} \label{GHZ2GHZ2} & 4 & $\ket{\rm{GHZ}_2}\otimes \ket{\rm{GHZ}_2}$ & $2\times2\times2\times2$ & $10^8$ & 48.617 \\
& 4 & $\ket{\rm{GHZ}_2}\otimes \ket{\rm{GHZ}_2}$ & $3\times2\times2\times2$ & $10^7$ & 65.887 \\ \cline{3-6} & 4 & $\ket{\rm{GHZ}_3}\otimes \ket{0}$ & $2\times2\times2\times2$ & $10^8$ & 74.683 \\ & 4 & $\ket{\rm{GHZ}_3}\otimes \ket{0}$ & $3\times2\times2\times2$ & $10^7$ & 90.134 \\ \cline{3-6} \label{GHZ4} & 4 & $\ket{\rm{GHZ}_4}$ & $2\times2\times2\times2$ & $10^8$ & 94.240 \\
& 4 & $\ket{\rm{GHZ}_4}$ & $3\times2\times2\times2$ & $10^8$ & 98.352 \\
& 4 & $\ket{\rm{GHZ}_4}$ & $4\times2\times2\times2$ & $10^7$ & 99.339 \\
& 4 & $\ket{\rm{GHZ}_4}$ & $3\times3\times2\times2$ & $10^7$ & 99.624 \\
& 4 & $\ket{\rm{GHZ}_4}$ & $4\times3\times2\times2$ & $10^6$ & 99.867 \\
& 4 & $\ket{\rm{GHZ}_4}$ & $4\times4\times2\times2$ & $10^5$ & 99.937 \\
& 4 & $\ket{\rm{GHZ}_4}$ & $3\times3\times3\times2$ & $10^7$ & 99.934 \\
& 4 & $\ket{\rm{GHZ}_4}$ & $4\times3\times3\times2$ & $10^5$ & 99.981 \\
& 4 & $\ket{\rm{GHZ}_4}$ & $4\times4\times3\times2$ & $10^5$ & 99.989 \\
& 4 & $\ket{\rm{GHZ}_4}$ & $4\times4\times4\times2$ & $10^5$ & 99.993 \\
& 4 & $\ket{\rm{GHZ}_4}$ & $3\times3\times3\times3$ & $10^6$ & 99.995 \\
& 4 & $\ket{\rm{GHZ}_4}$ & $4\times3\times3\times3$ & $10^5$ & 99.999 \\
& 4 & $\ket{\rm{GHZ}_4}$ & $4\times4\times3\times3$ & $10^5$ & 99.999 \\
& 4 & $\ket{\rm{GHZ}_4}$ & $4\times4\times4\times3$ & $10^5$ & 100.00 \\
& 4 & $\ket{\rm{GHZ}_4}$ & $4\times4\times4\times4$ & $10^4$ & 100.00 \\ \cline{3-6} \label{W4} & 4 & $\ket{\rm{W}_4}$ & $2\times2\times2\times2$ & $10^8$ & 85.920 \\
& 4 & $\ket{\rm{W}_4}$ & $3\times2\times2\times2$ & $10^7$ & 95.129 \\
& 4 & $\ket{\rm{W}_4}$ & $4\times2\times2\times2$ & $10^7$ & 97.969 \\
& 4 & $\ket{\rm{W}_4}$ & $5\times2\times2\times2$ & $10^6$ & 99.013 \\
& 4 & $\ket{\rm{W}_4}$ & $3\times3\times2\times2$ & $10^7$ & 98.757 \\
& 4 & $\ket{\rm{W}_4}$ & $3\times3\times3\times2$ & $10^7$ & 99.767 \\
& 4 & $\ket{\rm{W}_4}$ & $3\times3\times3\times3$ & $10^6$ & 99.966 \\
& 4 & $\ket{\rm{W}_4}$ & $4\times4\times4\times2$ & $10^5$ & 99.999 \\ \cline{3-6} & 4 & $\ket{\rm{D}_4^2}$ & $2\times2\times2\times2$ & $10^8$ & 83.577 \\ & 4 & $\ket{\rm{D}_4^2}$ & $3\times2\times2\times2$ & $10^7$ & 94.065 \\ & 4 & $\ket{\rm{D}_4^2}$ & $4\times2\times2\times2$ & $10^7$ & 97.315 \\ & 4 & $\ket{\rm{D}_4^2}$ & $3\times3\times2\times2$ & $10^7$ & 98.428 \\ & 4 & $\ket{\rm{D}_4^2}$ & $3\times3\times3\times2$ & $10^7$ & 99.716 \\ & 4 & $\ket{\rm{D}_4^2}$ & $3\times3\times3\times3$ & $10^6$ & 99.964 \\ & 4 & $\ket{\rm{D}_4^2}$ & $4\times4\times4\times2$ & $10^5$ & 99.996 \\ \cline{3-6} & 4 & $\ket{\psi_4^-}$ & $2\times2\times2\times2$ & $10^8$ & 74.943 \\ & 4 & $\ket{\psi_4^-}$ & $3\times2\times2\times2$ & $10^7$ & 89.604 \\ & 4 & $\ket{\psi_4^-}$ & $4\times2\times2\times2$ & $10^7$ & 94.918 \\ & 4 & $\ket{\psi_4^-}$ & $3\times3\times2\times2$ & $10^7$ & 96.621 \\ & 4 & $\ket{\psi_4^-}$ & $3\times3\times3\times2$ & $10^7$ & 99.344 \\ & 4 & $\ket{\psi_4^-}$ & $3\times3\times3\times3$ & $10^6$ & 99.908 \\ & 4 & $\ket{\psi_4^-}$ & $4\times4\times4\times2$ & $10^5$ & 99.991 \\ \cline{3-6} \label{cluster2222} & 4 & $\ket{\rm{Cluster}_4}$ & $2\times2\times2\times2$ & $10^8$ & 97.283 \\
& 4 & $\ket{\rm{Cluster}_4}$ & $3\times2\times2\times2$ & $10^8$ & 99.275 \\
& 4 & $\ket{\rm{Cluster}_4}$ & $4\times2\times2\times2$ & $10^7$ & 99.705 \\
& 4 & $\ket{\rm{Cluster}_4}$ & $3\times3\times2\times2$ & $10^7$ & 99.884 \\
& 4 & $\ket{\rm{Cluster}_4}$ & $3\times3\times3\times2$ & $10^7$ & 99.976 \\
& 4 & $\ket{\rm{Cluster}_4}$ & $3\times3\times3\times3$ & $10^6$ & 99.997 \\
& 4 & $\ket{\rm{Cluster}_4}$ & $4\times4\times4\times2$ & $10^5$ & 99.999 \\ \cline{3-6} \label{Smolin2222} & 4 & $\rho_{\rm{Smolin}_4}$ & $2\times2\times2\times2$ & $10^8$ & 0.023 \\ \label{Smolin3222} & 4 & $\rho_{\rm{Smolin}_4}$ & $3\times2\times2\times2$ & $10^8$ & 0.068 \\
& 4 & $\rho_{\rm{Smolin}_4}$ & $4\times2\times2\times2$ & $10^7$ & 0.127 \\
& 4 & $\rho_{\rm{Smolin}_4}$ & $5\times2\times2\times2$ & $10^7$ & 0.195 \\
& 4 & $\rho_{\rm{Smolin}_4}$ & $3\times3\times2\times2$ & $10^7$ & 0.197 \\
& 4 & $\rho_{\rm{Smolin}_4}$ & $3\times3\times3\times2$ & $10^7$ & 0.601 \\ \label{Smolin3333} & 4 & $\rho_{\rm{Smolin}_4}$ & $3\times3\times3\times3$ & $10^7$ & 2.009 \\ \cline{2-6} \label{GHZ5} & 5 & $\ket{\rm{GHZ}_5}$ & $2\times2\times2\times2\times2$ & $10^7$ & 99.601 \\ & 5 & $\ket{\rm{GHZ}_5}$ & $3\times2\times2\times2\times2$ & $10^6$ & 99.900 \\ \cline{3-6} & 5 & $\ket{\rm{W}_5}$ & $2\times2\times2\times2\times2$ & $10^7$ & 98.311 \\ \cline{3-6} & 5 & $\ket{\rm{D}_5^2}$ & $2\times2\times2\times2\times2$ & $10^7$ & 99.254 \\ \cline{3-6} & 5 & $\frac{1}{2}(\rho_{\rm{D}_5^2}+\rho_{\rm{D}_5^3})$ & $2\times2\times2\times2\times2$ & $10^7$ & 0.047 \\ \cline{3-6} & 5 & $\ket{\rm GHZ}_4 \ket{0} $ & $2\times2\times2\times2\times2$ & $10^7$ & 94.240 \\ \cline{3-6} \label{32}& 5 & $\ket{\rm GHZ}_3 \ket{00} $ & $2\times2\times2\times2\times2$ & $10^7$ & 74.688 \\ \cline{3-6} \label{23}& 5 & $\ket{\rm GHZ}_2 \ket{000} $ & $2\times2\times2\times2\times2$ & $10^7$ & 28.318 \\ \cline{3-6} & 5 & $\ket{\rm{L}_5}$ \cite{PhysRevA.73.022303} & $2\times2\times2\times2\times2$ & $10^6$ & 99.782 \\ \cline{3-6} \label{R5} & 5 & $\ket{\rm{R}_5}$ \cite{PhysRevA.73.022303}& $2\times2\times2\times2\times2$ & $10^6$ & 99.957 \\ \hline \hline \end{longtable} \end{footnotesize}
\begin{footnotesize} \begin{table} \caption{\label{tab-random} Frequencies of violation of local realism $\mathpzc{p}_V$ observed statistically with random measurements on random qubit states} \begin{tabular}{>{\the\numexpr\value{rowNo}}>{\refstepcounter{rowNo}}ccccc} \hline \hline \multicolumn{1}{c}{No.} & $N$ & Settings & Stat. & $\mathpzc{p}_V$ \\ \hline\hline & 3 & $2\times2\times2$ & $10^8$ & 12.396 \\ & 3 & $2\times2\times2$ & $10^8$ & 33.893 \\ & 3 & $2\times2\times2$ & $10^8$ & 38.959 \\ & 3 & $2\times2\times2$ & $10^8$ & 45.186 \\ & 3 & $2\times2\times2$ & $10^8$ & 43.505 \\ & 3 & $2\times2\times2$ & $10^8$ & 4.812 \\ & 3 & $2\times2\times2$ & $10^8$ & 59.824 \\ & 3 & $2\times2\times2$ & $10^8$ & 35.197 \\ & 3 & $2\times2\times2$ & $10^9$ & 43.602 \\ & 3 & $2\times2\times2$ & $10^8$ & 43.747 \\ \cline{2-5} & 4 & $2\times2\times2\times2$ & $10^7$ & 95.016 \\ & 4 & $2\times2\times2\times2$ & $10^7$ & 93.104 \\ & 4 & $2\times2\times2\times2$ & $10^7$ & 95.630 \\ & 4 & $2\times2\times2\times2$ & $10^7$ & 90.957 \\ & 4 & $2\times2\times2\times2$ & $10^7$ & 92.616 \\ \cline{2-5} \label{rand5a}& 5 & $2\times2\times2\times2\times2$ & $10^7$ & 99.862 \\ & 5 & $2\times2\times2\times2\times2$ & $10^7$ & 99.857 \\ & 5 & $2\times2\times2\times2\times2$ & $10^7$ & 99.900 \\ & 5 & $2\times2\times2\times2\times2$ & $10^7$ & 99.889 \\ & 5 & $2\times2\times2\times2\times2$ & $10^7$ & 99.913 \\ & 5 & $2\times2\times2\times2\times2$ & $10^7$ & 99.878 \\ & 5 & $2\times2\times2\times2\times2$ & $10^7$ & 99.884 \\ & 5 & $2\times2\times2\times2\times2$ & $10^7$ & 99.880 \\ & 5 & $2\times2\times2\times2\times2$ & $10^7$ & 99.861 \\ \label{rand5j}& 5 & $2\times2\times2\times2\times2$ & $10^7$ & 99.878 \\ \hline \hline \end{tabular} \end{table} \end{footnotesize}
\begin{footnotesize} \begin{longtable}{>{\the\numexpr\value{rowNo}}>{\refstepcounter{rowNo}}cccccc} \caption{\label{tab-qutrits} Frequencies of violation of local realism $\mathpzc{p}_V$ observed statistically with random measurements on qutrit states} \endfirsthead \endhead \hline \hline \multicolumn{1}{c}{No.} & $N$ & State & Settings & Stat. & $\mathpzc{p}_V$ \\ \hline\hline
& 2 & $\ket{\rm{GHZ}^{d=3}(0^\circ)}_2$ & $2\times2$ & $10^9$ & 9.925 \\ & 2 & $\ket{\rm{GHZ}^{d=3}(5^\circ)}_2$ & $2\times2$ & $10^9$ & 9.801 \\ & 2 & $\ket{\rm{GHZ}^{d=3}(10^\circ)}_2$ & $2\times2$ & $10^8$ & 10.021 \\ & 2 & $\ket{\rm{GHZ}^{d=3}(15^\circ)}_2$ & $2\times2$ & $10^7$ & 11.609 \\ & 2 & $\ket{\rm{GHZ}^{d=3}(20^\circ)}_2$ & $2\times2$ & $10^8$ & 15.057 \\ & 2 & $\ket{\rm{GHZ}^{d=3}(25^\circ)}_2$ & $2\times2$ & $10^8$ & 19.363 \\ \label{asymmGHZqt2} & 2 & \bf{$\ket{\rm{GHZ}^{d=3}(29.24^\circ)}_2$ [asym]} & $2\times2$ & $10^9$ & 22.317 \\ \label{symmGHZqt2} & 2 & \bf{$\ket{\rm{GHZ}^{d=3}(35.26^\circ)}_2$ [sym]} & $2\times2$ & $10^9$ & 24.011 \\ & 2 & \bf{$\ket{\rm{GHZ}^{d=3}(35.26^\circ)}_2$ [sym]} & $3\times3$ & $10^7$ & 78.667 \\ & 2 & \bf{$\ket{\rm{GHZ}^{d=3}(35.26^\circ)}_2$ [sym]} & $4\times4$ & $10^7$ & 98.229 \\ & 2 & $\ket{\rm{GHZ}^{d=3}(40^\circ)}_2$ & $2\times2$ & $10^8$ & 22.980 \\ & 2 & $\ket{\rm{GHZ}^{d=3}(45^\circ)}_2$ & $2\times2$ & $10^8$ & 19.763 \\ & 2 & $\ket{\rm{GHZ}^{d=3}(50^\circ)}_2$ & $2\times2$ & $10^8$ & 15.054 \\ & 2 & $\ket{\rm{GHZ}^{d=3}(55^\circ)}_2$ & $2\times2$ & $10^8$ & 10.153 \\ & 2 & $\ket{\rm{GHZ}^{d=3}(60^\circ)}_2$ & $2\times2$ & $10^8$ & 6.329 \\ & 2 & $\ket{\rm{GHZ}^{d=3}(65^\circ)}_2$ & $2\times2$ & $10^8$ & 3.638 \\ & 2 & $\ket{\rm{GHZ}^{d=3}(70^\circ)}_2$ & $2\times2$ & $10^8$ & 1.818 \\ & 2 & $\ket{\rm{GHZ}^{d=3}(75^\circ)}_2$ & $2\times2$ & $10^8$ & 0.714 \\ & 2 & $\ket{\rm{GHZ}^{d=3}(80^\circ)}_2$ & $2\times2$ & $10^8$ & 0.174 \\ & 2 & $\ket{\rm{GHZ}^{d=3}(85^\circ)}_2$ & $2\times2$ & $10^8$ & 0.012 \\ & 2 & $\ket{\rm{GHZ}^{d=3}(90^\circ)}_2$ & $2\times2$ & $10^8$ & 0.000 \\ \cline{2-6} & 3 & $\ket{\rm{GHZ}^{d=3}_3(0^\circ)}$ & $2\times2\times2$ & $10^8$ & 53.360 \\\cline{3-6} & 3 & $\ket{\rm{GHZ}^{d=3}_3(35.26^\circ)}$ & $2\times2\times2$ & $10^8$ & 82.720 \\\cline{3-6} & 3 & $\ket{\rm{A}^-}_3$ & $2\times2\times2$ & $10^8$ & 72.328 \\\cline{3-6} & 3 & $\ket{\rm{Q}^1_3}$ & $2\times2\times2$ & $10^8$ & 31.371 \\\cline{3-6} & 3 & $\ket{\rm{Q}^2_3}$ & $2\times2\times2$ & $10^8$ & 48.506 \\\cline{3-6} & 3 & $\ket{\rm{Q}^3_3}$ & $2\times2\times2$ & $10^8$ & 48.564 \\ \hline \hline \end{longtable} \end{footnotesize}
\section{Closing remarks}
In this paper we employed linear programming as a useful tool to analyze the nonclassical properties of quantum states. We checked how many randomly generated sets of observables allow for violation of local realism. Most of the conclusions were presented in the previous sections. Here we want to stress that the overall message of the obtained results is that either for many particles or many measurement settings we observe a conflict with local realism for almost any choice of observables (the probability of violation is greater than $99\%$) for typical families of quantum states.
Concerning the nonclassicality of two qutrits, our results are compatible with those presented in \cite{PhysRevA.92.030101}, that is, maximally entangled and maximally nonclassical states coincide. It is worth mentioning that, in addition, we addressed the apparently paradoxical result obtained in \cite{VERT-LASK-WIES}. It amounts to the observation, that the products of $k$-qubit GHZ states and $(N-k)$ pure single qubit states are more nonclassical than the $N$ qubit GHZ state, if we employ the robustness of correlations against white noise admixture as a measure of nonclassicality. Our numerical method shows that the probability of violation of local realism for such product states (for $N = 3,4,5$ and $k = 1,..., N$) is the same as for $k$-qubit GHZ state and thus strictly smaller than for the $N$ qubit GHZ state. This suggests that resistance against noise, although relevant, is not a good quantifier of nonclassicality.
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\BibitemOpen
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{\bibfield {journal} {\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo
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{D\"ur}}, \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont {Vidal}}, \ and\
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{journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {62}},\ \bibinfo
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\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.~J.}\ \bibnamefont
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{Raussendorf}},\ }\href {\doibase 10.1103/PhysRevLett.86.910} {\bibfield
{journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo
{volume} {86}},\ \bibinfo {pages} {910} (\bibinfo {year} {2001})}\BibitemShut
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\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
{Cabello}},\ }\href {\doibase 10.1103/PhysRevLett.89.100402} {\bibfield
{journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo
{volume} {89}},\ \bibinfo {pages} {100402} (\bibinfo {year}
{2002})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Laskowski}\ \emph {et~al.}(2014)\citenamefont
{Laskowski}, \citenamefont {Ryu},\ and\ \citenamefont {\.Zukowski}}]{DickeQt}
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{Laskowski}}, \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Ryu}}, \
and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
{\.Zukowski}},\ }\href
{http://stacks.iop.org/1751-8121/47/i=42/a=424019} {\bibfield {journal}
{\bibinfo {journal} {Journal of Physics A: Mathematical and Theoretical}\
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\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~F.}\ \bibnamefont
{Clauser}}, \bibinfo {author} {\bibfnamefont {M.~A.}\ \bibnamefont {Horne}},
\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Shimony}}, \ and\
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
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{Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {88}},\ \bibinfo {pages}
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {N.}~\bibnamefont
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{\bibfield {journal} {\bibinfo {journal} {Journal of Physics A:
Mathematical and Theoretical}\ }\textbf {\bibinfo {volume} {48}},\ \bibinfo
{pages} {465301} (\bibinfo {year} {2015})}\BibitemShut {NoStop} \end{thebibliography}
\appendix
\section{Proof}
\label{proofpartialent} The following observation is proven below: The probability of violation $\mathpzc{P}_V$ for any two-qubit state and two binary-outcome measurements cannot be greater than $2(\pi-3)$.
In order to prove it, first recall that $\mathpzc{P}_V=2(\pi-3)$ for the $\ket{\rm{GHZ_2}}$ state using two binary-outcome settings per party~\cite{PhysRevLett.104.050401}. Let \begin{equation} \label{partialent} \ket{\rm{GHZ}(\alpha)}_2=\sin \alpha \ket{00}+\cos \alpha \ket{11} \end{equation} stand for the two-qubit pure partially entangled state with $\alpha\in\{0,\pi/4\}$ written in the Schmidt bases. Clearly, $\alpha=\pi/4$ recovers the two-qubit maximally entangled state. Let $\mathpzc{P}_V(\alpha)$ denote the probability of violation corresponding to the state $\ket{\rm{GHZ}(\alpha)}_2$. Note that mixed two-qubit states cannot provide higher probability of violation, therefore we can restrict our attention to the probabilities $\mathpzc{P}_V(\alpha)$.
Firstly we prove the following lemma: if the CHSH inequality is violated using a state $\ket{\rm{GHZ}(\alpha)}_2$ and some projective measurements, at least the same violation occurs with the maximally entangled state $\ket{\rm{GHZ}}_2$ using the same measurements. \begin{proof} Let us write the measurement observables $A$ and $B$ as \begin{align} A &= a_x\sigma_x + a_y\sigma_y + a_z\sigma_z,\nonumber\\ B &= b_x\sigma_x + b_y\sigma_y + b_z\sigma_z, \end{align} where $\sigma_{x,y,z}$ denote Pauli matrices, and the coefficients of Alice measurements $a_x, a_y, a_z$ square to 1 (and similarly for Bob). Then we have the joint correlator \begin{equation} \label{AB} \langle AB\rangle = a_zb_z + \sin2\alpha(a_xb_x -a_yb_y). \end{equation} On the other hand, the CHSH expression reads \begin{equation} \label{CHSH} \textrm{CHSH} = \langle A_1B_1\rangle + \langle A_1B_2\rangle + \langle A_2B_1\rangle - \langle A_2B_2\rangle. \end{equation} Using formula~(\ref{AB}), we get \begin{equation} \label{CHSH_C} \textrm{CHSH}(\alpha) = C_z + \sin(2\alpha)(C_x - C_y), \end{equation} where $\alpha\in\{0,\pi/4\}$ and \begin{equation} C_i = a_{1i}b_{1i} + a_{1i}b_{2i} +a_{2i}b_{1i} -a_{2i}b_{2i}, \end{equation} where $i$ can take $x$, $y$, and $z$. Notice that $C_z\le2$, therefore a CHSH value greater than 2 in equation~(\ref{CHSH_C}) implies that $C_x-C_y>0$. This in turn implies that in case of violation of the CHSH inequality (that is $\rm{CHSH}>2$), we have $\rm{CHSH}(\pi/4)\ge \rm{CHSH}(\alpha)$ for all $\alpha\in\{0,\pi/4\}$. \end{proof}
Given the above lemma, it is not difficult to see that $\mathpzc{P}_V(\alpha)\le\mathpzc{P}_V=2(\pi-3)$ for all $\alpha\in\{0,\pi/4\}$. Indeed, notice that the classically attainable region of the 2-setting 2-outcome scenario is completely characterized by eight different versions of the CHSH expressions (see e.g.~\cite{PhysRevLett.104.050401}). However, after suitable relabeling of the inputs and flipping of the outcomes they all end up in the standard CHSH defined by equation \ref{CHSH}. Hence, violation of any of the versions of the CHSH inequality using a partially entangled state~(\ref{partialent}) along with some projective measurements entails at least the same violation of this version using the maximally entangled state and the same measurements. This implies the relation $p_V(\alpha) \le p_V = 2(\pi-3)$ we set out to prove.
\end{document} | arXiv |
Let $a,$ $b,$ and $c$ be the roots of $x^3 - 3x - 5 = 0.$ Find $abc.$
By Vieta's formulas, $abc = \boxed{5}.$ | Math Dataset |
The St. Čapek Crowd Control Academy, June, 2115.
Commander Gall is worried. On one hand, the graduating class this year is the most promising in the institution's history, every cadet's reasoning skills honed to perfection. On the other, General Domin is visiting and has insisted on writing and administering the robostanchion exam himself, despite the fact that robostanchion squads as a crowd-control technique came after his time, and he doesn't always grasp their limitations.
A cadet can deploy an inactive robostanchion, commanding it to teleport itself to any unoccupied point on the field—which happens to be an infinite integer lattice—and become active.
A cadet can move an active robostanchion, commanding it to teleport to a different, unoccupied point on the field. The robostanchion remains active.
A cadet can recall an active robostanchion, commanding it to teleport back to the hangar and become inactive.
Ordinarily, there are only two ways to fail a robostanchion exam: (1) damaging the squad by running out of power or (2) not achieving the requested formation within the generous time limit. Now the general, to gratify his ego, has decided that his version of the exam should be more challenging, and he has added (3) ever not using all of the available power.
Commander Gall, appropriately mortified, has, after much obsequious persuasion, gotten the general to agree to some concessions. First, the general will allow cadets four commands before he enforces his additional requirement. Second, the general has agreed to only ask for connected formations, i.e., formations where every robostanchion is joined to the others by at least one laser barrier.
While further promises from the general are improbable, there is one more fact working in the cadets' favor: To prevent cheating, school policy is that cadets are always given squads of distinct sizes. For instance, if one cadet makes the exam with a squad of 90, no other cadet will have exactly 90 robostanchions at their command.
Unsure whether these facts are enough to contain the general's caprice, Gall consults with Chief Engineer Fabry, asking for a worst-case analysis. What does Fabry reply? That is, in the worst case possible, how many cadets can Gall expect to fail?
Clarifications: General Domin is sensible enough not to ask for formations that require more robostanchions than are in the squad, nor for formations that violate his condition (3).
one. The only impossible formation is a $3\times3$ square of robots with the center removed.
Since all legal moves after the first four are reversible, in order to show a formation can be created from nothing, it suffices to show that it can be reduced to a small square of four robots.
Call a robot with exactly one robot next to it a loner. We first show that any formation with loners can be reduced to a small square. Starting with such a formation, remove all loners, and continue removing newly created loners until there are none. What remains is a loop. Unless the formation is now a small square, the top edge of the formation must have 3 robots in a row, say at (0,0), (1,0) and (2,0). Deploy one of the removed robots to (1,1), then move (0,0)'s other neighbor to (0,1). This creates a small square of robots: after removing all loners until there are none, only this square will remain.
So, assume there are no loners. Suppose without loss of generality that the lowest row in the formation has a $y$-coordinate of $0$, and on that row, the lowest $x$ coordinate is $0$. This implies that there are robots at (0,0), (1,0), and (0,1). If there is a robot at (1,1), that means we have the small square, so assume there isn't one. Furthermore, (1,0) can't have his second neighbor to the south, so there must be a robot at (2,0).
Case 1: Neither (2,1) nor (1,2) have a robot. Move (2,0) to (1,1). This means that (2,0)'s old neighbor at (3,0) will now be a loner, so the formation is now reducible.
Case 2: Exactly one of (2,1) or (1,2) has a robot. Move that robot to (1,1). The moved robot's two neighbors are now loners.
Case 3: Both (1,2) and (2,1) have robots.
(1,2) must have two neighbors. If they are both at (0,2) and (2,2), then this is the bad formation, so assume that only one of (0,2) and (2,2) is filled. By symmetry, we can assume (0,2) is unfilled. This means that the other neighbor of (0,1) must be at (-1,1). If there was a robot at (-1,0), that would mean there was a small square of four robots, contradicting that the entire formation was a loop. So, (-1,0) is empty.
Case 3a: (-2,0) has a robot. Move that robot to (-1,0). The two neighbors that (-2,0) had are now loners.
Case 3b: (-2,0) doesn't have a robot. Move (2,0) to (-1,0). The two neighbors that (2,0) had are now loners.
Thus, in all cases except for the $3\times 3$ square without its center, the formation can be made. To see why this formation cannot be made, you can check that no moves can be made from it.
If you were only given the exact number of robots needed, no deployments are possible. No moves are possible: a move destroys two lasers, so it must create two as well, but every spot you can move a robot would make it have 0,1,3 or 4 lasers. Finally, no recalls are possible, since this will destroy two lasers and only one robot.
Not the answer you're looking for? Browse other questions tagged mathematics strategy story graph-theory or ask your own question. | CommonCrawl |
Filippo Antonio Revelli
Filippo Antonio Revelli (1716 – 1801) was an Italian mathematician.[1]
Life
He was professor of geometry for 26 years at the University of Turin.[2]
He had among his pupils Joseph-Louis Lagrange.[1][3]
His son Vincenzo Antonio Revelli (1764-1835) was a philosopher and painter.[4][3]
Works
• Elementi dell'aritmetica universale e della geometria piana e solida (in Italian). Vol. 1. Turin: Giammichele Briolo. 1778.
• Elementi dell'aritmetica universale e della geometria piana e solida (in Italian). Vol. 2. Turin: Giammichele Briolo. 1778.
References
1. Paolo Revelli (1918). Un maestro del Lagrange: Filippo Antonio Revelli (1716-1801) (in Italian). Genoa: E. Oliveri.
2. M. Boisset fils (1779). Lettre contenant l'histoire, et un essai d'analyse des eaux de La Boisse (in French). Turin: Jean-Michel Briolo. p. 71.
3. Bollettino della Società piemontese di archeologia e belle arti (in Italian). Turin: Società piemontese di archeologia e belle arti. 1917. p. 101.
4. Vittorio Spreti (1935). Enciclopedia storico-nobiliare italiana (in Italian). Vol. 5. Bologna: Forni. p. 662.
| Wikipedia |
Key Formulas
Exponent Laws and Logarithm Laws
Trig Formulas and Identities
Differentiation Rules
Trig Function Derivatives
Table of Derivatives
Table of Integrals
Jump to solved problems
Evaluating Limits
Limits at Infinity
Limits at Infinity with Square Roots
Calculating Derivatives
Equation of a Tangent Line
Mean Value Theorem & Rolle's Theorem
Least expensive open-topped can
Printed poster
Related Rates
Snowball melts
Snowball melts, area decreases at given rate
How fast is the ladder's top sliding
Angle changes as a ladder slides
Lamp post casts shadow of man walking
Water drains from a cone
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Home » Calculus 1
Mean Value Theorem & Rolle's Theorem: Problems and Solutions
Are you trying to use the Mean Value Theorem or Rolle's Theorem in Calculus? Let's introduce the key ideas and then examine some typical problems step-by-step so you can learn to solve them routinely for yourself.
CALCULUS SUMMARY: Mean Value Theorem & Rolle's Theorem
Click to Show/Hide Summary
Mean Value Theorem (MVT):
If $f(x)$ is continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$, then there is a number $c$ in $(a,b)$ such that
$$\bbox[yellow,5px]{f'(c) = \frac{f(b) – f(a)}{b-a}}$$
or, equivalently,
$$\bbox[yellow,5px]{f(b) – f(a) = f'(c)(b-a)}$$
In words: there is at least one value $c$ between $a$ and $b$ where the tangent line is parallel to the secant line that connects the interval's endpoints. (See the figures.)
[Click on a figure to see a larger version.]
Rolle's Theorem:
In Calculus texts and lecture, Rolle's theorem is given first since it's used as part of the proof for the Mean Value Theorem (MVT). You can easily remember it, though, as just a special case of the MVT: it has the same requirements about continuity on $[a,b]$ and differentiability on $(a,b)$, and the additional requirement that $f(a) = f(b)$. In that case, the MVT says that
f(b) – f(a) &= f'(c)(b-a) \\
0 &= f'(c)(b-a) \\
Since $b \ne a$ (or there's no interval), we know $b-a \ne 0.$
Hence when $f(a) = f(b),$ we must have
\[\bbox[yellow,5px]{f'(c) = 0} \] \[\text{for some number $c$ in the open interval $(a,b)$} \] In words: when $f(a) = f(b),$ the slope of the secant line connecting the endpoints is zero, and hence there is at least one value $c$ between $a$ and $b$ where the tangent line has zero slope. (See the figures.)
The problems below illustrate some typical uses of the Mean Value Theorem and Rolle's Theorem.
Problem #1: Straightforward Application of Rolle's Theorem
Consider the function $f(x) = 9 – (x-3)^2$ on the interval $[0, 6]$. (Note that $f(0) = f(6) = 0$.) Find the value(s) of $c$ that satisfy Rolle's Theorem.
Click to View Calculus Solution
Since the function $f(x) = 9- (x-3)^2$ is a polynomial, it is continuous on the interval $[0, 6]$ and differentiable on the interval $(0,6)$. Furthermore, as the question states, $f(0) = f(6)$, and so Rolle's Theorem applies. We're looking for a value of $c$ such that $f'(c) = 0$:
f'(x) &= \frac{d}{dx}\left[9- (x-3)^2 \right] \\[8px] &= -2(x-3)
We want to find $c$ such that $f'(c) = 0$:
f'(c) &= -2(c-3) = 0 \\[8px] c &= 3 \quad \cmark
The point $c = 3$ has a horizontal tangent line, satisfying Rolle's Theorem since $0 < c < 6$.
Problem #2: Straightforward Application of the Mean Value Theorem
Consider the function $f(x) = x^2$ on the interval $[1, 4]$. Find the value(s) of $c$ that satisfy the Mean-Value Theorem.
Since the function $f(x) = x^2$ is a polynomial, it is continuous on the interval $[1, 4]$ and differentiable on the interval $(1,4)$, and so the Mean-Value Theorem applies.
We're looking for a value of $c$ such that
f'(c) &= \frac{f(4) – f(1)}{4-1} \\[8px] &= \frac{16 – 1}{3} \\[8px] &= \frac{15}{3} = 5
Now we can compute $f'(x)$ starting with $f(x) = x^2$:
f'(x) &= \frac{d}{dx}x^2 \\[8px] &= 2x
f'(c) &= 2c = 5 \\[8px] c &= 2.5 \quad \cmark
The tangent line at the point $c = 2.5$ is parallel to the secant line that connects the endpoints of the interval.
Problem #3: Prove if f '(x) > 0, then f(x) is an increasing function
Use the Mean Value Theorem to prove that if $f(x)$ is differentiable and $f'(x) > 0$ for all $x$, then $f(x)$ is an increasing function.
This question might seem silly: "Prove that if a function has positive slope, then it is increasing." But mathematics is partly about using theorems you've already proven to prove other things, no matter how obvious they may seem, so let's just do as the question asks.
First, note that since f(x) is differentiable for all x, it must be continuous for all x, and so the Mean Value Theorem (MVT) applies. The problems says to use the MVT, so let's start there, and consider an interval $(a,b)$. The MVT tells us that there exists a c, $a < c < b$, such that $$f(b) - f(a) = f'(c) (b-a)$$ Since $f'(x) > 0$ for all $x$, we have $f'(c) > 0$.
Furthermore, since $b > a$, $b-a > 0$.
$$f(b) – f(a) > 0$$
$$f(b) > f(a)$$
That is, $f(x)$ is increasing. $\quad \cmark$
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Convergence of the series $\sum\ln(1+\frac{(-1)^n}{n+1})$
I want to show that the series whose nth term is $a_n=\ln(1+\frac{(-1)^n}{n+1})$ is convergent. I wanted to use the limit comparison test to compare it to the $p$ series but $a_n$ is not positive. I thought of writing the power series representation of $a_n$ using the power series representation of $\ln(1+x)$ with $x=b_n=\frac{(-1)^n}{n+1}$ we find that $$a_n=b_n-\frac{1}{2}b_n^2+\frac{1}{3}b_n^3-\frac{1}{4}b_n^4+\cdots$$ Now the seris $\sum b_n$ is convergent by the alternating series test and the other terms are all terms of absolutely convergent series but it is an infinte sum, can I say so ? I mean is the infinite sum of convergent series a convergent series ? Is this correct and is there any other way to do it ?
sequences-and-series
paliopalio
$\begingroup$ Use $\lvert \ln (1+x) - x \rvert \leqslant \lvert x\rvert^2$ for $\lvert x\rvert \leqslant \frac{1}{2}$. $\endgroup$ – Daniel Fischer Feb 14 '16 at 21:40
$\begingroup$ The alternating series test also applies to $a_n$, which alternates in sign and converges to 0. $\endgroup$ – Matt Samuel Feb 14 '16 at 21:42
$\begingroup$ @Matt Samuel How I write $a_n=(-1)^nc_n$ what is $c_n$ ?Do you mean that I write $c_n=|\ln (1+\frac{(-1)^n}{n+1}|$ ? $\endgroup$ – palio Feb 14 '16 at 21:53
$\begingroup$ @MattSamuel It needs to have an absolute value decreasing (non-increasing, rather) to zero, convergence is not enough. $\endgroup$ – Clement C. Feb 14 '16 at 21:53
$\begingroup$ Let $a_n=\ln\left(1+\frac{(-1)^n}{n+1}\right)$ for every $n\geqslant1$, then $a_n\to0$ and $a_{2n-1}+a_{2n}=0$ hence the series $\sum\limits_{n\geqslant1}a_n$ converges to $a_1=-\ln2$. If one starts at $n=0$, the sum is $0$. $\endgroup$ – Did Feb 14 '16 at 22:36
One may use the Taylor series expansion, as $x\to 0$, $$ \log(1+x)=x+O(x^2) $$ giving, for some great $n_0$ and all $N$ greater than $n_0$, $$ \sum_{n_0 \leq n\leq N}\ln(1+\frac{(-1)^n}{n+1})=\sum_{n_0 \leq n\leq N}\frac{(-1)^n}{n+1}+\sum_{n_0 \leq n\leq N} O\left(\frac1{(n+1)^2}\right) $$ then conclude to the convergence of the initial series.
Olivier OloaOlivier Oloa
$\begingroup$ That's exactly what i want to use but without the big O notation what should be the procedure ? Thank you for your help! $\endgroup$ – palio Feb 14 '16 at 21:48
$\begingroup$ @OlivierOloa: it's technically correct, but seeing $\frac{1}{n+1}$ instead of $\frac{1}{n}$ seems... morally wrong. $\frac{(-1)^n}{n} + O(\frac{1}{n^2})$? $\endgroup$ – Clement C. Feb 14 '16 at 21:51
$\begingroup$ @palio You have $\displaystyle \left|O\left(\frac1{(n+1)^2}\right)\right|\leq \frac{C}{(n+1)^2}$ giving the sought convergence. Thanks. $\endgroup$ – Olivier Oloa Feb 14 '16 at 21:52
$\begingroup$ @ClementC. I see what you mean... $\endgroup$ – Olivier Oloa Feb 14 '16 at 21:54
$\begingroup$ It does not really matter here... actually, not at all here :) It's really more a question of habit and not "letting one's guard down" (otherwise, sneaky things based on differences of expressions like $e^{\frac{1}{n}}$ and $e^{\frac{1}{n+1}}$ can appear). $\endgroup$ – Clement C. Feb 14 '16 at 21:56
This series is alternating and the positive parts of the terms decrease to 0. It therefore converges. Note this little experiment and you will see why.
import math
def f(n):
return math.log(1 + (-1)**n/n)
for k in range(2,20):
print ("%s:%7.8f"% (k, f(k)))
2:0.40546511
3:-0.40546511
10:0.09531018
11:-0.09531018
ncmathsadistncmathsadist
$\begingroup$ How do you prove the second point elegantly, though? $\endgroup$ – Clement C. Feb 14 '16 at 21:52
$\begingroup$ $t\mapsto 1/(1+t)$ is increasing on (-1, 1).. $\endgroup$ – ncmathsadist Feb 14 '16 at 21:54
$\begingroup$ $\lvert a_n\rvert = \left\lvert \ln\left(1+\frac{(-1)^n}{n+1}\right)\right\rvert$. The jump to monotonicity of $t\mapsto \frac{1}{t+1}$ is not that straightforward, and may require at least a couple lines? $\endgroup$ – Clement C. Feb 14 '16 at 21:58
$\begingroup$ separate the even and odd terms first. It is a bit of delicate affair. $\endgroup$ – ncmathsadist Feb 14 '16 at 22:10
$\begingroup$ As pointed out by Did in another comment, the simplest option to show monotonicity of $\lvert a_n\rvert$ is probably to note that $$0 < a_{2n} = \ln\left(1+\frac{1}{2n}\right)=\ln\left(\frac{2n+1}{2n}\right)= -\ln\left(\frac{2n}{2n+1}\right) = -\ln\left(1-\frac{1}{2n+1}\right) = -a_{2n+1}.$$ $\endgroup$ – Clement C. Feb 14 '16 at 23:10
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Poincaré Conjecture and Homotopy
Poincaré conjecture is the most recent major proven theorem. Posited a century ago by Henri Poincaré, this major conjecture of topology was solved by Gregori Perelman. It has revolutionized our understanding of space and raised intriguing questions regarding the global structure of our Universe.
March 25, 2013 ArticleHomotopy, Mathematics, TopologyLê Nguyên Hoang 8607 views
Of the seven millenium prize problems, Poincaré conjecture is the only solved one. Conjectured by French mathematician Henri Poincaré (who, by the way, went through the same University as I did!!!), it was solved by Russian mathematician Gregori Perelman in 2003. This article explains what the conjecture says.
I wrote an article on P versus NP, which is another of the seven millenium prize problems.
Homeomorphism
Poincaré conjecture is a problem of topology, which is a subfield of mathematics which studies properties of connectedness of spaces. In particular, topology aims at continuously deforming spaces into simpler better-known spaces. For instance, in the animation from Wikipedia on the right, a mug is continuously deformed into a doughnut. This continuous deformation is called a homeomorphism.
So a homeomorphism is a continuous mapping between two spaces?
Not exactly. In order to have the topological properties preserved, topologists also want homeomorphisms to be invertible, and that the inversion is continuous too. This means that, not only do we need to be able to deform continuously the mug into a doughnut, but we also need to be able to invert continuously the doughnut back into a mug. In more precise terms, a homeomorphism is a continuous one-to-one mapping between two topological sets whose inverse is continuous too.
If you want to know more about the basics of topology, like what it means for a mapping to be continuous, you should read my article on basic topology. But you don't need the rigorous definition to follow the major ideas of this article.
This sounds complicated…
Visually, it's very simple. Basically, homeomorphisms involve stretchings and contractions, but forbid cuts and pastes. This means, for instance, that disconnected spaces cannot be connected by a homeomorphisms. Below are several examples of homeomorphic spaces.
As a bit of a geography geek, I would have loved to show you homeomorphisms of countries. But, as explained in CGPGrey's video, except for islands or disconnected countries, only Italy and South Africa aren't homeomorphic to a disk. And they are not even homeomorphic to each other, as Italy has 2 holes, while South Africa only has one…
Notice that the two spaces above are not homeomorphic to the two spaces below. This can proved by remarking that the two spaces above have one connected boundary. On the other hand, the two spaces below have a disconnected boundary. Since borders cannot be cut nor pasted, there is no way of deforming these of the above spaces into these of the below spaces.
So I guess we can classify spaces up to homeomorphism by the number of connected components of both the space and its boundary…
This definitely enables to tell non-homeomorphic spaces apart. But two spaces with equal number of connected components, and whose boundaries are also made of the same number of connected components, are not necessarily homeomorphic. To better understand this, Poincaré focused on connected spaces with no boundary. Also he only considered bounded spaces, as opposed to spaces which go to infinity like a plane. Technically, these spaces are defined as compact spaces with a dimension n such that, if you zoom enough around any of point, the space looks like a n-dimensional vector space. They are called closed n-manifolds. A 1-manifold is a curve, while a 2-manifold is a surface.
I'm not dwelling on the concept of compactness, although it is a fundamental concept in topology. For our purpose, because we only work on finite dimensional spaces, compactness corresponds to boundedness. This means that the space is in a limited volume.
Can you give an example?
Sure. In fact, the examples I'm going to give will be essential to discuss Poincaré conjecture. First is the simplest closed 2-manifold, namely, the sphere. If you zoom in on a sphere, the surface of the sphere looks like a plane, which is the 2-dimensional vector space. A sphere is thus a 2-manifold. Plus, it is compact since it is bounded.
What's the other?
The other important closed 2-manifold is the torus, which is basically the surface of the doughnut I mentioned earlier. What's important to note is that, even though the torus and the sphere are both closed 2-manifolds, they are not homeomorphic!
Why aren't they homeomorphic?
This is an awesome question to leads me to Poincaré's key insight.
Poincaré's key insight was to consider loops made on a figure. This study of loops is called homotopy. Listen to myself explaining this idea in this extract from a Trek through 20th Century Mathematics.
So a loop is a path on the surface which goes back to the starting point?
Yes! Exactly. But one thing to notice is that some loops are essentially equivalent. This is the case whenever one can be deformed continuously into the other. We say that such loops are homotopic. For instance, in the following figure, the purple loop displays the continuous deformation of the blue loop into the green loop. Thus, the blue and green loops are homotopic.
In fact, you can even notice that these loops can be deformed into a point. They are thus homotopic to points. We say that these loops are contractible. They sort of represent the zero of homotopy.
Aren't all loops contractible?
Well, it depends on the structure of the space we study. In the holed disk on the right, the blue and green loops cannot be deformed into one another. They are not homotopic. In particular, the blue loop is contractible, while the green loop isn't.
So what does homeomorphism have to do with homotopy?
Homeomorphism preserves homotopy.
Could you translate in understandable words?
Sure! Consider a disk and a polygon, and a homeomorphism between them. Let a green and a blue loop of the disk. Each point of each loop is thus uniquely mapped with a point of the polygon. Overall, each loop of the disk is uniquely mapped with a loop of the polygon. Thus, the green and blue loops of the disk are mapped with a green and a blue loop of the polygon.
When I say that homeomorphism preserves homotopy, I mean that the green and blue loops of the disk are homotopic if and only if their images in the polygon are homotopic.
Why would that be true?
Let's prove it! Assume that, in the disk, the green and blue loops are homotopic. Then there are purple loops which are intermediate loops from the blue to the green loop. Each purple loop of the disk is mapped with a purple loop of the polygon. The obtained purple loops of the polygon now form a transition from the blue to the green loop in the polygon. Thus, the image of homotopic loops by the homeomorphism are homotopic too. A similar reasoning shows that homotopic loops of the polygon are associated with homotopic loops of the disk by the inverse homeomorphism.
So, to prove that a sphere is not homeomorphic to a torus…
We only need to prove that there are loops on a torus which do not correspond to any loop on a sphere. The key argument is that all loops on a sphere are contractible, while the torus has the two non-contractible blue and green loops drawn below. If there were a homeomorphism, then the image of the blue and green loops in the sphere would be non-contractible, which contradicts the fact that loops on a sphere are all contractible. Thus, the sphere and the torus are not homeomorphic.
I'm not sure I see it clearly…
Surfaces in 3 dimensions are hard to visualize indeed. Fortunately, there is a much simpler way to describe two-dimensional surfaces using planar representations and… glue.
Planar Representations
Glue?
Yes! You have probably learned at school that you could make a cube using a net like the one on the right, and by then gluing the sides in the right way. This right way is represented by colors on the edge. Same colors must be glued together. We can describe loops on the cube by describing the loops on the net. Isn't this planar representation much simpler to visualize?
Yes! But what about the sphere and the torus? Are there nets for these?
Yes! For instance, the Earth is usually represented by a 2-dimensional map. However, the gluing of the map is a bit more complicated, as the whole top of the map should be glued together as the North Pole. Now, when I took a flight from Toronto to Beijing, the plane went through the North pole. This can be nicely represented on the map:
So are you going to show us that all loops are contractible on a sphere with the map above?
No. Although common to us all, the map drawn above is a bit too complicated in terms of gluing, as there are four sides with different properties. Rather, let's work on the most simple representation of the sphere, which is a disk centered on the North pole. The edge of the disk corresponds to the South pole, and should thus be all glued together.
By the way, this representation is great to visualize the fact that the shortest path from Toronto to Beijing does go through the North pole. Find out more about map making with Scott's article on non-Euclidean geometry.
OK… How do we now deduce that all loops are contractible on a sphere?
All the points of the edge of the disk are one single point. Thus, a path disappearing at one end and reappearing at another end can be deformed into a path going along the edge, as done below.
The deformed loop can then be contracted to a point.
I'm sort of convinced for this particular loop. But what about the general case of any loop?
Consider any loop on the sphere. We can deform it such that it never crosses the South pole. This deformed loop doesn't cross the edge of the disk. It can then thus be contracted to a point by a homothety centered on the North pole. You can learn about homothety with my article on symmetries.
What about the torus?
Hehe… The torus is actually a square with opposite sides glued together, exactly like in one of History's greatest video game, PAC-MAN. If this is missing of your culture, I feel sorry for you! But I'll explain it nevertheless. PAC-MAN is played on a screen. Whenever you go out of the screen on the right, you reappear on the left. Similarly, when you go out out of the screen on the top, you reappear on the bottom.
I'm not convinced that gluing the sides yields a torus…
As we first glue the top with the bottom, we obtain a cylinder. We now needs to glue the end sections of the cylinder, which can be done by curving the cylinder. We obtain the following figure:
This shows that a torus equals $\mathbb R^2/\mathbb Z^2$, or, equivalently, $\mathbb S\times \mathbb S$, where $\mathbb S = \mathbb R/\mathbb Z$ is the one-dimensional circle. Amusingly, this also indicates that tori are much simpler than spheres.
Check out this animation by Youtube channel Geometric Animations of the transformation I described above:
So what are non-contractible loops like in the square representation?
Non-contractible loops in the square representation are those which cross the edges. The two non-contractible loops we represented on the torus earlier are now represented as follows:
Why would these loops be non-contractible?
Let's consider the green loop. The reasoning for the blue one will be similar.
OK… So why is the green loop non-contractible?
That's because it crosses the top edge. In fact, for any continuous deformation of the green loop, we can track the intersection of the deformed loop with the top edge, which proves that it cannot disappear. Similarly, we can draw a horizontal line in the middle of the square, and any deformation of the loop would also have to cross this horizontal line. Thus, any deformation of the green loop must intersect both the top edge and the middle horizontal line. It can therefore not be contracted to a point. This proves that the green loop is not contractible.
So the torus has non-contractible loops while the sphere doesn't, and this proves that they are not homeomorphic?
Exactly! More generally, we now know that any space with non-contractible loop is not homeomorphic to a sphere.
Is that Poincaré conjecture?
No. Poincaré conjecture concerns the reciprocal phrase. Namely, it asks whether closed surfaces with no non-contractible loop are homeomorphic to a sphere…
Poincaré Conjecture
Poincaré gave a name to these surfaces and spaces with no non-contractible loop. He called them simply connected. Poincaré managed to prove the result for two-dimensional manifolds, also known as surfaces. His theorem is one of these amazingly simple ones, as it can be written with 10 words:
Theorem: Any closed simply-connected surface is homeomorphic to a sphere.
How did he prove it?
Unfortunately, the only proofs I know, which involve the classification of 2-dimensional closed surfaces with technics of cutting and gluing, would be too long to explain here. These are nice proofs with no really technical difficulty. You can check it out in this great lecture by Norman Wildberger.
If you know a simpler proof which could be presented here, please tell me!
But this is not Poincaré conjecture, is it?
No. Poincaré conjecture concerns higher dimension manifolds. In dimension n greater or equal to 2, Poincaré conjecture is stated as follows:
Theorem: Any closed simply connected n-manifold is homeomorphic to a n-sphere.
You should now be able to understand every single word here except one…
What's a n-sphere?
That's the one! A n-sphere is defined as the set of points at distance 1 to the origin in a (n+1)-dimensional normed vector space.
So how do we represent these n-spheres?
Hummm… Tough one. I give up for high dimensions, but I can provide a representation for dimension 3. A 3-sphere is the interior of the 2-sphere whose boundary is glued as a single point, in a similar way that we have described the 2-sphere with the interior of a 1-sphere. The interior of a n-sphere is known as a (n+1)-ball. So, in other words, a 3-sphere is a 3-ball whose boundary is glued into a single point.
Does this work?
It does! One construction consists in noting that a point on the 3-sphere lying in the 4 dimensional space is nearly known by 3 of its coordinates. In fact, if the 3 coordinates form a vector which belongs to a 3-ball, there is exactly 2 possible values for the last coordinate, which sort of correspond to the North and the South hemi-3-sphere. And if the 3 coordinates belong to the 2-sphere, then the 4th coordinate is uniquely defined. This corresponds to a sort of 2-dimension equator which belongs to both hemi-3-sphere. As a result, the 3-sphere can be thought as the union of two 3-balls which are to be glued along their boundaries.
I'm not sure I see what you mean…
Check this nice animation by Patrick Massot where the two hemi-3-sphere of the 3-sphere are displayed separately.
Patrick Massot wrote this great article on Poincaré conjecture on images.math.cnrs.fr and poincare.fr. It's in French though… But if you can read French, you should definitely check it out (as well as other articles of these two websites!). I'd like to personally thank him for sending me these videos and allowing me to include them here.
And we can glue these boundaries by inverting one of the 3-balls as follows:
Note that the center of the second 3-ball is inverted into the boundary of the final 3-ball. The final 3-ball can then be resized such that his radius equals 1. We eventually obtain a 3-ball whose boundary should be all glued together as one single point.
The homeomorphism from the 3-ball to the sphere can be written as a mapping of a point (x1, x2, x3) with (x1, x2, x3, sgn(z) sqrt(|z|)), where z = 1-2(x12+x22+x32). The value of the 4th coordinate thus goes from 1 to -1 as we go from the center of the 3-ball to its edge.
So I guess we can use a similar reasoning as for the 2-sphere to prove that the 3-sphere is simply connected…
Yes! In fact, this construction is more general and works for any n-sphere. Any n-sphere is thus a n-ball whose boundary is a single point. By deforming loops such that they never reach this point, we can then prove that any loop is homotopic to a loop which belongs to the n-ball, and can thus be contracted using a mere homothety. This proves that any n-sphere is simply connected.
This is good, but Poincaré conjecture is about the reciprocal, right?
Yes, but this is a much more difficult problem. In fact, in 1904, Poincaré himself wrote: this question would take us too far (cette question nous entraînerait trop loin). He was definitely right! It took mathematicians a century to finally solve it entirely. Progresses have been made steadily, as the theorem was first proven in the 1960s for dimensions 7 and higher. Amusingly, the smaller the dimension, the harder the problem. In 1982, dimension 4 was solved, and there was only the problem in dimension 3 left.
And then Perelman solved it in dimension 3 in 2003…
Yes. And this came as a surprise, as he methods were definitely not conventional in topology.
Topologists mainly use technics of cutting and pasting like those I've presented in this article. But Perelman's method rather consisted in using the dynamics of heated objects to prove Poincaré conjecture.
The dynamics of heated objects?
Yes. The idea is that, as objects get heated, they become rounder and rounder. More curved portions get straightened quicker. Eventually, they become spheres. By using this idea to deform the geometry of spaces, Perelman proved that all 3-manifolds could be deformed into one of the fundamental classes of closed 3-manifolds.
I'm not sure I understand…
So let's see another of Patrick Massot's animations:
In particular, if the initial 3-manifold is simply connected, it gets deformed into the 3-sphere. Which proves the Poincaré conjecture.
We're getting to topics which are way beyond my expertise. If you can, please write about this! Note though that you can find nice animations on this page, which, unfortunately, I haven't been able to include here. The first video represents the 3-sphere as the union of 2 3-balls glued together (the left hand side of the figure above). The third one illustrates the deformation of a manifold with Perelman's heating technics.
Let's Conclude
Poincaré conjecture has been a cornerstone for the classification of closed manifolds. While closed surfaces have been classified mainly by Bernhard Riemann, the classification of closed 3-manifolds had a deep connection with Poincaré conjecture. In 1982, William Thurston, probably the greatest topologist of the 20th century, conjectured that the geometry of any closed 3-manifold could be classified as one of 8 possible geometries. Perelman's proof, along with Poincaré conjecture, answered Thurston's conjecture positively.
There is a lot more to be said in topology of closed manifolds. For one thing, it has strong connections with graph theory and polyhedra, as you can read it in my article on the utilities problem. Also, I plan on writing an article soon on the amazing non-orientable surfaces, which involve the Möbius band, the projective plane or the Klein bottle. Stay tuned!
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Homotopy Type Theory and Higher Inductive Types Homotopy Type Theory and Higher Inductive Types
In this article, we explore the possibilities allowed by higher inductive types. They enable a much more intuitive formalization of integers and new mind-blowing definitions of the (homotopical) circle and sphere.
Poincaré Conjecture on Numberphile.
Who Cares About Poincaré? on Slate. | CommonCrawl |
\begin{document}
\title{Packing Plane Spanning Trees and Paths in Complete Geometric Graphs \thanks{This work was presented at the 26th Canadian Conference on Computational Geometry (CCCG 2014), Halifax, Nova Scotia, Canada, 2014. The journal version appeared in Information Processing Letters, 124 (2017), 35--41, \url{https://doi.org/10.1016/j.ipl.2017.04.006}.} }
\author{Oswin Aichholzer \thanks{Institute of Software Technology, Graz University of Technology, Austria. \texttt{[email protected]}} \and Thomas Hackl \thanks{Institute of Software Technology, Graz University of Technology, Austria. \texttt{[email protected]}} \and Matias Korman\thanks{ Tohoku University, Sendai, Japan. \texttt{[email protected]}} \and Marc van Kreveld\thanks{ Department of Information and Computing Sciences, Utrecht University, the Netherlands. \texttt{[email protected]}} \and Maarten L\"offler\thanks{ Department of Information and Computing Sciences, Utrecht University, the Netherlands. \texttt{[email protected]}} \and Alexander Pilz\thanks{ Department of Computer Science, ETH Zurich, Switzerland. \texttt{[email protected]}} \and Bettina Speckmann\thanks{ Department for Mathematics and Computer Science, TU Eindhoven, the Netherlands. \texttt{[email protected]}} \and Emo Welzl\thanks{ Department of Computer Science, ETH Zurich, Switzerland. \texttt{[email protected]}} }
\maketitle
\begin{abstract} We consider the following question: How many edge-disjoint plane spanning trees are contained in a complete geometric graph $GK_n$ on any set $S$ of $n$ points in general position in the plane? We show that this number is in $\Omega(\sqrt{n})$. Further, we consider variants of this problem by bounding the diameter and the degree of the trees (in particular considering spanning paths). \end{abstract}
\section{Introduction} \label{sec:intro}
A \emph{geometric graph} $G = (S, E)$ consists of a set of vertices $S$, which are points in general position in the plane, and a set of edges $E$ which are straight-line connections between two of these points. A long-standing open question is the following: Does every complete geometric graph with $2n$ vertices have a partition of its edges into $n$ plane spanning trees?
For \emph{complete convex geometric graphs} (where all vertices lie in convex position), a positive answer to this question follows from a result by Bernhart and Kainen~\cite{book_embeddings} (see~\cite{partitions_into_trees}). Bose et al.~\cite{partitions_into_trees} gave a characterization of the solutions; for complete convex geometric graphs all spanning trees can, but do not have to, be spanning paths. They also described a sufficient condition generalizing the convex case and considered a relaxation where the trees are not required to be spanning.
We consider a closely related question: How many edge-disjoint plane spanning trees are contained in a \emph{complete geometric graph} $GK_n$ on any set $S$ of $n$ points in general position in the plane?
In Section~\ref{sec:trees} we show how to combine a construction by Bose et al.~\cite{partitions_into_trees} with a result by Aronov {et al.}~\cite{crossing_families} to prove that $GK_n$ contains $\Omega(\sqrt{n})$ edge-disjoint plane spanning trees. Furthermore, if the convex hull of $S$ contains $h$ vertices then we can argue that $GK_n$ contains at least $\left\lfloor\frac{h}{2}\right\rfloor$ edge-disjoint plane spanning trees. We also show that $GK_n$ contains at least 2 plane edge-disjoint spanning trees if $n \geq 4$ and at least 3 edge-disjoint spanning trees if $n \geq 6$.
In Section~\ref{sec:paths} we study the special case of spanning paths. In particular, we first consider the ``regular wheel configuration'', that is, a set of points $W_{2n}$ which consists of $2n-1$ points regularly spaced on a circle $C$ and a point at the center of $C$. Let $GW_{2n}$ be the complete geometric graph on $W_{2n}$. We can argue that $GW_{2n}$ can be partitioned into $n$ spanning trees. But surprisingly, if $n \geq 3$ then none of these trees can be paths. If the ``hub'' of the wheel is moved close to the convex hull, then all $n$ spanning trees can be paths. This raises the following interesting open question: When does this transition happen and is it gradual? That is, does the number of spanning paths increase whenever the hub passes over certain diagonals? Note, though, that spanning paths can of course be used in packings which are not partitions. More specifically, $GW_{2n}$ always contains $n-1$ spanning paths. Only when we ask for a complete partition of the edges we cannot use even a single spanning path.
On the positive side we argue that $GK_n$ contains at least 2 edge-disjoint spanning paths if $n \geq 4$. Obviously it would be desirable to extend our argument to $3$ or more paths or to develop a different line of reasoning to prove that $GK_n$ always contains many paths. Alternatively, it would be very interesting to find point sets which contain only few edge-disjoint plane spanning paths.
We also study packings of edge-disjoint planar spanning trees that have bounded vertex degree and bounded diameter. In particular, in Section~\ref{sec:degree} we show that for any $k \leq \sqrt{n/12}$ any set of $n$ points has $k$ edge-disjoint plane spanning trees with maximum vertex degree $O(k^2)$ and diameter $O(\log(n/k^2))$.
\paragraph{Related work} A classic related problem in extremal graph theory is the following. For general geometric graphs, what is the maximum number $f(k,n)$ such that there exists a geometric graph~$G$ of~$n$ vertices and $f(k,n)$ edges such that $G$ contains no $k$ disjoint edges? Erd\H{o}s~\cite{disjoint_pair} showed that for all $n \geq 3$, $f(2,n) = n$, i.e., any geometric graph with $n+1$ edges contains a disjoint pair.
For general $k$, T\'oth and Valtr~\cite{toth_valtr} gave the lower and upper bounds of $3/2(k-1)n - 2k^2 \leq f(k+1, n) \leq k^3(n+1)$, and also showed that $4n-9 \leq f(4,n) \leq 8.5n$. \v{C}ern\'y~\cite{cerny} proved $f(3,n) \leq \floor{2.5n}$. More specifically, the existence of certain plane subgraphs has been investigated. K\'arolyi, Pach, and T\'oth~\cite{monochromatic_pst} showed that any edge 2-coloring of a complete geometric graph $GK_n$ admits a monochromatic plane spanning tree.
\v{C}ern\'y et al.~\cite{noncrossing_hamiltonian} also considered the existence of plane spanning trees in geometric graphs. They showed that after removing any set of at most $(1/2 \sqrt{2})\sqrt{n}$ edges from any $GK_n$, the resulting graph still contains a plane spanning path.
Aichholzer et al.~\cite{noncrossing_configurations} considered perfect matchings, subtrees and triangulations as plane subgraphs; further references to similar results can be found in~\cite{noncrossing_configurations}.
For any geometric graph~$G$, Rivera-Campo~\cite{five_points} showed that if any subgraph of $G$ induced by five vertices has a plane spanning tree, then $G$ as well has a plane spanning tree. Keller et al.~\cite{keller} gave a characterization of the smallest subgraphs of any~$GK_n$ that share at least one edge with any plane spanning tree of~$GK_n$ (so-called \emph{blockers}). They showed that if a subgraph~$G$ is a blocker for all plane spanning trees of diameter at most four, then $G$ blocks all plane spanning subgraphs; if the vertices of $GK_n$ are in convex position, the result already holds for a diameter of at most three.
Also the number of plane spanning trees attracted interest, analogously to classic results on the number of spanning trees (the \emph{tree density}) in general graphs. Nash-Williams~\cite{nash_williams} and Tutte~\cite{tutte} independently showed that a graph~$G$ has a tree density of $k$ if $|E_P(G)| \geq k (|P|-1)$ for every partition $P$ of $V(G)$, where $E_P(G)$ denotes the set of edges between different members of~$P$. This was used by Kundu~\cite{kundu} to relate the tree density in general graphs to their edge-connectivity: any $k$-edge-connected graph has at least $\ceil{k-1/2}$ edge-disjoint spanning trees.
Our problem is also closely related to the concept of \emph{$k$-book embeddings} of topological graphs, where, informally, the vertices are considered to be on the spine of a book and each edge of the graph is either on the spine or on exactly one of the $k$ pages, such that no two edges cross. The \emph{book thickness} of a graph~$G$ is the smallest number~$k$ for which there exists a $k$-book embedding of~$G$. Bernhart and Kainen~\cite[Theorem~3.4]{book_embeddings} showed that, for $n \geq 4$ vertices, the book thickness of the complete graph is $\ceil{n/2}$. Their construction of $\floor{n/2}$ edge-disjoint paths directly carries over to packing the same amount of plane spanning paths in the complete convex geometric graph~\cite{partitions_into_trees}.
A concept between graph-theoretical thickness and book thickness was later developed by Dillencourt, Eppstein, and Hirschberg~\cite{thickness}: given an abstract graph~$G$, the \emph{geometric thickness} of~$G$ is the smallest number~$k$ such that there exists a straight-line drawing of the graph that can be partitioned into $k$ plane subgraphs. They showed that the geometric thickness of the (abstract) complete graph is between $\ceil{(n/5.646)+0.342}$ and $\ceil{n/4}$.
Since the initial presentation of this work, the problem has attracted further attention. Most prominently, the lower bound on the number of plane edge-disjoint spanning trees has been improved to~$\lfloor n/3 \rfloor$ by Garc{\'\i}a~\cite{g-nepstg-15}. Schnider~\cite{double_stars} considers the special case of double stars (i.e., trees with only two interior nodes), showing that a partition into such trees does not always exist, and provides necessary as well as sufficient conditions for its existence.
\section{Packing Spanning Trees}\label{sec:trees}
Recall that $GK_{n}$ is the complete geometric graph on any set $S$ of $n$ points in general position in the plane.
\begin{theorem}\label{thm_sqrtlayers} $GK_{n}$ contains $\Omega(\sqrt{n})$ edge-disjoint plane spanning trees. \end{theorem} \begin{proof} Let $S$ be a set of $n$ points in the plane, and let $F$ be a set of $k$ edges (pairs of points of $S$) such that each pair of edges in $F$ has an interior crossing. The set $F$ is called a \emph{crossing family}.
We claim that there exists a set of $k$ edge-disjoint plane spanning trees on $S$. We use a construction similar to the \emph {double stars} by Bose {et al.}{}~\cite{partitions_into_trees}. For each edge $e = \overline{pq} \in F$, let $\ell_e$ be the supporting line of $e$. We connect all points to the left of $\ell_e$ to $p$, and all points to the right of $\ell_e$ to $q$. These edges together with $e$ form a tree~$T_e$ (see Figure~\ref{fig:doublestar}).
To see that this yields $k$ edge-disjoint trees, consider two trees $T_{\overline{pq}}$ and $T_{\overline{rs}}$. Suppose some edge is in both trees. Then one of its endpoints must be $p$ or $q$, and the other endpoint must be $r$ or $s$. However, if $r$ lies to the left of $\ell_{\overline{pq}}$, then $\overline {pr}$ and $\overline {qs}$ are in $T_{\overline{pq}}$ and $\overline {ps}$ and $\overline {qr}$ are in $T_{\overline{rs}}$, and vice versa if $r$ lies to the right of $\ell_{\overline{pq}}$.
Aronov {et al.}~\cite{crossing_families} showed that any set of $n$ points contains a crossing family of size $\sqrt{n/12}$. The theorem follows immediately. \end{proof}
\begin{figure}
\caption{A set of $15$ points with $4$ pairwise crossing edges.}
\label{fig:doublestar}
\end{figure}
\noindent In a set of $h$ points in convex position, there is always a crossing family of size $\floor{h/2}$. The proof of Theorem~\ref{thm_sqrtlayers} therefore immediately implies the following.
\begin{corollary} The complete graph of a set $S$ of $n$ points, of which $h$ are in convex position, contains at least $\floor{h/2}$ edge-disjoint plane spanning trees. \end{corollary}
\begin{theorem}\label{thm:2trees} If $n \geq 4$ then $GK_{n}$ contains at least 2 edge-disjoint plane spanning trees. \end{theorem} \begin{proof} \begin{figure}
\caption{The two cases for constructing two edge-disjoint plane
spanning trees on $S$.}
\label{fig:2trees}
\end{figure} Let $S$ be a set of $n$ points in the plane and let $e=rb$ be an edge spanned by $S$ having exactly 2 points ($p$ and $q$) of $S$ on one side (i.e., on one side of the straight line supporting $e$). The set $\{p,q,r,b\}$ is either in convex position (Case~1; see \figurename~\ref{fig:2trees}~(left)) or forms a triangle with one interior point (Case~2; see \figurename~\ref{fig:2trees}~(right)). Note that $e$ has to be an edge of the convex hull of $\{p,q,r,b\}$. W.l.o.g., let $pqrb$ be the convex polygon in Case~1 and let $q$ be the point inside the triangle $prb$ in Case~2. In both cases we construct two edge-disjoint spanning trees on $\{p,q,r,b\}$, $\spath{q,r,p,b}$ (blue) and $\spath{p,q,b,r}$ (red). To get two edge-disjoint spanning trees on $S$ we connect all points of $S\setminus\{p,q,r,b\}$ with $b$ (for the blue tree) and with $r$ (for the red tree). \end{proof}
Note that the proof of Theorem~\ref{thm_sqrtlayers} also immediately implies Theorem~\ref{thm:2trees} for $n\geq5$, because then there always exists a pair of crossing edges in $GK_{n}$. For $n=4$ the two cases for $\{p,q,r,b\}$ shown in \figurename~\ref{fig:2trees} serve as a proof.
\begin{lemma}\label{lem:6points} $GK_{6}$ contains 3 edge-disjoint plane spanning trees. \end{lemma} \begin{proof} For $n=6$ there exist 16 combinatorially different point sets (order types)~\cite{aak-eotsp-01a}.
It is easy to check that each of these 16 cases allows for 3 edge-disjoint plane spanning trees packed on $GK_{6}$ (see \figurename~\ref{fig:6points}). \end{proof}
Using the order type database for small point sets~\cite{a-otdb-06} it can be easily checked that $GK_{8}$ and $GK_{9}$ each contain 4 edge-disjoint plane spanning trees, and that $GK_{10}$ contains 5 edge-disjoint plane spanning trees. (The latter has been obtained by reducing the set of order types to so-called crossing-maximal ones, as characterized in~\cite{pw-oo-15}.)
\begin{figure}
\caption{Two examples depicting the construction of three edge-disjoint plane
spanning trees on $S$.}
\label{fig:3trees}
\end{figure}
\begin{theorem} If $n \geq 6$ then $GK_{n}$ contains at least 3 edge-disjoint plane spanning trees. \end{theorem} \begin{proof}
Let $S$ be a set of $n$ points in the plane and let $e=rb$ be an edge spanned by $S$ having exactly 4 points of $S$ on one side (i.e., on one side of the straight line $\ell_e$ supporting $e$). Let $S'$ be the set of 6 points containing $r$, $b$, and the exactly 4 points on one side of $e$.
By Lemma~\ref{lem:6points}, $S'$ contains 3 edge-disjoint plane spanning trees. For simplicity we call them red, blue, and green. W.l.o.g., assume that $e$ is part of the red tree. Note that each point of $S'$ is incident to all three trees, and that $r$ and $b$ are extremal points for $S\setminus (S'\setminus\{r,b\})$. We construct a red and a blue plane spanning tree by connecting $r$ and $b$, respectively, with all points in $S\setminus S'$.
Next we construct the third (green) plane spanning tree on $S$. Note that the green plane spanning tree on $S'$ can be completed to a triangulation $T$. Let $q$ be the point of $S'\setminus\{r,b\}$ such that $qrb$ is a triangle in $T$.
Observe that any edge incident to $q$ and crossing $e$ does not cross a green edge.
Assume that there exists a point $q'\in(S\setminus S')$ such that the edge $qq'$ crosses~$e$. Then we connect $q$ and $q'$ with a green edge and complete the green plane spanning tree by connecting all points in $S\setminus(S'\cup\{q'\})$ with $q'$. See \figurename~\ref{fig:3trees}~(left).
If such a point $q'$ does not exist, then there has to exist an edge $e'$ of the convex hull of $S$, such that $e'$ crosses $\ell_e$. Denote by $p$ the endpoint of $e'$ in $S\setminus S'$. We color $e'$ green and complete the green plane spanning tree by connecting all points in $S\setminus(S'\cup\{p\})$ with $p$. See \figurename~\ref{fig:3trees}~(right). \end{proof}
\section{Packing Spanning Paths}\label{sec:paths}
Let $W_{2n}$ be a set of $2n$ points in the ``regular wheel configuration'' in the plane. $W_{2n}$ consists of $2n-1$ points regularly spaced on a circle $C$ and a point at the center of $C$. Let $GW_{2n}$ be the complete geometric graph on $W_{2n}$.
\begin{figure*}
\caption{The graph $GW_{2n}$ cannot have plane spanning paths if it is partitioned into plane spanning trees.}
\label{fig:wheel}
\end{figure*}
\begin{theorem} $GW_{2n}$ can be partitioned into $n$ spanning trees. If $n\geq 3$ then none of these trees can be a path. \end{theorem} \begin{proof} In the following, we color the edges of $GW_{2n}$ that each class is plane and spanning. Let $v_{0}$ be the central vertex and let the other vertices be $v_1,\ldots, v_{2n-1}$ in cyclic order. The complete graph has edges of varying length between the vertices $v_1,\ldots, v_{2n-1}$, and we can use $E_1,\ldots,E_{n-1}$ to denote the length classes of the edges, from short to long. The edges involving $v_{0}$ are called the radial edges. There are $2n-1$ edges in each length class and also $2n-1$ radial edges.
We first consider the length class $E_{n-1}$, then the radial edges, and then $E_{n-2},\ldots,E_1$, and see how we must color these edges to produce plane spanning trees.
Given that there are $2n-1$ edges in $E_{n-1}$, to be divided over $n$ colors, and every non-adjacent pair of edges intersect, we will get these edges in $n-1$ pairs and one singleton, see Figure~\ref{fig:wheel}(a). Call the color of the singleton edge in $E_{n-1}$ red. The pairs must be two adjacent edges (they have a shared vertex), forming a wedge with point $v_{0}$ in between and at least one point to each side of the wedge if there are at least six points. This immediately shows that all spanning trees with non-red color are not paths. To show that a red spanning tree also cannot be a path, we observe that $v_0$ can have at most one edge in each non-red color (otherwise we make a cycle or an intersection within that color). Therefore, it must have $n$ incident red edges, showing that the red spanning tree is not a path either if $n\geq 3$ (Figure~\ref{fig:wheel}(b)).
We proceed to show that the geometric graph contains $n$ plane spanning trees. We color the radial edges by using the red color $n$ times. There are two options when we do not have crossings or cycles, and they are symmetric. The remaining radial edges get the other $n-1$ colors, one for each, and such that a path of length $3$ appears in each color. Then we assign the edges in $E_{n-2},\ldots,E_1$ a color at once. We make $2n-1$ fans, one for each of $v_1,\ldots, v_{2n-1}$, consisting of one edge of each length class (there are two choices: clockwise and counterclockwise), see Figure~\ref{fig:wheel}(c) for the two fans of one color. Each fan can be assigned a color so that all spanning trees are isomorphic balanced double stars, completing the partitioning into $n$ plane spanning trees (Figure~\ref{fig:wheel}(d)). \end{proof}
\begin{figure}
\caption{$GW_{2n}$ contains $n-1$ plane spanning paths.}
\label{fig:pathsinwheel}
\end{figure}
Interestingly, $GW_{2n}$ contains $n-1$ plane spanning paths, via the zigzag construction used for points in convex position (as described in~\cite{partitions_into_trees}). When the path passes the center point, it picks it up using two radial edges instead of a long edge, see Figure~\ref{fig:pathsinwheel}. But to get one more plane spanning tree in $GW_{2n}$, all paths must be trees.
We now return to $GK_{n}$, the complete geometric graph on any set $S$ of $n$ points in general position in the plane.
\begin{theorem} If $n \geq 4$ then $GK_{n}$ contains at least 2 edge-disjoint plane spanning paths. \end{theorem} \begin{proof}
Let $S$ be a set of $n$ points in the plane and let $p$ be an extremal point of $S$. Order the points of $S\setminus\{p\}$ clockwise around $p$. Partition $S\setminus\{p\}$ into two (disjoint) sets $A$ and $B$, such that $A\cup B = S\setminus\{p\}$ and $|B|-1 \leq |A| \leq |B|$.
We denote by $\ell$ a line through $p$ (but no other point of $S$) that is separating $A$ from $B$ (see \figurename~\ref{fig:partition}).
\begin{figure}
\caption{Partition of $S\setminus\{p\}$ and the two edge-disjoint
spanning paths. Left: $q\neq b_l$. Right: $q=b_l$.}
\label{fig:partition}
\end{figure}
We will construct the two edge-disjoint paths, for simplicity call them red and blue.
The red path (${\cal R}=G(V,E_1)$) we simply construct as a plane zigzag path starting at $p$, with a point $q$ in $B$ as a second point, and with every edge of $\cal R$, except $pq$, intersecting $l$. (An algorithm for constructing such a zigzag path is described by Hershberger and Suri~\cite{hershberger_suri}, see also Abellanas et al.~\cite{bipartite_embeddings}.)
The blue path (${\cal B}=G(V,E_2)$) consists of two subpaths, ${\cal B}_A$ and ${\cal B}_B$, joined at $p$.
Observe that no red edge (edge of $\cal R$) connects two points of $A\cup\{p\}$ or two points of $B$.
Thus, any (blue) path completely contained in $A\cup\{p\}$ is edge-disjoint to $\cal R$. We choose the path starting at $p$ and connecting the points of $A$ in clockwise order around $p$ for ${\cal B}_A$.
Let $b_f$ and $b_l$ be the first and last, respectively, point of $B$ in clockwise order around $p$.
If $q=b_l$ then we connect $p$ with $b_f$ and continue on the points of $B\setminus\{b_f\}$ in clockwise order around $p$ for ${\cal B}_B$ (see \figurename~\ref{fig:partition}~(right)).
Otherwise, we construct ${\cal B}_B$ with $pb_l$ as the first edge and then finish the path by connecting the points of $B\setminus\{b_l\}$ in counter clockwise order around $p$ (see \figurename~\ref{fig:partition}~(left)).
Connecting ${\cal B}_A$ and ${\cal B}_B$ at $p$ results in the plane spanning path ${\cal B}$ that is edge-disjoint to the plane spanning path ${\cal R}$. \end{proof}
\section{Packing Spanning Trees with low Degree}\label{sec:degree}
The edge-disjoint plane spanning trees we studied in the previous sections are somehow extreme in terms of vertex degree. The trees constructed in Section~\ref{sec:trees} always contain at least one vertex of degree $\Omega(n)$, while in Section~\ref{sec:paths} we consider spanning paths. Thus the question arises if intermediate results are possible.
In the following, we obtain a trade-off between the number of edge-disjoint spanning trees and the maximum degree of each vertex.
\begin{figure}
\caption{The hierarchical clustering strategy.}
\label{fig_decomp}
\end{figure}
\begin{figure*}
\caption{The 16 combinatorially different point sets
for $n=6$~\cite{aak-eotsp-01a,a-otdb-06}, with 3 edge-disjoint plane spanning trees each.}
\label{fig:6points}
\end{figure*}
\begin{theorem}\label{theo_hierarch} For any set $S$ of $n$ points and $k\leq \sqrt{n/12}$ there exist $k$ edge-disjoint plane spanning trees $T_1,\ldots, T_k$ on~$S$ such that the maximum degree of any tree is in $O(k^2)$. Also, the diameter of each tree is in $O(\log(n/k^2))$. \end{theorem} \begin{proof} The general idea of the proof is to ``peel off'' small clusters of points and connect each of the clusters with $k$ edge-disjoint spanning trees independently. Consider a $(12k^2-2)$-edge, i.e., an edge $uv$, $u,v \in S$, such that exactly $12k^2-2$ points of~$S$ are strictly to the left of the directed line~$\ell$ through~$uv$. Consider the set $C_1$ of these $12k^2$ points and construct $k$ edge-disjoint plane spanning trees of~$C_1$ using Theorem~\ref{thm_sqrtlayers}. Now consider the midpoint between $u$ and $v$. Let $\ell'$ be a line through that midpoint that splits the remaining point set $S \setminus C_1$ into two subsets $S_u$ and $S_v$, each containing at most $\ceil{(n-12k^2)/2}$ points.
Since the two subsets are separated by $\ell'$, we can recursively repeat a similar process in the two subsets independently. That is, pick a $(12k^2-2)$-edge $u'v'$ of $S_u \cup \{u\}$ such that $u$ is contained among the $12k^2$ points separated by $u'v'$ but is not an endpoint of the edge (such an edge must always exist). We construct $k$ plane spanning trees on this subset, which are connected to the spanning trees of~$C_1$ via~$u$. We treat $S_v \cup \{v\}$ analogously (see Figure~\ref{fig_decomp}). The recursion stops when we are not able to partition the remaining points into two sets of size at least $12k^2-1$; here, we simply add the remaining points of the subset to the last cluster. Note that this cluster must have between $12k^2$ and $36k^2-3$ points, thus we can still create $k$ edge-disjoint spanning trees using Theorem~\ref{thm_sqrtlayers}.
We construct the $k$ spanning trees of $S$ by assigning one of the spanning trees of each cluster arbitrarily to each of the trees $T_1, \ldots, T_k$. We claim that the resulting trees are indeed spanning: By construction, each tree is spanning in the cluster; hence points of the same cluster will be connected in $T_i$ (for all $i\leq k$). Moreover, the hierarchical construction certifies that each cluster shares a point with the cluster constructed in the previous step of induction. Likewise, planarity of each tree is guaranteed.
We obtain at most $N = \floor{n/(12k^2-1)}$ clusters which are arranged such that they form a balanced binary tree with $C_1$ as root. Note that the spanning trees constructed in the proof of Theorem~\ref{thm_sqrtlayers} have diameter~$3$. Thus, the diameter of each spanning tree is at most $6\ceil{\log_2 N}$. The degree bound follows from the fact that any point of $S$ can only belong to at most two clusters (and each cluster has $\Theta(k^2)$ points).
\end{proof}
\paragraph{Acknowledgments.} Research was initiated during the 10th European Research Week on Geometric Graphs (GGWeek 2013), Illgau, Switzerland. We would like to thank all participants of the 10th European Research Week on Geometric Graphs for fruitful discussions. T.~H.\ was supported by the Austrian Science Fund (FWF): P23629-N18. M.~K.\ was partially supported by MEXT KAKENHI No.~17K12635. A.~P.\ is supported by an Erwin Schr\"odinger fellowship, Austrian Science Fund (FWF): J-3847-N35.
\small
\end{document} | arXiv |
I am a postdoc in the Math Department at University of Hawaii (2016–2017). In fall 2017 I will move to University of Colorado, Boulder.
I have worked at Iowa State University (2014–2016) and University of South Carolina (2012–2014).
My primary research area is universal algebra; current projects focus on lattice theory, computational complexity, and universal algebraic approaches to constraint satisfaction problems. Other research interests include logic, computability theory, type theory, category theory, functional programming, dependent types, and proof-carrying code.
Most of my research papers are posted on my Math arXiv page or my CS arXiv page. A more comprehensive collection of my work resides in my Github repositories.
This post describes the steps I use to update or install multiple versions of the Java Development Kit (JDK) on Linux machines.
The alt-comm repository contains a note giving an alternate decription of the (non-modular) commutator that yields a polynomial time algorithm for computing it. This is inspired by the alternate description of the commutator given by Kearnes in .
It is not hard to see that 3-SAT reduces to the problem of deciding whether all coatoms in a certain partition lattice are contained in the union of a collection of certain principal filters. Therefore, the latter problem, which we will call the covered coatoms problem (CCP), is NP-complete. In this post we simply define CCP. Later we check that 3-SAT reduces to CCP, and then develop some ideas about constructing a feasible algorithm to solve CCP.
The questions below appeared on an online test administered by Jane Street Capital Management to assess whether a person is worthy of a phone interview.
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I made this version for my own reference, while working through the tutorial, making some revisions and additions, and a few corrections. You may prefer the original.
Consider two finite algebras that are the same except for a relabeling of the operations. Our intuition tells us that these are simply different names for the same mathematical structure. Formally, however, not only are these distinct mathematical objects, but also they are nonisomorphic.
Isotopy is another well known notion of "sameness" that is broader than isomorphism. However, I recently proved a result that shows why isotopy also fails to fully capture algebraic equivalence.
Claim: If the three $\alpha_i$'s pairwise permute, then all pairs in the lattice permute.
I started a GitHub repository called TypeFunc in an effort to get my head around the massive amount of online resources for learning about type theory, functional programming, category theory, $\lambda$-calculus, and connections between topology and computing. | CommonCrawl |
\begin{document}
\title{Spectral Variational Integrators}
\author[J.~Hall]{James Hall} \address{Department of Mathematics\\ University of California, San Diego\\ 9500 Gilman Drive \#0112\\ La Jolla, California 92093-0112, USA} \email{[email protected]}
\author[M.~Leok]{Melvin Leok} \address{Department of Mathematics\\ University of California, San Diego\\ 9500 Gilman Drive \#0112\\ La Jolla, California 92093-0112, USA} \email{[email protected]}
\begin{abstract} In this paper, we present a new variational integrator for problems in Lagrangian mechanics. Using techniques from Galerkin variational integrators, we construct a scheme for numerical integration that converges geometrically, and is symplectic and momentum preserving. Furthermore, we prove that under appropriate assumptions, variational integrators constructed using Galerkin techniques will yield numerical methods that are in a certain sense optimal, converging at the same rate as the best possible approximation in a certain function space. We further prove that certain geometric invariants also converge at an optimal rate, and that the error associated with these geometric invariants is independent of the number of steps taken. We close with several numerical examples that demonstrate the predicted rates of convergence. \end{abstract}
\maketitle
\allowdisplaybreaks
\section{Introduction}
There has been significant recent interest in the development of structure-preserving numerical methods for variational problems. One of the key points of interest is developing high-order symplectic integrators for Lagrangian systems. The generalized Galerkin framework has proven to be a powerful theoretical and practical tool for developing such methods. This paper presents a high-order Galerkin variational integrator that exhibits geometric convergence to the true flow of a Lagrangian system. In addition, this method is symplectic, momentum-preserving, and stable even for very large time steps.
Galerkin variational integrators fall into the general framework of discrete mechanics. For a general and comprehensive introduction to the subject, the reader is referred to \citet{MaWe2001}. Discrete mechanics develops mechanics from discrete variational principles, and, as Marsden and West demonstrated, gives rise to many discrete structures which are analogous to structures found in classical mechanics. By taking these structures into account, discrete mechanics suggests numerical methods which often exhibit excellent long term stability and qualitative behavior. Because of these qualities, much recent work has been done on developing numerical methods from the discrete mechanics viewpoint. See, for example, \citet{HaLuWa2006} for a broad overview of the field of geometric numerical integration, and \citet{MuOr2004, MaWe2001, PaCu2009} discuss the error analysis of variational integrators. Various extensions have also been considered, including, \citet{LaWe2006, LeZh2009} for Hamiltonian systems; \citet{FeMaOrWe2003} for nonsmooth problems with collisions; \citet{MaPaSh1998, LeMaOrWe2003} for Lagrangian PDEs; \citet{CoMa2001,McPe2006, FeZe2005} for nonholonomic systems; \citet{BoOw2009, BoOw2010} for stochastic Hamiltonian systems; \citet{LeLeMc2007, LeLeMc2009, BoMa2009} for problems on Lie groups and homogeneous spaces.
The fundamental object in discrete mechanics is the discrete Lagrangian \(L_{d}: Q \times Q \times \mathbb{R} \rightarrow \mathbb{R}\), where \(Q\) is a configuration manifold. The discrete Lagrangian is chosen to be an approximation to the action of a Lagrangian over the time step \(\left[0,h\right]\),
\begin{align*} L_{d}\lefri{q_{0},q_{1},h} \approx \ext_{\truqargsf{q_{0}}{q_{1}}} \int_{0}^{h} L\lefri{q,\dot{q}} \dt, \end{align*}
or simply \(L_{d}\lefri{q_{0},q_{1}}\) when \(h\) is assumed to be constant. Discrete mechanics is formulated by finding stationary points of a discrete action sum based on the sum of discrete Lagrangians,
\begin{align*} \mathbb{S}\lefri{\left\{q_{k}\right\}_{k=1}^{n}} = \sum_{k=1}^{n-1} L_{d}\lefri{q_{k},q_{k+1}} \approx \int_{t_{1}}^{t_{2}} L\lefri{q,\dot{q}}\dt. \end{align*}
For Galerkin variational integrators specifically, the discrete Lagrangian is induced by constructing a discrete approximation of the action integral over the interval \(\left[0,h\right]\) based on a finite-dimensional function space and quadrature rule. Once this discrete action is constructed, the discrete Lagrangian can be recovered by solving for stationary points of the discrete action subject to fixed endpoints, and then evaluating the discrete action at these stationary points,
\begin{align} L_{d}\lefri{q_{0},q_{1},h} = \ext_{\galargsf{q_{0}}{q_{1}}} h\sum_{j=1}^{m} b_{j}L\lefri{q\lefri{c_{j}h},\dot{q}\lefri{c_{j}h}}. \label{discaction} \end{align}
Because the rate of convergence of the approximate flow to the true flow is related to how well the discrete Lagrangian approximates the true action, this type of construction gives a method for constructing and analyzing high-order methods. The hope is that the discrete Lagrangian inherits the accuracy of the function space used to construct it, much in the same way as standard finite-element methods. We will show that for certain Lagrangians, Galerkin constructions based on high-order approximation spaces do in fact result in correspondingly high order methods.
Significant work has already been done constructing and analyzing these types of Galerkin variational integrators. In \citet{Le04}, a number of different possible constructions based on the Galerkin framework are presented. In \citet{LeSh2011}, Hermite polynomials are used to construct globally smooth high-order methods. What separates this work from the work that precedes it is the use of a spectral approximation paradigm, which induces methods that exhibit geometric convergence. This type of convergence is established theoretically and demonstrated through numerical examples.
\subsection{Discrete Mechanics}
Before discussing the construction and convergence of spectral variational integrators, it is useful to review some of the fundamental results from discrete mechanics that are used in our analysis. We have already introduced the \emph{discrete Lagrangian} \(L_{d}:Q \times Q \times \mathbb{R} \rightarrow \mathbb{R}\),
\begin{align*} L_{d} \lefri{q_{0},q_{1},h} \approx \ext_{\truqargsf{q_{0}}{q_{1}}} \int_{0}^{h}L\lefri{q,\dot{q}}\dt. \end{align*}
and the \emph{discrete action sum},
\begin{align*} \mathbb{S}\lefri{\left\{q_{k}\right\}_{k=1}^{n}} = \sum_{k=1}^{n-1} L_{d}\lefri{q_{k},q_{k+1}} \approx \int_{t_{1}}^{t_{2}} L\lefri{q,\dot{q}}\dt. \end{align*}
Taking variations of the discrete action sum and using discrete integration by parts leads to the discrete Euler-Lagrange equations,
\begin{align} D_{2}L_{d}\lefri{q_{k-1},q_{k}} + D_{1}L_{d}\lefri{q_{k},q_{k+1}} = 0, \label{DEL} \end{align}
where \(D_{1}\) denotes differentiation with respect to the first argument and \(D_{2}\) denotes differentiation with respect to the second argument. Given \(\lefri{q_{k-1},q_{k}}\), these equations implicitly define an update map, known as the \textit{discrete Lagrangian flow map}, \(F_{L_{d}}: Q \times Q \rightarrow Q \times Q\), given by \(F_{L_{d}}\lefri{q_{k-1},q_{k}} = \lefri{q_{k},q_{k+1}}\), where \(\lefri{q_{k-1},q_{k}},\lefri{q_{k},q_{k+1}}\) satisfy (\ref{DEL}). Furthermore, the discrete Lagrangian defines the \textit{discrete Legendre transforms}, \(\mathbb{F}^{\pm}L_{d}: Q\times Q \rightarrow T^{*}Q\):
\begin{align*} \mathbb{F}^{+}L_{d}&:\lefri{q_{0},q_{1}} \rightarrow \lefri{q_{1},p_{1}} = \lefri{q_{1},D_{2}L_{d}\lefri{q_{0},q_{1}}}, \\ \mathbb{F}^{-}L_{d}&:\lefri{q_{0},q_{1}} \rightarrow \lefri{q_{0},p_{0}} = \lefri{q_{0},-D_{1}L_{d}\lefri{q_{0},q_{1}}}. \end{align*}
Using the discrete Legendre transforms, we define the \textit{discrete Hamiltonian flow map}, \(\tilde{F}_{L_{d}}: T^{*}Q \rightarrow T^{*}Q\),
\begin{align*} \tilde{F}_{L_{d}} &: \lefri{q_{0},p_{0}} \rightarrow \lefri{q_{1},p_{1}} = \mathbb{F}^{+}L_{d}\lefri{\lefri{\mathbb{F}^{-}L_{d}}^{-1}\lefri{q_{0},p_{0}}}. \end{align*}
The following commutative diagram illustrates the relationship between the discrete Hamiltonian flow map, discrete Lagrangian flow map, and the discrete Legendre transforms,
\begin{align*} \xymatrix{ & \lefri{q_{k},p_{k}} \ar[rr]^{\tilde{F}_{L_{d}}} & & \lefri{q_{k+1},p_{k+1}} & \\ & & & & \\ \lefri{q_{k-1},q_{k}} \ar[uur]^{\mathbb{F}^{+}L_{d}} \ar[rr]_{F_{L_{d}}} & & \lefri{q_{k},q_{k+1}} \ar[rr]_{F_{L_{d}}} \ar[uur]^{\mathbb{F}^{+}L_{d}} \ar[uul]_{\mathbb{F}^{-}L_{d}}& &\lefri{q_{k+1},q_{k+2}} \ar[uul]_{\mathbb{F}^{-}L_{d}} } \end{align*}
We now introduce the \textit{exact discrete Lagrangian} \(L^{E}_{d}\),
\begin{align*} L^{E}_{d} \lefri{q_{0},q_{1},h} = \ext_{\truqargs{q_{0}}{q_{1}}}\int_{0}^{h}L\lefri{q,\dot{q}}\dt. \end{align*}
An important theoretical result for the error analysis of variational integrators is that the discrete Hamiltonian and Lagrangian flow maps associated with the exact discrete Lagrangian produces an exact sampling of the true flow, as was shown in \citet{MaWe2001}. Using this result, \citet{MaWe2001} shows that there is a fundamental relationship between how well a discrete Lagrangian \(L_{d}\) approximates the exact discrete Lagrangian \(L_{d}^{E}\) and how well the corresponding discrete Hamiltonian flow maps, discrete Lagrangian flow maps and discrete Legendre transforms approximate each other. Since the exact discrete Lagrangian produces an exact sampling of the true flow, this in turn leads to the following theorem regarding the error analysis of variational integrators, also found in \citet{MaWe2001}:
\begin{theorem} \emph{(Variational Error Analysis)} \label{MarsConv} Given a regular Lagrangian \(L\) and corresponding Hamiltonian \(H\), the following are equivalent for a discrete Lagrangian \(L_{d}\): \begin{enumerate} \item the discrete Hamiltonian flow map for \(L_{d}\) has error \(\mathcal{O}\lefri{h^{p+1}}\), \item the discrete Legendre transforms of \(L_{d}\) have error \(\mathcal{O}\lefri{h^{p+1}}\), \item \(L_{d}\) approximates the exact discrete Lagrangian with error \(\mathcal{O}\lefri{h^{p+1}}\). \end{enumerate} \end{theorem}
\noindent We will make extensive use of this theorem later when we analyze the convergence of spectral variational integrators.
In addition, in \citet{MaWe2001}, it is shown that integrators constructed in this way, which are referred to as \textit{variational integrators}, have significant geometric structure. Most importantly, variational integrators always conserve the canonical symplectic form, and a discrete Noether's Theorem guarantees that a discrete momentum map is conserved for any continuous symmetry of the discrete Lagrangian. The preservation of these discrete geometric structures underlie the excellent long term behavior of variational integrators.
\section{Construction}
\subsection{Generalized Galerkin Variational Integrators}
\begin{figure}
\caption{A visual schematic of the curve \(\galqn\lefri{t} \in \FdFSpace\). The points marked with (\(\times\)) represent the quadrature points, which may or may not be the same as interpolation points \(d_{i}h\). In this figure we have chosen to depict a curve constructed from interpolating basis functions, but this is not necessary in general.}
\label{GalCurveVis}
\end{figure}
The construction of spectral variational integrators falls within the framework of generalized Galerkin variational integrators, discussed in \citet{Le04} and \citet{MaWe2001}. The motivating idea is that we try to replace the generally non-computable exact discrete Lagrangian \(\EDL{q_{k}}{q_{k+1}}\) with a computable discrete analogue, \(\GDL{q_{k}}{q_{k+1}}\). Galerkin variational integrators are constructed by using a finite-dimensional function space to discretize the action of a Lagrangian. Specifically, given a Lagrangian \(L:TQ \rightarrow \mathbb{R}\), to construct a Galerkin variational integrator:
\begin{enumerate} \item choose an \(n\)-dimensional function space \(\FdFSpace \subset \CQ\), with a finite set of basis functions \(\left\{\phi_{i}\lefri{t}\right\}_{i=1}^{n}\), \item choose a quadrature rule \(\mathcal{G}\lefri{\cdot}:F\lefri{\left[0,h\right],\mathbb{R}}\rightarrow\mathbb{R}\), so that \(\mathcal{G}\lefri{f} = h\sum_{j=1}^{m} b_{j}f\lefri{c_{j}h} \approx \int_{0}^{h} f\lefri{t} \dt\), where \(F\) is some appropriate function space, \end{enumerate}
and then construct the discrete action \(\mathbb{S}_{d}\lefri{\left\{q^{i}_{k}\right\}_{i=1}^{n}}:\prod_{i=1}^{n} Q_{i} \rightarrow \mathbb{R}\), (not to be confused with the discrete action sum \(\mathbb{S}\lefri{\left\{q_{k}\right\}_{k=1}^{\infty}}\)),
\begin{align*} \mathbb{S}_{d}\lefri{\left\{q_{k}^{i}\right\}_{i=1}^{n}} = \mathcal{G}\lefri{L\lefri{\sum_{i=1}^{n} q_{k}^{i}\phi_{i}\lefri{t}, \sum_{i=1}^{n}q_{k}^{i}\dot{\phi}_{i}\lefri{t}}} = h\sum_{j=1}^{m} b_{j} L\lefri{\sum_{i=1}^{n} q_{k}^{i} \phi_{i}\lefri{c_{j}h}, \sum_{i=1}^{n} q_{k}^{i} \dot{\phi}_{i}\lefri{c_{j}h}}, \end{align*}
where we use superscripts to index the weights associated with each basis function, as in \citet{MaWe2001}.
Once the discrete action has been constructed, a discrete Lagrangian can be induced by finding stationary points \(\tilde{q}_{n}\lefri{t} = \sum_{i=1}^{n}q_{k}^{i}\phi_{i}\lefri{t}\) of the action under the conditions \(\tilde{q}_{n}\lefri{0} = \sum_{i=1}^{n} q_{k}^{i} \phi_{i}\lefri{0} = q_{k}\) and \(\tilde{q}_{n}\lefri{h} = \sum_{i=1}^{n} q_{k}^{i} \phi_{i}\lefri{h} = q_{k+1}\) for some given \(q_{k}\) and \(q_{k+1}\),
\begin{align*} L_{d}\lefri{q_{k},q_{k+1},h} = \ext_{\substack{\tilde{q}_{n}\lefri{0} = q_{k}\\ \tilde{q}_{n}\lefri{h} = q_{k+1}}} \mathbb{S}_{d}\lefri{\left\{q_{k}^{i}\right\}_{i=1}^{n}} = \ext_{\substack{\tilde{q}_{n}\lefri{0} = q_{k}\\ \tilde{q}_{n}\lefri{h} = q_{k+1}}} h\sum_{j=1}^{m} b_{j}L\lefri{\tilde{q}_{n}\lefri{c_{j}h},\dot{\tilde{q}}_{n}\lefri{c_{j}h}}. \end{align*}
A discrete Lagrangian flow map that result from this type of discrete Lagrangian is referred to as a Galerkin variational integrator.
\subsection{Spectral Variational Integrators}
There are two defining features of spectral variational integrators. The first is the choice of function space \(\FdFSpace\), and the second is that convergence is achieved not by shortening the time step \(h\), but by increasing the dimension \(n\) of the function space.
\subsubsection{Choice of Function Space}
Restricting our attention to the case where \(Q\) is a linear space, spectral variational integrators are constructed using the basis functions \(\phi_{i}\lefri{t} = l_{i}\lefri{t}\), where \(l_{i}\lefri{t}\) are Lagrange interpolating polynomials based on the points \(h_{i} = \frac{h}{2}\cos\lefri{\frac{i \pi}{n}} + \frac{h}{2}\) which are the Chebyshev points \(t_{i} = \cos\lefri{\frac{i \pi}{n}}\), rescaled and shifted from \(\left[-1,1\right]\) to \(\left[0,h\right]\). The resulting finite dimensional function space \(\FdFSpace\) is simply the polynomials of degree at most \(n\) on \(Q\). However, the choice of this particular set of basis functions offer several advantages over other possible bases for the polynomials:
\begin{enumerate} \item the restriction on variations \(\sum_{i=1}^{n} \delta q_{k}^{i} \phi_{i}\lefri{0} = \sum_{i=1}^{n} \delta q_{k}^{i} \phi_{i}\lefri{h} = 0\) reduces to \(\delta q_{k}^{1} = \delta q_{k}^{n} = 0\), \item the condition \(\galqn\lefri{0} = q_{k}\) reduces to \(q_{k}^{1} = q_{k}\), \item the induced numerical methods have generally better stability properties because of the excellent approximation properties of the interpolation polynomials at the Chebyshev points. \end{enumerate}
Using this choice of basis functions, for any chosen quadrature rule, the discrete Lagrangian becomes,
\begin{align*} L_{d}\lefri{q_{k},q_{k+1},h} = \ext_{\galargsvf{q_{k}^{1}}{q_{k}}{q_{k}^{n}}{q_{k+1}}} h\sum_{j=1}^{m} b_{j} L\lefri{\galqn\lefri{c_{j}h},\dgalqn\lefri{c_{j}h}}. \end{align*}
Requiring the curve \(\galqn\lefri{t}\) to be a stationary point of the discretized action provides \(n-2\) internal stage conditions:
\begin{align} h\sum_{j=1}^{m} b_{j}\lefri{\dLdq \lefri{\galqn\lefri{c_{j}h},\dgalqn\lefri{c_{j}h}}\phi_{p}\lefri{c_{j}h} + \dLddq \lefri{\galqn \lefri{c_{j}h}, \dgalqn\lefri{c_{j}h}}\dot{\phi}_{p}\lefri{c_{j}h}} &= 0, & p = 2,...,n-1. \label{InterStage} \end{align}
Combining these internal stage conditions with the discrete Euler-Lagrange equations,
\begin{align*} D_{1}L_{d}\lefri{q_{k-1},q_{k}} + D_{2}L_{d}\lefri{q_{k},q_{k+1}} = 0, \end{align*}
and the continuity condition \(q_{k}^{1} = q_{k}\) yields the following set of \(n\) nonlinear equations,
\begin{align} q_{k}^{1} &= q_{k}, & \nonumber \\ h\sum_{j=1}^{m} b_{j}\lefri{\dLdq \lefri{\galqn\lefri{c_{j}h},\dgalqn\lefri{c_{j}h}}\phi_{p}\lefri{c_{j}h} + \dLddq \lefri{\galqn \lefri{c_{j}h}, \dgalqn\lefri{c_{j}h}}\dot{\phi}_{p}\lefri{c_{j}h}} &= 0, & p = 2,...,n-1, \nonumber\\ h\sum_{j=1}^{m} b_{j}\lefri{\dLdq \lefri{\galqn\lefri{c_{j}h},\dgalqn\lefri{c_{j}h}}\phi_{1}\lefri{c_{j}h} + \dLddq \lefri{\galqn \lefri{c_{j}h}, \dgalqn\lefri{c_{j}h}}\dot{\phi}_{1}\lefri{c_{j}h}} &= p_{k-1}, & \label{momcon} \end{align}
which must be solved at each time step \(k\), and the momentum condition:
\begin{align*} h\sum_{j=1}^{m} b_{j}\lefri{\dLdq \lefri{\galqn\lefri{c_{j}h},\dgalqn\lefri{c_{j}h}}\phi_{n}\lefri{c_{j}h} + \dLddq \lefri{\galqn \lefri{c_{j}h}, \dgalqn\lefri{c_{j}h}}\dot{\phi_{n}}\lefri{c_{j}h}} &= p_{k}, \end{align*}
which defines (\ref{momcon}) for the next time step. Evaluating \(q_{k+1} = \galqn\lefri{h}\) defines the next step for the discrete Lagrangian flow map:
\begin{align*} F_{L_{d}}\lefri{q_{k-1},q_{k}} = \lefri{q_{k},q_{k+1}}, \end{align*}
and because of the choice of basis functions, this is simply \(q_{k+1} = q_{k}^{n}\).
\subsubsection{\(n\)-Refinement}
As is typical for spectral numerical methods (see, for example, \citet{Bo2001, Tr2000}), convergence for spectral variational integrators is achieved by increasing the dimension of the function space, \(\FdFSpace\). Furthermore, because the order of the discrete Lagrangian also depends on the order of the quadrature rule \(\mathcal{G}\), we must also refine the quadrature rule as we refine \(n\). Hence, for examining convergence, we must also consider the quadrature rule as a function of \(n\), \(\mathcal{G}_{n}\). Because of the dependence on \(n\) instead of \(h\), we will often examine the discrete Lagrangian \(L_{d}\) as a function of \(Q \times Q \times \mathbb{N}\),
\begin{align*} L_{d}\lefri{q_{k},q_{k+1},n} = \ext_{\galargsvf{q_{k}^{1}}{q_{k}}{q_{k}^{n}}{q_{k+1}}} \mathcal{G}_{n}\lefri{L\lefri{\galqn\lefri{t},\dgalqn\lefri{t}}} = \ext_{\galargsvf{q_{k}^{1}}{q_{k}}{q_{k}^{n}}{q_{k+1}}} h\sum_{j=1}^{m_{n}} \bnj L\lefri{\galqn\lefri{\cnjh},\dgalqn\lefri{\cnjh}}, \end{align*}
as opposed to the more conventional
\begin{align*} L_{d}\lefri{q_{k},q_{k+1},h} = \ext_{\galargsvf{q_{k}^{1}}{q_{k}}{q_{k}^{n}}{q_{k+1}}} \mathcal{G}\lefri{L\lefri{\galqn\lefri{t},\dgalqn\lefri{t}}} = \ext_{\galargsvf{q_{k}^{1}}{q_{k}}{q_{k}^{n}}{q_{k+1}}} h\sum_{j=1}^{m} b_{j} L\lefri{\galqn\lefri{c_{j}h},\dgalqn\lefri{c_{j}h}}. \end{align*}
This type of refinement is the foundation for the exceptional convergence properties of spectral variational integrators.
\section{Existence, Uniqueness and Convergence}
In this section, we will discuss the major important properties of Galerkin variational integrators and spectral variational integrators. The first will be the existence of unique solutions to the internal stage equations (\ref{InterStage}) for certain types of Lagrangians. The second is the convergence of the one-step map that results from the Galerkin and spectral variational constructions, which will be shown to be optimal in a certain sense. The third and final is the convergence of continuous approximations to the Euler-Lagrange flow which can easily be constructed from Galerkin and spectral variational integrators, and the behavior of geometric invariants associated with the approximate continuous flow. We will show a number of different convergence results associated with these quantities, which demonstrate that Galerkin and spectral variational integrators can be used to compute continuous approximations to the exact solutions of the Euler-Lagrange equations which have excellent convergence and geometric behavior.
\subsection{Existence and Uniqueness}
In general, demonstrating that there exists a unique solution to the internal stage equations for a spectral variational integrator is difficult, and depends on the properties of the Lagrangian. However, assuming a Lagrangian of the form
\begin{align*} L\lefri{q,\dot{q}} = \frac{1}{2} \dot{q}^{T}M\dot{q} - V\lefri{q}, \end{align*}
it is possible to show the existence and uniqueness of the solutions to the implicit equations for the one-step method under appropriate assumptions.
\begin{theorem}\label{ExisUniq}\emph{(Existence and Uniqueness of Solutions to the Internal Stage Equations)} Given a Lagrangian \(L:TQ \rightarrow \mathbb{R}\) of the form
\begin{align*} L\lefri{q,\dot{q}} = \frac{1}{2}\dot{q}^{T}M\dot{q} - V\lefri{q}, \end{align*}
if \(\nabla V\) is Lipschitz continuous, \(\bj > 0\) for every \(j\) and \(\sum_{i=1}^{m} \bj = 1\), and \(M\) is symmetric positive-definite, then there exists an interval \(\left[0,h\right]\) where there exists a unique solution to the internal stage equations for a spectral variational integrator. \end{theorem}
\begin{proof} We will consider only the case where \(q\lefri{t}\in \mathbb{R}\), but the argument generalizes easily to higher dimensions. To begin, we note that for a Lagrangian of the form,
\begin{align*} L\lefri{q,\dot{q}} = \frac{1}{2} \dot{q}^{T}M\dot{q} - V\lefri{q} \end{align*}
the equations
\begin{align*} q_{k}^{1} =& q_{k}, & \\ h\sum_{j=1}^{m} b_{j}\lefri{\dLdq \lefri{\galqn\lefri{c_{j}h},\dgalqn\lefri{c_{j}h}}\phi_{p}\lefri{c_{j}h} + \dLddq \lefri{\galqn \lefri{c_{j}h}, \dgalqn\lefri{c_{j}h}}\dot{\phi}_{p}\lefri{c_{j}h}} =& 0, & p = 2,...,n-1, \\ h\sum_{j=1}^{m} b_{j}\lefri{\dLdq \lefri{\galqn\lefri{c_{j}h},\dgalqn\lefri{c_{j}h}}\phi_{1}\lefri{c_{j}h} + \dLddq \lefri{\galqn \lefri{c_{j}h}, \dgalqn\lefri{c_{j}h}}\dot{\phi}_{1}\lefri{c_{j}h}} =& p_{k-1}, & \end{align*}
take the form
\begin{align} Aq^{i} - f\lefri{q^{i}} = 0, \label{InternalDEL} \end{align}
where \(q^{i}\) is the vector of internal weights, \(q^{i} = \lefri{q_{k}^{1},q_{k}^{2},...,q_{k}^{n}}^{T}\), \(A\) is a matrix with entries defined by
\begin{align} A_{1,1} =& 1, & & \label{AMatrix1}\\ A_{1,i} =& 0, & i = 2,...,n, & \label{AMatrix2}\\ A_{p,i} =& h\sum_{j=1}^{m} b_{j}M\dot{\phi}_{i}\lefri{c_{j}h}\dot{\phi}_{p}\lefri{c_{j}h}, & p = 2,...,n; & i = 1,...,n, \label{AMatrix3} \end{align}
and \(f\) is a vector valued function defined by
\begin{align*} f\lefri{q^{i}} = \left(\begin{array}{c} q_{k} \\ h\sum_{j=1}^{m} b_{j} \nabla V\lefri{\sum_{i=1}^{n}q_{k}^{i} \phi_{i}\lefri{c_{j}h}}\phi_{2}\\ \vdots \\ h\sum_{j=1}^{m} b_{j} \nabla V\lefri{\sum_{i=1}^{n}q_{k}^{i} \phi_{i}\lefri{c_{j}h}}\phi_{n-1}\\ p_{k-1} \end{array}\right). \end{align*}
It is important to note that the entries of \(A\) depend on \(h\). For now we will assume \(A\) is invertible, and that \(\left\|A^{-1}\right\| < \left\|A_{1}^{-1}\right\|\), for where \(A_{1}\) is the matrix \(A\) generated on the interval \(\left[0,1\right]\). Of course, the properties of \(A\) depend on the choice of basis functions \(\left\{\phi_{i}\right\}_{i=1}^{n}\), but we will establish these properties for the polynomial basis later. Defining the map:
\begin{align*} \Phi\lefri{q^{i}} = A^{-1}f\lefri{q^{i}}, \end{align*}
it is easily seen that (\ref{InternalDEL}) is satisfied if and only if \(q^{i} = \Phi\lefri{q^{i}}\), that is, \(q^{i}\) is a fixed point of \(\Phi\lefri{\cdot}\). If we establish that \(\Phi\lefri{\cdot}\) is a contraction mapping,
\begin{align*}
\left\|\Phi\lefri{w^{i}} - \Phi\lefri{v^{i}}\right\|_{\infty} \leq k \left\|w^{i} - v^{i}\right\|_{\infty}, \end{align*}
for some \(k < 1\), we can establish the existence of a unique fixed point, and thus show that the steps of the one step method are well-defined. Here, and throughout this section, we use \(\MNorm{\cdot}{p}\) to denote the vector or matrix \(p\)-norm, as appropriate.
To show that \(\Phi\lefri{\cdot}\) is a contraction mapping, we consider arbitrary \(w^{i}\) and \(v^{i}\):
\begin{align*}
\left\|\Phi\lefri{w^{i}} - \Phi\lefri{v^{i}}\right\|_{\infty} &= \left\|A^{-1}f\lefri{w^{i}} - A^{-1}f\lefri{v^{i}}\right\|_{\infty} \\
&= \left\|A^{-1}\lefri{f\lefri{w^{i}} - f\lefri{v^{i}}}\right\|_{\infty} \\
&\leq \left\|A^{-1}\right\|_{\infty} \left\|f\lefri{w^{i}} - f\lefri{v^{i}}\right\|_{\infty}. \end{align*}
Considering \(\left\|f\lefri{w^{i}} - f\lefri{v^{i}}\right\|_{\infty}\), we see that
\begin{align}
\left\|f\lefri{w^{i}} - f\lefri{v^{i}}\right\|_{\infty} &= \left| \sum_{j=1}^{m} b_{j} \left[ \nabla V\lefri{\sum_{i=1}^{n}w_{k}^{i} \phi_{i}\lefri{c_{j}h}} -\nabla V\lefri{\sum_{i=1}^{n}v_{k}^{i} \phi_{i}\lefri{c_{j}h}}\right]\phi_{p^{*}}\lefri{c_{j}h} \right|, \label{maxele} \end{align}
for some appropriate index \(p^{*}\). Note that the first and last terms of \(\MNorm{f\lefri{w^{i}} - f\lefri{v^{i}}}{\infty}\) will vanish, so the maximum element must take the form of (\ref{maxele}). Let \(\phi^{i}\lefri{t} = \lefri{\phi_{1}\lefri{t},\phi_{2}\lefri{t},...,\phi_{n}\lefri{t}}\). Let \(\LipC\) be the Lipschitz constant for \(\nabla V\lefri{q}\). Now
\begin{align*}
\MNorm{f\lefri{w^{i}} - f\lefri{v^{i}}}{\infty} &=\left| h\sum_{j=1}^{m} b_{j} \left[\nabla V\lefri{\sum_{i=1}^{n} w_{k}^{i} \phi_{i}\lefri{c_{j}h}} - \nabla V\lefri{\sum_{i=1}^{n} v_{k}^{i} \phi_{i}\lefri{c_{j}h}}\right]\phi_{p^{*}}\lefri{c_{j}h} \right| \\
&\leq h\sum_{j=1}^{m}\left|b_{j}\right| \left|\left[\nabla V\lefri{\sum_{i=1}^{n} w_{k}^{i} \phi_{i}\lefri{c_{j}h}} - \nabla V\lefri{\sum_{i=1}^{n} v_{k}^{i} \phi_{i}\lefri{c_{j}h}}\right]\right|\left|\phi_{p^{*}}\lefri{c_{j}h} \right| \\
&\leq h\sum_{j=1}^{m} b_{j} \LipC\left|\sum_{i=1}^{n} w_{k}^{i} \phi_{i}\lefri{c_{j}h} - \sum_{i=1}^{n} v_{k}^{i} \phi_{i}\lefri{c_{j}h} \right| \left|\phi_{p^{*}}\lefri{c_{j}h} \right| \\
&= h\sum_{j=1}^{m} b_{j} \LipC\left|\sum_{i=1}^{n} \lefri{w_{k}^{i} - v_{k}^{i}} \phi_{i}\lefri{c_{j}h}\right|\left|\phi_{p^{*}}\lefri{c_{j}h} \right| \\
&\leq h \sum_{j=1}^{m} b_{j} \LipC \left\|w^{i} - v^{i}\right\|_{\infty} \left\| \phi^{i}\lefri{c_{j}h} \right\|_{1} \left|\phi_{p^{*}}\lefri{c_{j}h}\right| \\
&\leq h \sum_{j=1}^{m} b_{j} \LipC \max_{j}\lefri{\left\|\phi^{i}\lefri{c_{j}h}\right\|_{1}\left|\phi_{p^{*}}\lefri{c_{j}h}\right|} \left\|w^{i} - v^{i}\right\|_{\infty}\\
&= h \LipC \max_{j}\lefri{\left\|\phi^{i}\lefri{c_{j}h}\right\|_{1}\left|\phi_{p^{*}}\lefri{c_{j}h}\right|} \left\|w^{i} - v^{i}\right\|_{\infty}. \end{align*}
Hence, we derive the inequality
\begin{align*}
\left\| \Phi\lefri{w^{i}} - \Phi\lefri{v^{i}} \right\|_{\infty} &\leq h \left\|A^{-1}\right\|_{\infty} \LipC \max_{j}\lefri{\left\|\phi^{i}\lefri{c_{j}h}\right\|_{1}\left|\phi_{p^{*}}\lefri{c_{j}h}\right|} \left\|w^{i} - v^{i}\right\|_{\infty}, \end{align*}
and since by assumption \(\MNorm{A^{-1}}{\infty} \leq \MNorm{A_{1}^{-1}}{\infty}\), \begin{align*}
\MNorm{\Phi\lefri{w^{i}} - \Phi\lefri{v^{i}}}{\infty} &\leq h \left\|A_{1}^{-1}\right\|_{\infty} \LipC \max_{j}\lefri{\left\|\phi^{i}\lefri{c_{j}h}\right\|_{1}\left|\phi_{p^{*}}\lefri{c_{j}h}\right|} \left\|w^{i} - v^{i}\right\|_{\infty}. \end{align*}
Thus if:
\begin{align*}
h < \lefri{\left\|A_{1}^{-1}\right\|_{\infty} \LipC \max_{j}\lefri{\left\|\phi^{i}\lefri{c_{j}h}\right\|_{1}\left|\phi_{p^{*}}\lefri{c_{j}h}\right|}}^{-1}, \end{align*}
then
\begin{align*}
\left\|\Phi\lefri{w^{i}} - \Phi\lefri{v^{i}}\right\|_{\infty} \leq k \left\|w^{i} - v^{i} \right\|_{\infty}, \end{align*}
where \(k < 1\), which establishes that \(\Phi\lefri{\cdot}\) is a contraction mapping, and establishes the existence of a unique fixed point, and thus the existence of unique steps of the one step method. \end{proof}
A critical assumption made during the proof of existence and uniqueness is that the matrix \(A\) is nonsingular. This property depends on the choice of basis functions \(\phi_{i}\). However, using a polynomial basis, like Lagrange interpolation polynomials, it can be shown that \(A\) is invertible.
\begin{lemma}\emph{(\(A\) is invertible)}\label{AInvert} If \(\left\{\phi_{i}\right\}_{i=1}^{n}\) is a polynomial basis of \(P_{n}\), the space of polynomials of degree at most \(n\), M is symmetric positive-definite, and the quadrature rule is order at least \(2n + 1\), then \(A\) defined by \eqref{AMatrix1} -- \eqref{AMatrix3} is invertible. \end{lemma}
\begin{proof} We begin by considering the equation:
\begin{align*} Aq^{i} = 0. \end{align*}
Let \(\galqn\lefri{t} = \sum_{i=1}^{n} q_{k}^{i} \phi_{i}\lefri{t}\). Considering the definition of \(A\), \(Aq^{i} = 0\) holds if and only if the following equations hold:
\begin{align} \galqn\lefri{0} &= 0, & \nonumber \\ h\sum_{j=1}^{m}b_{j} M\dgalqn\lefri{c_{j}h}\dot{\phi}_{p}\lefri{c_{j}h} &= 0, & p = 1,...,(n-1). \label{innerproductcond} \end{align}
It can easily be seen that \(\left\{\dot{\phi}_{i}\right\}_{i=1}^{n-1}\) is a basis of \(P_{n-1}\). Using the assumption that the quadrature rule is of order at least \(2n-1\) and that \(M\) is symmetric positive-definite, we can see that (\ref{innerproductcond}) implies:
\begin{align*} \int_{0}^{h} M\dgalqn\lefri{t}\dot{\phi}_{p}\lefri{t} \dt &= 0, & p = 1,...,(n-1), \end{align*}
but,
\begin{align*} \int_{0}^{h} M\dgalqn\lefri{t} \dot{\phi}_{i}\lefri{t} \dt = 0 \end{align*}
implies
\begin{align*} \left<\dgalqn, \dot{\phi}_{p}\right> = 0, \end{align*}
where \(\left<\cdot,\cdot\right>\) is the standard \(L^{2}\) inner product on \(\left[0,h\right]\). Since \(\left\{\dot{\phi}_{i}\right\}_{i=1}^{n-1}\) forms a basis for \(P_{n-1}\), \(\dgalqn \in P_{n-1}\), and \(\left<\cdot,\cdot\right>\) is non-degenerate, this implies that \(\dgalqn\lefri{t} = 0\). Thus,
\begin{align*} \galqn\lefri{0} = 0 \\ \dgalqn\lefri{t} = 0 \end{align*}
which implies that \(\galqn\lefri{t} = 0\) and hence \(q^{i} = 0\). Thus, \(Aq^{i} = 0\) then \(q^{i} = 0\), from which it follows that \(A\) is non-singular. \end{proof}
Another subtle difficulty is that the matrix \(A\) is a function of \(h\). Since we assumed that \(\left\|A^{-1}\right\|_{\infty}\) is bounded to prove Theorem \ref{ExisUniq}, we must show that for any choice of \(h\), the quantity \(\left\|A^{-1}\right\|_{\infty}\) is bounded. We will do this by establishing \(\left\|A^{-1}\right\|_{\infty} \leq \MNorm{A_{1}^{-1}}{\infty}\), where \(A_{1}\) is \(A\) defined with \(h=1\). By Lemma \ref{AInvert}, we know that \(\MNorm{A_{1}^{-1}}{\infty} < \infty\), which establishes the upper bound for \(\MNorm{A^{-1}}{\infty}\). This argument is easily generalized for a higher upper bound on \(h\).
\begin{lemma}\emph{(\(\left\|A^{-1}\right\|_{\infty} \leq \left\|A_{1}^{-1}\right\|_{\infty})\)} For the matrix \(A\) defined by \eqref{AMatrix1} -- \eqref{AMatrix3}, if \(h < 1\), \(\left\|A^{-1}\right\|_{\infty} < \left\|A_{1}^{-1}\right\|_{\infty}\) where \(A_{1}\) is \(A\) defined on the interval \(\left[0,1\right]\). \end{lemma}
\begin{proof} We begin the proof by examining how \(A\) changes as a function of \(h\). First, let \(\left\{\phi_{i}\right\}_{i=1}^{n}\) be the basis for the interval \(\left[0,1\right]\). Then for the interval \(\left[0,h\right]\), the basis functions are
\begin{align*} \phi^{h}_{i} \lefri{t} = \phi_{i}\lefri{\frac{t}{h}} \end{align*}
and hence the derivatives are:
\begin{align*} \dot{\phi}^{h}_{i}\lefri{t} = \frac{1}{h} \dot{\phi}_{i}\lefri{\frac{t}{h}}. \end{align*}
Thus, if \(A_{1}\) is the matrix defined by \eqref{AMatrix1} -- \eqref{AMatrix3} on the interval \(\left[0,1\right]\), then for the interval \(\left[0,h\right]\),
\begin{align*} A = \left(\begin{array}{cc} 1 & 0\\ 0 & \frac{1}{h}I_{\lefri{n-1}\times \lefri{n-1}}\end{array}\right) A_{1}, \end{align*}
where \(I_{n\times n}\) is the \(n\times n\) identity matrix. This gives
\begin{align*} A^{-1} = A^{-1}_{1} \left(\begin{array}{cc} 1 & 0 \\ 0 & h I_{\lefri{n-1} \times \lefri{n-1}}\end{array}\right) \end{align*}
which gives
\begin{align*}
\left\|A^{-1}\right\|_{\infty} = \left\|A^{-1}_{1} \left(\begin{array}{cc} 1 & 0 \\ 0 & h I_{\lefri{n-1} \times \lefri{n-1}}\end{array}\right)\right\|_{\infty} \leq \left\|A^{-1}_{1}\right\|_{\infty} \left\| \left(\begin{array}{cc} 1 & 0 \\ 0 & h I_{\lefri{n-1} \times \lefri{n-1}}\end{array}\right)\right\|_{\infty} = \left\|A^{-1}_{1}\right\|_{\infty}, \end{align*}
which proves the statement. \end{proof}
\subsection{Order Optimal and Geometric Convergence}\label{ConSection}
To determine the rate of convergence for spectral variational integrators, we will utilize Theorem \ref{MarsConv} and a simple extension of Theorem \ref{MarsConv}:
\begin{theorem} \emph{(Extension of Theorem \ref{MarsConv} to Geometric Convergence)} \label{MarsConvExt}
Given a regular Lagrangian \(L\) and corresponding Hamiltonian \(H\), the following are equivalent for a discrete Lagrangian \(L_{d}\lefri{q_{0},q_{1},n}\):
\begin{enumerate} \item there exists a positive constant \(K\), where \(K < 1\), such that the discrete Hamiltonian map for \(L_{d}\) has error \(\mathcal{O}\lefri{K^{n}}\), \item there exists a positive constant \(K\), where \(K < 1\), such that the discrete Legendre transforms of \(L_{d}\) have error \(\mathcal{O}\lefri{K^{n}}\), \item there exists a positive constant \(K\), where \(K < 1\), such that \(L_{d}\) is equivalent to a discrete Lagrangian with error \(\mathcal{O}\lefri{K^{n}}\). \end{enumerate}
\end{theorem}
This theorem provides a fundamental tool for the analysis of Galerkin variational methods. Its proof is almost identical to that of Theorem \ref{MarsConv}, and can be found in the appendix. The critical result is that the order of the error of the discrete Hamiltonian flow map, from which we construct the discrete flow, has the same order as the discrete Lagrangian from which it is constructed. Thus, in order to determine the order of the error of the flow generated by spectral variational integrators, we need only determine how well the discrete Lagrangian approximates the exact discrete Lagrangian. This is a key result which greatly reduces the difficulty of the error analysis of Galerkin variational integrators.
Naturally, the goal of constructing spectral variational integrators is constructing a variational method that has geometric convergence. To this end, it is essential to establish that Galerkin type integrators inherit the convergence properties of the spaces which are used to construct them. The order optimality result is related to the problem of $\Gamma$-convergence (see, for example, \citet{Da1993}), as the Galerkin discrete Lagrangians are given by extremizers of an approximating sequence of variational problems, and the exact discrete Lagrangian is the extremizer of the limiting variational problem. The $\Gamma$-convergence of variational integrators was studied in \citet{MuOr2004}, and our approach involves a refinement of their analysis. We now state our results, which establish not only the geometric convergence of spectral variational integrators, but also order optimality of all Galerkin variational integrators under appropriate smoothness assumptions.
\begin{theorem} \emph{(Order Optimality of Galerkin Variational Integrators)} \label{OptConv}
Given an interval \(\left[0,h\right]\) and a Lagrangian \(L:TQ \rightarrow \mathbb{R}\), let \(\truq\) be the exact solution to the Euler-Lagrange equations subject to the conditions \(\truq\lefri{0} = q_{0}\) and \(\truq\lefri{h} = q_{h}\), and let \(\galqn\) be the stationary point of a Galerkin variational discrete action, i.e. if \(\GDLn:Q\times Q \times \mathbb{R} \rightarrow \mathbb{R}\), \begin{align*} \GDLh{q_{0}}{q_{h}}{h} = \ext_{\galargsf{q_{0}}{q_{h}}} \mathbb{S}_{d}\lefri{\left\{q_{i}\right\}_{i=1}^{n}} = \ext_{\galargsf{q_{0}}{q_{h}}}h\sum_{j=1}^{m} b_{j} L\lefri{q_{n}\lefri{c_{j}h},\dot{q}_{n}\lefri{c_{j}h}}, \end{align*}
then
\begin{align*} \galqn = \argmin_{\galargsf{q_{0}}{q_{h}}}h\sum_{j=1}^{m} b_{j} L\lefri{q_{n}\lefri{c_{j}h},\dot{q}_{n}\lefri{c_{j}h}}. \end{align*}
If:
\begin{enumerate}
\item there exists a constant \(\ApproxC\) independent of \(h\), such that, for each \(h\), there exists a curve \(\optqn \in \FdFSpace\), such that,
\begin{align*}
\left|\lefri{\optqn\lefri{t},\doptqn\lefri{t}} - \lefri{\truq\lefri{t},\dtruq\lefri{t}}\right| & \leq \ApproxC h^{n}, \end{align*}
\item there exists a closed and bounded neighborhood \(U \subset TQ\), such that \(\lefri{\truq\lefri{t},\dtruq\lefri{t}} \in U\), \(\lefri{\optqn\lefri{t},\doptqn\lefri{t}} \in U\) for all \(t\), and all partial derivatives of \(L\) are continuous on \(U\),
\item for the quadrature rule \(\mathcal{G}\lefri{f} = h\sum_{j=1}^{m} b_{j}f\lefri{c_{j}h} \approx \int_{0}^{h}f\lefri{t}\dt\), there exists a constant \(\QuadC\), such that,
\begin{align*}
\left|\int_{0}^{h} L\lefri{q_{n}\lefri{t}, \dot{q}_{n}\lefri{t}}\dt - h\sum_{j=1}^{m} b_{j} L\lefri{q_{n}\lefri{c_{j}h},\dot{q}_{n}\lefri{c_{j}h}}\right| \leq \QuadC h^{n+1}, \end{align*}
for any \(q_{n} \in \mathbb{M}^{n}\lefri{\left[0,h\right],Q}\), \item and the stationary points \(\truq\), \(\galqn\) minimize their respective actions,
\end{enumerate}
then
\begin{align*}
\left|\EDLh{q_{0}}{q_{h}}{h} - \GDLh{q_{0}}{q_{h}}{h}\right| \leq \OptC h^{n+1}, \end{align*}
for some constant \(\OptC\) independent of \(h\), i.e. discrete Lagrangian \(L_{d}\) has error \(\mathcal{O}\lefri{h^{n+1}}\), and hence the discrete Hamiltonian flow map has error \(\mathcal{O}\lefri{h^{n+1}}\).
\end{theorem}
\begin{proof}
First, we rewrite both the exact discrete Lagrangian and the Galerkin discrete Lagrangian:
\begin{align*}
\left|\EDLh{q_{0}}{q_{h}}{h} - \GDLh{q_{0}}{q_{h}}{h}\right| &= \left|\int_{0}^{h}L\lefri{\truq\lefri{t},\dtruq\lefri{t}}\dt - \mathcal{G}\lefri{L\lefri{\galqn\lefri{t},\dgalqn\lefri{t}}}\right| \\
&= \left|\int_{0}^{h}L\lefri{\truq\lefri{t},\dtruq\lefri{t}}\dt - h\sum_{j=1}^{m}b_{j} L\lefri{\galqn\lefri{c_{j}h},\dgalqn\lefri{c_{j}h}}\right| \\
&= \left|\int_{0}^{h}L\lefri{\truq,\dtruq}\dt - h\sum_{j=1}^{m}b_{j} L\lefri{\galqn,\dgalqn}\right|, \end{align*}
where in the last line, we have suppressed the \(t\) argument, a convention we will continue throughout the proof. Now we introduce the action evaluated on the \(\optqn\) curve, which is an approximation with error \(\mathcal{O}\lefri{h^{n}}\) to the exact solution \(\truq\):
\begin{subequations} \begin{align}
\left|\int_{0}^{h}L\lefri{\truq,\dtruq}\dt - h\sum_{j=1}^{m}b_{j} L\lefri{\galqn,\dgalqn}\right| &=
\left|\int_{0}^{h}L\lefri{\truq,\dtruq}\dt - \int_{0}^{h} L\lefri{\optqn, \doptqn} \dt \right. \nonumber\\
& \hspace{10em} + \left.\int_{0}^{h} L\lefri{\optqn,\doptqn}\dt - h\sum_{j=1}^{m}b_{j} L\lefri{\galqn,\dgalqn}\right| \nonumber \\
& \leq \left|\int_{0}^{h}L\lefri{\truq,\dtruq}\dt - \int_{0}^{h} L \lefri{\optqn, \doptqn}\dt\right| \label{FirstTerm} \\
& \hspace{10em} +\left|\int_{0}^{h} L \lefri{\optqn,\doptqn} \dt - h\sum_{j=1}^{m}b_{j}L\lefri{\galqn,\dgalqn}\right|. \label{SecondTerm} \end{align} \end{subequations}
Considering the first term (\ref{FirstTerm}):
\begin{align*}
\left|\int_{0}^{h}L\lefri{\truq,\dtruq}\dt - \int_{0}^{h} L \lefri{\optqn, \doptqn}\dt\right| &= \left|\int_{0}^{h}L\lefri{\truq,\dtruq} - L \lefri{\optqn, \doptqn} \dt\right| \\
& \leq \int_{0}^{h} \left|L\lefri{\truq,\dtruq} - L\lefri{\optqn,\doptqn}\right|\dt. \end{align*}
By assumption, all partials of \(L\) are continuous on \(U\), and since \(U\) is closed and bounded, this implies \(L\) is Lipschitz on \(U\). Let \(\LagLipC\) denote that Lipschitz constant. Since, again by assumption, \(\lefri{\truq,\dtruq} \in U\) and \(\lefri{\optqn,\doptqn} \in U\), we can rewrite:
\begin{align*}
\int_{0}^{h}\left|L\lefri{\truq,\dtruq} - L \lefri{\optqn,\doptqn}\right|\dt & \leq \int_{0}^{h} \LagLipC \left|\lefri{\truq,\dtruq} - \lefri{\optqn,\doptqn}\right| \dt \\ &\leq \int_{0}^{h} \LagLipC \ApproxC h^{n} \dt\\ &= \LagLipC \ApproxC h^{n+1}, \end{align*}
where we have made use of the best approximation estimate. Hence,
\begin{align}
\left|\int_{0}^{h} L\lefri{\truq, \dtruq} \dt - \int_{0}^{h} L\lefri{\optqn,\doptqn} \dt\right| \leq L_{\alpha}C_{1}h^{n+1}. \label{FirstTermIneq} \end{align}
Next, considering the second term (\ref{SecondTerm}),
\begin{align*}
\left|\int_{0}^{h} L\lefri{\optqn,\doptqn} \dt - h\sum_{j=1}^{m} b_{j} L\lefri{\galqn,\dgalqn} \right|, \end{align*}
since \(\galqn\), the stationary point of the discrete action, minimizes its action and \(\optqn \in \mathbb{M}^{n}\lefri{\left[0,h\right],Q}\),
\begin{align} h\sum_{j=1}^{m} b_{j} L\lefri{\galqn,\dgalqn} \leq h\sum_{j=1}^{m} b_{j} L \lefri{\optqn,\doptqn} \leq \int_{0}^{h} L\lefri{\optqn,\doptqn} \dt + \QuadC h^{n+1} \label{UpperBound} \end{align}
where the inequalities follow from the assumptions on the order of the quadrature rule. Furthermore,
\begin{align} h\sum_{j=1}^{m} b_{j} L \lefri{\galqn,\dgalqn} &\geq \int_{0}^{h} L\lefri{\galqn,\dgalqn} \dt - \QuadC h^{n+1} \nonumber \\ &\geq \int_{0}^{h} L\lefri{\truq,\dtruq} \dt - \QuadC h^{n+1} \nonumber\\ &\geq \int_{0}^{h} L\lefri{\optqn,\doptqn} \dt - \LagLipC \ApproxC h^{n+1} - \QuadC h^{n+1}, \label{LowerBound} \end{align}
where the inequalities follow from (\ref{FirstTermIneq}), the order of the quadrature rule, and the assumption that \(\truq\) minimizes its action. Putting (\ref{UpperBound}) and (\ref{LowerBound}) together, we can conclude:
\begin{align}
\left|\int_{0}^{h}L\lefri{\optqn,\doptqn}\dt - h\sum_{j=1}^{m} b_{j} L\lefri{\galqn,\dgalqn}\right| \leq \lefri{\LagLipC \ApproxC + \QuadC}h^{n+1} \label{SecondTermIneq}. \end{align}
Now, combining the bounds (\ref{FirstTermIneq}) and (\ref{SecondTermIneq}) in (\ref{FirstTerm}) and (\ref{SecondTerm}), we can conclude
\begin{align*}
\left|\EDLh{q_{0}}{q_{h}}{h} - \GDLh{q_{0}}{q_{h}}{h}\right| \leq \lefri{2\LagLipC \ApproxC + \QuadC}h^{n+1} \end{align*}
which, combined with Theorem \ref{MarsConv}, establishes the order of the error of the integrator. \end{proof}
The above proof establishes a significant convergence result for Galerkin variational integrators, namely that for sufficiently well behaved Lagrangians, Galerkin variational integrators will produce discrete approximate flows that converge to the exact flow as \(h\rightarrow0\) with the highest possible order allowed by the approximation space, provided the quadrature rule is of sufficiently high order.
We will discuss assumption 4 in \S \ref{MinAction}. While in general we cannot assume that stationary points of the action are minimizers, it can be shown that for Lagrangians of the canonical form
\begin{align*} L\lefri{q,\dot{q}} = \dot{q}^{T}M\dot{q} - V\lefri{q}, \end{align*}
under some mild assumptions on the derivatives of \(V\) and the accuracy of the quadrature rule, there always exists an interval \(\left[0,h\right]\) over which stationary points are minimizers. In \S \ref{MinAction} we will show the result extends to the discretized action of Galerkin variational integrators. A similar result was established in \citet{MuOr2004}.
Geometric convergence of spectral variational integrators is not strictly covered under the proof of order optimality. While the above theorem establishes convergence of Galerkin variational integrators by shrinking \(h\), the interval length of each discrete Lagrangian, spectral variational integrators achieve convergence by holding the interval length of each discrete Lagrangian constant and increasing the dimension of the approximation space \(\FdFSpace\). Thus, for spectral variational integrators, we have the following analogous convergence theorem:
\begin{theorem} \emph{(Geometric Convergence of Spectral Variational Integrators)} \label{SpecConv}
Given an interval \(\left[0,h\right]\) and a Lagrangian \(L:TQ \rightarrow \mathbb{R}\), let \(\truq\) be the exact solution to the Euler-Lagrange equations, and \(\galqn\) be the stationary point of the spectral variational discrete action: \begin{align*} \SDLN{q_{0}}{q_{h}}{n} = \ext_{\galargsf{q_{0}}{q_{h}}} \mathbb{S}_{d}\lefri{\left\{q_{i}\right\}_{i=1}^{n}}= \ext_{\galargsf{q_{0}}{q_{h}}} h\sum_{j=0}^{m_{n}} \bnj L\lefri{q_{n}\lefri{\cnjh},\dot{q}_{n}\lefri{\cnjh}}. \end{align*}
If:
\begin{enumerate}
\item there exists constants \(\ApproxC,\ApproxK\), \(\ApproxK < 1\), independent of \(n\) such that, for each \(n\), there exists a curve \(\optqn \in \mathbb{M}^{n}\lefri{\left[0,h\right],Q}\), such that,
\begin{align*}
\left|\lefri{\truq,\dtruq} - \lefri{\optqn,\doptqn}\right| & \leq \ApproxC \ApproxK^{n}, \end{align*}
\item there exists a closed and bounded neighborhood \(U \subset TQ\), such that, \(\lefri{\truq\lefri{t},\dtruq\lefri{t}} \in U\) and \(\lefri{\optqn\lefri{t},\doptqn\lefri{t}} \in U\) for all \(t\) and \(n\), and all partial derivatives of \(L\) are continuous on \(U\),
\item for the sequence of quadrature rules \(\mathcal{G}_{n}\lefri{f} = \sum_{j=1}^{m_{n}} \bnj f\lefri{\cnjh} \approx \int_{0}^{h}f\lefri{t}\dt\), there exists constants \(\QuadC\), \(\QuadK\), \(\QuadK < 1\), independent of \(n\) such that
\begin{align*}
\left|\int_{0}^{h} L\lefri{q_{n}\lefri{t}, \dot{q}_{n}\lefri{t}}\dt - h\sum_{j=1}^{m_{n}} \bnj L\lefri{q_{n}\lefri{\cnjh},\dot{q}_{n}\lefri{\cnjh}}\right| \leq \QuadC \QuadK^{n}, \end{align*}
for any \(q_{n}\in \mathbb{M}^{n}\lefri{\left[0,h\right],Q}\),
\item and the stationary points \(\truq\), \(\galqn\) minimize their respective actions,
\end{enumerate}
then
\begin{align}
\left|\EDL{q_{0}}{q_{1}} - \SDLN{q_{0}}{q_{1}}{n}\right| \leq \SpecC\SpecK^{n} \label{GeoBound} \end{align}
for some constants \(\SpecC,\SpecK\), \(\SpecK < 1\), independent of \(n\), and hence the discrete Hamiltonian flow map has error \(\mathcal{O}\lefri{\SpecK^{n}}\).
\end{theorem} The proof of the above theorem is very similar to that of order optimality, and would be tedious to repeat here. It can be found in the appendix. The main differences between the proofs are the assumption of the sequence of converging functions in the increasingly high-dimensional approximation spaces, and the assumption of a sequence of increasingly high-order quadrature rules. These assumptions are used in the obvious way in the modified proof.
\subsection{Minimization of the Action} \label{MinAction}
One of the major assumptions made in the convergence theorems (\ref{OptConv}) and (\ref{SpecConv}) is that the the stationary points of both the continuous and discrete actions are minimizers over the interval \(\left[0,h\right]\). This type of minimization requirement is similar to the one made in the paper on \(\Gamma\)-convergence of variational integrators by \citet{MuOr2004}. In fact, the results in \citet{MuOr2004} can easily be extended to demonstrate that for sufficiently well-behaved Lagrangians of the form
\begin{align*} L\lefri{q,\dot{q}} = \frac{1}{2}\dot{q}^{T}M\dot{q} - V\lefri{q}, \end{align*}
where \(q \in \CQ\), there exists an interval \(\left[0,h\right]\), such that stationary points of the Galerkin action are minimizers.\\
\begin{theorem}
Consider a Lagrangian of the form
\begin{align*} L\lefri{q,\dot{q}} = \frac{1}{2}\dot{q}^{T}M\dot{q} - V\lefri{q} \end{align*}
where \(q \in \CQ\) and each component \(q^{d}\) of \(q\), \(q^{d} \in \CQ\), is a polynomial of degree at most \(s\). Assume \(M\) is symmetric positive-definite and all second-order partial derivatives of \(V\) exist, and are continuous and bounded. Then, there exists a time interval \(\left[0,h\right]\) such that stationary points of the discrete action,
\begin{align*} \mathbb{S}_{d}\lefri{\left\{q_{k}^{i}\right\}_{i=1}^{n}} = h\sum_{j=1}^{m} b_{j} \lefri{\frac{1}{2} \dgalqn\lefri{c_{j}h}^{T}M\dgalqn\lefri{c_{j}h} - V\lefri{\galqn\lefri{c_{j}h}}}, \end{align*}
on this time interval are minimizers if the quadrature rule used to construct the discrete action is of order at least \(2s+1\).
\end{theorem} We quickly note that the assumption that each component of \(q\), \(q^{d}\), is a polynomial of degree \emph{at most} \(s\) allows for discretizations where different components of the configuration space are discretized with polynomials of different degrees. This allows for more efficient discretizations where slower evolving components are discretized with lower-degree polynomials than faster evolving ones.
\begin{proof}
Let \(\galqn\) be a stationary point of the discrete action \(\mathbb{S}_{d}\lefri{\cdot}\), and let \(\delta q\) be an arbitrary perturbation of the stationary point \(\galqn\), under the conditions \(\delta q^{d} \in P_{S_{d}}\), \(\delta q\lefri{0} = \delta q\lefri{h} = 0\), which is uniquely defined by \(\left\{\delta q_{k}^{i}\right\}_{i=1}^{n} \subset Q\). Then,
\begin{align*} &\mathbb{S}_{d}\lefri{\left\{q_{k}^{i} + \delta q_{k}^{i}\right\}_{i=1}^{n}} - \mathbb{S}_{d}\lefri{\left\{q_{k}^{i}\right\}_{i=1}^{n}} \\ &= h\sum_{j}^{m} b_{j} \lefri{\frac{1}{2} \lefri{\dgalqn + \delta \dot{q}}^{T}M\lefri{\dgalqn + \delta \dot{q}} - V\lefri{\galqn + \delta q}} - h\sum_{j}^{m} b_{j} \lefri{\frac{1}{2} \dgalqn^{T}M\dgalqn - V\lefri{\galqn}}\\ &= h\sum_{j}^{m} b_{j} \lefri{\frac{1}{2} \lefri{\dgalqn + \delta \dot{q}}^{T}M\lefri{\dgalqn + \delta \dot{q}} - V\lefri{\galqn + \delta q} - \frac{1}{2} \dgalqn^{T}M\dgalqn + V\lefri{\galqn}}. \end{align*}
Making use of Taylor's remainder theorem, we expand:
\begin{align*}
V\lefri{\galqn + \delta q} = V\lefri{\galqn} + \nabla V\lefri{\galqn}\cdot \delta q + \frac{1}{2}\delta \galqn^{T} R \delta \galqn, \end{align*}
where \(\left|R_{lm}\right| \leq \sup_{l,m}\left|\frac{\partial^{2}V}{\partial q_{l} \partial q_{m}}\right|\). Using this expansion, we rewrite
\begin{align*} \mathbb{S}_{d}\lefri{\left\{q_{k}^{i} + \delta q_{k}^{i}\right\}_{i=1}^{n}} - \mathbb{S}_{d}\lefri{\left\{q_{k}^{i}\right\}_{i=1}^{n}} &= h\sum_{j}^{m} b_{j} \left(\frac{1}{2} \lefri{\dgalqn + \delta \dot{q}}^{T}M\lefri{\dgalqn + \delta \dot{q}} - V\lefri{\galqn} - \nabla V\lefri{\galqn} \cdot \delta q \right. \\ & \hspace{5em} -\left. \frac{1}{2} \delta q^{T} R \delta q - \lefri{\frac{1}{2} \dgalqn^{T}M\dgalqn + V\lefri{\galqn}}\right) \end{align*}
which, given the symmetry in \(M\), rearranges to:
\begin{align*} \mathbb{S}_{d}\lefri{\left\{q_{k}^{i} + \delta q_{k}^{i}\right\}_{i=1}^{n}} - \mathbb{S}_{d}\lefri{\left\{q_{k}^{i}\right\}_{i=1}^{n}} &= h\sum_{j}^{m} b_{j} \lefri{\dgalqn^{T}M\delta\dot{q} - \nabla V\lefri{\galqn} \cdot \delta q + \frac{1}{2} \delta \dot{q}^{T}M\delta \dot{q} - \frac{1}{2} \delta q^{T} R \delta q}. \end{align*}
Now, it should be noted that the stationarity condition for the discrete Euler-Lagrange equations is
\begin{align*} h\sum_{j=1}^{m} b_{j}\lefri{\dgalqn^{T}M\delta \dot{q} - \nabla V\lefri{\galqn} \cdot \delta q} = 0 \end{align*}
for arbitrary \(\delta q\), which allows us to simplify the expression to
\begin{align*}
\mathbb{S}_{d}\lefri{\left\{q_{k}^{i} + \delta q_{k}^{i}\right\}_{i=1}^{n}} - \mathbb{S}_{d}\lefri{\left\{q_{k}^{i}\right\}_{i=1}^{n}} = h\sum_{j}^{m} b_{j} \lefri{\frac{1}{2} \delta \dot{q}^{T}M\delta \dot{q} - \frac{1}{2} \delta q^{T} R \delta q}. \end{align*}
Now, using the assumption that the partial derivatives of \(V\) are bounded, \(\left|R_{lm}\right| \leq \left|\frac{\partial^{2} V}{\partial q_{l}\partial q_{m}}\right| < C_{R}\), and standard matrix inequalities, we get the inequality:
\begin{align}
\delta q^{T} R \delta q \leq \left\|R\delta q\right\|_{2} \left\|\delta q\right\|_{2} \leq \left\|R\right\|_{2} \left\|\delta q\right\|_{2}^{2} \leq \left\|R\right\|_{F} \left\|\delta q\right\|^{2}_{2} \leq DC_{R}\left\|\delta q\right\|^{2}_{2} = DC_{R}\delta q^{T}\delta q, \label{HBound} \end{align}
where \(D\) is the number of spatial dimensions of \(Q\). Thus
\begin{align*} h\sum_{j}^{m} b_{j} \lefri{\frac{1}{2} \delta \dot{q}^{T}M\delta \dot{q} - \frac{1}{2} \delta q^{T} R \delta q} \geq h\sum_{j}^{m} b_{j} \lefri{\frac{1}{2} \delta \dot{q}^{T}M\delta \dot{q} - \frac{1}{2} DC_{R}\delta q^{T}\delta q}. \end{align*}
Because \(M\) is symmetric positive-definite, there exists \(m > 0\) such that \(x^{T}Mx \geq mx^{T}x\) for any \(x\). Hence,
\begin{align*} h\sum_{j}^{m} b_{j} \lefri{\frac{1}{2} \delta \dot{q}^{T}M\delta \dot{q} - \frac{1}{2} DC_{R}\delta q^{T}\delta q} \geq h\sum_{j}^{m} b_{j} \lefri{\frac{1}{2} m\delta\dot{q}^{T}\delta\dot{q} - \frac{1}{2} DC_{R}\delta q^{T}\delta q}. \end{align*}
Now, we note that since each component of \(\delta q\) is a polynomial of degree at most \(s\), \(\delta q^{T} \delta q\) and \(\delta \dot{q}^{T} \delta \dot{q}\) are both polynomials of degree less than or equal to \(2s\). Since our quadrature rule is of order \(2s+1\), the quadrature rule is exact, and we can rewrite
\begin{align*} h\sum_{j}^{m} b_{j} \lefri{\frac{1}{2} m\delta \dot{q}^{T}\delta \dot{q} - \frac{1}{2} DC_{R} \delta q^{T} \delta q} &= \frac{1}{2}\int_{0}^{h} m \delta \dot{q}^{T}\delta \dot{q} - DC_{R} \delta q^{T} \delta q\dt \\
&= \frac{1}{2}\lefri{\int_{0}^{h} m \delta \dot{q}^{T}\delta \dot{q} \dt - \int_{0}^{h} DC_{R} \delta q^{T} \delta q \dt}. \end{align*}
From here, we note that \(\delta q \in \HoQ\), and make use of the Poincar\'{e} inequality to conclude
\begin{align*} \frac{1}{2}\lefri{\int_{0}^{h}m \delta \dot{q}^{T}\delta \dot{q} \dt - \int_{0}^{h} nC_{R} \delta q^{T} \delta q \dt} & \geq \frac{1}{2} \lefri{m\frac{\pi^{2}}{h^{2}}\int_{0}^{h} \delta q^{T}\delta q \dt - DC_{R}\int_{0}^{h} \delta q^{T} \delta q \dt} \\
&= \frac{1}{2}\lefri{\frac{m\pi^{2}}{h^{2}} - DC_{R}} \int_{0}^{h} \delta q^{T} \delta q \dt. \end{align*}
Since \(\int_{0}^{h} \delta q^{T} \delta q \dt > 0\),
\begin{eqnarray*}
\mathbb{S}_{d}\lefri{\left\{q_{k}^{i} + \delta q_{k}^{i}\right\}_{i=1}^{n}} - \mathbb{S}_{d}\lefri{\left\{q_{k}^{i}\right\}_{i=1}^{n}} \geq \frac{1}{2}\lefri{\frac{m\pi^{2}}{h^{2}} - DC_{R}} \int_{0}^{h} \delta q^{T} \delta q \dt > 0 \end{eqnarray*} so long as \(h < \sqrt{\frac{m \pi^{2}}{DC_{R}}}\). \end{proof}
\subsection{Convergence of Galerkin Curves and Noether Quantities}
\subsubsection{Galerkin Curves}
In order to construct the one-step method, spectral variational integrators determine a curve,
\begin{align*} \galqn\lefri{t} = \sum_{i=1}^{n}q^{i}_{k}\phi_{i}\lefri{t}, \end{align*}
which satisfies
\begin{align*} \galqn\lefri{t} &= \argmin_{\galargsf{q_{k}}{q_{k+1}}} h\sum_{j=1}^{m}b_{j}L\lefri{\galqn\lefri{c_{j}h},\dgalqn\lefri{c_{j}h}}. \end{align*}
Evaluating this curve at \(h\) defines the next step of the one-step method, \(q_{k+1} = \galqn\lefri{h}\), but the curve itself has many desirable properties which makes it a good continuous approximation to the true solution of the Euler Lagrange equations \(\truq\lefri{t}\). In this section, we will examine some of the favorable properties of \(\galqn\lefri{t}\), hereafter referred to as the \emph{Galerkin curve}.
However, before discussing the properties of the Galerkin curve, it is useful review the different curves with which we are working. We have already defined the Galerkin curve, \(\galqn\lefri{t}\), and we will also be making use of the local solution to the Euler-Lagrange equations \(\truq\lefri{t}\), where
\begin{align*} \truq\lefri{t} = \argmin_{\truqargsf{q_{k}}{q_{k+1}}} \int_{0}^{h}L\lefri{q\lefri{t},\dot{q}\lefri{t}}\dt. \end{align*}
However, while for each interval \(\truq\) satisfies the Euler-Lagrange equations exactly, it is not the exact solution of the Euler-Lagrange equations globally, as \(q_{k} \neq \Phi_{kh}\lefri{q_{0},\dot{q}_{0}}\), where \(\Phi_{t}\lefri{q_{0},\dot{q}_{0}}\) is the flow of the Euler-Lagrange vector field. This is particularly important when discussing invariants, where the invariants of \(\truq\) remain constant within a time-step, but not from time-step to time-step.
The first property of the Galerkin curve that we will examine is its rate of convergence to the true flow of the Euler-Lagrange vector field. There are two general sources of error that affect the convergence of these curves, the first being the accuracy to which the curves approximate the local solution to the Euler-Lagrange equations over the interval \(\left[0,h\right]\) with the boundary \(\lefri{q_{k},q_{k+1}}\), and the second being the accuracy of the boundary conditions \(\lefri{q_{k},q_{k+1}}\) as approximations to a true sampling of the exact flow. Numerical experiments will show that often the second source of error dominates the first, causing the Galerkin curves to converge at the same rate as the one-step map. However, the accuracy to which the Galerkin curves approximate the true minimizers independent of the error of the boundary can also be established under appropriate assumptions about the action. Two theorems which establish this convergence are presented below.
Before we state the theorems, we quickly recall the definitions of the \emph{Sobolev Norm} \(\SobNorm{\cdot}{p}\),
\begin{align*}
\SobNorm{f}{p} = \lefri{\LNorm{f}{p}^{p} + \LNorm{\dot{f}}{p}^{p}}^{\frac{1}{p}} = \lefri{\int_{0}^{h}\left|f\right|^{p}\dt + \int_{0}^{h}\left|\dot{f}\right|^{p}\dt}^{\frac{1}{p}}. \end{align*}
We will make extensive use of this norm when examining convergence of Galerkin curves.
\begin{theorem}\emph{(Geometric Convergence of Galerkin Curves with \(n\)-Refinement)}\label{GeoGalerk} Under the same assumptions as Theorem \ref{SpecConv}, if at \(\truq\), the action is twice Frechet differentiable, and if the second Frechet derivative of the action \(\mbox{D}^{2}\mathfrak{S}\lefri{\cdot}\left[\cdot,\cdot\right]\) is coercive in a neighborhood \(U\) of \(\truq\), that is,
\begin{align*} D^{2}\mathfrak{S}\lefri{\nu}\left[\delta q, \delta q\right] \geq \CoerC\SobNorm{\delta q}{1}^{2}, \end{align*}
for all curves \(\delta q \in \HoQ \) and all \(\nu \in U\), then the curves which minimize the discrete action converge to the true solution geometrically with \(n\)-refinement with respect to \(\SobNorm{\cdot}{1}\). Specifically, if the discrete Hamiltonian flow map has error \(\mathcal{O}\lefri{\SpecK^{n}}\), \(\SpecK < 1\), then the Galerkin curves have error \(\mathcal{O}\lefri{{\sqrt{\SpecK}}^{n}}\). \end{theorem}
\begin{proof} We start with the bound (\ref{GeoBound}) given at the end of Theorem \ref{SpecConv},
\begin{align*}
\left|\EDL{q_{k}}{q_{k+1}} - \SDLN{q_{k}}{q_{k+1}}{n}\right| \leq \SpecC\SpecK^{n} \end{align*}
and expand using the definitions of \(\EDL{q_{k}}{q_{k+1}}\) and \(\SDLN{q_{k}}{q_{k+1}}{n}\), as well as the assumed accuracy of the quadrature rule \(\mathcal{G}_{n}\) to derive
\begin{align}
\SpecC\SpecK^{n} &\geq \left|\EDL{q_{k}}{q_{k+1}} - \SDLN{q_{k}}{q_{k+1}}{n}\right| \label{ReconIneq1} \\
&= \left|\int_{0}^{h}L\lefri{\truq,\dtruq}\dt - h\sum_{j=1}^{m_{n}} \bnj L\lefri{\galqn\lefri{\cnjh},\galqn\lefri{\cnjh}}\right| \nonumber \\
&\geq \left|\int_{0}^{h}L\lefri{\truq,\dtruq}\dt - \int_{0}^{h} L\lefri{\galqn,\galqn}\dt\right| - \QuadC\QuadK^{n} \label{ReconIneq2}\\
& = \left| \mathfrak{S}\lefri{\galqn} - \mathfrak{S}\lefri{\truq} \right| - \QuadC \QuadK^{n}\nonumber \end{align}
which implies:
\begin{align*}
\lefri{\SpecC + \QuadC}\SpecK^{n} \geq& \left| \mathfrak{S}\lefri{\galqn} - \mathfrak{S}\lefri{\truq} \right| \end{align*}
because \(\SpecK \geq \QuadK\), (see the proof of Theorem \ref{SpecConv} in the appendix). Using this inequality, we make use of a Taylor expansion of \(\mathfrak{S}\lefri{\galqn}\),
\begin{align*} \mathfrak{S}\lefri{\galqn} = \mathfrak{S}\lefri{\truq} + \mbox{D}\mathfrak{S}\lefri{\truq}\left[\galqn - \truq\right] + \frac{1}{2}\mbox{D}^{2}\mathfrak{S}\lefri{\nu}\left[\galqn - \truq,\galqn - \truq\right], \end{align*}
for some \(\nu \in U\), to see that
\begin{align*}
\lefri{\SpecC + \QuadC}\SpecK^{n} &\geq \left| \mathfrak{S}\lefri{\galqn} - \mathfrak{S}\lefri{\truq} \right| \\
&= \left|\mathfrak{S}\lefri{\truq} + \mbox{D}\mathfrak{S}\lefri{\truq}\left[\galqn - \truq\right] + \frac{1}{2}\mbox{D}^{2}\mathfrak{S}\lefri{\truq}\left[\galqn - \truq, \galqn - \truq \right] - \mathfrak{S}\lefri{\truq}\right|. \end{align*}
But
\begin{align*} D\mathfrak{S}\lefri{\truq}\left[\galqn -\truq\right] &= \int_{0}^{h}\dLdq\lefri{\truq,\dtruq}\lefri{\galqn - \truq} + \dLddq\lefri{\truq,\dtruq}\lefri{\dgalqn - \dtruq} \dt \\ &= \int_{0}^{h} \lefri{\dLdq\lefri{\truq,\dtruq} - \frac{\mbox{d}}{\dt} \dLddq\lefri{\truq,\dtruq}} \cdot \lefri{\galqn - \truq} \dt\\ &= 0, \end{align*}
because \(\galqn\lefri{0} = \truq\lefri{0}\) and \(\galqn\lefri{h} = \truq\lefri{h}\) by definition (note that this implies \(\lefri{\galqn - \truq} \in H^{1}_{0}\lefri{\left[0,h\right],Q}\)). Then
\begin{align*}
\lefri{\SpecC + \QuadC}\SpecK^{n} &\geq \left|\mbox{D}^{2}\mathfrak{S}\lefri{\nu}\left[\galqn - \truq,\galqn - \truq \right] \right| \\ &\geq \CoerC \SobNorm{\galqn - \truq}{1}^{2} \\ C\sqrt{\SpecK}^{n} &\geq \SobNorm{\galqn - \truq}{1}^{2} \end{align*}
where \(C = \frac{\SpecC + \QuadC}{\CoerC}\).\end{proof}
This result shows that Galerkin curves converge to the true solution geometrically with \(n\)-refinement, albeit with a larger geometric constant, and hence a slower rate. By simply replacing the bounds (\ref{ReconIneq1}) and (\ref{ReconIneq2}) from Theorem \ref{SpecConv} with those from Theorem \ref{OptConv} and the term \(\SpecC\SpecK^{n}\) with \(\OptC h^{p}\), an identical argument shows that Galerkin curves converge at half the optimal rate with \(h\)-refinement.
\begin{theorem}\emph{(Convergence of Galerkin Curves with \(h\)-Refinement)}\label{OptGalerk} Under the same assumptions as Theorem \ref{OptConv}, if at \(\truq\), the action is twice Frechet differentiable, and if the second Frechet derivative of the action \(\mbox{D}^{2}\mathcal{S}\lefri{\cdot}\left[\cdot,\cdot\right]\) is coercive with a constant \(\CoerC\) independent of \(h\) in a neighborhood \(U\) of \(\truq\), for all curves \(\delta q \in H^{1}_{0}\lefri{\left[0,h\right],Q}\), then if the discrete Lagrange map has error \(\mathcal{O}\lefri{h^{p+1}}\), the Galerkin curves have error at most \(\mathcal{O}\lefri{h^{\frac{p+1}{2}}}\) in \(\SobNorm{\cdot}{1}\). If \(\CoerC\) is a function of \(h\), this bound becomes \(\mathcal{O}\lefri{\CoerC\lefri{h}^{-1}h^{\frac{p+1}{2}}}\). \end{theorem}
Like the requirement that the stationary points of the actions are minimizers, the requirement that the second Frechet derivative of the action is coercive may appear quite strong at first. Again, the coercivity will depend on the properties of the Lagrangian \(L\), but we can establish that for Lagrangians of the canonical form,
\begin{align*} L\lefri{q,\dot{q}} = \frac{1}{2}\dot{q}^{T}M\dot{q} - V\lefri{q}, \end{align*}
there exists a time step \(\left[0,h\right]\) over which the action is coercive on \(\HoQ\).
\begin{theorem}\emph{(Coercivity of the Action)} For Lagrangian of the form
\begin{align*} L\lefri{q,\dot{q}} &= \frac{1}{2}\dot{q}^{T}M\dot{q} - V\lefri{q}, \end{align*}
where \(M\) is symmetric positive-definite, and the second derivatives of \(V\lefri{q}\) are bounded, there exists an interval \(\left[0,h\right]\) over which the action is coercive over \(\HoQ\), that is,
\begin{align*} \mbox{D}^{2}\mathfrak{S}\lefri{\nu}\left[\delta q, \delta q\right] \geq \CoerC \SobNorm{\delta q}{1}^{2}, \end{align*}
for any \(\delta q \in \HoQ\) and any \(\nu \in \CQ\). \end{theorem}
\begin{proof} First, we note that if
\begin{align*} \mathfrak{S}\lefri{\nu} &= \int_{0}^{h} \frac{1}{2}\dot{\nu}^{T}M\dot{\nu} - V\lefri{\nu}, \end{align*}
then
\begin{align*} \mbox{D}^{2}\mathfrak{S}\lefri{\nu}\left[\delta q, \delta q\right] &= \int_{0}^{h} \delta\dot{q}^{T}M\delta\dot{q} - \delta q^{T}H\lefri{\nu}\delta q \dt\\ &= \int_{0}^{h} \delta \dot{q}^{T}M\delta\dot{q} \dt - \int_{0}^{h} \delta q^{T}H\lefri{\nu} \delta q \dt \end{align*}
where \(H\lefri{\nu}\) is the Hessian of \(V\lefri{\nu}\) at the point \(\nu\). Since \(M\) is symmetric positive-definite, and the second derivatives of \(V\lefri{\cdot}\) are bounded, then there exists \(C_{r}\) and \(m\) such that:
\begin{align} \int_{0}^{h} \delta \dot{q}^{T}M\delta \dot{q} \dt &\geq \int_{0}^{h}m\delta\dot{q}^{T}\delta\dot{q} \dt \nonumber \\ \int_{0}^{h} \delta q^{T}H\lefri{\nu}\delta q \dt &\leq \int_{0}^{h}DC_{r}\delta q^{T} \delta q \dt, \label{SecondCoerIneq} \end{align}
(see (\ref{HBound}) for a derivation of (\ref{SecondCoerIneq})). Hence,
\begin{align} \mbox{D}^{2}\mathfrak{S}\lefri{\nu}\left[\delta q, \delta q\right] &\geq \int_{0}^{h} m\delta\dot{q}^{T}\delta\dot{q} \dt - \int_{0}^{h} DC_{r}f^{T}f \dt \nonumber \\ &= \frac{1}{2}m\int_{0}^{h} \delta\dot{q}^{T}\delta\dot{q} \dt + \frac{1}{2}m \int_{0}^{h} \delta\dot{q}^{T}\delta\dot{q} \dt - DC_{r}\int_{0}^{h} \delta q^{T}\delta q\dt. \label{CoerTwoTerms} \end{align}
Considering the last two terms in (\ref{CoerTwoTerms}), and noting that \(\delta q \in \HoQ\), we make use of the Poincar\'{e} inequality to derive:
\begin{align} \frac{1}{2}m \int_{0}^{h} \delta\dot{q}^{T}\delta\dot{q} \dt - DC_{r}\int_{0}^{h} \delta q^{T}\delta q \dt &\geq \frac{m \pi^{2}}{2h^{2}} \int_{0}^{h} \delta q^{T}\delta q \dt - nC_{r}\int_{0}^{h}\delta q^{T}\delta q \dt \nonumber \\ &\geq \lefri{\frac{m \pi^{2}}{2h^{2}} - DC_{r}} \int_{0}^{h}\delta q^{T}\delta q\dt. \label{CoerPCI} \end{align}
Thus, substituting (\ref{CoerPCI}) in for the last two terms of (\ref{CoerTwoTerms}), we conclude:
\begin{align*}
\mbox{D}^{2}\mathfrak{S}\lefri{q,\dot{q}}\left[\delta q,\delta q\right] &\geq \lefri{\frac{m \pi^{2}}{2h^{2}} - DC_{r}} \int_{0}^{h} \delta q^{T}\delta q\dt + \frac{m}{2} \int_{0}^{h} \delta\dot{q}^{T}\delta\dot{q} \dt \\ &\geq \min\lefri{\frac{m}{2},\lefri{\frac{m\pi^{2}}{2h^{2}} - DC_{r}}}\lefri{\int_{0}^{h}\delta q^{T}\delta q\dt + \int_{0}^{h}\delta\dot{q}^{T}\delta\dot{q} \dt}\\ &= \min\lefri{\frac{m}{2},\lefri{\frac{m\pi^{2}}{2h^{2}} - DC_{r}}}\lefri{\LNorm{\delta q}{2}^{2} + \LNorm{\delta \dot{q}}{2}^{2}}, \end{align*}
and making use of H\"{o}lder's inequality, we see that \(\LNorm{\delta q}{2} \geq h^{\frac{1}{2}}\LNorm{\delta q}{1}\), thus
\begin{align*} \mbox{D}^{2}\mathfrak{S}\lefri{q,\dot{q}}\left[\delta q,\delta q\right] &\geq \min\lefri{\frac{m}{2},\lefri{\frac{m\pi^{2}}{2h^{2}} - DC_{r}}}\lefri{h\LNorm{\delta q}{1}^{2} + h\LNorm{\delta \dot{q}}{1}^{2}}\\ &\geq \min\lefri{\frac{mh}{2},\lefri{\frac{m\pi^{2}}{2h} - hDC_{r}}}\frac{1}{2}\lefri{\LNorm{\delta q}{1} + \LNorm{\delta \dot{q}}{1}}^{2}\\ &= \min\lefri{\frac{mh}{4},\lefri{\frac{m\pi^{2}}{4h} - hDC_{r}}}\SobNorm{\delta q}{1}^{2} \end{align*}
which establishes the coercivity result. \end{proof}
\subsubsection{Noether Quantities}
\begin{figure}
\caption{Conserved and approximately conserved Noether quantities and the resulting constrained solution space. Suppose that both \(p^{T}q = 1\) and \(p^{2} + q^{2} = 5\) were conserved quantities for a certain Lagrangian. Then the solutions of the Euler-Lagrange equations would be constrained to the intersections of these two constant surfaces in phase space; in the above diagram, this is the intersection of the dashed and solid lines. If these quantities were conserved up to a fixed error along a numerical solution, then the numerical solution would be constrained to the intersection of the shaded regions in the above figure. The constraint of the numerical solution to these regions is what leads to the many excellent qualities of variational integrators.}
\end{figure}
One of the great advantages of using variational integrators for problems in geometric mechanics is that by construction they have a rich geometric structure which helps lead to excellent long term and qualitative behavior. An important geometric feature of variational integrators is the preservation of discrete Noether quantities, which are invariants that are derived from symmetries of the action. These are analogous to the more familiar Noether quantities of geometric mechanics in the continuous case. We quickly recall Noether's theorem in both the discrete and continuous case, which will also help define the notation used throughout the proofs that follow. The proofs of both these theorems can be found in \citet{HaLuWa2006}.
\begin{theorem}\emph{(Noether's Theorem)} Consider a system with Hamiltonian \(H\lefri{p,q}\) and Lagrangian \(L\lefri{q,\dot{q}}\). Suppose \(\left\{g_{s}:s\in\mathbb{R}\right\}\) is a one-parameter group of transformations which leaves the Lagrangian invariant. Let
\begin{align*}
a\lefri{q} = \left.\frac{d}{d\mbox{s}}\right|_{s=0}g_{s}\lefri{q} \end{align*}
be defined as the vector field with flow \(g_{s}\lefri{q}\), referred to as the infinitesimal generator, and define the canonical momentum
\begin{align*} p = \dLddq\lefri{q,\dot{q}}. \end{align*}
Then
\begin{align*} I\lefri{p,q} = p^{T}a\lefri{q} \end{align*}
is a first integral of the Hamiltonian system. \end{theorem}
\begin{theorem}\emph{(Discrete Noether's Theorem)} Suppose the one-parameter group of transformations leaves the discrete Lagrangian \(L_{d}\lefri{q_{k},q_{k+1}}\) invariant for all \(\lefri{q_{k},q_{k+1}}\). Then:
\begin{align*} p_{k+1}^{T}a\lefri{q_{k+1}} = p_{k}^{T}a\lefri{q_{k}} \end{align*}
where
\begin{align*} p_{k} &= -D_{1}L_{d}\lefri{q_{k},q_{k+1}}, \\ p_{k+1} &= D_{2}L_{d}\lefri{q_{k},q_{k+1}}. \end{align*} \end{theorem}
For the remainder of this section, we will refer to \(I\lefri{q,p}\) as the \emph{Noether quantity} and \(p_{n}^{T}a\lefri{q_{n}} = p_{n+1}^{T}a\lefri{q_{n+1}}\) as the \emph{discrete Noether quantity}.
For Galerkin variational integrators, it is possible to bound the error of the Noether quantities along the Galerkin curve from the behavior of the analogous discrete Noether quantities of the discrete problem and, more importantly, this bound is independent of the number of time steps that are taken in the numerical integration. This is significant because it offers insight into the excellent behavior of spectral variational integrators even over long periods of integration.
The proof of convergence and near preservation of Noether quantities is broken into three major parts. First, we note that on step \(k\) of a numerical integration the discrete Noether quantity arises from a function of the Galerkin curve and the initial point of the one-step map \(\lefri{q_{k-1},q_{k}}\), and that a bound exists for the difference of this discrete Noether quantity evaluated on the Galerkin curve and evaluated on the local exact solution to the Euler-Lagrange equations \(\truq\). Second, we show that a bound exists for the difference of the discrete Noether quantity on the local exact solution of the Euler-Lagrange equations and the value of the Noether quantity of the local exact solution, which is conserved along the flow of the Euler-Lagrange vector field. Finally, we show that under certain smoothness conditions, there exists a point-wise bound between the Noether quantity evaluated on the Galerkin curve and the Noether quantity evaluated on the local exact solution. Thus, we establish a point-wise bound between the Noether quantity evaluated on the Galerkin curve and the discrete Noether quantity, and a bound between the discrete Noether quantity and the Noether quantity, which leads to a point-wise bound between the Noether quantity evaluated on the Galerkin curve, and the Noether quantity which is conserved along the global flow of the Euler-Lagrange vector field.
Throughout this section we will make the simplifying assumptions that
\begin{align*} \galqn = \sum_{i=1}^{n} q_{k}^{i}\phi_{i} \end{align*}
where \(q_{k}^{1} = q_{k}\), and thus
\begin{align*} \frac{\partial \galqn}{\partial q_{k}} = \phi_{1}. \end{align*}
This assumption significantly simplifies the analysis.
We begin by bounding the discrete Noether quantity by a function of the local exact solution of the Euler-Lagrange equations.
\begin{lemma}\emph{(Bound on Discrete Noether Quantity)} \label{DNBound} Define the Galerkin Noether map as:
\begin{align*} I_{d}\lefri{q\lefri{t},q_{k}} &= -\lefri{h\sum_{j=1}^{n}b_{j}\left[\dLdq\lefri{q,\dot{q}}\phi_{1} + \dLddq\lefri{q,\dot{q}}\dot{\phi}_{1}\right]}^{T}a\lefri{q_{k}} \end{align*}
and note that the discrete Noether quantity is given by
\begin{align*} I_{d}\lefri{\galqn,q_{k}} = p_{n}^{T}a\lefri{q_{k}}. \end{align*}
Assuming the quadrature accuracy of Theorem (\ref{SpecConv}) with \(n\)-refinement and Theorem (\ref{OptConv}) with \(h\)-refinement, if \(\dLdq\lefri{q,\dot{q}}\), \(\dLddq\lefri{q,\dot{q}}\) and \(\frac{\mbox{d}}{\dt}\dLddq\) are Lipschitz continuous, \(\LNorm{\phi_{1}}{\infty}\) is bounded with \(n\) refinement, and \(\SobNorm{\galqn - \truq}{1}\) is bounded below by the quadrature error, then
\begin{align*}
\left|I_{d}\lefri{\galqn,q_{k}} - I_{d}\lefri{\truq,q_{k}}\right| \leq C \left|a\lefri{q_{k}}\right|\lefri{\SobNorm{\galqn - \truq}{1} + \LNorm{\galqn - \truq}{\infty} + \LNorm{\dgalqn - \dtruq}{\infty}} \end{align*} for some \(C\) independent of \(n\) and \(h\). \end{lemma}
\begin{proof} We begin by expanding the definitions of the discrete Noether quantity: \begin{align*}
\left|I_{d}\lefri{\galqn,q_{k}} - I_{d}\lefri{\truq,q_{k}}\right|=& \left|h\lefri{\sum_{j=1}^{m} b_{j}\left[\dLdq\lefri{\galqn,\dgalqn}\phi_{1} + \dLddq \lefri{\galqn,\dgalqn}\dot{\phi}_{1}\right]}^{T}a\lefri{q_{k}} \right. \\
& \hspace{5em} - \left. \lefri{h\sum_{j=1}^{m} b_{j}\left[\dLdq\lefri{\truq,\dtruq}\phi_{1} + \dLddq \lefri{\truq,\dtruq}\dot{\phi}_{1}\right]}^{T}a\lefri{q_{k}}\right| \\
=& \left|\lefri{h\sum b_{j} \left[\lefri{\dLdq\lefri{\galqn,\dgalqn} - \dLdq\lefri{\truq,\dtruq}}\phi_{1} - \lefri{\dLddq\lefri{\galqn,\dgalqn} - \dLddq\lefri{\truq,\dtruq}}\dot{\phi_{1}}\right]}^{T}a\lefri{q_{k}}\right| \\
\leq & \left|h\sum_{j=1}^{m} b_{j} \left[\lefri{\dLdq\lefri{\galqn,\dgalqn} - \dLdq\lefri{\truq,\dtruq}}\phi_{1} - \lefri{\dLddq\lefri{\galqn,\dgalqn} - \dLddq\lefri{\truq,\dtruq}}\dot{\phi_{1}}\right] \right|\left|a\lefri{q_{k}}\right|. \end{align*}
Now we introduce the function \(\quaderr{\cdot}{\cdot}\) which gives the error of the quadrature rule, and thus
\begin{align*}
\left|I_{d}\lefri{\galqn,q_{k}} - I_{d}\lefri{\truq,q_{k}}\right| \leq & \left|\int_{0}^{h}\lefri{\dLdq\lefri{\galqn,\dgalqn} - \dLdq\lefri{\truq,\dtruq}}\phi_{1} - \lefri{\dLddq\lefri{\galqn,\dgalqn} - \dLddq\lefri{\truq,\dtruq}}\dot{\phi}_{1} \dt\right. \\
& \hspace{5em} \left. \vphantom{\int_{0}^{h}\dLddq} + \quaderr{\galqn - \truq}{\dgalqn - \dtruq}\right| \left|a\lefri{q_{k}}\right|. \end{align*}
Integrating by parts, we get:
\begin{align*}
\left|I_{d}\lefri{\galqn,q_{k}} - I_{d}\lefri{\truq,q_{k}}\right| \leq & \left|\int_{0}^{h}\lefri{\dLdq\lefri{\galqn,\dgalqn} - \dLdq\lefri{\truq,\dtruq}}\phi_{1} - \frac{\mbox{d}}{\dt}\lefri{\dLddq\lefri{\galqn,\dgalqn} - \dLddq\lefri{\truq,\dtruq}}\phi_{1} \dt\right. \\
& \hspace{5em} \left. \vphantom{\int_{0}^{h}\dLddq} + \left.\lefri{\dLddq\lefri{\galqn,\dgalqn} - \dLddq\lefri{\truq,\dtruq}}\phi_{1}\right|_{0}^{h} + \quaderr{\galqn - \truq}{\dgalqn - \dtruq}\right| \left|a\lefri{q_{k}}\right|. \end{align*}
Introducing the Lipschitz constants \(L_{1}\) for \(\dLdq\), \(L_{2}\) for \(\dLddq\), and \(L_{3}\) for \(\frac{d}{dt}\dLddq\),
\begin{align*}
\left|I_{d}\lefri{\galqn,q_{k}} - I_{d}\lefri{\truq,q_{k}}\right| \leq& \left(\int_{0}^{h} \lefri{L_{1} + L_{3}}\left|\lefri{\galqn,\dgalqn} - \lefri{\truq,\dtruq}\right|\left|\phi_{1}\right| \dt + 2L_{2}\lefri{\LNorm{\phi_{1}}{\infty}} \right. \\
& \hspace{5em} \left. \vphantom{\int_{0}^{h}} \left(\LNorm{\galqn - \truq}{\infty} + \LNorm{\dgalqn - \dtruq}{\infty}\right) + \quaderr{\galqn - \truq}{\dgalqn - \dtruq} \right) \left|a\lefri{q_{k}}\right|\\
& \leq \lefri{L_{1} + L_{3}}\LNorm{\phi_{1}}{\infty}\left|a\lefri{q_{k}}\right|\lefri{\int_{0}^{h} \left|\lefri{\galqn,\dgalqn} - \lefri{\truq,\dtruq}\right|\dt} \\
& \hspace{5em} + 2L_{2}\lefri{\LNorm{\phi_{1}}{\infty}}\left|a\lefri{q_{k}}\right|\lefri{\LNorm{\galqn - \truq}{\infty} + \LNorm{\dgalqn - \dtruq}{\infty}} \\
& \hspace{5em} + \quaderr{\galqn - \truq}{\dgalqn - \dtruq}\left|a\lefri{q_{k}}\right|. \end{align*}
We now make the simplification that the quadrature error \(\left|\quaderr{\cdot}{\cdot}\right|\) serves as a lower bound for \(\SobNorm{\galqn - \truq}{1}\). While this may not strictly hold, all of our estimates on the convergence for \(\galqn\) imply this bound, and hence it is a reasonable simplification for establishing convergence in this case. Now, note that \(\LNorm{\phi_{1}}{\infty}\) is invariant under \(h\) rescaling, and let
\begin{align*} C = \max\lefri{L_{1} + L_{3},2L_{2}}\LNorm{\phi_{1}}{\infty} + 1 \end{align*}
to get
\begin{align*}
\left|I_{d}\lefri{\galqn,q_{k}} - I_{d}\lefri{\truq,q_{k}}\right| \leq C\left|a\lefri{q_{k}}\right|\lefri{\SobNorm{\galqn - \truq}{1} + \LNorm{\galqn - \truq}{\infty} + \LNorm{\dgalqn-\dtruq}{\infty}} \end{align*}
which establishes the result. \end{proof}
Lemma \ref{DNBound} establishes a bound between the discrete Noether quantity and \(I_{d}\lefri{\truq,q_{k}}\). The next step is to establish a bound between \(I_{d}\lefri{\truq,q_{k}}\) and the Noether quantity.
\begin{lemma}\label{DNandTNBound}\emph{(Error Between Discrete Noether Quantity and True Noether Quantity)} Assume that \(\phi_{1}\lefri{0} = 1\) and \(\phi_{1}\lefri{h} = 0 \), and that the sequence \(\left\{\left|a\lefri{q_{k}}\right|\right\}_{k=1}^{N}\) is bounded by a constant \(\akC\) which is independent of \(N\). Let
\begin{align*} \trup\lefri{t} = \dLddq\lefri{\truq\lefri{t},\dtruq\lefri{t}}. \end{align*}
Once again, let the error of the quadrature rule be given by \(\quaderr{\cdot}{\cdot}\). Then
\begin{align*}
\left|I^{d}\lefri{\truq,q_{k}} - I\lefri{\trup\lefri{t},\truq\lefri{t}}\right| &\leq \akC \left|\quaderr{\truq}{\dtruq}\right| \end{align*}
for any \(t \in \left[0,h\right]\). \end{lemma}
\begin{proof} First, we note that since \(\truq\) solves the Euler-Lagrange equations exactly, \(I\lefri{\trup\lefri{t},\truq\lefri{t}}\) is a conserved quantity along the flow, so it suffices to show the inequality holds for \(t=0\). We begin by expanding: \begin{align*}
\left|I^{d}\lefri{\truq,q_{k}} - I\lefri{\trup\lefri{0},\truq\lefri{0}}\right| &= \left|-h\lefri{\sum_{j=1}^{m}b_{j}\dLdq\lefri{\truq,\dtruq}\phi_{1} + \dLddq\lefri{\truq,\dtruq}\dot{\phi}_{1}}^{T}a\lefri{q_{k}} - \trup\lefri{0}^{T}a\lefri{\truq\lefri{0}}\right| \\
&= \left|-\lefri{\int_{0}^{h}\dLdq\lefri{\truq,\dtruq}\phi_{1} + \dLddq\lefri{\truq,\dtruq}\dot{\phi}_{1}\dt + \quaderr{\truq}{\dtruq}}^{T}a\lefri{q_{k}} - \trup\lefri{0}^{T}a\lefri{\truq\lefri{0}} \right|\\
&= \left|-\left(\int_{0}^{h}\lefri{\dLdq\lefri{\truq,\dtruq} - \frac{d}{d\mbox{t}}\dLddq\lefri{\truq,\dtruq}}\phi_{1}\dt + \dLddq\lefri{\truq\lefri{h},\dtruq\lefri{h}}\phi_{1}\lefri{h} \right. \right. \\
& \left. \left. \hspace{5em} - \vphantom{\int_{0}^{h}} \dLddq\lefri{\truq\lefri{0},\dtruq\lefri{0}}\phi_{1}\lefri{0} + \quaderr{\truq}{\dtruq}\right)^{T} a\lefri{q_{k}} - \trup\lefri{0}^{T}a\lefri{\truq\lefri{0}} \right| \end{align*}
Since \(\truq\lefri{t}\) solves the Euler-Lagrange equations, \(\phi_{1}\lefri{0} = 1\) and \(\phi_{1}\lefri{h} = 0\), and \(\truq\lefri{0} = q_{k}\),
\begin{align*}
\left|I^{d}\lefri{\truq,q_{k}} - I\lefri{\trup\lefri{0},\truq\lefri{0}}\right| =& \left|\lefri{\dLddq\lefri{\truq\lefri{0},\dtruq\lefri{0}}}^{T}a\lefri{q_{k}} + \lefri{\quaderr{\truq}{\dtruq}}^{T}a\lefri{q_{k}} - \trup\lefri{0}^{T}a\lefri{q_{k}}\right|\\
=& \left|\lefri{\trup\lefri{0}}^{T}a\lefri{q_{k}} + \lefri{\quaderr{\truq}{\dtruq}}^{T}a\lefri{q_{k}} - \lefri{\trup\lefri{0}}^{T}a\lefri{q_{k}}\right|\\
=& \left|\quaderr{\truq}{\dtruq}^{T}a\lefri{q_{k}}\right|\\
\leq & \left|\quaderr{\truq}{\dtruq}\right| \left|a\lefri{q_{k}}\right|\\
\leq & \akC \left|\quaderr{\truq}{\dtruq}\right| \end{align*}
which yields the desired bound. \end{proof}
Once again, if we assume that the quadrature error serves as a lower bound for the Sobolev error, combining the bounds from (\ref{DNBound}) and (\ref{DNandTNBound}) yields:
\begin{align*}
\left|I_{d}\lefri{\galqn,q_{k}} - I\lefri{\trup\lefri{t},\truq\lefri{t}}\right| & \leq 2C\akC\lefri{\SobNorm{\galqn - \truq}{1} + \LNorm{\galqn - \truq}{\infty} + \LNorm{\dgalqn - \dtruq}{\infty}}. \end{align*}
This bound serves two purposes; the first is to establish a bound between the discrete Noether quantity and the Noether quantity computed on the local exact solution \(\truq\). The second is to establish a bound between the discrete Noether quantity after one step and the Noether quantity computed on the initial data:
\begin{align*}
\left|I_{d}\lefri{\galqn,q_{1}} - I\lefri{p\lefri{0},q\lefri{0}}\right| &\leq \ 2C\akC\lefri{\SobNorm{\galqn - \truq}{1} + \LNorm{\galqn - \truq}{\infty} + \LNorm{\dgalqn - \dtruq}{\infty}}, \end{align*}
since for \(\lefri{q_{1},q_{2}}\), \(\truq\) is the \emph{global} exact flow of the Euler-Lagrange equations.
The difference between these two bounds is subtle but important; by establishing a bound between the discrete Noether quantity and the Noether quantity associated with the initial conditions, on any step of the method we can bound the error between the discrete Noether quantity and the Noether quantity associated with the \emph{global} exact flow. By establishing the bound between the discrete Noether quantity and the Noether quantity associated with \(\truq\) at any step, we can bound the error between the Noether quantity associated with the local exact flow \(\truq\) and the true Noether quantity conserved along the global exact flow:
\begin{align}
\left|I\lefri{\trup\lefri{t},\truq\lefri{t}} - I\lefri{p\lefri{0},q\lefri{0}}\right| &\leq \left|I\lefri{\trup\lefri{t},\truq\lefri{t}} - I_{d}\lefri{\galqn,q_{k}}\right| + \left|I_{d}\lefri{\galqn,q_{k}} - I\lefri{p\lefri{0},q\lefri{0}}\right| \nonumber \\ &\leq 4C\akC\lefri{\SobNorm{\galqn - \truq}{1} + \LNorm{\galqn - \truq}{\infty} + \LNorm{\dgalqn - \dtruq}{\infty}} \label{SuperImportantBound} \end{align}
for any \(t_{0} \in \left[0,h\right]\) on any time step \(k\). Because the local exact flow \(\truq\) is generated from boundary conditions \(\lefri{q_{k},q_{k+1}}\) which only approximate the boundary conditions of the true flow, there is no guarantee that the Noether quantity associated with \(\truq\) will be the same step to step, only that it will be conserved within each time step. However, because there is a bound between the Noether quantity associated with \(\truq\) and the discrete Noether quantity at every time step, the discrete Noether quantity and the Noether quantity associated with the exact flow, and because the Noether quantity is conserved point-wise along \(\truq\) on each time step, there exists a bound between the Noether quantity associated with each point of the local exact flow and the Noether quantity associated with the true solution.
We finally arrive at the desired result, which is a theorem that bounds the error between the Noether quantity along the Galerkin curve and the true Noether quantity. It is significant because not only does it bound the error of the Noether quantity, but the bound is independent of the number of steps taken, and hence will not grow even for extremely long numerical integrations.
\begin{theorem}\emph{(Convergence of Conserved Noether Quantities)}\label{ConvNQ} Define
\begin{align*} \galpn = \dLddq\lefri{\galqn,\dgalqn}. \end{align*}
Under the assumptions of Lemmas (\ref{DNBound} - \ref{DNandTNBound}), if the Noether map \(I\lefri{p,q}\) is Lipschitz continuous in both its arguments, then there exists a constant \(\NoeC\) independent \(N\), the number of method steps, such that:
\begin{align*}
\left|I\lefri{p\lefri{0},q\lefri{0}} - I\lefri{\tilde{p}_{n}\lefri{t},\galqn\lefri{t}}\right| \leq \NoeC\lefri{\SobNorm{\galqn - \truq}{1} + \LNorm{\galqn - \truq}{\infty} + \LNorm{\dgalqn - \dtruq}{\infty}}. \end{align*}
for any \(t \in \left[0,Nh\right]\). \end{theorem}
\begin{proof}We begin by introducing the Noether quantity evaluated at \(t\) on the local exact flow, \(\truq\):
\begin{align}
\left|I\lefri{p\lefri{0},q\lefri{0}}-I\lefri{\galpn\lefri{t},\galqn\lefri{t}}\right| \leq& \left|I\lefri{\galpn\lefri{t},\galqn\lefri{t}} - I\lefri{\trup\lefri{t},\truq\lefri{t}}\right| \label{NoeTwoTerm}\\
& \hspace{5em} + \left|I\lefri{\trup\lefri{t},\truq\lefri{t}} - I\lefri{p\lefri{0},q\lefri{0}}\right|. \nonumber \end{align}
Considering the first term in (\ref{NoeTwoTerm}), let \(L_{4}\) be the Lipschitz constant for \(I\lefri{\cdot,\cdot}\). Then
\begin{align}
\left|I\lefri{\galpn\lefri{t},\galqn\lefri{t}} - I\lefri{\trup\lefri{t},\truq\lefri{t}}\right| &\leq L_{4}\left|\lefri{\galpn\lefri{t},\galqn\lefri{t}} - \lefri{\trup\lefri{t},\truq\lefri{t}}\right| \nonumber \\
& \leq L_{4} \lefri{\left|\galpn\lefri{t} - \trup\lefri{t}\right| + \left|\galqn\lefri{t} - \trup\lefri{t}\right|} \nonumber \\
& = L_{4} \lefri{\left|\dLddq\lefri{\galqn\lefri{t},\dgalqn\lefri{t}} - \dLddq\lefri{\truq\lefri{t},\dtruq{\lefri{t}}}\right| + \left|\galqn\lefri{t} - \truq\lefri{t}\right|} \nonumber \\
& \leq L_{4}\lefri{L_{2}\left|\dgalqn\lefri{t} - \dtruq\lefri{t}\right| + \lefri{L_{2}+1}\left|\galqn\lefri{t} - \truq\lefri{t}\right|} \nonumber\\ & \leq L_{4}\lefri{L_{2} + 1}\lefri{\LNorm{\galqn - \truq}{\infty} + \LNorm{\dgalqn - \dtruq}{\infty}}. \label{LipBound} \end{align}
The second term in (\ref{NoeTwoTerm}) is exactly the bound given by (\ref{SuperImportantBound}) and thus combining (\ref{LipBound}) and (\ref{SuperImportantBound}) in (\ref{NoeTwoTerm}) and defining \(\NoeC = 4C\akC + L\lefri{L_{2}+1}\), we have:
\begin{align*}
\left|I\lefri{\galpn\lefri{t},\galqn\lefri{t}} - I\lefri{\trup\lefri{t},\truq\lefri{t}}\right| \leq \NoeC\lefri{\SobNorm{\galqn - \truq}{1} + \LNorm{\galqn - \truq}{\infty} + \LNorm{\dgalqn - \dtruq}{\infty}} \end{align*}
which completes the result. \end{proof}
The convergence and bounds of the Noether quantity evaluated on the Galerkin curve to that of the true solution is hampered by one issue. While Theorems (\ref{GeoGalerk}) and (\ref{OptGalerk}) provide estimates for convergence in the Sobolev norm \(\SobNorm{\cdot}{1}\), Theorem (\ref{ConvNQ}) requires estimates in the \(L^{\infty}\) norm. We can establish a bound for \(\LNorm{\galqn\lefri{t} - \truq\lefri{t}}{\infty}\), but it is much more difficult to establish a general estimate for \(\LNorm{\dgalqn\lefri{t} - \dtruq\lefri{t}}{\infty}\).
\begin{lemma} \emph{(Bound on \(L^{\infty}\) Norm from Sobolev Norm)}\label{LinSobNorm} For any \(t \in \left[0,h\right]\), the following bound holds:
\begin{align*}
\left|q\lefri{t}\right| \leq \max\lefri{\frac{1}{h},1} \SobNorm{q}{1} \end{align*}
and thus
\begin{align*} \LNorm{q}{\infty} \leq \max\lefri{\frac{1}{h},1} \SobNorm{q}{1}. \end{align*} \end{lemma}
\begin{proof} This is a basic extension of the arguments from Lemma A.1. in \citet{LaTh2003}, generalizing the lemma from the interval \(\left[0,1\right]\) to an interval of arbitrary length, \(\left[0,h\right]\). We note that for any \(t, s \in \left[0,h\right]\), \(q\lefri{t} = q\lefri{s} + \int_{s}^{t} \dot{q}\lefri{u}\mbox{d}u\). Thus:
\begin{align*}
\left|q\lefri{t}\right| \leq& \left|q\lefri{s}\right| + \int_{0}^{h} \left|\dot{q}\lefri{u}\right| \mbox{d}u \\
\leq& \left|q\lefri{s}\right| + \LNorm{\dot{q}}{1}. \end{align*}
Now, we integrate with respect to \(s\):
\begin{align*}
\int_{0}^{h} \left|q\lefri{t}\right| \mbox{d}s &\leq \int_{0}^{h}\left|q\lefri{s}\right| \mbox{d}s + \int_{0}^{h}\LNorm{\dot{q}}{1}\mbox{d}s \\
h\left|q\lefri{t}\right| &\leq \lefri{\LNorm{q}{1} + h\LNorm{\dot{q}}{1}}. \end{align*}
which yields the desired result. \end{proof} Under certain assumptions about the behavior of \(\dgalqn - \dtruq\), it is possible to establish bounds on the point-wise error of \(\dgalqn\) from the Sobolev error \(\SobNorm{\galqn - \truq}{1}\). For example, if the length of time that the error is within a given fraction of the max error is proportional to the length of the interval \(\left[0,h\right]\), i.e. there exists \(C_{1},C_{2}\) independent of \(h\): i.e.,
\begin{align*}
m\lefri{\left\{t\left|\left\|\lefri{\dgalqn\lefri{t} - \dtruq\lefri{t}}\right\| \geq C_{1}\left\|\dgalqn\lefri{t} - \dtruq\lefri{t}\right\|_{\infty}\right.\right\}} \geq C_{2}h, \end{align*}
where \(m\) is the Lebesque measure, then it can easily be seen that:
\begin{align*}
\SobNorm{\galqn - \truq}{1} \geq \int_{0}^{h} \left\|\dgalqn\lefri{t} - \dtruq\lefri{t}\right\| \dt \geq C_{1}C_{2}h\LNorm{\dgalqn - \dtruq}{\infty}. \end{align*}
While we will not establish here that the \(\dgalqn\) converges in the \(L^{\infty}\) norm with the same rate that the Galerkin curve converges in the Sobolev norm, our numerical experiments will show that the Noether quantities tend to converge at the same rate as the Galerkin curve.
\section{Numerical Experiments}
To support the results in this paper, as well as to investigate the efficiency and stability of spectral variational integrators, several numerical experiments were conducted by applying spectral variational techniques to well-known variational problems. For each problem, the spectral variational integrator was constructed using a Lagrange interpolation polynomials at \(n\) Chebyshev points with a Gauss quadrature rule at \(2n\) points. Convergence of both the one-step map and the Galerkin curves was measured using the \(\ell^{\infty}\) and \(L^{\infty}\) norms respectively, although we record them on the same axis using labeled \(L^{\infty}\) error in a slight abuse of notation. The experiments strongly support the results of this paper, and suggest topics for further investigation.
\afterpage{
}
\begin{figure}
\caption{Geometric convergence of the spectral variational integration of the harmonic oscillator problem, for 100 steps at step size \(h=20.0\).}
\label{fig:HarmOscNRefine}
\end{figure}
\begin{figure}
\caption{Geometric convergence of the energy error of the spectral variational integration of the harmonic oscillator problem for 100 steps at step size \(h=20.0\).}
\label{fig:HarmOscEnergyNRefine}
\end{figure}
\subsection{Harmonic Oscillator}
\begin{figure}
\caption{Energy Stability of the spectral variational integration of the harmonics oscillator problem. This energy was computed for the integration using \(n=14\) for step size \(h=20.0\).}
\label{fig:HarmOscEnergyStability}
\end{figure}
The first and simplest numerical experiment conducted was the harmonic oscillator. Starting from the Lagrangian,
\begin{align*} L\lefri{q,\dot{q}} = \frac{1}{2}\dot{q}^{2} - \frac{1}{2}q^{2}, \end{align*}
where \(q\in \mathbb{R}\), the induced spectral variational discrete Euler-Lagrange equations are linear. Choosing the large time step \(h=20\) over 100 steps yields the expected geometric convergence, and attains very high accuracy, as can be seen in Figure \ref{fig:HarmOscNRefine}. In addition, the max error of the energy converges geometrically, see Figure \ref{fig:HarmOscEnergyNRefine}, and does not grow over the time of integration, see Figure \ref{fig:HarmOscEnergyStability}.
\subsection{N-body Problems}
We now turn our attention towards Kepler \(N\)-body problems, which are both good benchmark problems and are interesting in their own right. The general form of the Lagrangian for these problems is
\begin{align*}
L\lefri{q,\dot{q}} &= \frac{1}{2} \sum_{i=1}^{N} \dot{q}_{i}^{T}M\dot{q}_{i} + G\sum_{i=1}^{N} \sum_{j=0}^{i-1} \frac{m_{i}m_{j}}{\left\|q_{i}-q_{j}\right\|}, \end{align*}
where \(q_{i} \in \mathbb{R}^{D}\) is the center of mass for body \(i\), \(G\) is a gravitational constant, and \(m_{i}\) is a mass constant associated with the body described by \(q_{i}\).
\subsubsection{2-Body Problem}
\begin{figure}\label{fig:Kepler2bodyNRefinement}
\end{figure}
\begin{figure}\label{fig:Kepler2bodyNRefinement_Energy}
\end{figure}
\begin{figure}
\caption{Geometric convergence of the angular momentum of the Kepler 2-body problem with eccentricity 0.6 over 100 steps of \(h = 2.0\). Again, the error is of the same order as it was the Galerkin curves.}
\label{fig:Kepler2bodyNRefinement_Angular}
\end{figure}
\begin{figure}\label{fig:Kepler2bodyhRefinement}
\end{figure}
\begin{figure}
\caption{Convergence of the Kepler 2-body problem energy with eccentricity 0.6 over 10 steps with \(h\) refinement.}
\label{fig:Kepler2bodyhRefinement_energy}
\end{figure}
\begin{figure}
\caption{Convergence of the angular momentum Kepler 2-body problem with eccentricity 0.6 over 100 steps of \(h = 2.0\).}
\label{fig:Kepler2bodyhRefinement_ang}
\end{figure}
\begin{figure}
\caption{Stability of energy for Kepler 2-body problem.}
\label{fig:Kepler2bodyEnergyStable}
\end{figure}
The first experiment we will examine is the choice of parameters \(D = 2\), \(m_{1} = m_{2} = 1\). Centering the coordinate system at \(q_{1}\), we choose \(q_{2}\lefri{0} = \lefri{0.4,0}\), \(\dot{q}_{2}\lefri{0} = \lefri{0, 2}\), which has a known closed form solution which is a stable closed elliptical orbit with eccentricity \(0.6\). Knowing the closed form solution allows us to examine the rate of convergence to the true solution, and when solved with the large time step \(h = 2.0\), over 100 steps, the error of the one-step map is \(\mathcal{O}\lefri{0.56^{n}}\) with \(n\)-refinement and \(\mathcal{O}\lefri{h^{2\left\lceil\frac{n}{2}\right\rceil}}\) with \(h\)-refinement, as can be seen in Figure \ref{fig:Kepler2bodyNRefinement} and Figure \ref{fig:Kepler2bodyhRefinement}, respectively. The numerical evidence suggests that our bound for the one-step map with \(h\)-refinement is not sharp, as the convergence of the one-step map is always even. Interestingly, it is also possible to observe the different convergence rates of the one-step map and the Galerkin curves with \(n\)-refinement, as eventually the Galerkin curves have error approximately \(\mathcal{O}\lefri{0.74^{n}}\) while the one-step map has error approximately \(\mathcal{O}\lefri{0.56^{n}}\), and \(\sqrt{0.56} \approx 0.7483\). However, it appears that the error from the one step map dominates until very high choices of \(n\), and thus it is difficult to observe the error of the Galerkin curves directly with \(h\)-refinement, round off error becomes a problem before the error of the Galerkin curves does for smaller choices of \(n\).
The N-body Lagrangian is invariant under the action of \(\mbox{SO}\lefri{D}\), which yields the conserved Noether quantity of angular momentum. For the two body problem this is:
\begin{align*} L\lefri{q,\dot{q}} = q_{x}\dot{q}_{y} - q_{y}\dot{q}_{x} \end{align*}
where \(q = \lefri{q_{x},q_{y}}\). Numerical experiments show that the error of the angular momentum does not grow with the number of steps taken in the integration, Figure \ref{fig:Kepler2bodyLStable}, but that the error is of the same order as the error of Galerkin curve with \(n\)-refinement in Figure \ref{fig:Kepler2bodyNRefinement_Angular}. With \(h\)-refinement, the angular momentum appears to have error \(\mathcal{O}\lefri{h^{\frac{n}{2}} + 2}\) in Figure \ref{fig:Kepler2bodyhRefinement_ang}. This is interesting because the theoretical bound on the error of the Galerkin curves is \(\mathcal{O}\lefri{h^\frac{n}{2}}\), and the error of the Noether quantities is theoretically a factor \(C\lefri{h}\) times the error of the Galerkin curves, where \(C\) is the factor that arises in the proof of the convergence of the conserved Noether quantities. Numerical experiments suggest \(C\) is \(\mathcal{O}\lefri{h^{2}}\) for this problem, but that the Galerkin curves do converge at a rate of \(\mathcal{O}\lefri{h^{\frac{n}{2}}}\), which is consistent with of the Galerkin curve error estimate. While this evidence is not conclusive, it is suggestive that the error analysis provides a plausible bound. A careful analysis of the factor \(C\) would be an interesting direction for further investigation.
\subsubsection{The Solar System}
To illustrate the excellent stability proprieties of spectral variational integrators, we let \(D = 3\), \(N = 10\), and use the velocities, positions, and masses of the sun, 8 planet, and the dwarf planet Pluto on January 1, 2000 (as provided by the JPL Solar System ephemeris \cite{JPLEph}) as initial configuration parameters for the Kepler system. Taking \(100\) time steps of \(h = 100\) days, the \(N = 25\) spectral variational integrator produces a highly stable flow in Figure \ref{fig:InnerSolar}. It should be noted that orbits are closed, stable, and exhibit almost none of the ``precession'' effects that are characteristic of symplectic integrators, even though the time step is larger than the orbital period of Mercury. Additionally, considering just the outer solar system (Jupiter, Saturn, Uranus, Neptune, Pluto), and aggregating the inner solar system (Sun, Mercury, Venus, Earth, Mars) to a point mass, an \(N = 25\) spectral variational integrator taking 100 time steps \(h = 1825\) days (5 year steps) produces the orbital flow seen in Figure \ref{fig:OuterSolar}. Again, these are highly stable, precession free orbits. As can be clearly seen, the spectral variational integrator produces extremely stable flows, even for very large time steps.
\begin{figure}
\caption{Stability of angular momentum for Kepler 2-body problem.}
\label{fig:Kepler2bodyLStable}
\end{figure}
\afterpage{
}
\begin{figure}
\caption{Orbital diagram for the inner solar produced by the spectral variational integrator using all 8 planets, the sun, and Pluto with 100 time steps with \(h=100\) days.}
\label{fig:InnerSolar}
\end{figure}
\begin{figure}
\caption{Orbital diagram for the inner solar produced by the spectral variational integrator using the 4 outer planets, Pluto, with the sun and 4 inner planets aggregated to a point with 100 time steps at \(h=1825\) days.}
\label{fig:OuterSolar}
\end{figure}
\section{Conclusions and Future Work}
In this paper a new numerical method for variational problems was introduced, specifically a symplectic momentum-preserving integrator that exhibits geometric convergence to the true flow of a system under the appropriate conditions. These integrators were constructed under the general framework of Galerkin variational integrators, and made use of the global function paradigm common to many different spectral methods.
Additionally, a general convergence theorem was established for Galerkin type variational integrators, establishing the important result that, under suitable hypotheses, Galerkin variational integrators will inherit the optimal order of convergence permitted by the underlying approximation space used in their construction. This result provides a powerful tool for both constructing and analyzing variational integrators, it provides a methodology for constructing methods of very high order of accuracy, and it also establishes order of convergence for methods that can be viewed as Galerkin variational integrators. It was shown that from the one step map, a continuous approximation to the solution of the Euler-Lagrange equations can be easily recovered over each time step. The error of these continuous approximations was shown to be related to the error of the one step map. Furthermore, the Noether quantities along this continuous approximation approximate the true Noether quantity up to a small error which does not grow with the number of steps taken. It was also shown that the error of the Noether quantities converges to zero with \(n\) or \(h\) refinement at a predictable rate.
In addition to the convergence results, another interesting feature of spectral variational integrators is the construction of very high order methods that remain stable and accurate using time steps that are orders of magnitude larger than can be tolerated by traditional integrators. The trade off is that the computational effort required to compute each time step is also orders of magnitude larger than that of other methods, which are a major trade off in terms of the practicality of spectral variational integrators. However, a mitigating factor of this trade off is that the approach of solving a short sequence of large problems, as opposed to a large sequence of small problems, lends itself much better to parallelization and computational acceleration. The literature on methods for acceleration of the construction and solution of structured systems of linear and nonlinear problems for PDE problems is extensive, and it is likely that such methods could be applied to spectral variational integrators to greatly improve their computation cost.
\subsection{Future Work}
Future directions for this work are numerous. Because of generality of the construction of Galerkin variational integrators, there exists many possible directions of further exploration.
\subsubsection{Lie Group Spectral Variational Integrators} Following the approach of \citet{LeSh2011b} or \citet{BoMa2009}, it is relatively straight forward to extend spectral variational integrators to Lie groups using natural charts. A systematic investigation of the resulting Lie group methods, including convergence and near conservation of Noether quantities, would be a natural extension of the work done here.
\subsubsection{Novel Variational Integrators} The power of the Galerkin variational framework is its high flexibility in the choice of approximation spaces and quadrature rules used to construct numerical methods. This flexibility allows for the construction of novel methods specifically tailored to certain applications. One immediate example is the use of periodic functions to construct methods for detecting choreographies in Kepler problems, which would be closely related to methods already used with great success to detect choreographies. Another interesting application would be the use of high order polynomials to develop integrators for high-order Lie group problems, such as the construction of Riemannian splines, which has a variety of applications in motion planning. Enriching traditional polynomial approximation spaces with highly oscillatory functions could be used to develop methods for problems with dynamics evolving on radically different time scales, which are also very challenging for traditional numerical methods.
\subsubsection{Multisymplectic Variational Integrators} Multisymplectic geometry (see \citet{MaPaSh1998}) has become an increasingly popular framework for extending much of the geometric theory from classical Lagrangian mechanics to Lagrangian PDEs. The foundations for a discrete theory have been laid, and there have been significant results achieved in geometric techniques for structured problems such as elasticity, fluid mechanics, non-linear wave equations, and computational electromagnetism. However, there is still significant work to be done in the areas of construction of numerical methods, analysis of discrete geometric structure, and especially error analysis. Galerkin type methods have become a standard method in classical numerical PDE methods, such as Finite-Element Methods, Spectral, and Pseudospectral methods. The variational Galerkin framework could provide a natural framework for extending these classical methods to structure preserving geometric methods for PDEs, and the analysis of such methods will rely on the notion of the boundary Lagrangian (see \citet{VaLiLe2011}), which is the PDE analogue of the exact discrete Lagrangian.
\appendix \section{Proofs of Geometric Convergence of Spectral Variational Integrators}
As stated in \S\ref{ConSection}, it can be shown that spectral variational integrators converge geometrically to the true flow associated with a Lagrangian under the appropriate conditions. The proof is similar to that of order optimality, and is offered below.
However, before we offer a proof of the theorem, we must establish a result that extends Theorem \ref{MarsConv}. Specifically, we must show: \begin{theorem} \emph{(Extension of Theorem \ref{MarsConv} to Geometric Convergence)} \label{MarsExt}
Given a regular Lagrangian \(L\) and corresponding Hamiltonian \(H\), the following are equivalent for a discrete Lagrangian \(L_{d}\lefri{q_{0},q_{1},n}\):
\begin{enumerate} \item there exist a positive constant \(K\), where \(K < 1\), such that the discrete Hamiltonian map for \(L_{d}\lefri{q_{0},q_{h},n}\) has error \(\mathcal{O}\lefri{K^{n}}\), \item there exists a positive constant \(K\), where \(K < 1\), such that the discrete Legendre transforms of \(L_{d}\lefri{q_{0},q_{h},n}\) have error \(\mathcal{O}\lefri{K^{n}}\), \item there exists a positive constant \(K\), where \(K < 1\), such that \(L_{d}\lefri{q_{0},q_{h},n}\) approximates the exact discrete Lagrangian \(L_{d}^{E}\lefri{q_{0},q_{h},h}\) with error \(\mathcal{O}\lefri{K^{n}}\). \end{enumerate}
\end{theorem}
The proof of this theorem is a simple modification of the proof of Theorem \ref{MarsConv}, and is included here for completeness. For details, the interested reader is referred to \cite{MaWe2001}. \begin{proof} Since we are assuming that the time step \(h\) is being held constant, we will suppress it as an argument to the exact discrete Lagrangian, writing \(L^{E}_{d}\lefri{q_{0},q_{h}}\) for \(L^{E}_{d}\lefri{q_{0},q_{h},h}\). First, we will assume that \(L_{d}\lefri{q_{0},q_{h},n}\) approximates \(L_{d}\lefri{q_{0},q_{h}}\) with error \(\mathcal{O}\lefri{K^{n}}\) and show this implies the discrete Legendre transforms have error \(\mathcal{O}\lefri{K^{n}}\). By assumption, if \(L_{d}\lefri{q_{0},q_{h},n}\) has error \(\mathcal{O}\lefri{K^{n}}\), there exists a function which is smooth in its first two arguments \(e_{v}: Q \times Q \times \mathbb{N} \rightarrow \mathbb{R}\) such that:
\begin{align*} L_{d}\lefri{q_{0},q_{h},n} = L_{d}^{E}\lefri{q_{0},q_{h}} + K^{n}e_{v}\lefri{q_{0},q_{h},n}, \end{align*}
with \(\left|e_{v}\lefri{q_{0},q_{h},n}\right| \leq C_{v}\) on \(U_{v}\). Taking derivatives with respect to the first argument yields:
\begin{align*} \mathbb{F}^{-}L^{n}_{d}\lefri{q_{0},q_{h}} = \mathbb{F}^{-}L_{d}^{E}\lefri{q_{0},q_{h}} + K^{n}D_{1}e_{v}\lefri{q_{0},q_{h},n}, \end{align*}
and with respect to the second yields:
\begin{align*} \mathbb{F}^{+}L^{n}_{d}\lefri{q_{0},q_{h}} = \mathbb{F}^{+}L_{d}^{E}\lefri{q_{0},q_{h}} + K^{n}D_{2}e_{v}\lefri{q_{0},q_{h},n}. \end{align*}
Since \(e_{v}\) is smooth and bounded over the closed set \(U\), so are \(D_{1}e_{v}\) and \(D_{2}e_{v}\), yielding that the discrete Legendre transforms have error \(\mathcal{O}\lefri{K^{n}}\). Now, to show that if the discrete Legendre transforms have error \(\mathcal{O}\lefri{K^{n}}\), the discrete Lagrangian has error \(\mathcal{O}\lefri{K^{n}}\), we write: \begin{align*} e_{v}\lefri{q_{0},q_{h},n} &= \frac{1}{K^{n}}\left[L_{d}\lefri{q_{0},q_{h},n} - L^{E}_{d}\lefri{q_{0},q_{h}}\right],\\ D_{1}e_{v}\lefri{q_{0},q_{h},n} &= \frac{1}{K^{n}}\left[\mathbb{F}^{-}L_{d}\lefri{q_{0},q_{h},n} - \mathbb{F}^{-}L^{E}_{d}\lefri{q_{0},q_{h}}\right], \\ D_{2}e_{v}\lefri{q_{0},q_{h},n} &= \frac{1}{K^{n}}\left[\mathbb{F}^{+}L_{d}\lefri{q_{0},q_{h},n} - \mathbb{F}^{+}L^{E}_{d}\lefri{q_{0},q_{h}}\right]. \end{align*} Since \(D_{1}e_{v}\) and \(D_{2}e_{v}\) are smooth and bounded on a bounded set, this implies there exists a function \(d\lefri{n}\) such that \begin{align*}
\left\|e_{v}\lefri{q\lefri{0},q\lefri{h},n} - d\lefri{n}\right\| \leq C_{v}, \end{align*} for some constant \(C_{v}\). This shows that the discrete Lagrangian is equivalent to a discrete Lagrangian with error \(\mathcal{O}\lefri{K^{n}}\). We note that the equivalence is a consequence of the fact that one can add a function of \(h\) or \(n\) to any discrete Lagrangian and the resulting discrete Euler-Lagrange equations and discrete Legendre Transforms are unchanged, hence the function \(d\lefri{n}\).
To show the equivalence of the discrete Hamiltonian map having error \(\mathcal{O}\lefri{K^{n}}\) and the discrete Legendre transforms having error \(\mathcal{O}\lefri{K^{n}}\), we recall expressions for the discrete Hamiltonian map for the discrete Lagrangian \(L_{d}\) and exact discrete Lagrangian \(L^{E}_{d}\): \begin{align*} F_{L_{d}} &= \mathbb{F}^{+}L_{d} \circ \lefri{\mathbb{F}^{-}L_{d}}^{-1}, \\ F_{L^{E}_{d}} &= \mathbb{F}^{+}L^{E}_{d} \circ \lefri{\mathbb{F}^{-}L^{E}_{d}}^{-1}. \end{align*} Now, we make use of the following consequence of the implicit function theorem: If we have smooth functions \(g_{1},g_{2}\) and the sequences of functions \(\left\{f_{1_{n}}\right\}_{n=1}^{\infty}\), \(\left\{f_{2_{n}}\right\}_{n=1}^{\infty}\), \(\left\{e_{1_{n}}\right\}_{n=1}^{\infty}\) and \(\left\{e_{2_{n}}\right\}_{n=1}^{\infty}\) such that \begin{align*} f_{1_{n}}\lefri{x} &= g_{1}\lefri{x} + K^{n}e_{1_{n}}\lefri{x}, \\ f_{2_{n}}\lefri{x} &= g_{2}\lefri{x} + K^{n}e_{2_{n}}\lefri{x}, \end{align*}
where \(\sup{\left\{\left\|e_{1_{n}}\right\|\right\}_{n=1}^{\infty}} < C_{1}\) and \(\sup{\left\{\left\|e_{2_{n}}\right\|\right\}_{n=1}^{\infty}} < C_{2}\) on compact sets, then \begin{align} f_{2_{n}}\lefri{f_{1_{n}}\lefri{x}} &= g_{2}\lefri{g_{1}\lefri{x}} + K^{n}e_{12_{n}}\lefri{x} \label{ImpFun1}\\ f_{1_{n}}^{-1}\lefri{y} &= g_{1}^{-1}\lefri{y} + K^{n}\bar{e}_{1_{n}}\lefri{y} \label{ImpFun2} \end{align}
for some sequences of functions \(\left\{e_{12_{n}}\right\}_{n=1}^{\infty}\), \(\left\{\bar{e}_{1_{n}}\right\}_{n=1}^{\infty}\) where \(\sup{\left\{\left\|e_{12_{n}}\right\|\right\}_{n=1}^{\infty}} < C_{1}\) and \(\sup{\left\{\left\|\bar{e}_{1_{n}}\right\|\right\}_{n=1}^{\infty}} < C_{2}\) on compact sets.
It follows from (\ref{ImpFun1}) and (\ref{ImpFun2}) that if the discrete Legendre transforms have error \(\mathcal{O}\lefri{K^{n}}\), the discrete Hamiltonian map does as well. Finally, if we have a discrete Hamiltonian map has error \(\mathcal{O}\lefri{K^{n}}\), we use the identity \begin{align*} \lefri{\mathbb{F}^{-}L_{d}}^{-1}\lefri{q_{0},p_{0}} &= \lefri{q_{0}, \pi_{Q}\circ F_{L_{d}}\lefri{q_{0},p_{0}}} \end{align*} where \(\pi_{Q}\) is the projection map \(\pi_{Q}:\lefri{q,p} \rightarrow q\) and (\ref{ImpFun2}) to see that \(\mathbb{F}^{-}L_{d}\) is has error \(\mathcal{O}\lefri{K^{n}}\), and the identity: \begin{align*} \mathbb{F}^{+}L_{d} = F_{L_{d}} \circ \mathbb{F}^{-}L_{d}, \end{align*} along with (\ref{ImpFun1}) to establish that \(F^{+}L_{d}\) also has error \(\mathcal{O}\lefri{K^{n}}\), which completes the proof. \end{proof}
This simple extension is a critical tool for establishing the geometric convergence of spectral variational integrators, and leads to the following theorem concerning the accuracy of spectral variational integrators.
\begin{theorem} \emph{(Geometric Convergence of Spectral Variational Integrators)} \label{SpecConv_App}
Given an interval \(\left[0,h\right]\) and a Lagrangian \(L:TQ \rightarrow \mathbb{R}\), let \(\truq\) be the exact solution to the Euler-Lagrange equations, and \(\galqn\) be the stationary point of the spectral variational discrete action \begin{align*} \SDLN{q_{0}}{q_{h}}{n} = \ext_{\galargsf{q_{0}}{q_{h}}} \mathbb{S}_{d}\lefri{\left\{q_{i}\right\}_{i=1}^{n}}= \ext_{\galargsf{q_{0}}{q_{h}}} h\sum_{j=0}^{m_{n}} \bnj L\lefri{q_{n}\lefri{\cnjh},\dot{q}_{n}\lefri{\cnjh}}. \end{align*}
If:
\begin{enumerate}
\item there exists constants \(\ApproxC,\ApproxK\), \(\ApproxK < 1\), independent of \(n\), such that, for each \(n\), there exists a curve \(\optqn \in \mathbb{M}^{n}\lefri{\left[0,h\right],Q}\) such that,
\begin{align*}
\left\|\lefri{\truq,\dtruq} - \lefri{\optqn,\doptqn}\right\| & \leq \ApproxC \ApproxK^{n}, \end{align*}
\item there exists a closed and bounded neighborhood \(U \subset TQ\) such that \(\lefri{\truq\lefri{t},\dtruq\lefri{t}} \in U\) and \(\lefri{\optqn\lefri{t},\doptqn\lefri{t}} \in U\) for all \(t\) and \(n\), and all partial derivatives of \(L\) are continuous on \(U\),
\item for the sequence of quadrature rules \(\mathcal{G}_{n}\lefri{f} = h\sum_{j=1}^{m_{n}} \bnj f\lefri{\cnjh} \approx \int_{0}^{h}f\lefri{t}\dt\) there exists constants \(\QuadC\), \(\QuadK\), \(\QuadK < 1\), independent of \(n\) such that:
\begin{align*}
\left|\int_{0}^{h} L\lefri{q_{n}\lefri{t}, \dot{q}_{n}\lefri{t}}\dt - h\sum_{j=1}^{m_{n}} \bnj L\lefri{q_{n}\lefri{\cnjh},\dot{q}_{n}\lefri{\cnjh}}\right| \leq \QuadC \QuadK^{n}, \end{align*}
for any \(q_{n}\in \mathbb{M}^{n}\lefri{\left[0,h\right],Q}\),
\item and the stationary points \(\truq\), \(\galqn\) minimize their respective actions,
\end{enumerate}
then
\begin{align}
\left|\EDL{q_{0}}{q_{1}} - \SDLN{q_{0}}{q_{1}}{n}\right| \leq \SpecC\SpecK^{n} \label{GeoBound_App} \end{align}
for some constants \(\SpecC,\SpecK\), \(\SpecK < 1\), independent of \(n\), and hence the discrete Hamiltonian flow map has error \(\mathcal{O}\lefri{\SpecK^{n}}\).
\end{theorem}
\begin{proof}
As before, we rewrite both the exact discrete Lagrangian and the spectral discrete Lagrangian:
\begin{align*}
\left|\EDL{q_{0}}{q_{1}} - \SDLN{q_{0}}{q_{1}}{n}\right| &= \left|\int_{0}^{h}L\lefri{\truq\lefri{t},\dtruq\lefri{t}}\dt - \mathcal{G}_{n}\lefri{L\lefri{\galqn\lefri{t},\dgalqn\lefri{t}}}\right| \\
&= \left|\int_{0}^{h}L\lefri{\truq\lefri{t},\dtruq\lefri{t}}\dt - h\sum_{j=1}^{m_{n}}\bnj L\lefri{\galqn\lefri{\cnjh},\dgalqn\lefri{\cnjh}}\right| \\
&= \left|\int_{0}^{h}L\lefri{\truq,\dtruq}\dt - h\sum_{j=1}^{m_{n}}\bnj L\lefri{\galqn,\dgalqn}\right|, \end{align*}
with suppression of the \(t\) argument. We introduce the action evaluated on the curve \(\optqn\):
\begin{align}
\left|\int_{0}^{h}L\lefri{\truq,\dtruq}\dt - h\sum_{j=1}^{m_{n}}\bnj L\lefri{\galqn,\dgalqn}\right| & = \left|\int_{0}^{h}L\lefri{\truq,\dtruq}\dt - \int_{0}^{h} L\lefri{\optqn, \doptqn} \dt \right. \\
& \hspace{10em} \left. + \int_{0}^{h} L\lefri{\optqn,\doptqn}\dt - h\sum_{j=1}^{m_{n}}\bnj L\lefri{\galqn,\dgalqn}\right| \nonumber \\
&\leq \left|\int_{0}^{h}L\lefri{\truq,\dtruq}\dt - \int_{0}^{h} L \lefri{\optqn, \doptqn}\dt\right| \\
& \hspace{10em} + \left|\int_{0}^{h} L \lefri{\optqn,\doptqn} \dt - h\sum_{j=1}^{m}\bnj L\lefri{\galqn,\dgalqn}\right|. \label{STwoTerms} \end{align}
Considering the first term in (\ref{STwoTerms}):
\begin{align*}
\left|\int_{0}^{h}L\lefri{\truq,\dtruq}\dt - \int_{0}^{h} L \lefri{\optqn, \doptqn}\dt\right| &= \left|\int_{0}^{h}L\lefri{\truq,\dtruq} - L \lefri{\optqn, \doptqn} \dt\right| \\
& \leq \int_{0}^{h} \left|L\lefri{\truq,\dtruq} - L\lefri{\optqn,\doptqn}\right|\dt. \end{align*}
By assumption, all partials of \(L\) are continuous on \(U\), and since \(U\) is closed and bounded, this implies \(L\) is Lipschitz on \(U\), so let \(\LagLipC\) denote the Lipschitz constant. Since, again by assumption, \(\lefri{\truq,\dtruq} \in U\) and \(\lefri{\optqn,\doptqn} \in U\), we can obtain:
\begin{align*}
\int_{0}^{h}\left|L\lefri{\truq,\dtruq} - L \lefri{\optqn,\doptqn}\right|\dt \leq& \int_{0}^{h} \LagLipC \left|\lefri{\truq,\dtruq} - \lefri{\optqn,\doptqn}\right| \dt \\ \leq& \int_{0}^{h} \LagLipC \ApproxC \ApproxK^{n} \dt \\ =& h\LagLipC\ApproxC\ApproxK^{n}. \end{align*} Hence, \begin{align}
\left|\int_{0}^{h} L\lefri{\truq, \dtruq} \dt - \int_{0}^{h} L\lefri{\optqn,\doptqn} \dt\right| \leq h \LagLipC \ApproxC \ApproxK^{n}. \label{SFirstTermIneq} \end{align}
Next, considering the second term in (\ref{STwoTerms}),
\begin{align*}
\left|\int_{0}^{h} L\lefri{\optqn,\doptqn} \dt - \sum_{j=1}^{m} h\bnj L\lefri{\galqn,\dgalqn} \right|, \end{align*}
since \(\galqn\) minimizes its action,
\begin{align} h\sum_{j=1}^{m_{n}} \bnj L\lefri{\galqn,\dgalqn} \leq h\sum_{j=1}^{m_{n}} \bnj L \lefri{\optqn,\doptqn} \leq \int_{0}^{h} L\lefri{\optqn,\doptqn} \dt + \QuadC \QuadK^{n} \label{SUpperBound} \end{align}
where the inequalities follow from the assumptions on the order of the quadrature rule and (\ref{SFirstTermIneq}). Furthermore,
\begin{align} h\sum_{j=1}^{m_{n}} \bnj L \lefri{\galqn,\dgalqn} \geq \int_{0}^{h} L\lefri{\galqn,\dgalqn} \dt - \QuadC\QuadK^{n} &\geq \int_{0}^{h} L\lefri{\truq,\dtruq} \dt - \QuadC \QuadK^{n} \nonumber\\ &\geq \int_{0}^{h} L\lefri{\optqn,\doptqn} \dt - h\LagLipC \ApproxC \ApproxK^{n} - \QuadC \QuadK^{n}, \label{SLowerBound} \end{align}
where the inequalities follow from (\ref{SFirstTermIneq}), the order of the sequence of quadrature rules, and the assumption that \(\truq\) minimizes its action. Putting (\ref{SUpperBound}) and (\ref{SLowerBound}) together, we can conclude: \begin{align}
\left|\int_{0}^{h}L\lefri{\optqn,\doptqn}\dt - h\sum_{j=1}^{m_{n}} \bnj L\lefri{\galqn,\dgalqn}\right| \leq \lefri{h\LagLipC\ApproxC + \QuadC}\SpecK^{-n} \label{SSecondTermIneq}. \end{align}
where \(\SpecK = \max\lefri{\ApproxK,\QuadK}\). Now, combining the bounds (\ref{SFirstTermIneq}) and (\ref{SSecondTermIneq}) in (\ref{STwoTerms}), we can conclude \begin{align*}
\left|\EDL{q_{0}}{q_{1}} - \SDLN{q_{0}}{q_{1}}{n}\right| \leq \lefri{2h\LagLipC\ApproxC + \QuadC}\SpecK^{-n} \end{align*}
which, combined with Theorem \ref{MarsExt}, establishes the rate of convergence. \end{proof}
\end{document} | arXiv |
At what distance from the Sun can planetary moons exist?
Mercury and Venus are theorized to have no moons because they are so close to the sun.
Is there a theoretical distance in which moons tend to exist based on simulations?
natural-satellites mathematics
RonJohn
WilliamWilliam
$\begingroup$ You want to take into account the Hill Sphere of the body and check its long-term stability via an nbody simulation $\endgroup$
– fasterthanlight
There are several factors determining the inner limit to moons. Perhaps the simplest is that it needs to stay inside the Hill sphere, the region around the planet where the planet's gravity dominates over the sun's. If the planet's orbit has semi-major axis $a$ and eccentricity $e$ the farthest the moon can orbit is $$r_H \approx a(1-e)\sqrt[3]{\frac{m}{3M}}$$ where $m$ is the planet mass and $M$ the sun mass.
The closest a satellite can orbit a planet is the Roche limit, $$r_R = r_m\sqrt[3]{\frac{2m}{m_m}}$$ where $r_m$ and $m_m$ is the radius and mass of the moon. Equating $r_H=r_R$ and assuming $e=0$ to get the minimal $a$ where a moon is possible gives $$ a_{min}= r_m\sqrt[3]{\frac{6M}{m_m}}.$$
For a Moon with $r_m=1737.4$ km and $M/m_m=27090711$ (e.g. our moon), this is 0.006 AU (948,179 km), 1.36 solar radii! This is still just barely outside the Roche limit for an Earth-sized planet relative to the sun.
(See (Donnison 2010) for a more careful estimation for the full three-body problem applied to moons. (Domingos, Winter & Yokoyama 2006) found the rough limits $a_{crit}\approx 0.4895(1-1.0305e_{planet}-0.2738e_{sat})r_H$ for prograde satellites and about twice this limit for retograde satellites.)
However, while this shows that in principle you can have moons extremely close to stars, in practice they are not going to occur.
The most obvious problem is that very close planets will become tidally locked to the sun, and that will make the moon spiral inward since it will dissipate orbital energy through tidal deformation of the planet. The effect becomes bigger for heavier satellites. (Barnes & O'Brien 2002) calculate the following allowed region in a 4.6 Gy old system around a 1 Jupiter mass planet: The curve scales as $m_m \propto a^{13/2} m^{8/3}r^{-5}$; for an Earth-like primary the corresponding masses have to be 2.7% of the Jupiter case (although the different tidal properties of the planets makes this estimate somewhat dodgy).
There are other destabilizing factors for small bodies close to the sun such as the Yarkovsky effect. So while a tiny moon can reside close to the sun, it will likely not remain there long.
The opposite question, if there is an outer limit to planets holding satellites, presumably can be answered negatively. Clearly there are less and less disturbances the further you go outward, and the only issue is if a planet can accrete or capture a satellite. Given the common presence of satellites around trans-Neptunian objects this seems fairly common.
Anders SandbergAnders Sandberg
$\begingroup$ I'm a little surprised that the size of the moon (its radius) matters (you did set e=0). Can't one assume it's a point particle and come up with a similar estimate for a? Why does $r_m$ matter? $\endgroup$
– Buck Thorn
$\begingroup$ @BuckThorn The Roche limit formula factors in the relative density of both bodies. Hence you need mass and volume (and volume derives from radius for a roughly spherical body). $\endgroup$
– Tonny
$\begingroup$ Note that the Barnes & O'Brien 2002 paper assumes that even large moons are still much smaller than the planet, such that the tidal torque exerted on the planet by the moon is negligible compared to solar tides. I'm not sure if (or to what extent) their conclusions can be applied to systems where lunar and solar tides are comparable in magnitude, like the Earth–Moon system. In particular, off the top of my head I cannot say for sure that there couldn't be some critical moon size above which the system would become more stable again. $\endgroup$
– Ilmari Karonen
$\begingroup$ Your "Domingos, Winter & Yokoyama 2006" link appears to have some (expired) session information in it. $\endgroup$
– Peter - Reinstate Monica
$\begingroup$ @Peter-ReinstateMonica - Oops, replaced the link with a stable one. $\endgroup$
– Anders Sandberg
Short Answer:
There is an inner limit to how clase a palnet could orbit to its star and keep a moonin orbit around it. But I don't know how to calculate it. As far as I know there is no outer limit to how far a planet could be from its star and have moons. A rogue planet in interstellar space far from a star could have moons.
Every planet whch orbits a star has an inner and outer distance between which any moons would have to orbit.
The inner distance partially depends on the characteristics of the specific moon. It is call the Roche radius or Roche limit. Any object, that is held together mostly by its gravity, like a star, planet, or moon, which passes with the Roche limit of a more massive object will be torn apart by tidal forces. The Roch elimit depends of the relative densiites of the two bodies and whether the smaller object is rigid or fluid.
A moon can not form out of smaller objects if it is within the Roche limit of its planet, and if a moon passes within the Roche limit of its planet it will break up. So there should be no moons within the Roche limits of their planets.
Formulas for calculating the Roche limits of astronomical bodies can be found at:
https://en.wikipedia.org/wiki/Roche_limit[1]
The outer limit at which a moon can orbit a planet is call the Hill radius or Hill sphere. The size of the hill sphere depends on the masses of the planet and its star and the distance between them, since the stronger the gravity of the star is relative to the planet, the smaller the Hill sphere of the planet will be.
There are formulas for calculating an object's Hill Sphere at:
https://en.wikipedia.org/wiki/Hill_sphere[2]
However, it turns out that a moon orbiting in the outer parts of a Hill sphere will not have an orbit stable for long periods of astronomical time.
The Hill sphere is only an approximation, and other forces (such as radiation pressure or the Yarkovsky effect) can eventually perturb an object out of the sphere. This third object should also be of small enough mass that it introduces no additional complications through its own gravity. Detailed numerical calculations show that orbits at or just within the Hill sphere are not stable in the long term; it appears that stable satellite orbits exist only inside 1/2 to 1/3 of the Hill radius. The region of stability for retrograde orbits at a large distance from the primary is larger than the region for prograde orbits at a large distance from the primary. This was thought to explain the preponderance of retrograde moons around Jupiter; however, Saturn has a more even mix of retrograde/prograde moons so the reasons are more complicated.3
https://en.wikipedia.org/wiki/Hill_sphere#True_region_of_stability[3]
The Hill sphere of Earth extends to about 1,500,000 kilometers, so the region of truely stable orbits around Earth extends only to about 500,000 to 750,000 kilometers from Earth.
The Roche limit for Earth is 1.49 radii for rigid objects and 2.88 radii for fluid objects. Since the radius of Earth is 6,371 kilometers, the Roche limit for rigid objects is 9,492.79 kilometers and for fluid objects is 18,348.48 Kilometers.
The larger and more massive a planet is, the greater the radii of its Roche limit and Hill sphere will be. The farther a planet is from its star, the greter the radii of its Hill sphere will be.
Mercury and Venus are less massive than the Earth, and so their Roche limits are smaller, which is good for not destroying close moons.. They are closer to the Sun than the Earth, so the Sun's gravity is stronger where they orbit, and since they are less massive than Earth, their mases and distances from the Sun combine to make their HIll spheres much smaller than Earth's, which is bad for keeping moons.
The Hill sphere of Mercury has a radius of only 175,300 kilometers, so the true region of stability should have an outer edge at only 58,432 to 87,650 kilometers.
The Hill Spere of Venus is 1,004,200 kilometers, so the true region of stability should have an outer edge at only 334,733.3 to 502,100 kilometers, much larger than Mercury's but smaller than Earth's.
But there ae other problems with Mercury and Venus having moons. The orbit of a moona round tis planet will chang eover time, as the moon moves closer to or farther from the planet. So if a moon starts out between the Roche limit and the Hill sphere of its planet, it could move out of the zone where stable orbits are possible.
If a planet and its moon form together, the moon will have a prograde orbit. Most of the objects in the Solar System rotate in the same direction that they orbit the sun in, whch is called a prograde orbit. If a moon forms with its planet, what is called a regular moon, it will orbit the planet in the same direction as the planet rotates, which will be prograde relative to the planet. Since most planets rotate in a prograde direction relative to their orbit, most of the regular moons formed with their planets orbit in a prograde direction relative to the plant's rbit around the Sun.
Planets can also capture passing objects and make them their moons. The capture process can result in either a prograde orbit, or an orbit in the opposite direction, a retrograde orbit. Therer are a number of captured moons in the Solar System, some with prograd eorbits and some with retrograde orbits.
All retrograde satellites experience tidal deceleration to some degree. The only satellite in the Solar System for which this effect is non-negligible is Neptune's moon Triton. All the other retrograde satellites are on distant orbits and tidal forces between them and the planet are negligible.
https://en.wikipedia.org/wiki/Retrograde_and_prograde_motion#Natural_satellites_and_rings[4]
Triton's revolution around Neptune has become a nearly perfect circle with an eccentricity of almost zero. Viscoelastic damping from tides alone is not thought to be capable of circularizing Triton's orbit in the time since the origin of the system, and gas drag from a prograde debris disc is likely to have played a substantial role.4 Tidal interactions also cause Triton's orbit, which is already closer to Neptune than the Moon's is to Earth, to gradually decay further; predictions are that 3.6 billion years from now, Triton will pass within Neptune's Roche limit.[25] This will result in either a collision with Neptune's atmosphere or the breakup of Triton, forming a new ring system similar to that found around Saturn.[25]
https://en.wikipedia.org/wiki/Triton_(moon)#Orbit_and_rotation[5]
So if Mercury or Venus captured a moon in an orbit retrograde to their rotation, that moon would gradually spiral down to within the Roche limit and be destroyed after millions or billions of years.
Moons in prograde orbits around their planets will either move away from the planets or move in toward their planets.
If a moon is in a prograde orbit above the synchronous orbit level, it will move away from the planet due to tidal acceleration. Thus, after millions or billions or psosibly trillions of years, it might pass outside the Hill sphere of the planet and go into orbit around their star. MOs tof the moons is the Solar System orbit farther than the synchronous orbits of their planets, and so a gradually receeding form them.
https://en.wikipedia.org/wiki/Tidal_acceleration#Other_cases_of_tidal_acceleration[6]
Some moons orbit their planets below the synchronous orbit level. Their orbital periods are less than one day of the planet. Those moons experience tidal deceleration and slowly spiral inward toward their planets, eventually reaching the Roche limits of their planets and breaking up.
https://en.wikipedia.org/wiki/Tidal_acceleration#Tidal_deceleration[7]
The sidereal days, the period it takes them to rotate 360 degrees with respect to the disant stars, of Mercury and Venus are 58.646 Earth days and 243.0226 Earth days respectively. So any moons orbiting Mercury and Venus at the synchrnous orbits would be orbiting very far from their planets.
Mercury and Venus are believed to have no satellites chiefly because any hypothetical satellite would have suffered deceleration long ago and crashed into the planets due to the very slow rotation speeds of both planets; in addition, Venus also has retrograde rotation.
For a long time astronomers believed that Mercury, and possibly Venus, were tidally locked to the Sun, with rotation periods equal to their oribital periods, so that a day would be equal toa year, and one side of theplanet would always face the Sun and the other side always face away from the Sun. That is called a 1:1 resonance. It is now known that Mercury and Venus do not have 1:1 resonances, but their rotation periods are very long compared to their years of 87.97 Earth days and 224.7 Earth days.
When the gravity of a star is too strong at the orbit of a planet - which depends on the mas sof hte star and the orbital distance of the planet from the star - that planet will become tidally locked to the Star, either with a 1:1 resonance or a resonance with simple integers, such as the 3:2 resonance of the planet Mercury - Mercury rotates three times in two Mercurian years - or some other simple resonance.
So if a planet is too deep in the gravity well of its star, the planet's rotation will be slowed down a lot, and the synchronous orbit of the planet will be very far from the planet, possbily beyond the Hill sphere. So a moon outside the Hill spehre would be lost into space, and a moon below the synchronous orbit would have a decaying orbit and eventually reach the roche limit of the planet.
So if the mass of a star is known, the distance at which a planet would have its day greatly slowed down, thus making it hard or impossible for it to keep its moon(s) can be calculated.
And of course the shorter years of planets orbiting close to their stars is another problem for any potential moons.
The longest possible length of a satellite's day compatible with Hill stability has been shown to be about P∗p/9, P∗p being the planet's orbital period about the star (Kipping 2009a)
https://arxiv.org/ftp/arxiv/papers/1209/1209.5323.pdf[8]
And the source is given as:
https://academic.oup.com/mnras/article/392/1/181/1071655[9]
So this claims that a mooon can not have a stable orbit around its planets unless the orbital of the moon is less than one nineth (0.111111) theorbital period of the planet around the star.
The greater the mass of a star, the shorter the year of a lanet oriting it at a specific distance will be, since the increased gravity of the star means the planet will have to orbit faster to stay in orbit. The closer a planet orbits to a star, the faster it will have to orbit, due to the stronger gravity at tht distance, and the shorter its year will be.
So the closer a planet orbit sits star, and the more massive the star is, the shorter will be the year of the planet.
The shortest orbital period or year of any known exoplanet orbiting a main sequence star is that of K2-137b, I think, whichis only 4.31 hours long.
https://academic.oup.com/mnras/article/474/4/5523/4604789[10]
So if a moon of K2-137b had a stable orbt it around it, that orbit would proably have to be less than about 0.4788 hours long. Which should be far below the Roche limit of K2-137b and probably also far below the surface of K2-137b.
So it should be possible to create a formula showing how close a planet could get to a star of a specific mass and still retain a moon.
Of course there is no outer limit to how far away a planet with a moon could get from its star.
M. A. GoldingM. A. Golding
$\begingroup$ I gave the question a +1 but was I really looking for math formulas. $\endgroup$
– William
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Jupiter's asteroid-like moons and planetary-systems around sub-brown dwarfs | CommonCrawl |
Encyclopedia Magnetica
barkhausen_noise
Barkhausen noise
Magnetisation process
Single Barkhausen jumps
Measurement of BN
Analysis of BN activity
RMS of BN signal
Total sum of amplitudes (TSA)
Total number of peaks (TNP)
Power spectrum
Other types of analysis
Magneto-acoustic emissions
Stan Zurek, Barkhausen noise, Encyclopedia Magnetica
https://E-Magnetica.pl/barkhausen_noise
Barkhausen noise (BN) - the phenomenon of rapid changes of positions of domain walls during the process of magnetisation of a ferromagnetic material.1)2)3)
Barkhausen noise is caused by rapid changes of flux density B due to domain wall movements - this causes high-frequency noise-like changes in induced voltage V
S. Zurek, E-Magnetica.pl, CC-BY-4.0
These sudden jumps (also referred to as Barkhausen jumps) can be made audible by suppressing the large voltage induced in a search coil (with a high-pass filter) and amplifying the frequencies in the audible range (see the recording of audible noise with the animation).1)4)
Barkhausen noise was discovered by Heinrich Barkhausen in 1919.5)
A phenomenon similar to magnetic Barkhausen noise is also present in ferroelectric materials, which have ferroelectric domains and hence ferroelectric domain walls.6)7)
Recording of a real Barkhausen noise in grain-oriented electrical steel (with sound)
barkhausen_noise_magnetica.mp4
(link to video file)
Barkhausen noise (BN) activity is greater at intervals during which the changes of flux density B are the most rapid
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Magnetisation process involves changes in the configuration of magnetic domains4)
Ferromagnetism is synonymous with spontaneous magnetisation of a material. Each part of the volume is magnetised to saturation and each such partial volume is known as a magnetic domain. The magnetic alignments of domains can point in different or opposing directions so that globally their contributions cancel partially or fully. Thus, the net volume magnetisation can be significantly smaller than saturation, and even zero for a demagnetised body (even though the individual domains remain saturated).2)
Magnetic domain structure (lancet combs switching during magnetisation) in high-permeability grain-oriented electrical steel. Bar domains are visible in the upper part. Copyright © Oles Hostanar
The magnetisation process, for example by applying external magnetic field, involves changes in the configuration of magnetic domains, which is accomplished by movement of the domain walls which separate domains. These movements can be impeded in several ways: crystal defects, grain boundaries, non-magnetic inclusions and precipitates, surface defects, etc.8) The phenomenon of "sticking" to local energy minima is called domain wall pinning.
Simplified animation of a domain wall crossing a non-magnetic void8)
The magnetisation process generates an effective pressure on the given domain wall to move. However, such domain wall can be pinned to one of the above-mentioned defect and therefore it will require additional pressure to overcome to pinning force. Once the pressure to move exceeds the pinning force the domain wall will be suddenly unpinned and it will move rapidly, until the forces are equalised, or for example when the wall encounters the next defect.
As a result, the process of magnetisation is not smooth, but comprises jittery jumps of domain walls. But a sudden movement of a domain wall is synonymous with a rapid change of local magnetisation M and hence also of the local flux density B. And according to the Faraday's law changes in B generate changes in electric field and thus to the voltage induced in a coil magnetically coupled to such material.
Typical Co-Fe amorphous microwires
There are magnetic materials which are magnetically very soft (very low coercivity) and because of their geometry can have just a single domain present (at least in some part of the volume).
Co-Fe amorphous core (red arrow) in glass coating (blue arrows and translucent tip), 0.1 mm diameter
This is the case for example in microwires made from amorphous cobalt. Glass coating is added during the manufacturing process to help with the wire production, obtaining amorphous phase and applying internal stress. The magnetic domain structure is such that there is an inner core with the single-domain system, and the outer core with a cylindrical9) or multiple closure domains.10)
Closure domains can remain at the ends of the inner core. Once external magnetic field is applied, the main domain wall can rapidly change its position (even faster than 1000 m/s) travelling from one end of the wire to the other. Such a rapid transition constitutes a single Barkhausen jump.11)
When plotted as a B-H loop the rapid reversal of magnetisation makes the loop appear rectangular, because once the coercive field HC threshold is exceeded the corresponding B changes its polarity.12)
Structure of magnetic domains in a microwire10)
Single Barkhausen jump in a microwire creates a rectangular B-H loop12) by T. Charubin, M. Nowicki, R. Szewczyk, CC-BY-4.0
Domain wall velocity in Co-based amorphous microwire11)
Simple search coil for detecting Barkhausen noise13) by F. Bohn, G. Durin, M.A. Correa, N.R. Machado, R.D. Della Pace, C. Chesman, R.L. Sommer, CC-BY-4.0
Barkhausen noise is generated by changes of magnetisation M and hence also by flux density B and it can be detected by a search coil (pick-up coil) whose operation is based on the Faraday's law of induction.
Any changes in B induce corresponding voltage V at the terminals of the search coil.
Voltage in a Barkhausen coil sensor
$$ V = N · A · \frac{dB}{dt}$$ (V)
$V$ - voltage induced in the coil (V), $N$ - number of turns of the coil (unitless), $A$ - active cross-sectional area of the coil (m2), $dB/dt$ - derivative of flux density $B$ (T) with respect to time $t$ (s)
However, typical changes of B (apart from the single-domain materials) comprise large-amplitude slower changes which induce low-frequency and high amplitude of the associated voltage. But the very fast Barkhausen events are of much smaller amplitude and thus create much smaller signal, which is superimposed on the slower large signal.
Therefore, before the BN can be analysed it is necessary either to filter out the slower, large amplitude signal, or to compensate it out. This can be achieved either by high-pass or band-pass filter in signal processing electronics, or by arranging the pick-up coils in such a way that the slow large signal is eliminated or not induced at all. Typical band-pass filtering can be from 300 Hz to 300 kHz.4)13)
For example, it is possible to use two search coil connected in series opposition. Barkhausen noise is quite random locally so noise detected at two different locations will just add to each other. But the slow large components of voltage will be similar in both coils and thus these will compensate out each other, leaving only the BN noise signature in the output voltage of such two coils.
Another approach is to use a pick-up coil on a ferromagnetic core positioned perpendicularly to the surface of the sample under test. The activity in the main sample will magnetically couple to the small magnetic core and thus it can be detected without the large voltage being induced in it. The additional magnetic core should be made of a material which has much lower Barkhausen noise activity than the main sample.4)
Typical ways of detecting Barkhausen noise:4) with two opposing B-coils and with a single B-coil perpendicular to the surface (with a small cylindrical magnetic core)
Barkhausen noise is stochastic (random) in nature and its analysis is not straightforward, because the induced noise depends on many factors, including the frequency, amplitude and waveshape of the magnetic excitation (e.g. magnetising current).
Many methods were devised by researchers internationally4), with some examples given below. However, there is no standardised method for performing such measurements so the numerical values from different publications cannot be compared in the absolute sense.
RMS value of Barkhausen noise measured with two opposing B-coils (separated by 5 mm or 40 mm) at 50 Hz for a sample of grain-oriented electrical steel4)
RMS of the noise signal can be calculated after some high-pass or band-pass filtering. The RMS calculation follows the same method as measurement of RMS (root mean square) of any other signal, but it is applied to the Barkhausen noise waveform, typically digitised, with the calculations performed by a computer, for example over one cycle of magnetisation.
If the gain of the signal processing is calibrated, then the RMS of BN can be expressed in absolute units, which are typically quite small, e.g. less than 1 mV (as illustrated).
RMS of Barkhausen noise
(expressed
as integral) $$ V_{BN,RMS} = \sqrt{\frac{1}{T} · \int_0^T {(V_{BN}(t))^2} dt } $$ (V)
as sum
of samples) $$ V_{BN,RMS} = \sqrt{\frac{1}{N_{BN}} · \sum_{i=0}^{N_{BN}-1} {(V_{BN,i})^2} } $$ (V)
where: $V_{BN}(t)$ - voltage signal after filtering (V), $V_{BN,i}$ - sampled (digitised) single value of voltage after filtering (V), $T$ - time interval (s), $N_{BN}$ - total number of sample (unitless), $i$ - index (unitless), $t$ - time (s)
Total sum of amplitudes (TSA) of Barkhausen noise for grain-oriented electrical steel, measured at 50 Hz excitation4)
The total sum of amplitudes (TSA) is a method in which all the instances of digitised signal are added up to produce a single value. Absolute values are used in order to include the negative numbers.
The TSA values are not comparable between different measurement systems, because they depend on the sampling frequency (more data points produces higher values, even if the amplitude of the noise is similar).
Total sum of amplitudes TSA
$$ V_{BN,TSA} = \sum_{i=0}^{N_{BN}-1} { | V_{BN,i} | } $$ (V)
where: $V_{BN,i}$ - sampled (digitised) single value of voltage after filtering (V), $N_{BN}$ - total number of sample (unitless), $i$ - index (unitless)
Total number of peaks (TNP) of Barkhausen noise for non-oriented electrical steel, measured at 50 Hz excitation4)
Total number of peaks (TNP), as the name implies, is calculated simply as the number of detectable peaks in the filtered voltage. The result depends on the type of sensing, filtering, and criteria used for peak detection. Larger Barkhausen events which cause avalanches can cause fewer peaks.
The TNP value is unitless, because it reports the integer number of items.
Total number of peaks TNP
$$ TNP = \sum_{i=0}^{N_{BN}-1} { Peak_{V_{BN},i} } $$ (unitless)
where: $Peak_{V_{BN},i}$ - an instant of peak in the filtered voltage (unitless), $N_{BN}$ - total number of sample (unitless), $i$ - index (unitless)
Power spectrum of Barkhausen noise at lower frequencies for grain-oriented electrical steel, measured at 50 Hz excitation4)
In the power spectrum method, the Barkhausen noise (after filtering) is processed by a Fourier transform, which detects the frequency spectrum of the noise.
By definition, the spectrum will be limited by the filtering used in the analogue and digital processing. In the illustration showing an example of such spectrum, the values reduce to zero below 50 Hz, which is cause by the high-pass filter characteristics of the signal processing.
With digital processing, the maximum frequency that can be detected is limited by the Shannon-Nyquist limit of the data acquisition device.
Also, the possibility of aliasing has to be considered, because the Barkhausen noise can extend up to MHz frequencies. Therefore, some analogue anti-aliasing has to be employed. As a consequence, the signal is processed with band-pass characteristics, because high-pass filter is required to suppress the high-amplitude low-frequency induced voltage, and low-pass filter is required for anti-aliasing. This is one the main reasons for digital methods to have limited upper frequency of BN processing.
Kurtosis of Barkhausen noise for grain-oriented electrical steel, measured at 50 Hz excitation4)
Kurtosis is a method of statistical analysis of a given population of samples. It can quantify the "peakedness" or "flatness" of a statistical distribution. Using this method it is possible to compare the kurtosis value for example to that of ideal Gaussian distribution curve, thus studying the "randomness" of Barkhausen events.
The value of kurtosis has the units of V4 (volt to the power of 4).
Kurtosis K
$$ K = \frac{1}{N_{BN}} · \sum_{i=0}^{N_{BN}-1} { ( V_{BN,i} - V_{BN,mean} )^4 } $$ (V4)
where: $V_{BN,i}$ - subsequent voltage values (V), $V_{BN,mean}$ - mean value of voltage (V), $N_{BN}$ - total number of sample (unitless), $i$ - index (unitless)
Duration of Barkhausen events is correlated with their amplitude. One local Barkhausen jump can initiate others and the whole such sequence is sometimes referred to as Barkhausen avalanche.13)
The duration of avalanches can vary, and they can be analysed from the viewpoint of duration or frequency components. A whole range of analyses can be used even within the same study of the Barkhausen noise phenomenon.13)
Statistical analysis of Barkhausen avalanches in polycrystalline NiFe films of different thicknesses, from 20 to 1000 nm: a) distributions of avalanche sizes measured at 50 mHz, b) similar plot for the distributions of avalanche durations, c) average size as a function of the avalanche duration, d) power spectra.13) by F. Bohn, G. Durin, M.A. Correa, N.R. Machado, R.D. Della Pace, C. Chesman, R.L. Sommer, CC-BY-4.0
If the moving domain walls separate domains which are not in the opposing directions (0-180°), but for example at 90° to each other, then the changes in domain wall position can cause changes of dimensions of the material due to magnetostriction.
Such low-amplitude local vibrations of the material are known as magnetoacoustic emissions (MAE). The frequency spectrum for studying such phenomenon is similar to the Barkhausen noise, and also the type of analysis is similar, for example by plotting the power spectrum. However, the detection is carried out with a very sensitive microphone or acceleration sensor, rather than an inductive coil.4)
Simplified block diagram of signal processing for magneto-acoustic emissions4)
Barkhausen noise activity is affected by crystallographic structure and defects in the given material. Materials exposed to mechanical stress can deform thus increasing the number of internal defects. Also other processes such as neutron irradiation in nuclear plants can degrade the crystallographic arrangements in the steel exposed to such radiation.
The image below shows an example of Barkhausen noise activity in two samples exposed to different mechanical stress, so that the elastic deformation was ε=2.5% and 15%, respectively. In the sample with larger deformation the BN activity is visibly reduced, and this can be correlated with the amount of damage sustained by the given steel.14)
Reduced Barkhausen noise activity in material exposed to larger deformation14) by M. Pitoňák, M. Neslušan, P. Minárik, J. Čapek, K. Zgútová, M. Jurkovič, T. Kalina, CC-BY-4.0
System for measuring residual mechanical stresses in rails by means of Barkhausen noise15) by Y.-I. Hwang, Y.-I. Kim, D.-C. Seo, M.-K. Seo, W.-S. Lee, S. Kwon, K.-B. Kim, CC-BY-4.0
Detection of mechanical properties through measurement of Barkhausen noise is beneficial, because it can be carried out on the surface of the material, without the need for cutting out a sample - hence it belongs to the class of non-destructive testing. The applicability of the method is limited, because the Barkhausen noise cannot be measured or correlated to the material damage in an absolute way.
Nonetheless, there are commercial devices capable of performing non-destructive measurements in an automated way. The excitation is typically applied by a small U-shaped magnetising yoke, and the sensing is carried out by pick-up coils, with processing and filtering similar to as described above. Parameters such as degradation in strength, increase in hardness or embrittlement can be automatically quantified to some extent.
XYZ scanner with transducer: a) block diagram, b) photo, c) 3D view of the transducer16) by M. Maciusowicz, G. Psuj, CC-BY-4.0
However, the correlation between Barkhausen noise and the mechanical properties of a given magnetic sample is not strict, and cannot be quantified independently of a material. It is therefore not possible to calibrate such system for a generic measurement.
Instead, a comparative measurement has to be carried out, when a known "good" sample is available for calibration. For example, degradation of surface of gears made of magnetic steel can be detected.17) In such applications the quality and thermal pre-processing is well known for the "good" steel and degradation with the Barkhausen noise system can give reliable results.
The BN method can be used for assessment of a large surface area for example by employing the scanning methods.16) A small-size detection head can be automatically moved around a large surface to perform the "scanning" action, and a computerised system can collate, analyse and display all the data accordingly.
Magnetic domain
Magnetic domain wall
1), 1) Sławomir Tumański, Handbook of magnetic measurements, CRC Press / Taylor & Francis, Boca Raton, FL, 2011, ISBN 9780367864958
2), 2) David C. Jiles, Introduction to Magnetism and Magnetic Materials, Second Edition, Chapman & Hall, CRC, 1998, ISBN 9780412798603
3) Richard M. Bozorth, Ferromagnetism, Wiley-IEEE Press, 1993, ISBN 0780310322
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5) Heinrich Barkhausen (1919), Zwei mit Hilfe der neuen Verstärker entdeckte Erscheinungen, Phys. Z., 20, pp. 401–403
6) Yangyang Xu, Dezhen Xue, Yumei Zhou, Tong Su, Xiangdong Ding, Jun Sun, and E. K. H. Salje, "Avalanche dynamics of ferroelectric phase transitions in BaTiO3 and 0.7Pb(Mg2∕3Nb1∕3)O3-0.3PbTiO3 single crystals", Appl. Phys. Lett. 115, 022901 (2019) https://doi.org/10.1063/1.5099212
7) Keisuke Yazawa, Benjamin Ducharne, Hiroshi Uchida, Hiroshi Funakubo, and John E. Blendell, "Barkhausen noise analysis of thin film ferroelectrics", Appl. Phys. Lett. 117, 012902 (2020) https://doi.org/10.1063/5.0012635
8), 8) B.D. Cullity, C.D. Graham, Introduction to Magnetic Materials, 2nd edition, Wiley, IEEE Press, 2009, ISBN 9780471477419
9) Alekhina, I.; Kolesnikova, V.; Rodionov, V.; Andreev, N.; Panina, L.; Rodionova, V.; Perov, N. An Indirect Method of Micromagnetic Structure Estimation in Microwires. Nanomaterials 2021, 11, 274, https://doi.org/10.3390/nano11020274
10), 10) J. Olivera et al., "Temperature Dependence of the Magnetization Reversal Process and Domain Structure in Fe(77.5-x)Ni(x)Si(7.5)B(15) Magnetic Microwires," IEEE Transactions on Magnetics, vol. 44, no. 11, pp. 3946-3949, Nov. 2008, doi: 10.1109/TMAG.2008.2002194
11), 11) H. Chiriac, T. Ovari and M. Tibu, "Domain Wall Propagation in Nearly Zero Magnetostrictive Amorphous Microwires," in IEEE Transactions on Magnetics, vol. 44, no. 11, pp. 3931-3933, Nov. 2008, doi: 10.1109/TMAG.2008.2001326
12), 12) Charubin, T.; Nowicki, M.; Szewczyk, R. Influence of Torsion on Matteucci Effect Signal Parameters in Co-Based Bistable Amorphous Wire. Materials 2019, 12, 532. https://doi.org/10.3390/ma12030532
13), 13), 13), 13), 13) Bohn, F., Durin, G., Correa, M.A. et al. Playing with universality classes of Barkhausen avalanches. Sci Rep 8, 11294 (2018). https://doi.org/10.1038/s41598-018-29576-3
14), 14) Pitoňák, M.; Neslušan, M.; Minárik, P.; Čapek, J.; Zgútová, K.; Jurkovič, M.; Kalina, T. Investigation of Magnetic Anisotropy and Barkhausen Noise Asymmetry Resulting from Uniaxial Plastic Deformation of Steel S235. Appl. Sci. 2021, 11, 3600. https://doi.org/10.3390/app11083600
15) Hwang, Y.-I.; Kim, Y.-I.; Seo, D.-C.; Seo, M.-K.; Lee, W.-S.; Kwon, S.; Kim, K.-B. Experimental Consideration of Conditions for Measuring Residual Stresses of Rails Using Magnetic Barkhausen Noise Method. Materials 2021, 14, 5374. https://doi.org/10.3390/ma14185374
16), 16) Maciusowicz, M.; Psuj, G. Use of Time-Frequency Representation of Magnetic Barkhausen Noise for Evaluation of Easy Magnetization Axis of Grain-Oriented Steel. Materials 2020, 13, 3390. https://doi.org/10.3390/ma13153390
17) Stresstech, Barkhausen Noise Equipment, Non-destructive (NDT) measurement solutions for grinding burn and heat treatment defect testing, {accessed 2021-11-08}
Barkhausen noise, Magnetic domain walls, Magnetic domains, Ferromagnetism, Counter
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\begin{document}
\title{Cloning of Gaussian states by linear optics} \author{Stefano Olivares} \email{[email protected]} \author{Matteo G.~A.~Paris} \affiliation{Dipartimento di Fisica dell'Universit\`a degli Studi di Milano, Italia.} \author{Ulrik L.~Andersen} \email{[email protected]} \affiliation{Institut f\" ur Optik, Information und Photonik, Max-Planck Forschungsgruppe, Universit\" at Erlangen-N\" urnberg, G\" unther-Scharowsky str.~1, 91058, Erlangen, Germany}
\begin{abstract} We analyze in details a scheme for cloning of Gaussian states based on linear optical components and homodyne detection recently demonstrated by U.~L.~Andersen {\em et al.}~[Phys.~Rev.~Lett.~{\bf 94}, 240503 (2005)]. The input-output fidelity is evaluated for a generic (pure or mixed) Gaussian state taking into account the effect of non-unit quantum efficiency and unbalanced mode-mixing. In addition, since in most quantum information protocols the covariance matrix of the set of input states is not perfectly known, we evaluate the average cloning fidelity for classes of Gaussian states with the degree of squeezing and the number of thermal photons being only partially known. \end{abstract} \date{\today} \pacs{03.67.Hk, 03.65.Ta, 42.50.Lc} \keywords{Quantum cloning, Gaussian states, linear optics} \maketitle
\section{Introduction}\label{s:intro} The generation of perfect copies of an unknown quantum state is impossible according to the very nature of quantum mechanics. This is succinctly formulated by the no-cloning theorem~\cite{wooters82.nat,dieks82.pla,cl3,cl4}. It is, however, possible to make approximate copies of a quantum state by using a quantum cloning machine~\cite{buzek96.pra}. Originally, such a machine was proposed for cloning of qubits and has later been demonstrated experimentally~\cite{dv:experiment}. Shortly after this development, a continuous variable (CV)~\cite{braunstein05.rev} analog of the qubit quantum cloner was proposed~\cite{cl:cerf,cerf:PRA:2000} and recently it was shown that a CV optimal Gaussian cloner of coherent states can be implemented using an appropriate combination of beam splitters and a single phase insensitive parametric amplifier~\cite{braunstein01.prl,fiurasek01.prl}. Although this proposal sounds experimentally promising, the implementation of an efficient phase insensitive amplifier operating at the fundamental limit is a challenging task. This problem was solved by Andersen et al.~\cite{andersen05.prl}, who proposed and experimentally realized a much simpler configuration for optimal cloning of coherent states. The realization relies on simple linear optical components and a feed-forward loop. As a consequence of the simplicity, as well as the high quality of the optical devices used in this experiment, performances close to optimal ones were attained. In turn, the resulting cloning machine represents a highly versatile tool for further investigations on transformation of quantum information from a single system to many systems. \par A commonly used figure of merit to quantify the performance of cloning machines is the fidelity which is a measure of similarity between the hypothetically perfect clone, {\em i.e} the input state, and the actual clone. If the cloning fidelity is independent on the initial state the machine is referred to as a {\em universal} cloner. On the other hand, if the efficiency of the cloning action depends on the input state, then the proper measure in order to assess the performances of the machine is the average fidelity, which weight the fidelities associated to possible input states with the corresponding occurrence probability. In other words, for non-universal cloners, the alphabet of input states, and the distribution thereof, must be taken into account while evaluating the fidelity. Such an average fidelity has been considered in \cite{cochrane04.pra,braunstein00.mod,hammerer05.prl}. However, in all these references it is assumed that the input alphabet is only consisting of coherent states, hereby keeping the covariance matrix of all the possible input states constant. On the other hand, in some experimental realizations, the covariance matrix is not perfectly known due to uncontrollable fluctuations, and therefore it is important to include this uncertainty into the analysis. \par The aim of this paper is two-fold. At first we present a thorough theoretical description of the cloning machine described in Ref.~\cite{andersen05.prl} using a suitable phase-space analysis. In this way the full quantum dynamics of the machine can be taken into account; in particular we include the effect of losses in the detection scheme, as well as variations in the setups beam splitter ratios. The second topic of the paper is to investigate the average fidelity of the cloning machine for different ensembles of input states such as sets made of displaced squeezed or displaced thermal states with the squeezing parameter, or the number of thermal photons, distributed according to predefined distributions. \par The paper is structured as follows: in Sec.~\ref{s:scheme} we review the main components of the cloning machine based on linear optics, whereas in Sec.~\ref{s:GScloning} we calculate the input-output fidelities for the case of generic Gaussian states, and for specific classes including coherent, displaced squeezed and displaced thermal states. Finally, Sec.~\ref{s:outro} closes the paper with some concluding remarks.
\section{The linear cloning machine}\label{s:scheme}
Optimal Gaussian cloning can be realized using a phase insensitive amplifier and a beam splitter~\cite{braunstein01.prl,fiurasek01.prl}. However, it has been recently shown, theoretically and experimentally, that the parametric amplifier can be replaced by a simpler scheme involving only linear optical components, homodyne detection and a feed-forward loop~\cite{andersen05.prl}. This scheme, which is schematically depicted in Fig.~\ref{f:cl:scheme}, will be referred to as the linear cloning machine throughout the paper.
\begin{figure}
\caption{ Cloning of Gaussian states by linear optics: the input state $\varrho_{\rm in}$ is mixed with the vacuum $\varrho_{0}$ at a beam splitter (BS) of transmissivity $\tau_{1}$. The reflected beam is measured by double-homodyne detection and the outcome of the measurement $x + i y$ is forwarded to a modulator, which imposes a displacement $g (x + iy)$ on the transmitted beam, $g$ being a suitable amplification factor. Finally, the displaced state is impinged onto a second beam splitter of transmissivity $\tau_{2}$. The two outputs, $\varrho_1$ and $\varrho_2$, from the beam splitter represents the two clones.}
\label{f:cl:scheme}
\end{figure}
\par The input state, denoted by the density operator $\varrho_{\rm in}$, is mixed with the vacuum at a beam splitter (BS) with transmittivity $\tau_1$. On the reflected part, double-homodyne detection is performed using two detectors with equal quantum efficiencies $\eta$: this measurement is executed by splitting the state at a balanced beam splitter and, then, measuring the two conjugate quadratures $\hat x = \frac{1}{\sqrt{2}}(\hat{a}+\hat{a}^{\dag})$ and $\hat y = \frac{1}{i\sqrt{2}}(\hat{a} - \hat{a}^{\dag})$, with $\hat{a}$ and $\hat{a}^\dagger$ being the field annihilation and creation operator. The outcome of the double-homodyne detector gives the complex number $\alpha = x + i y$. According to these outcomes, the transmitted part of the input state undergoes a displacement by an amount $g \alpha$, where $g$ is a suitable electronic amplification factor, and, finally, the two output states, denoted by the density operators $\varrho_1$ and $\varrho_2$, are obtained by dividing the displaced state using another beam splitter with transmittivity $\tau_2$. When $\tau_1 = \tau_2 = 1/2$, $g = 1$ and $\eta = 1$, the scheme reduces to that of Ref.~\cite{andersen05.prl}, which was shown to be optimal for Gaussian cloning of coherent states on the basis of a description in the Heisenberg picture. Here we apply a different approach which captures all the essential features of the machine. Towards this aim, in the following we carry out a thorough description of the machine using the characteristic function approach. \par The characteristic function $\chi_{\rm in}(\bmLambda_1) \equiv \chi[\varrho_{\rm in}](\bmLambda_1)$ associated with a generic Gaussian input state $\varrho_{\rm in}$ of mode $1$ reads: \begin{equation}\label{rho:in} \chi_{\rm in}(\bmLambda_1) = \exp\left\{ -\mbox{$\frac12$} \bmLambda_1^T \bmsigma_{\rm in}\,\bmLambda_1 - i \bmLambda_1^T \bmX_{\rm in}\right\}\,, \end{equation} where $\bmLambda_1 = (\sfx_1, \sfy_1)^T$, $(\cdots)^T$ denotes the transposition operation, and \begin{equation}\label{m:covarianza} \bmsigma_{\rm in} = \left( \begin{array}{cc} \gamma_{11} & \gamma_{12}\\ \gamma_{21} & \gamma_{22} \end{array} \right)\,, \end{equation} with $\gamma_{12}=\gamma_{21}$, is the covariance matrix. $\bmX_{\rm in} = {\rm Tr}[\varrho_{\rm in}\, (\hat x, \hat y)^T]$ is the vector of mean values, $\hat x$ and $\hat y$ being the quadrature operators defined above. The vacuum state $\varrho_0 = \ket{0}\bra{0}$ of mode $2$ is described by the (Gaussian) characteristic function \begin{equation} \chi_{0}(\bmLambda_2) \equiv \chi[\varrho_0](\bmLambda_2) = \exp\left\{ -\mbox{$ \frac12 $} \bmLambda_2^T \bmsigma_0\, \bmLambda_2 \right\}\,, \end{equation} where $\bmsigma_0 = \frac12 \mathbbm{1}_2$, $\mathbbm{1}_2$ being the $2 \times 2$ identity matrix. In turn, the initial two-mode state $\varrho = \varrho_{\rm in} \otimes \varrho_0$ is Gaussian and its two-mode characteristic function reads: \begin{equation} \chi[\varrho](\bmLambda) = \exp\left\{ -\mbox{$\frac12$} \bmLambda^T \tilde{\bmsigma} \,\bmLambda - i \bmLambda^T \tilde{\bmX} \right\}\,, \end{equation} with \begin{equation} \tilde{\bmsigma} = \left(
\begin{array}{c|c} \bmsigma_{\rm in} & {\boldsymbol 0} \\ \hline {\boldsymbol 0} & \bmsigma_0 \end{array} \right)\,,\qquad \tilde{\bmX} = (\bmX_{\rm in}, {\boldsymbol 0})^T\,, \end{equation} and $\bmLambda = (\bmLambda_1, \bmLambda_2)^T$. Under the action of the first BS the state $\chi[\varrho](\bmLambda)$ preserves its Gaussian form, namely \begin{equation} \chi[\varrho](\bmLambda) \rightsquigarrow \chi[\varrho'](\bmLambda) = \exp\left\{ -\mbox{$\frac12$} \bmLambda^T \bmsigma \,\bmLambda - i \bmLambda^T \bmX \right\}\,, \end{equation} where $\varrho' = U_{{\rm BS},1}\,\varrho_{\rm in}\otimes\varrho_0\,U_{{\rm BS},1}^{\dag}$, while its covariance matrix and mean values transform as~\cite{FOP:napoli:05}: \begin{align} &\tilde{\bmsigma} \rightsquigarrow \bmsigma \equiv {\bmS}_{{\rm BS},1}^T\, \tilde{\bmsigma}\,{\bmS}_{{\rm BS},1} = \left(
\begin{array}{c|c} \bmA & \bmC\\ \hline \bmC^T & \bmB \end{array} \right)\,,\label{transf:cvm}\\ &\tilde{\bmX} \rightsquigarrow \bmX \equiv {\bmS}_{{\rm BS},1}^T\,\tilde{\bmX} = (\bmX_1,\bmX_2)^T\,, \label{transf:ave} \end{align} $\bmA$, $\bmB$, and $\bmC$ are $2 \times 2$ matrices, and \begin{equation}\label{symp:BS} \bmS_{{\rm BS},1} = \left(
\begin{array}{c|c} \sqrt{\tau_1}\, \mathbbm{1}_2 & \sqrt{1-\tau_1}\, \mathbbm{1}_2 \\ \hline -\sqrt{1-\tau_1}\, \mathbbm{1}_2 & \sqrt{\tau_1}\, \mathbbm{1}_2 \end{array} \right)\,, \end{equation} is the symplectic transformation associated with the evolution operator $U_{{\rm BS},1}$ of the BS with transmission $\tau_1$. Note that $\varrho'$ is an entangled state if the set of states to be cloned consists of nonclassical states, {\em i.e.} states with singular Glauber P-function or negative Wigner function \cite{visent,wang}. \par The subsequent step is to describe double-homodyne detection with quantum efficiency $\eta$ on the reflected beam. This action can be described by the following positive operator-valued measure (POVM): \begin{equation}\label{DH:POVM} \Pi_{\eta}(\alpha) = \int_{\mathbb{C}} d^2\xi\,
\frac{1}{\pi \sigma_\eta}\exp\left\{ -\frac{|\alpha - \xi|^2} {\sigma_\eta} \right\} \frac{\ket{\xi}\bra{\xi}}{\pi}\,, \end{equation} where $\sigma_\eta = (1-\eta)/\eta$ and $\ket{\xi}$ is a coherent state. Eq.~(\ref{DH:POVM}) describes a Gaussian measurement, the characteristic function associated with $\Pi_{\eta}(\alpha)$ has the form \begin{equation} \chi[\Pi_{\eta}(\alpha)](\bmLambda_2) =\frac{1}{\pi} \exp\left\{ -\mbox{$\frac12$} \bmLambda_2^T\,\bmsigma_{\rm M}\,\bmLambda_2 - i \bmLambda_2^T\,\bmX_{\rm M} \right\}\,, \end{equation} with $\bmX_{\rm M} = \left({\rm Re}[\alpha],{\rm Im}[\alpha]\right)^T$ and \begin{equation}\label{sigma:M} \bmsigma_{\rm M} = \Delta^2\,\mathbbm{1}_2, \qquad \Delta^2 = \frac12 + \sigma_{\eta} = \frac{2 - \eta}{2\eta}\,. \end{equation} The probability of obtaining the outcome $\alpha$ is given by \begin{align} p_{\eta}(\alpha) &= {\rm Tr}_{12}[\varrho'\, \mathbb{I}\otimes\Pi_{\eta}(\alpha)]\\
&= \frac{1}{(2\pi)^2} \int_{\mathbb{R}^4}\!\!\! d^4\bmLambda\, \chi[\varrho'](\bmLambda)\, \chi[\mathbb{I}\otimes\Pi_{\eta}(\alpha)](-\bmLambda)\\ &=\frac{ \exp\left\{ -\mbox{$\frac12$}(\bmX_{\rm M}-\bmX_{2})^T\, \bmSigma^{-1}\,(\bmX_{\rm M}-\bmX_{2})\right\}} {\pi \sqrt{{\rm Det}[\bmSigma]}}\,, \end{align} where $\chi[\mathbb{I}\otimes\Pi_{\eta}(\alpha)](\bmLambda)\equiv \chi[\mathbb{I}](\bmLambda_1)\, \chi[\Pi_{\eta}(\alpha)](\bmLambda_2)$, $\chi[\mathbb{I}](\bmLambda_1) = 2\pi \delta^{(2)}(\bmLambda_1)$ and $\delta^{(2)}(\zeta)$ is the complex Dirac's delta function. We also introduced the $2\times 2$ matrix $\bmSigma = \bmB + \bmsigma_{\rm M}$. \par The conditional state $\varrho_{\rm c}$ of the transmitted beam, obtained when the outcome of the measurement is $\alpha$, i.e., \begin{equation} \varrho_{\rm c} = \frac{{\rm Tr}_{2}[\varrho'\,\Pi_{\eta}(\alpha)]}{p_{\eta}(\alpha)}\,, \end{equation} has the following characteristic function (for the sake of clarity we explicitly write the dependence on $\bmLambda_1$ and $\bmLambda_2$)
\begin{align} \chi[\varrho_{\rm c}](\bmLambda_1) =&
\int_{\mathbb{R}^2}\!\!\! d^2\bmLambda_2\, \frac{ \chi[\varrho'](\bmLambda_1,\bmLambda_2)\, \chi[\Pi_{\eta}(\alpha)](-\bmLambda_2)} {p_{\eta}(\alpha)}\\ =&\exp\left\{ -\mbox{$\frac12$}\bmLambda_1^T \left[ \bmA - \bmC\bmSigma^{-1}\bmC^T \right] \bmLambda_1 \right.\nonumber\\ &\left. -\mbox{$\frac12$}\bmX_2^T\,\bmSigma^{-1}\, \bmX_2 +i \bmLambda_1^T \left[ \bmC\bmSigma^{-1}\,\bmX_2 - \bmX_1 \right] \right\}\nonumber\\ &\times \exp\left\{ -\mbox{$\frac12$}\bmX_{\rm M}^T\,\bmSigma^{-1}\, \bmX_{\rm M} \right.\nonumber\\ &\hspace{0.2cm} \left. +i \bmX_{\rm M}^T \left[ i\bmSigma^{-1}\,\bmX_2 +\bmSigma^{-1}\bmC^T \bmLambda_1 \right]\right\}\,. \end{align}
Now, the conditional state $\varrho_{\rm c}$ is displaced by the amount $g \alpha$ resulting from the measurement amplified by a factor $g$. By averaging over all possible outcomes of the double-homodyne detection, we obtain the following output state: \begin{equation}\label{rho:d} \varrho_{\rm d} = \int_{\mathbb{C}}d^2 \alpha\, p_{\eta}(\alpha)\,D(g \alpha)\, \varrho_{c} \,D^{\dag}(g \alpha)\,, \end{equation} with $D(z)$ being the displacement operator. In turn, the characteristic function reads as follows: \begin{equation} \chi[\varrho_{\rm d}](\bmLambda_1) = 2\exp\left\{ -\mbox{$\frac12$} \bmLambda_1^T\,\bmsigma_{\rm d}\,\bmLambda_1 -i \bmLambda_1^T \bmX_{\rm d} \right\}\,, \end{equation} with $\bmsigma_{\rm d} = \bmA + g(\bmSigma + 2 \bmC^T)$ and $\bmX_{\rm d} = \bmX_1 + g \bmX_2$. The conditioned state (\ref{rho:d}) is then sent to a second beam splitter with transmission $\tau_2$ (see Fig.~\ref{f:cl:scheme}), where it is mixed with the vacuum $\varrho_0$, and finally the two clones are generated. Note that, in practice, the average over all the possible outcomes $\alpha$ in Eq. (\ref{rho:d}) should be performed at this stage, that is after the second beam splitter. On the other hand, because of the linearity of the integration, the results are identical, but performing the averaging just before the beam splitter simplifies the calculations. Since $\varrho_{\rm d}$ is still Gaussian, the two-mode state $\varrho_{\rm f} = \varrho_{\rm d} \otimes \varrho_0$ is a Gaussian with covariance matrix and mean given by \begin{equation} \bmsigma_{\rm f} = \left(
\begin{array}{c|c} \bmsigma_{\rm d} & {\boldsymbol 0} \\ \hline {\boldsymbol 0} & \bmsigma_0 \end{array} \right)\,,\qquad \bmX_{\rm f} = (\bmX_{\rm d}, {\boldsymbol 0})^T\,, \end{equation} respectively, which, as in the case of Eqs.~(\ref{transf:cvm}) and (\ref{transf:ave}), under the action of the BS transform as follows: \begin{align} &\bmsigma_{\rm f} \rightsquigarrow \bmsigma_{\rm out} \equiv {\bmS}_{{\rm BS},2}^T\, \bmsigma_{\rm f}\,{\bmS}_{{\rm BS},2} = \left(
\begin{array}{c|c} {\mathscr A}} \newcommand{\dbra}[1]{\langle\langle #1 \vert_1 & \Cop\\ \hline \Cop^T & {\mathscr A}} \newcommand{\dbra}[1]{\langle\langle #1 \vert_2 \end{array} \right)\,,\label{transf:cvm:fin}\\ &\bmX_{\rm f} \rightsquigarrow \bmX_{\rm out} \equiv {\bmS}_{{\rm BS},2}^T\,\bmX_{\rm f} = (\Xop_1,\Xop_2)^T\,, \label{transf:ave:fin} \end{align} where ${\mathscr A}} \newcommand{\dbra}[1]{\langle\langle #1 \vert_k$ and $\Cop$ are $2 \times 2$ matrices, and $\bmS_{{\rm BS},2}$ is the symplectic matrix given by Eq.~(\ref{symp:BS}) with $\tau_1$ replaced by $\tau_2$. Finally, the (Gaussian) characteristic function of the clone $\varrho_k$, $k=1,2$, is obtained by integrating over $\bmLambda_{h}$, $h\ne k$, the two-mode characteristic function $\chi[\varrho_{\rm out}](\bmLambda_1,\bmLambda_2)$, where $\varrho_{\rm out} = U_{{\rm BS},2}\, \varrho_{\rm f}\otimes \varrho_0 \,U_{{\rm BS},2}^{\dag}$, i.e., \begin{align} \chi[\varrho_k](\bmLambda_k) &= \frac{1}{2\pi} \int_{\mathbb{R}^2}\!\!\! d^2\bmLambda_h\, \chi[\varrho_{\rm out}](\bmLambda_1,\bmLambda_2)\\ &=\exp\left\{ -\mbox{$\frac12$}\bmLambda_{k}^T\,{\mathscr A}} \newcommand{\dbra}[1]{\langle\langle #1 \vert_k\,\bmLambda_{k} - i \bmLambda_k^T\, \Xop_k\right\}\,.\label{clone:k} \end{align} Let us now focus our attention on $\bmX_{\rm out}$: the explicit expressions of $\Xop_1$ and $\Xop_2$ are \begin{align} &\Xop_1 = \sqrt{\tau_2}\left( \sqrt{\tau_1} + g\sqrt{1-\tau_1}\right) \bmX_{\rm in}\,,\\ &\Xop_2 = \sqrt{1-\tau_2}\left( \sqrt{\tau_1} + g\sqrt{1-\tau_1}\right) \bmX_{\rm in}\,. \end{align} As a matter of fact, in order to have two output Gaussian states with the same means $\Xop_1 = \Xop_2$, one should put $\tau_2 = 1/2$; furthermore, if one also sets \begin{equation}\label{g:symm} g = g_{\rm s} \equiv \sqrt{\frac{2}{1-\tau_1}}- \sqrt{\frac{\tau_1}{1-\tau_1}} \,, \end{equation} then $\Xop_1=\Xop_2=\bmX_{\rm in}$, corresponding to unity gain cloning. On the other hand, ${\mathscr A}} \newcommand{\dbra}[1]{\langle\langle #1 \vert_{k}$ can be written in a compact form as follows: \begin{subequations}\label{A:k} \begin{align}{\mathscr A}} \newcommand{\dbra}[1]{\langle\langle #1 \vert_{1} &= \mbox{$\frac12$} (1-\tau_2)\,\mathbbm{1} + \tau_2\, \mathbf{\Gamma}(\bmsigma_{\rm in})\,,\\ {\mathscr A}} \newcommand{\dbra}[1]{\langle\langle #1 \vert_{2} &= \mbox{$\frac12$} \tau_2\,\mathbbm{1} + (1-\tau_2)\, \mathbf{\Gamma}(\bmsigma_{\rm in})\,, \end{align} \end{subequations} where
\begin{equation} \mathbf{\Gamma}(\bmsigma_{\rm in}) = \left( \begin{array}{cc} \Fop(\gamma_{11}) & \Gop(\gamma_{12})\\ \Gop(\gamma_{21}) & \Fop(\gamma_{22}) \end{array} \right)\,, \end{equation} with \begin{align} &\Fop(\gamma) = 1 - \tau_1 + g\left[\tau_1 - 2 \sqrt{(1-\tau_1)\tau_1}+\Delta^2\right] + \Gop(\gamma)\,,\\ &\Gop(\gamma) = \left[\tau_1 + g\left(1 - \tau_1 + 2\sqrt{(1-\tau_1)\tau_1}\right)\right]\gamma\,. \end{align} Now, if $\tau_2=1/2$ and $g=g_{\rm s}$, one has ${\mathscr A}} \newcommand{\dbra}[1]{\langle\langle #1 \vert_1 = {\mathscr A}} \newcommand{\dbra}[1]{\langle\langle #1 \vert_2$ and $\Xop_1=\Xop_2$, as we have seen above, i.e., the cloning becomes {\em symmetric}. Furthermore, when also $\tau_1 = 1/2$, thanks to Eqs.~(\ref{clone:k}) and (\ref{A:k}) we have that the cloning map for the scheme in Fig.~\ref{f:cl:scheme} is given by the following Gaussian map: \begin{equation}\label{cl:map} {\cal G}_{\sigma_{\rm GN}}(\varrho_{\rm in}) = \int_{\mathbb{C}} \frac{d^2\gamma}{\pi \sigma_{\rm GN}^2}\,
\exp\left\{ - \frac{|\gamma|}{\sigma_{\rm GN}^2} \right\}\, D(\gamma)\,\varrho_{\rm in}\,D^{\dag}(\gamma)\,, \end{equation} where $\sigma_{\rm GN}^2 = \frac12 + \Delta^2$. Finally, although $\Cop$ does not appear in Eq.~(\ref{clone:k}), for the sake of completeness, we give its analytic expression: \begin{equation} \Cop = \sqrt{(1-\tau_2)\tau_2}\left[ \mbox{$\frac12$}\mathbbm{1} -\mathbf{\Gamma}(\bmsigma_{\rm in})\right]\,. \end{equation} \par In the following we will analyze the input-output fidelities for a generic (pure or mixed) Gaussian state. In particular, we will consider three classes of Gaussian states, i.e. coherent, displaced squeezed and displaced thermal states.
\section{Cloning of Gaussian states}\label{s:GScloning} \subsection{Fidelity} Usually, the performance of cloning machines are quantified by the fidelity which is a measure of the similarity between the hypothetically perfect clone and the actual clone. In its most general form, the fidelity is given by the Uhlmann's transition probability \cite{uhlman:RepMP:76} \begin{equation}\label{fidelity} {F}(\varrho_{\rm in},\varrho_k) = \left( {\rm Tr}\left[ \sqrt{\sqrt{\varrho_{\rm in}}\,\varrho_{k}\,\sqrt{\varrho_{\rm in}}} \right] \right)^2\,, \end{equation} and satisfies the natural axioms \begin{itemize} \item ${F}(\varrho_{\rm in},\varrho_k)\le 1$ and ${F}(\varrho_{\rm in},\varrho_k) = 1$ if and only if $\varrho_{\rm in} = \varrho_k$; \item ${F}(\varrho_{\rm in},\varrho_k) = {F}(\varrho_k,\varrho_{\rm in})$; \item if $\varrho_{\rm in}$ is a pure state $\varrho_{\rm in} = \ket{\psi_{\rm in}}\bra{\psi_{\rm in}}$, then we have ${F}(\varrho_{\rm in},\varrho_k) = \bra{\psi_{\rm in}} \varrho_{k} \ket{\psi_{\rm in}}$; \item ${F}(\varrho_{\rm in},\varrho_k)$ is invariant under unitary transformations on the state space. \end{itemize} Furthermore, when $\varrho_{\rm in}$ and $\varrho_k$ are Gaussian states of the form (\ref{rho:in}) and (\ref{clone:k}), the fidelity (\ref{fidelity}) becomes \cite{scutaru:JPA:98,nha:2005} \begin{align}\label{gen:fid} &{F}_{\eta} \equiv {F}(\varrho_{\rm in},\varrho_k) = \frac{1} { \sqrt{{\rm Det}[\bmsigma_{\rm in}+{\mathscr A}} \newcommand{\dbra}[1]{\langle\langle #1 \vert_k]+\delta}-\sqrt{\delta} }\nonumber\\ &\times\exp\left\{ -\mbox{$\frac12$} (\bmX_{\rm in}-\Xop_k)^T(\bmsigma_{\rm in}+{\mathscr A}} \newcommand{\dbra}[1]{\langle\langle #1 \vert_{k})^{-1} (\bmX_{\rm in}-\Xop_k) \right\} \,, \end{align} where $\delta = 4({\rm Det}[\bmsigma_{\rm in}]-\frac14) ({\rm Det}[{\mathscr A}} \newcommand{\dbra}[1]{\langle\langle #1 \vert_k]-\frac14)$. Note that for pure Gaussian states ${\rm Det}[\bmsigma_{\rm in}] = \frac14$, and in turn $\delta = 0$. \par For {\em symmetric} cloning, i.e. for $\tau_2 = 1/2$ and $g = g_{\rm s}$ in Eq.~(\ref{g:symm}), then Eq.~(\ref{gen:fid}) reduces to \begin{equation}\label{gen:fid:symm} {F}_{\eta} = \frac{1} {\sqrt{{\rm Det}[\bmsigma_{\rm in}+{\mathscr A}} \newcommand{\dbra}[1]{\langle\langle #1 \vert_k]+\delta}-\sqrt{\delta}}\,. \end{equation}
\par In general, the cloning fidelity in (\ref{gen:fid}) is state dependent, and therefore the figure of merit to be considered is the mean cloning fidelity, averaged over the ensemble of possible input states. In order to evaluate this quantity, we parametrize the input ensemble (class) $\{ \varrho_{\rm in}(\boldsymbol\lambda) \}$ of different Gaussian states, by $\boldsymbol\lambda\in\Omega$ and consider each of them occurring with the {\em a priori} probability $p(\boldsymbol\lambda)$. The average fidelity then reads \begin{equation}\label{ave_fid} \oF_{\eta} = \int_{\Omega} d\boldsymbol\lambda\, p(\boldsymbol\lambda)\,{F}_{\eta}(\boldsymbol\lambda)\,. \end{equation} Within the set of possible states both the mean values as well as the covariance matrices may vary. Assuming that the probability distribution $p(\boldsymbol{\lambda})$ is factorisable into a distribution for the mean values $p(\alpha)$ and a distribution for the covariance matrix, $p(\boldsymbol\sigma_{\rm in})$, we may write $p(\boldsymbol{\lambda})=p(\alpha)\,p(\boldsymbol\sigma_{\rm in})$, and the average fidelity reads \begin{equation} \oF_{\eta} = \int_{\Omega} d\boldsymbol\sigma_{\rm in}\,d\alpha\, p(\alpha)\,p(\boldsymbol\sigma_{\rm in})\, {F}_{\eta}(\boldsymbol\sigma_{\rm in},\alpha)\,. \end{equation} In the extreme case where both $\bmsigma_{\rm in}$ and $\alpha$ are fixed, the input state is completely known and perfect cloning with unit fidelity is of course possible. A more interesting scenario is when the covariance matrix is fixed, as for example the case in which the set is made by coherent states, while the displacement (that is, the mean value) is random. In this case the average fidelity reduces to \begin{equation}\label{F:alpha} \oF_{\eta} = \int_{\mathbb C} d^2\alpha\,p(\alpha)\, {F}_{\eta}(\alpha) \end{equation} If $\tau_1=1/2$ and $g=g_s$, the map (\ref{cl:map}) is covariant with respect to displacements, meaning that if two input states are identical up to a displacement their respective clones should be identical up to the same displacement \cite{cerf:EPJ:2002}. Indeed, if the input state is of the form $\varrho_{\rm in}(\alpha) = D(\alpha)\,\varrho_{\rm s}\,D^{\dag}(\alpha)$, $\varrho_{\rm s}$ being a seed state, then the fidelity ${F}_{\eta}(\alpha)$ actually does not depend on complex parameter $\alpha$, and, as consequence, $\oF_{\eta} = {F}_{\eta}$. Therefore, in this case the noise added by the cloning process (\ref{cl:map}) is the same noise added in cloning of coherent states, i.e., the cloning is {\em optimal}. Notice that the corresponding optimal fidelity $\oF$ is not necessarily equal to $2/3$ [see Eq.~(\ref{gen:fid:symm})].
\subsection{Coherent states} Before addressing the general case let us reconsider cloning of (pure) coherent states. For this set of states our linear machine provides universal cloning, {\em i.e.} state independent fidelity. In Fig.~\ref{f:CS} we plot the fidelity as given by Eq.~(\ref{gen:fid:symm}), as a function of $\tau_1$, for different values of $\eta$ and for $\tau_2 = 1/2$, $g = g_{\rm s}$. In this case, corresponding to symmetric cloning, the machine yields the optimal fidelity $F=2/3$ predicted for universal Gaussian cloning of coherent states. Notice that the optimal fidelity is achieved with $\tau_1 = 1/2$ and $\eta = 1$; by expanding the fidelity up to the second order around $\tau_1=1/2$ we obtain \begin{align} \overline{F}_{\eta} \equiv {F}_{\eta} \simeq \frac{2\eta}{1+2\eta}\left[ 1- \frac{\left(\tau_1-\mbox{$\frac12$}\right)^2}{1+2\eta} \right] \nonumber \end{align} >From this expression we clearly see that the cloning machine proposed in \cite{andersen05.prl} is robust against fluctuations of the BS ratio. This conclusion can be also directly drawn from Fig.~\ref{f:CS}.
\begin{figure}
\caption{ Linear cloning fidelity ${F}_{\eta}$ for a coherent state as a function of the BS transmittivity $\tau_1$ for different values of the quantum efficiency $\eta$: from top to bottom $\eta = 1.0$, $0.75$, and $0.5$. We set $\tau_2 = 1/2$ and $g = g_{\rm s}$ (symmetric cloning). The dashed line is the value $2/3$. The fidelity does not depend on the coherent state amplitude.}
\label{f:CS}
\end{figure}
\par Let us however note that the fidelity value $F=2/3$ is the optimal one only if the input distribution of coherent states is flat, that is, if we have no {\em a priori} information about the amplitudes. If the set of coherent states is restricted such that the distribution of amplitudes is the Gaussian \begin{equation} p_{\rm a}(\alpha) =\frac{1}{\pi\sigma_{\rm a}^2}
\exp\left\{-\frac{|\alpha|^2}{\sigma_{\rm a}^2}\right\}\,, \end{equation} the average fidelity can be increased by choosing a different gain~\cite{cochrane04.pra}. However, in this scenario the cloning action becomes state dependent, and the integration in (\ref{F:alpha}) should be explicitly performed.
By optimizing the gain we find~\cite{cochrane04.pra} \begin{align} \oF& = \frac{2(1+\sigma_{\rm a}^2)}{1+3\sigma_{\rm a}^2}, &{\rm if} \quad \sigma_{\rm a}^2 \geq 1+\sqrt{2},\\ \oF& = \frac{2}{2+(3-2\sqrt{2})\sigma_{\rm a}^2}, &{\rm if} \quad \sigma_{\rm a}^2 < 1+\sqrt{2}\:. \end{align} \par We have now seen that by fixing the covariance matrix of the input states to coherent states, the fidelity is a function of the distribution (being delta, flat or Gaussian) of these states. This aspect has been investigated in the literature~\cite{cochrane04.pra}. In contrast, the case where the covariance matrix may fluctuate has not received much attention heretofore. In the following sections we therefore discuss the average cloning fidelity for classes of states with covariance matrix randomly distributed according to a predetermined distribution. We assume that the displacement of the input state is random and that the cloner is set to unity gain (that is invariant with respect to the displacement corresponding to $g=g_s$). In this case the average over the mean value is trivial and the average fidelity can be written as \begin{equation} \oF_{\eta} = \int_{\Sigma} d\bmsigma_{\rm in}\, p(\boldsymbol\sigma_{\rm in})\, {F}_{\eta}(\boldsymbol\sigma_{\rm in})\,. \end{equation}
\subsection{Squeezed states} When the input Gaussian state is the squeezed state $\ket{\alpha,\xi} = D(\alpha)S(\xi)\ket{0}$, where $D(\alpha) = \exp\{\alpha a^{\dag} - \alpha^* a\}$ and $S(\xi) = \exp\{\frac12 (\xi{a^{\dag}}^2 - \xi^* a^2)\}$ are the displacement and squeezing operator, respectively, the entries of the input covariance matrix (\ref{m:covarianza}) are \begin{subequations} \begin{align}
&\gamma_{11} = \mbox{$\frac12$}\left(\cosh 2|\xi|
+ \sinh 2|\xi|\,\cos \varphi\right)\,,\\
&\gamma_{22} = \mbox{$\frac12$}\left(\cosh 2|\xi|
- \sinh 2|\xi|\,\cos \varphi\right)\,,\\
&\gamma_{12} = \gamma_{21} = - \mbox{$\frac12$} \sinh 2|\xi|\,\sin \varphi\,, \end{align} \end{subequations}
where we put $\xi = |\xi|\,e^{i\varphi}$; obviously, when $\xi=0$ the squeezed state $\ket{\alpha,\xi}$ reduces to the coherent state $\ket{\alpha}$ and $\bmsigma_{\rm in} = \frac12 \mathbbm{1}_2$. Note that in this Section we are addressing the case of an {\em unknown} squeezing parameter $\xi$ (randomly distributed according to a given probability density): when it is known, the optimal strategy in the Gaussian regime is to perform the unsqueezing operation $S^{-1}(\xi)$ just before the cloning machine, proceed as in the case of coherent states, and, at the output stage, apply the squeezing operation $S(\xi)$ to both the clones which yields a fidelity of $2/3$ (independent on the amount of fixed squeezing) as in the coherent state case~\cite{braunstein01.prl}. \par However in the case of an unknown squeezing parameter the squeezing action $S(\xi)$ is not known. Therefore in the following, we investigate the cloning of unknown squeezed states using the cloning machine outlined in this paper. First we note that since the linear elements involved in the cloning machine do not affect the phase of the input state, the fidelity
${F}_{\eta}(\xi)$ depends only on $|\xi|$ and, without loss of generality, we may take $\xi$ as real. The fidelity for Gaussian squeezed input states, using the coherent state cloning machine, is given by \begin{equation}\label{F:r:eta:1:2} {F}_{\eta}(\xi) =
\frac{4}{\sqrt{(5+2\Delta^2)^2+16(1+2\Delta^2)\sinh^2 |\xi|}}\,. \end{equation} This fidelity is plotted in Fig.~\ref{f:SQfid} as a function of the squeezing parameter for different values of $\eta$. We clearly see that for coherent states (corresponding to $\xi$=0), the fidelity is $2/3$ while decreasing with the degree of squeezing, eventually reaching zero for highly squeezed input states.
\begin{figure}
\caption{ Plot of the fidelity ${F}_{\eta}(\xi)$ of the squeezed state
$\ket{\alpha, \xi}$ as a function of $|\xi|$ for different values of $\eta$: from top to bottom $\eta = 1.0$, $0.75$, and $0.5$. We set $\tau_1 = \tau_2 = 1/2$ and $g = g_{\rm s}$ (symmetric cloning). The dashed line is the value $2/3$. The fidelity does not depend on the displacement amplitude $\alpha$.}
\label{f:SQfid}
\end{figure}
\par In order to calculate the average fidelity, we assume that the squeezed state $\ket{\alpha,\xi}$ is drawn from an ensemble of states with {\em a priori} probability $p(\alpha, \xi)= p(\alpha)\,p(\xi)$. Above we mentioned that the cloning action with unity gain is independent on the distribution $p(\alpha)$, which can then be left undefined. The distribution of the squeezing factor is however quite important: it is clear that for completely unknown input squeezing (corresponding to a flat distribution) the average fidelity goes to zero. We therefore must restrict the set of input squeezed states to, say, a Gaussian distribution given by \begin{equation}\label{distribution} p_{\rm s}(\xi) =\frac{1}{\pi\sigma_{\rm s}^2}
\exp\left\{-\frac{|\xi|^2}{\sigma_{\rm s}^2}\right\}\,. \end{equation} As evident from this expression we assume the distribution to be centered at $\xi=0$ which corresponds to a coherent state. This means that the coherent state is the most likely member in the set of input states, and we therefore conjecture that our machine is optimal in the Gaussian scenario. If however the distribution is centered at a known squeezing amplitude, say $\xi=\xi_0$, then we believe that the optimal machine is the one mentioned above where the input states are unsqueezed [$S^{-1}(\xi_0)$] before the cloning machine and squeezed [$S(\xi_0)$] again after the cloning action. \par Using the polar coordinates, $d^2\xi = \rho\, d\rho\, d\phi$, $\xi = \rho\, e^{i\phi}$, and ${F}_{\eta}(\xi) =
{F}_{\eta}(|\xi|)$, the average fidelity now reads \begin{align} \oF_{\eta} &= \int_{\mathbb{C}} d^2\xi\,p(\xi)\,{F}_{\eta}(\xi)\\ &= 2 \int_{0}^{+\infty}\!\!\! d\rho\, \frac{\rho}{\sigma_{\rm s}^2} \exp\left\{-\frac{\rho^2}{2\sigma_{\rm s}^2}\right\} \,{F}_{\eta}(\rho) \label{ave:fid:squeezed} \end{align} This function is depicted in Fig.~\ref{f:ave:squeezed} as a function of $\sigma_{\rm s}$ for different values of $\eta$. If the standard deviation $\sigma _s=0$, the distribution in (\ref{distribution}) is a delta function and the input alphabet contains only coherent states. In this case it reduces to the case discussed in the previous section and the expected fidelity is 2/3 (for ideal detection efficiency) as seen in the figure. We also see that the fidelity degrades as the width of the distribution of the squeezing parameter increases, and eventually reaches zero when the a priori information is poor. At this point we should note that if one allows for non-Gaussian output clones the fidelity can be improved. E.g. it is known that the optimal cloner of coherent states and the optimal universal cloner employ non-Gaussian operations and they yield fidelities of 68.3\%~\cite{cerf05.prl} and 50\%~\cite{braunstein01.pra} respectively.
\begin{figure}
\caption{Plot of the average fidelity $\oF_{\eta}$ of a set of squeezed states as a function of $\sigma_{\rm s}$ (see text for details) and different values of the efficiency $\eta$: form top to bottom $\eta = 1.0$, $0.75$, and $0.5$. The dashed line corresponds to $2/3$, i.e., the optimal cloning fidelity of coherent states. We put $\tau_1 = \tau_2 = 1/2$ and $g = g_{\rm s}$.}
\label{f:ave:squeezed}
\end{figure}
\subsection{Thermal states} Another interesting class of Gaussian states is the set of displaced thermal states $\varrho_{{\rm th},\alpha} = D(\alpha)\,\nu_{\rm th}\, D^{\dag}(\alpha)$, which arise, for example, from the propagation of coherent states in a noisy environment \cite{ComEnt}. The thermal state $\nu_{\rm th}$ is given by \begin{equation} \nu_{\rm th} = \frac{1}{1+N}\, \sum_{m=0}^{\infty} \left(\frac{N}{1+N}\right)^m \ket{m}\bra{m}\,, \end{equation} where $N$ is the average number of thermal photons. Its covariance matrix is given by $\bmsigma_{\rm in} = (N+\frac12) \mathbbm{1}_2$. Since $\nu_{\rm th}$ and, in turn, $D(\alpha)\,\nu_{\rm th}\,D^{\dag}(\alpha)$ are not pure states, the cloning fidelity $F_{\rm \eta}(N)$ should be calculated using the full expression of Eq.~(\ref{gen:fid}), and the result is plotted in Fig.~\ref{f:thermal:2D} as a function of $N$ and different values of $\eta$. For the unity gain cloner and assuming the detection efficiency to be ideal ($\eta =1$), we derive the expression \begin{align} {F}_{\eta=1}(N)=&\bigg( \frac32 + N (3 + 2 N) \nonumber\\ &-\sqrt{N(2N+1)(2N^2+5N+3)} \bigg)^{-1}\,. \end{align} We see that the fidelity increases with the average number of thermal photons, that is, using the fidelity as a measure, the quality of the cloning action increases with the mixedness of the input states.
\begin{figure}\label{f:thermal:2D}
\end{figure}
\par
\begin{figure}
\caption{ Plot of the average fidelity $\oF_{\eta}$ of the set of thermal states distributed according to the top-hat distribution (\ref{top:hat}) as a function of the threshold value ${\cal N}$ and different values of the efficiency $\eta$: form top to bottom $\eta = 1.0$, $0.75$, and $0.5$. The dashed line corresponds to $2/3$. We put $\tau_1 = \tau_2 = 1/2$ and $g = g_{\rm s}$.}
\label{f:alphabet}
\end{figure}
\begin{figure}
\caption{ Plot of the average fidelity $\oF_{\eta}$ of the set of thermal states distributed according to a ``half-Gaussian'' distribution (\ref{half:gauss}) as a function of $\mu_N$ and different values of the efficiency $\eta$: form top to bottom $\eta = 1.0$, $0.75$, and $0.5$. The dashed line corresponds to $2/3$. We put $\tau_1 = \tau_2 = 1/2$ and $g = g_{\rm s}$.}
\label{f:gauss:alphabet}
\end{figure}
\par Let us now consider a different ensemble of displaced thermal states, with random displacement and average number of thermal photons $N$ distributed around zero either as a bounded flat, top-hat, distribution or as a ``half-Gaussian'' distribution. The average fidelity is \begin{equation} \oF_{\eta} = \int_{0}^{+\infty} \!\!\!\! d N\, p(N)\, {F}_{\eta}(N)\,, \end{equation} where \begin{equation}\label{top:hat} p(N) = \left\{ \begin{array}{ll} {\cal N}^{-1} & {\rm if} \quad N \in [0,{\cal N}] \\ 0 & {\rm otherwise} \end{array} \right. \end{equation} for a top-hat distribution, and \begin{equation}\label{half:gauss} p(N) =\frac{2}{\sqrt{2\pi \mu_N^2}} \exp\left\{-\frac{N^2}{2 \mu_N^2}\right\}\,, \qquad (N\geq 0) \end{equation} for a (re-normalized) ``half-Gaussian'' distribution. In Figs.~\ref{f:alphabet} and \ref{f:gauss:alphabet} we show the corresponding average fidelities, as functions of ${\cal N}$ and $\mu_N$, respectively, for different values of $\eta$. For the top-hat distribution the average fidelity monotonically increases as the threshold value ${\cal N}$ increases, whereas for the half-Gaussian one the average fidelity shows a maximum value depending on the value of $\eta$, as far as $\eta\gtrsim 0.7$.
\section{Conclusions}\label{s:outro} We have analyzed in details a recently demonstrated scheme for linear cloning of Gaussian states \cite{andersen05.prl}. Using a suitable phase-space analysis the input-output fidelity has been evaluated for a generic (pure or mixed) Gaussian state taking into account the effect of non-unit quantum efficiency of homodyne detection and fluctuations in the beam splitters transmittivity. Our results indicate that the linear cloning machine suggested in \cite{andersen05.prl} is robust against fluctuations of transmissivity and non-unit quantum efficiency. \par We have explicitly evaluated the cloning fidelity for specific classes of non coherent displaced states. We found that a fixed (unknown) squeezing of the input states degrades the fidelity with respect to the coherent level, as one may expect for cloning of highly nonclassical states, while, on the contrary, cloning of displaced thermal states may be achieved with larger fidelity. Using the above results we have evaluated the average cloning fidelity for classes of Gaussian states with fluctuating covariance matrix, as for example displaced squeezed or displaced thermal states with the degree of squeezing or the number of thermal photons randomly distributed according to a Gaussian or a uniform distribution. Results indicate that the average fidelity monotonically decreases as the squeezing dispersion increases, whereas the behaviour with respect to dispersion of thermal photons is not monotone.
\section*{Acknowledgments} Fruitful discussions with A.~Ferraro are kindly acknowledged. This work has been supported by MIUR through the project PRIN-2005024254-002 and by the EU project COVAQIAL no. FP6-511004.
\end{document} | arXiv |
\begin{document}
\articletype{Views}
\author[1]{Markus Rademacher}
\author[2]{James Millen}
\author*[3]{Ying Lia Li}
\runningauthor{...}
\affil[3]{Department of Physics \& Astronomy, University College London, London WC1E 6BT, UK, e-mail: [email protected]}
\affil[1]{Department of Physics \& Astronomy, University College London, London WC1E 6BT, UK}
\affil[2]{Department of Physics, King’s College London, Strand, London, WC2R 2LS, UK}
\title{Quantum sensing with nanoparticles for gravimetry: when bigger is better}
\runningtitle{...}
\subtitle{...}
\abstract{Following the first demonstration of a levitated nanosphere cooled to the quantum ground state in 2020~\cite{delic_cooling_2020}, macroscopic quantum sensors are seemingly on the horizon. The nanosphere's large mass as compared to other quantum systems enhances the susceptibility of the nanoparticle to gravitational and inertial forces. In this viewpoint we describe the features of experiments with optically levitated nanoparticles \cite{millen_optomechanics_2020} and their proposed utility for acceleration sensing. Unique to the levitated nanoparticle platform is the ability to implement not only quantum noise limited transduction, predicted by quantum metrology to reach sensitivities on the order of $10^{-15}$\,ms$^{-2}$\hl{{~\cite{Qvarfort2018}},} but also long-lived quantum spatial superpositions for enhanced gravimetry. This follows a global trend in developing sensors, such as cold atom interferometers, that exploit superposition or entanglement. Thanks to significant commercial development of these existing quantum technologies, we discuss the feasibility of translating levitated nanoparticle research into applications. }
\keywords{...}
\classification[PACS]{...}
\communicated{...}
\dedication{...}
\received{...}
\accepted{...}
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\section{Introduction} Quantum mechanics is a cornerstone of modern physics, and quantum behaviour, such as superposition and entanglement, have been extensively observed using subatomic particles, photons, and atoms since the early 1900's~\cite{davisson_reflection_1928}. A global effort is now underway to take these existing quantum experiments and devices out of the laboratory and into industry, in a process dubbed the `second quantum revolution'.
Technological advances are now enabling larger objects to enter the quantum regime, with 2010 heralding the first ground state cooling of the motion of a human-made object, \hl{specifically a micron-scale `quantum drum'} \cite{oconnell_quantum_2010}. Operating in the quantum regime with \emph{free} or \emph{levitated} particles would allow the generation of macroscopic quantum states, and enable greatly enhanced sensitivity to external forces. The state-of-the-art demonstration of a macroscopic superposition is currently provided by matter-wave interferometry with an engineered molecule of mass beyond 25,000\,Da \cite{fein_quantum_2019}. This year, the centre-of-mass (c.o.m.) motion of a 143\,nm diameter silica nanosphere, levitated within an optical cavity, was cooled to its zero point energy (average phonon occupancy $<1$) using the cavity optomechanical interaction \cite{delic_cooling_2020}. Significant developments in trapping, stabilisation and cooling techniques (see Section~\ref{implementation}) have enabled levitated systems to reach the quantum regime (as shown in Figure~\ref{fig:quantumoptomechanics}), bringing researchers closer towards generating macroscopic quantum states with solid nanoscale objects.
\begin{figure*}
\caption{Experimental results for cooling of macroscopic systems. Minimum phonon occupation number is plotted against the sideband-resolvability parameter $\omega_m/\kappa$. Blue solid line displays the minimum achievable phonon number using quantum limited passive cavity cooling. Blue data points represent experiments relying only on passive cavity cooling. Red data points are results using squeezed light to surpass the standard quantum limit imposed on cavity cooling. Purple data points present results using a feedback cooling scheme. Orange data points show recent results of cooling levitated nanoparticles using coherent scattering in a cavity. Green dashed line shows recent data of a nanoparticle feedback cooled in an optical tweezer using no cavity for cooling or read out purposes. EPFL '20:~\cite{qiu_laser_2020}; Vienna 2020:~\cite{delic_cooling_2020}; ETH 2020:~\cite{tebbenjohanns_motional_2020}; ETH 2019:~\cite{windey_cavity-based_2019}; Delft 2019:~\cite{guo_feedback_2019}; Florence 2019:~\cite{chowdhury_calibrated_2019}; Copenhagen 2018:~\cite{rossi_measurement-based_2018}; Boulder 2017:~\cite{clark_sideband_2017}; JILA 2016:~\cite{peterson_laser_2016}; Boulder '11:~\cite{teufel_sideband_2011}; Caltech '11:~\cite{chan_laser_2011}; EPFL '11:~\cite{riviere_optomechanical_2011}; MIT '11:~\cite{schleier-smith_optomechanical_2011}; Cornell '10:~\cite{oconnell_quantum_2010}; MPQ '09:~\cite{schliesser_resolved-sideband_2009}; Vienna 2009:~\cite{groblacher_demonstration_2009}; JILA 2008:~\cite{teufel_dynamical_2008};}
\label{fig:quantumoptomechanics}
\end{figure*}
A driving force for creating quantum states of more massive objects, beyond proving their feasibility, is to test theories of gravity. The gravitational interaction has so far presented itself as classical \cite{carney_tabletop_2019}. It is unknown whether gravity acts as a quantum interaction, \hl{for example, via virtual graviton exchange~{\cite{marshman2020}},} or if in fact gravity is responsible for wave function collapse. In the latter case, one extends the Schr\"odinger equation nonlinearly to account for gravitational self-interaction, formalised in the Schr\"odinger-Newton equation\hl{. Models of gravitationally induced wave function collapse} aim to define the timescale of collapse due to a superposition of two different space-time curvatures, whilst avoiding superluminal signalling \cite{penrose_gravitys_1996,penrose_gravitization_2014}. Nonlinear modifications to the Schr\"odinger equation are also studied in the continuous spontaneous localisation (CSL) model~\cite{ghirardi_markov_1990}\hl{. This} aims to justify quantum wave-function collapse by introducing a stochastic diffusion process driven by an unknown noise field that continuously counteracts the spread of the quantum wave function. \hl{Through experiment, these collapse models can be falsified to rule out a mass-limit on quantum superpositions due to gravitational or noise induced localization} ~\cite{bassi_models_2013,martin_cosmic_2020, nimmrichter_optomechanical_2014}. The CSL effect would be practically unobservable on the atomic level but strongly amplified for high-mass systems. Many collapse models can be discounted by the sheer act of observing matter-wave interference with increasingly massive objects, such as levitated nanoparticles~\cite{bateman_near-field_2014,arndt_testing_2014}.
For sensing applications, dense macroscopic systems offer an enhanced sensitivity to acceleration, such that a single quantum nanosphere is predicted to reach acceleration sensitivities $10^{5}$ times more sensitive than a cloud of cold atoms \hl{{\cite{Qvarfort2018}}}. If used for navigation applications, this improvement in sensitivity reduces the accumulated error in position caused by double integrating a less noisy acceleration signal. Similar to cold atoms, levitated nanospheres are \hl{well} isolated from environmental decoherence, resulting in long coherence times for matter-wave interferometry and the ability to perform free-fall experiments. Free-fall accelerometers are particularly suited for gravimetry applications aimed at resolving the temporal and spatial fluctuations of gravitational acceleration at the Earth's surface, which can vary roughly between 9.78\,ms$^{-2}$ and 9.83\,ms$^{-2}$~\cite{menoret_gravity_2018}. Through gravimetry, one can directly infer information about sub-surface mass distributions, including volcanic activity monitoring~\cite{carbone_added_2017}, ice mass changes~\cite{makinen_absolute_2007}, subsidence monitoring~\cite{van_camp_repeated_2011}, and the detection of underground cavities~\cite{romaides_comparison_2001}. The latter is of interest to the oil and gas industry, as well as the construction industry. Section~\ref{sec:sense} lists a range of quantum sensing proposals involving levitated nanospheres suitable for these types of applications.
\section{Implementation} \label{implementation} Macroscopic mechanical oscillators which are controlled using light belong to a field of study called optomechanics. The use of a resonant optical cavity can significantly enhance the light-matter interaction. At the heart of all cavity optomechanical systems is a dispersive interaction, where an optical resonance frequency is shifted due to mechanical motion. This governs the read out of zero-point fluctuations and any motion caused by forces acting on the system. Before explaining the benefits of such a sensing scheme, we first describe the main components of an optomechanical system, and the variety of mechanical modes and optical resonances employed by researchers.
\subsection{Typical features of cavity optomechanics} In general, a cavity optomechanical system consists of three main ingredients. Firstly, a mechanical mode, such as the centre-of-mass (c.o.m.) motion of the end-mirror of a Fabry-Perot cavity, as shown in Figure~\ref{fig:schematic}(A). It can also be the c.o.m. motion of a levitated nanoparticle \cite{chang_cavity_2010,kiesel_cavity_2013}, a membrane \cite{chan_laser_2011} or a cantilever structure \cite{metzger_cavity_2004}. Levitated (nanoparticle) optomechanics benefits from \hl{excellent} environmental isolation \cite{millen_optomechanics_2020} whereas some clamped systems possess exceptionally high frequency mechanical modes in the microwave (MW) frequency range~\cite{kippenberg_cavity_2008}. Typical c.o.m. oscillation frequencies of levitated nanoparticles range from a few \,kHz to several hundred \,kHz~\cite{kiesel_cavity_2013,delic_cooling_2020,millen_optomechanics_2020}. The mass of nanoparticles typically employed in levitated optomechanics varies from $10^{-19}$\,kg to $10^{-16}$\,kg~\cite{delic_cooling_2020,millen_cavity_2015,millen_optomechanics_2020}.~\footnote{For levitated systems, rotational and librational motion at higher frequencies is also studied \cite{kuhn_optically_2017,ahn_optically_2018,reimann_ghz_2018,rashid_precession_2018,rahman_laser_2017}. For clamped systems, internal mechanical modes such as radial breathing modes, which can possess GHz frequencies, are routinely used to demonstrate near-quantum ground state preparation \cite{schliesser_resolved-sideband_2009}. Higher order mechanical modes are also studied \cite{jiao_nonlinear_2016}.}
The second requirement is a cavity-confined optical mode which is coupled to the mechanical oscillator. Figure~\ref{fig:schematic}(A) illustrates a resonant standing wave within a Fabry-Perot cavity, which can be coupled to the motion of the cavity end-mirror via radiation pressure. Alternative optical modes include the evanescent fields of whispering gallery mode \cite{schliesser_cavity_2014} and photonic crystal \cite{eichenfield_optomechanical_2009,magrini_near-field_2018} resonances, which can be coupled to their own internal mechanical modes or to external mechanical oscillators. A narrow cavity resonance linewidth $\kappa$ enables one to reach the `resolved-sideband regime', where the mechanical oscillation frequency $\Omega_{m}$ is larger than $\kappa$. This allows energy transfer between the optical and mechanical modes in an anti-Stokes/Stokes process, enabling cooling of the mechanical oscillator. A cavity is not necessarily required to reach the ground state \cite{tebbenjohanns_motional_2020}, but a cavity provides resonant enhancement in read out and interaction strength\hl{. This reduces} the number of photons needed to interact with the mechanical oscillator, improving the signal to noise.
Thirdly, a sufficiently high optomechanical coupling rate $G$ (in units of Hz/m) is required. This encodes the shift in the optical cavity resonance frequency caused by the motion of the mechanical oscillator. Large $G$ is required to optically transduce, control or cool the oscillator, with the highest $G$ obtained when the overlap between the optical field and the displacement field is maximised \cite{camacho_characterization_2009}, i.e. by using a spatially confined optical mode. \begin{figure}\label{fig:schematic}
\end{figure}
\subsection{Ground state cooling} In this viewpoint, we focus on quantum enhanced sensing using levitated systems. Below, we explain the coherent scattering technique, and how it enables cooling of the c.o.m. mode of a macroscopic object to the ground state of an optical potential \cite{delic_cooling_2020}.
A macroscopic quantum state can be created by cooling the c.o.m. motion of a nanoparticle levitated within a harmonic potential, with a mechanical frequency $\Omega_{m}$. The position uncertainty of the particle is $\sigma_x = \sqrt{\hbar(1+2n)/2m\Omega_m}$, where the phonon occupancy $n$ is related to the c.o.m. temperature $T_{\rm{CM}}$ through $n = \sqrt{k_BT_{\rm{CM}}/\hbar\Omega_m}$. When cooled to the ground state, the particle has a position uncertainty, or zero-point fluctuation, of $\sigma_{\rm{zpf}} = \sqrt{\hbar/2m\Omega_m}$. If the particle is released from the levitating potential, this position spread \hl{grows approximately linearly in time{~\cite{romero-isart_coherent_2017}}}. Considering typical parameters for a levitated nanoparticle of $m = 10^{-18}\,$kg and $\Omega_m = 2\pi\times 10^5\,$rad/s, this yields $\sigma_{\rm{zpf}} \approx 10^{-11}\,$m, requiring hundreds of seconds of expansion until the quantum position spread is as large as the particle, a reasonable definition of a macroscopic quantum state. However, subsequent matter-wave interferometry can be used to boost the size of the quantum state~\cite{millen_optomechanics_2020}, discussed further in Section~\ref{sec:sense}.
Following the above discussion, the particle must be initially cooled near to the ground state of the levitating potential. A range of passive and active cooling methods to achieve this are described in multiple review papers \cite{aspelmeyer_cavity_2014}, with many techniques such as sideband resolved cooling derived from the cold atom community \cite{ritsch_cold_2013}. Here we focus on the `coherent scattering' protocol \cite{vuletic_laser_2000,delic_cavity_2019,windey_cavity-based_2019}, as illustrated in Figure~\ref{fig:schematic}(B), which is the most successful cooling technique for levitated nanoparticles.
The mechanical oscillator is the c.o.m. motion of a levitated nanoparticle, and it is held within an optical cavity using an optical tweezer, as shown in Figure~\ref{fig:schematic}(B). The frequency of the oscillator is set by the optical potential provided by this single-beam gradient force trap. The tweezer allows the optimal placement of the particle within the cavity mode; the coupling strength is at its highest when the particle is held at the cavity node.
The optical cavity is not pumped externally. The trapping optical tweezer frequency is stabilized relative to the cavity resonance, and light scattered out of the tweezer field by the nanosphere then populates the cavity mode, interacting coherently with the oscillator again. It is a key feature of the coherent scattering technique that the optical cavity is only pumped by the light scattered by the nanoparticle.
Consequently, each photon populating the cavity mode interacts with the particle, increasing the optomechanical coupling rate. As a result, the quantum cooperativity \footnote{The cooperativity is defined as a ratio of the optomechanical coupling strength and the product of the optical and mechanical decay rates.} of the experiment is well above 1000. To put this into perspective, a quantum cooperativity >1 is the benchmark for entering the quantum back-action regime~\cite{millen_optomechanics_2020}; a long sought-after goal in levitated optomechanics. A high cooperativity is also known in cold atom physics to produce a constant cooling rate for cavity assisted molecule cooling in dynamical potentials~\cite{ritsch_cold_2013,chang_quantum_2014}. Compared to externally pumped cavity cooling schemes~\cite{delic_levitated_2020}, the estimated improvement in cooperativity is $10^5$-fold~\cite{delic_thesis_2019} due to coherent scattering.
One of the biggest challenges that previously prevented ground state cooling is a method to circumvent heating due to scattering and phase noise in the optical cavity. Phase noise in the cavity field can be reduced by increasing optical power, with the trade-off that scattering noise increases. In the coherent scattering scheme, laser phase noise is almost completely evaded since optimal cooling of the nanoparticle occurs at the cavity node, where the intensity minimum of the cavity standing wave is located. Further noise reduction involves a balance between increasing the optomechanical coupling strength and decreasing the scattering noise, whilst still ensuring the particle is stably levitated.
Finally, the coherent scattering cooling scheme is inherently multidimensional. Although to date only one axis has been optimized for ground state cooling, when rotating the trap accordingly full 3D cooling is possible~\cite{windey_cavity-based_2019}. Varying the coupling of each c.o.m. degree of freedom to the cavity by moving and tuning the scattering plane of the optical tweezer makes the coherent scattering implementation flexible. In contrast, only strong one dimensional cooling can be achieved in cavity systems such as Figure~\ref{fig:schematic}(A), where the static intracavity trapping potential limits cooling to be along the cavity axis.
Recent theoretical work on how to treat the multidimensional cooling dynamics illustrate apparent 3D hybridisation effects in coherent scattering. These hybridising pathways act as a road map to engineer displacement sensing possibly surpassing the standard quantum limit (SQL)~\cite{toros_quantum_2020}.
\section{Inertial Sensing \& Gravimetry} \label{sec:sense} Macroscopic quantum objects offer significant sensing advantages over their lower-mass cold atom counterparts through an enhanced coupling to inertial and gravitational forces\hl{\footnote{\hl{This is generally true for inertial and relative gravity sensors based on mechanical oscillators. For free-fall absolute measurements of gravity, the mass component often cancels out due to the equivalence principle.}}}. Generally, the competitive edge of optomechanical sensors can be summarised through two sensing strategies; quantum limited transduction, and sensing which exploits either superposition or entanglement, the features of the so-called `second quantum revolution'. Levitation \hl{provides excellent} environmental isolation, ensuring long coherence times in which to perform sensitive measurements. For example, the coherent scattering experiment mentioned above achieves a coherence time of 7.6\,$\mu s$, corresponding to 15 coherent oscillations before the ground state is populated by even a single phonon~\cite{delic_cooling_2020}.
We will first discuss the limitations of continuous optical transduction of the oscillator position, before exploring the use of spatial superpositions to extract a gravitationally induced phase shift via matter-wave interferometry. We will describe proposals that increase this phase shift through coupling to spin, creating a spin-oscillator superposition where gravity or acceleration is read out using the spin state.
\subsection{Continuous optical sensing}
The dispersive interaction between a mechanical oscillator and an optical resonance allows for continuous read out of the oscillator motion through probing the optical field quadrature. When thermal motion is present, the acceleration sensitivity for frequencies below mechanical resonance remains bound by $a_{\rm{th}}=\sqrt{\frac{4k_{B}T_{\rm{CM}}\Omega_{\rm{m}}}{m Q_{\rm{m}}}}$, where $T_{\rm{CM}}$ is the cooled mode temperature, $\Omega_{\rm{m}}$ the oscillator frequency and $Q_{\rm{m}}$ the mechanical quality factor. During cooling, $T_{\rm{CM}}$ decreases proportionally with $Q_{\rm{m}}$, resulting in no net change to the noise floor $a_{\rm{th}}$ \cite{krause_high-resolution_2012}. However, cooling does reduce the classical thermomechanical noise at the mechanical frequency. With the mechanical oscillator prepared in the ground state, an optomechanical measurement of the position is no longer limited by thermal motion, but by the zero point fluctuations and added noise from the transduction (measurement). The sensing sensitivity is now set by the SQL where back-action noise caused by photon momentum kicks, and imprecision noise caused by phase fluctuations contribute equally, yielding a displacement read out sensitivity of $2\times \sigma_{\rm{zpf}}$.
However, the SQL does not correspond to a fundamental quantum limit \cite{mason_continuous_2019}. Methods to read out the mechanical motion whilst minimizing or evading the effects of back-action are known as quantum nondemolition measurements\hl{. In these cases, the measured observable commutes with the system Hamiltonian, as provided by} coupling to the velocity of a free mass \cite{purdue_practical_2002} or modification of the mechanical susceptibility through engineering an optical spring to establish a new SQL \cite{arcizet_beating_2006}. The SQL can also be surpassed by monitoring only one of the two non-commuting quadratures of the motion, known as a back-action-evading measurement\hl{. This} can squeeze either the optical or mechanical quadrature, and is achieved using pulsed optomechanics \cite{vanner_pulsed_2011}.
\subsection{Spatial superpositions for sensing} Superposition and entanglement are quantum effects, which have no classical analogue. Entanglement enables \hl{the distribution of quantum states between oscillators separated by a distance}, whilst superposition enables the oscillator to be in a linear sum of several motional states. When exploiting these effects for inertial sensing and gravimetry, additional sensitivity or resolution can be achieved. For example, multiple entangled quantum nanospheres could be used for distributed sensing. Such sensing schemes require additional experimental steps beyond cooling to prepare the quantum state. In this section we describe a select number of quantum sensing proposals involving levitated nanoparticles, which are now within grasp.
An alternative to performing continuous transduction of the nanoparticle's motion would be to prepare the particle in a spatial superposition and then perform matter-wave interferometry. To sense gravity would require the creation of a coherent superposition, such that the superposition is localized at two heights or regions of a varying local gravitational field. The two amplitudes of the wave-function evolve under the Newtonian gravitational potential, resulting in a relative phase difference which is then measured interferometrically. The phase difference is defined: \begin{equation}
\Delta\phi=\frac{1}{\hbar}\int_{0}^{t}\Delta U dt=\int_{0}^{t}\frac{mg\Delta z(t)\cos(\theta)}{\hbar}dt, \end{equation} where $\Delta U$ is the gravitational potential energy difference across a vertical spatial separation $\Delta z(t)$, $m$ is the mass of the oscillator, and $\theta$ the angle between the interferometer and the direction of acceleration $g$ (here, defined by local gravity). The integral $\int^{t}_{0}\Delta z(t)dt$ is the path difference between the trajectories of the two amplitudes of the wave-function. For a trapped system, discussed in Section~\ref{sec:levsense}, $\Delta z(t)$ is the maximum spatial superposition separation, limited by the period of the trap oscillation $2\pi/\Omega_{\rm{m}}$. For an oscillator under free-evolution, discussed in Section~\ref{sec:freefall}, the particle wave-function freely evolves for as long as it remains coherent, enabling the creation of larger spatial superpositions~\cite{brawley_nonlinear_2016}.
\subsubsection{Optically preparing spatial superpositions} \label{sec:optical-superposition} Generation of a quantum superposition requires a nonlinear interaction. To prepare a spatial superposition in an optical cavity a strong quadratic coupling to motion governed by $g_{2}~\hat{x}^{2}$, or a strong single-photon coupling utilising the nonlinearity of the radiation-pressure interaction, is required~\cite{brawley_nonlinear_2016}. For example, by using a laser pulse interaction that measures $\hat{x}^{2}$ via a homodyne measurement. This provides information of the nanosphere position relative to the cavity centre, but not the offset direction (left or right), creating a spatial superposition similar to a double slit. Technically, it is challenging to create either a strong linear or a strong quadratic optomechanical coupling.
Alternatively, superposition can be generated by entangling a photon with a modified optomechanical Michelson interferometer formed by adding an additional Fabry-Perot cavity at the unused port of the beamsplitter \hl{annotated with a star} in Figures~\ref{fig:schematic}(A,B). A superposition of the two optical cavity modes is formed by the beamsplitter interaction such that, without measurement, the photon enters both cavities at the same time. The radiation pressure of this photon causes a deflection of the mechanical oscillator of approximately the zero-point motion, thus creating a mechanical superposition where the mechanical oscillator is unperturbed and perturbed \cite{bose_scheme_1999}.
\subsubsection{Spatial superpositions through coupling to spin} \label{sec:levsense} A nonlinear interaction can also be mediated by a two-level-system. In contrast to measuring the nanoparticle position through the phase modulation of the optical cavity field, or through generation of matter-wave interferometry, coupling the motion to a two-level-system enables read out through the two level states. The advantage of this method, which relies on discrete variables such as spin, is an immunity to motional noise which relaxes the need for ground state preparation. For example, many levitated nanosphere proposals that utilise spin only require a thermal state with moderate cooling. In general, because discrete variables allow the use of heralded probabilistic protocols, they also benefit from high fidelity and resilience to background noise or detection \hl{losses}.
Proposals combining levitated particles with two level systems include levitated nanodiamonds with an embedded nitrogen-vacancy (NV) centre with an electron spin \hl{{~\cite{yin_large_2013,scala_matter-wave_2013,chen_high-precision_2018,bose_spin_2017,marletto_gravitationally_2017}},} and a superconducting ring resonator coupled to a qubit \cite{johnsson_macroscopic_2016}. Here, we consider stationary spatial superpositions of a levitated nanoparticle oscillator with embedded spin, which remains trapped by an optical tweezer throughout the sensing protocol, as illustrated in Figure~\ref{fig:gravimetry}A. The spin is manipulated by microwave pulses. The first pulse introduces Rabi oscillations between the spin eigenvalue states $S_{z}=+1$ and $S_{z}=-1$, such that when a magnetic field gradient is applied the oscillator wavepacket is delocalized. This spin-dependent spatial shift is given by $\pm\Delta z=\frac{g_{\rm{nv}}\mu_{\rm{B}} B_{\rm{z}}}{2m\Omega^{2}}$, where $B_{\rm{z}}$ is the magnetic field gradient along the z-direction, which is the same direction that gravity acts in~\cite{chen_high-precision_2018,johnsson_macroscopic_2016}\footnote{A spatial superposition can be prepared at an angle $\theta$ to the acceleration force by tilting the applied magnetic field direction~\cite{scala_matter-wave_2013}}, $g_{\rm{nv}}\approx 2$ is the Land{\'e} $g$ factor and $\mu_{B}$ is the Bohr magneton. This effectively splits the harmonic trapping potential, creating a spatial superposition with equilibrium positions governed by a spin-dependent acceleration. The spin-oscillator system now has states $\ket{+1}$ and $\ket{-1}$ in different gravitational potentials, accumulating a relative gravitational phase difference. A measurement at $t_{0}=\frac{2\pi}{\Omega_m}$ yields a phase shift \cite{scala_matter-wave_2013}: \begin{equation}
\Delta\phi=\frac{16 \Lambda\Delta\lambda t_{0} }{\hbar^{2}\Omega_{m}} \end{equation} where $\Delta\lambda=\frac{1}{2}mg\cos(\theta)\sigma_{\rm{zpf}}$ is the gravity induced displacement, $\theta$ the angle between the applied magnetic field gradient $B_{\rm{z}}$ and the direction the nanosphere is accelerated (defined here as the direction of local gravity $g$), and $\sigma_{\rm{zpf}}$ the zero-point fluctuations. The spin-oscillator coupling is given by $\Lambda=g_{\rm{nv}}\mu_{\rm{B}}B_{\rm{z}}\sigma_{\rm{zpf}}$, with $g_{\rm{nv}}$ and $\mu_{\rm{B}}$ defined previously. This phase difference can be measured by applying a microwave pulse that probes the population of the spin state $S_{z}=0$ via $P_{0}=\cos^{2}(\frac{\Delta\phi}{2})$.
The maximum spatial superposition~\footnote{The path trajectory of each wavepacket is given by $z_{\pm}(t)=\pm\Delta z(1-\cos(\Omega_{\rm{m}}t))+\frac{g}{\Omega_{\rm{m}}^{2}}$, where $\frac{g}{\Omega_{\rm{m}}^{2}}$ is a shift in the equilibrium position of the non-localised particle due to the force of gravity, $g$.} in the direction of gravity is given by $\Delta Z=2\frac{g_{\rm{nv}}\mu_{\rm{B}} B_{\rm{}z}}{m\Omega_m^{2}}$ which tends to be much smaller than the physical size of typical nanodiamonds~\cite{scala_matter-wave_2013,yin_large_2013}. This makes levitated sensing schemes unfriendly for resolving gravity gradients without the use of surveying or arrays.
\begin{figure}
\caption{Spin-oscillator coupling has been proposed as a gravimetry technique, whereby a spatial superposition is created through the interaction of an embedded two-level-system such as a nitrogen-vacancy (NV) centre with an external magnetic field gradient. Variations of (A) has been proposed in~\cite{yin_large_2013,scala_matter-wave_2013,chen_high-precision_2018} A microwave pulse can be applied to split the spin states of the internal NV center in a trapped quantum nanosphere (i), which in turn creates a spatial superposition that can be viewed as the splitting of the optical trap (ii) into a superposition (iii). Note that the cavity is used to prepare the nanosphere in the ground state and can be switched off during (ii and iii). (B) Allowing for free evolution increases the spatial size of the superposition, created and probed in a Ramsey-type interferometer, as proposed by~\cite{wan_free_2016}. Here, the coupled NV-nanosphere superposition is prepared with a microwave (MW) pulse at time $t_{1}$, undergoing free-fall until the spins are flipped at $t_{2}$ to enable matter-wave interferometry at $t_{3}$.}
\label{fig:gravimetry}
\end{figure}
\subsubsection{Free-fall measurements} \label{sec:freefall} Releasing the nanoparticle from the trap, such that it undergoes free-fall, allows its wave-function to evolve, increasing the position spread linearly in time. The ability to perform such a drop-test is unique to the levitated platform amongst optomechanical systems. Free-fall acceleration sensors are known as absolute gravimeters because they give a direct measure of gravity in units of ms$^{-2}$ traceable to metrological standards. Relative gravimeters are masses supported by a spring, for example, the stiffness of a cantilever or the optical trapping of a nanosphere. One must calibrate relative gravimeters by measuring the stiffness of the spring and placing the instrument in a location with a known gravitational acceleration. Absolute gravimeters are therefore required to calibrate relative ones.
Perhaps the most recognisable free-fall quantum measurement is the double slit experiment, with a particle falling through a diffractive element. This can be recreated with a nanoparticle launched ballistically, with two additional cavities positioned vertically along the free-fall path. The first is used to apply a laser pulse at time $T$ that measures the square of the position via a homodyne measurement to create a vertical spatial superposition. After the superposition evolves for a further $T$ seconds, the second cavity is used to measure the particle's c.o.m. position such that after many iterations, an interference pattern is formed. The effect of gravity is to introduce a vertical shifting of the entire interference pattern on the screen by $\delta y=\frac{g}{2}(2T)^{2}$. If one arm of the superposition experiences a differing force to the other, i.e. due to a slight difference in the gravitational field, this also has a shifting effect on the \hl{interference} pattern~\cite{romero-isart_large_2011,rasel_2012}. However, creating a clear interference pattern requires many repeated launches of the same nanoparticle, requiring capture and launch techniques more mature than currently achieved, discussed in Section~\ref{sec:sen-comm}.
For detecting transverse accelerations, i.e. to detect a nearby object placed perpendicular to the force of gravity, one can use a Talbot interferometer scheme proposed in~\cite{Geraci2015}.
In a Talbot interferometer, a light pulse grating is applied to a free-falling quantum nanosphere, causing the nanosphere to diffract and interfere with itself. This creates an image of the grating at a distance defined by the Talbot length, and at every integer of the Talbot length. At half the Talbot length the interference pattern is phase shifted by half a period. By positioning an object adjacent to the free-fall path at a distance $<10\,\mu$m, one can probe the gravitational force produced by the object based on the transverse shift of the interference pattern. A sensitivity of $10^{-8}$\,ms$^{-2}$ is predicted for a sphere 13\,nm in diameter where a phase difference of $\Delta\phi=\pi$ corresponds to an acceleration of $4\times 10^{-6}\,$ms$^{-2}$. The fringe pattern shift is given by $\delta x_{\phi}=-a\,T^{2}_{t}$ where $a$ is the transverse acceleration and $2T_{t}$ is the total flight time with $T_{t}$ the Talbot time.
Alternatively, one can avoid using matter interferometry, and instead, directly measure the shift in the nanosphere position caused by transverse acceleration $\delta x=\frac{at^{2}}{2}$. This can be achieved by dropping the nanosphere ballistically, such that after $t$ seconds, it falls into a cavity which then measures its shifted position. The sensitivity obtained for this ballistic scheme compared to the interferometric measurement scales with $\frac{\chi \sigma_{\rm{v}}t}{d}$, where a decrease in $\chi$, the fringe contrast, or an increase in $d$, the grating period, will reduce the advantage of the Talbot scheme over the ballistic one. The spread of the position in the ballistic set-up, given by $\sigma_{\rm{v}}t=\sqrt{\frac{\hbar\Omega_{\rm{m}}}{2m}}t$, grows during the free-fall time, contributing to the uncertainty in $\delta x$. This is another reason why the Talbot scheme is more sensitive. The initial momentum spread due to zero-point fluctuations does not influence the position of the interference fringes in a Talbot interferometer, instead, only affecting the envelope.
It is not straightforward to conclude when the Talbot interference scheme surpasses the ballistic scheme if parameters such as mass are varied, since this can result in a reduction of the time-of-flight due to decoherence. For example, ground state cooling of a 200\,nm diameter nanosphere enables a ballistic sensitivity 10 times higher than a Talbot measurement on a 12\,nm diameter particle ~\cite{Geraci2015}.
Lastly, the levitated spin-oscillator Ramsey interferometer scheme shown in Figure~\ref{fig:gravimetry}A can be modified for free-fall evolution, as shown in Figure~\ref{fig:gravimetry}B. Due to the long coherence time of spin states, the superposition persists even if the oscillator does not remain in a pure coherent state. The scale of the superposition is controllable through flight time and/or magnetic field gradient such that the acquired phase is given by \cite{wan_free_2016}: \begin{equation}
\Delta\phi=\frac{1}{16\hbar}g_{\rm{nv}}\mu_{\rm{B}}B_{\rm{x}}g t_{3}^{3}\cos(\theta),
\label{eq:freefall} \end{equation} where $g_{\rm{nv}}$, $\mu_{\rm{B}}$ are defined as above, $\theta$ is the angle between the applied magnetic field $B_{\rm{x}}$ and the direction of the gravitational acceleration $g$ and $t_{3}$ is the total free-fall time (i.e. when the wavepackets merge and interfere).
In contrast to the levitated Ramsey scheme, equation~\ref{eq:freefall} does not depend on the mass, \hl{as expected for a free-fall measurement. An} additional microwave pulse is required at time $t_{3}$ to reverse the propagation direction of the superposed wave packets. This pulse flips the spin state such that the spin dependent force will reverse. The split wavepackets eventually merge and interfere. The measurement time $t_{3}$ is therefore unconstrained and can be on the order of milliseconds, enabling spatial superpositions spanning~100\,nm, over $10^{3}$ times larger than if the nanoparticle was levitated \cite{wan_free_2016}, and comparable in scale to the size of the particle. The maximum spatial separation along the \emph{tilted} $x$-axis is $\Delta x_{\theta}=2\frac{g_{\rm{nv}}\mu_{\rm{B}}B_{\rm{x}}}{2m}(t_{3}/4)^{2}$, at $t=t_{3}/2$. Currently, coherent scattering cooling would achieve a maximum coherence time of 1.4\,$\mu s$ in free-fall, limited by background pressure, which only allows for an expansion of the wavepacket from 3.1\,pm to 10.2\,pm~\cite{delic_cooling_2020}. Further progress will require deep vacuum environments and likely cryogenic operating conditions.
\subsection{Comparison} Table~\ref{tab:comparison} shows a comparison of the predicted and achieved acceleration sensitivities obtained by quantum and classical research sensors, alongside the current commercial state-of-the-art. \hl{Also included is the achieved accuracy obtained by two sensors. Note that sensitivity, defined by the velocity random walk on an Allan deviation plot of the sensor output~\footnote{\hl{For pulsed measurements, common in atom interferometry, sensitivity does not imply a spectral density.}}, is a measure of precision, but does not guarantee accuracy which describes the trueness of the value. Academic devices that are not based on cold atoms require calibration, and therefore not as easily traceable to the International System of Units (SI).} We focus on devices which are suitable for gravimetry but would need to be used in a flywheel operation with a classical inertial measurement unit (IMU) for navigation applications. The latter requires sampling rates above 100\,Hz, incompatible with the time of flight used in free-fall experiments or the pulse sequence needed for Ramsey interferometry. Flywheel operation uses a classical IMU to provide inertial measurements in-between this deadtime, and is used by cold atom inertial sensor prototypes~\cite{battelier_development_2016}. In turn, the quantum measurement, which is less susceptible to drift, is used to reset the growing errors accrued by the IMU. The achievable sensitivities across all types of sensors are comparable, which is unsurprising considering the majority are operated classically, where optimisation of the detection noise and/or effective mass can still significantly improve sensitivity at the cost of bandwidth. However, all current sensors struggle to surpass a sensitivity better than $10^{-9}\,$ms$^{-2}$.
An interesting question is: what sets the fundamental limit in sensitivity at the quantum level? The field of quantum metrology seeks to find these fundamental limits through use of quantum Fisher information (QFI) which is a metric of intrinsic accuracy \hl{(assuming all unknown variables can be traced to SI units)}, and only depends on the input state. It is formally defined via the Cram{\'e}r-Rao bound as the inverse of the variance of a measurable property, in this case, phase. A high QFI ensures greater precision. The QFI cannot reveal the underlying measurement protocol to obtain such limits, but one may test a measurement protocol by computing its associated classical Fisher information (CFI). The CFI takes into account both the input state and the extractable information from the measurement scheme, and may or may not meet the QFI bound~\cite{armata_quantum_2017,schneiter_optimal_2020}.
\begin{table}
\begin{tabular}{>{\arraybackslash}m{2.8cm}|>{\arraybackslash}m{5.1cm}} \hl{Existing} System & Achieved sensitivity (ms$^{-2}\,$Hz$^{-1/2}$)\\ \hline Free-fall cube mirror$^{\dagger}$ & $1.5\times 10^{-7}$ \textmd{($10^{-9}$ms$^{-2}$ in 6.25hr)}\cite{lacoste}$^{\ddag}$ \\ Atom interferometer & $4.2\times 10^{-8}$ \textmd{$(3\times 10^{-9}$ms$^{-2}$ in 300s)}\cite{hu_2013}\\ \hl{Atom interferometer} & \hl{$9.6\times 10^{-8}$ \textmd{$(5\times 10^{-10}$ms$^{-2}$ in 2.8hr)}{\cite{Freier_2016}}}\\
Lev. optomechanics$^{\dagger}$ & $4\times 10^{-6}$ \textmd{($6.9\times 10^{-9}$ms$^{-2}$ in 3.8hr)}\cite{monteiro_optical_2017} \\ Lev. optomechanics$^{\dagger}$ & $9.3\times 10^{-7}$ \cite{monteiro_force_2020}$^{\clubsuit}$\\ Opto-MEMS$^{\dagger}$ & $7.8\times 10^{-8}$ \cite{krishnamoorthy_-plane_2008}\\ Capacitive-MEMS$^{\dagger}$ & $3\times 10^{-9}$ \cite{pike_2019}\\ \hline \hl{Existing System} & \hl{Achieved accuracy (ms$^{-2}$)}\\ \hline \hl{Free-fall cube mirror$^{\dagger}$} & $2\times 10^{-8}$~\cite{lacoste}\\ \hl{Atom interferometer} & \hl{$3.9\times 10^{-8}${~\cite{Freier_2016}}}\\ \hline
&\\ \hline \hl{Proposed System} & \hl{Predicted} sensitivity (ms$^{-2}\,$Hz$^{-1/2}$) \\ \hline Trapped cold atom* & $\times 10^{-10}$ \\ Lev. optomechanics* & $\times 10^{-15}$ \\ Lev. spin-mechanics & $2.2\times 10^{-9}$ \cite{johnsson_macroscopic_2016} \\ Talbot optomechanics & $\times 10^{-8}$ \cite{Geraci2015} \\ \hline \end{tabular} \caption{\label{tab:comparison}Table showing the achieved and predicted acceleration sensing sensitivities among various types of accelerometer. \hl{We include two measurements of sensor accuracy. Optomechanical systems require further work to define their accuracy as they are only calibrated against commercial sensors.} 'lev.' denotes levitated experiments, BEC is a Bose Einstein Condensate, and MEMS are micro-electro-mechanical systems. $^{\dagger}$ denotes classical measurements, noting that capacitive MEMS do not currently operate in the quantum regime and optomechanical devices have only recently entered the quantum regime. $^{\ddag}$ is the value from LaCoste's brochure whereas $5.6\times 10^{-7}$\,ms$^{-2}$\,Hz$^{-1/2}$ was measured in the laboratory~\cite{wang_shift_2018}. $^{\clubsuit}$ indicates the sensitivity whilst the c.o.m. temperature of the oscillator is cooled to $50\mu$K~\cite{monteiro_force_2020}. * is the predicted sensitivity using classical Fisher information assuming a homodyne detection scheme, obtaining a sensitivity close to that predicted by quantum Fisher information \hl{{\cite{Qvarfort2018}}}.} \end{table}
In \hl{{\cite{Qvarfort2018}}} the CFI for a homodyne measurement of a quantum levitated nanosphere using continuous optomechanical transduction is calculated, which they compare with the CFI for a typical atom interferometry set-up. Over five orders of magnitude improvement in sensitivity is predicted for a levitated optomechanical oscillator versus a cloud of $10^{5}$ atoms, shown in the lower portion of Table~\ref{tab:comparison}. This clearly highlights the potential competitive advantage in quantum levitated optomechanics, including the relative ease in preparing a single macro-sized object over a dense cloud of atoms. \hl{However, we stress that such a claim requires further work to improve the accuracy of optomechanical sensors, involving precise calibration of mass and frequency to SI standards. cold atom measurements are intrinsically traceable.}
\subsection{Road to commercialisation} \label{sec:sen-comm} For some time the optomechanics community has been prototyping classical accelerometers \cite{li_field_2018,huang_chip-scale_2020,krause_high-resolution_2012,guzman_cervantes_high_2014,monteiro_optical_2017,monteiro_force_2020,middlemiss_field_2017}. Such sensors rarely require the level of environmental isolation needed for long-lived quantum state preparation. Once the community is able to repeatedly demonstrate quantum state preparation of levitated nanospheres, now possible with coherent scattering techniques, they should use the advancement of commercial cold atom interferometers as a roadmap for developing application-ready tools.
Common to both quantum optomechanics and cold atom interferometry is the need for ultralow and stable vacuum pressures, preferably at~$10^{-12}$\,mbar or below, to prevent collisions with background gas molecules. Lowering the environmental temperature using cryostat technologies reduces the influence of thermal heating, although this is more crucial for clamped optomechanics experiments using cantilevers, membranes or MEMS structures\hl{. Levitation} minimizes thermal contact with the environment. For the coherent scattering set-up, they measure background collisions as their largest source of decoherence when at a pressure of $10^{-6}$\,mbar, requiring $10^{-11}$\,mbar and cryostat temperatures below 130\,K to sustain a wavepacket on the order of the particle radius~\cite{delic_cooling_2020}. The stability of the pressure, ambient temperature, and laser frequency and intensity require consideration, as these may be sources of decoherence or drift that skew or washout the measured signal. Shot-noise limited laser sources are best suited.
Any vibrational noise in the components will also create errors, with vibrations at low frequency the most critical. A baseline vibrational stability of nm\,Hz$^{-1/2}$ is recommended~\cite{Geraci2015}, achievable with modern commercial cyrostats. Another source of mechanical error is misalignment. For example, a vertical tilt when measuring gravity will create an offset, characterised as noise if the alignment varies per shot. Tilt fluctuations no higher than $0.5\,\mu$rad\,Hz$^{-1/2}$ are recommended~\cite{Geraci2015}.
Unique to levitated optomechanics is the need for a launch and recapture method, since nanospheres are not indistinguishable like atoms. So far, loading at low vacuum pressure has been achieved through (i) momentum imparted by either a piezo-speaker \cite{millen_cavity_2015} or a laser induced acoustic shock of a wafer covered by tethered nanorods \cite{kuhn_cavity-assisted_2015}, (ii) electrospray injection of particles \cite{bullier_characterisation_2020}, or (iii) conveyor-belt loading using optical \cite{grass_optical_2016} or electrical forces \cite{bullier_characterisation_2020}. Reliable recapture of a levitated nanosphere still remains a technical challenge, but can be built upon successful proof of principle demonstrations~\cite{hebestreit_sensing_2018}. Alternatively, sourcing nanospheres with high reproducibility enables continuous injection of particles, where an in-situ calibration could be performed to account for size variances. Nanoparticles also vary in their surface charges, requiring shielding from stray electric fields to avoid dephasing from Coulomb force interactions. Recently, it was demonstrated that levitated particles could be discharged using a high voltage wire that ionizes residual gas molecules, adding charges to environment~\cite{frimmer_controlling_2017}. Single elementary charge precision was achieved, which would enable zero net charged nanoparticles to be prepared when starting with a mixture of the number of charges~\cite{ranjit_attonewton_2015}.
Lastly, all sensors must pass certain environmental testing conditions to be deployed for space/aerospace, military and metrology use. These include high electromagnetic interference protection levels, operational temperatures between -40$^{o}$C to +85$^{o}$C and shock resistance (in some cases, up to 20,000\,ms$^{-2}$). Testing is conducted in shake and bake chambers that apply acceleration whilst cycling the ambient temperature defined by standards such as those used by the military (MIL-STD), aerospace (DO) or consumer use (CE marking in EU)~\cite{honeywell_standards}. Those who perform these tests must also obtain certification that they meet the requirements governed by the International Organization for Standardization (ISO). Reduction in size, weight and power (SWaP) alongside cost are also factors that require involvement with supply chains and industry. Through global quantum technology initiatives, such collaborations are already underway, resulting in miniature vacuum chambers and chipscale ion traps \cite{schwindt_highly_2016,rushton_contributed_2014,birkl_micro_2007} developed for cold atoms, and miniaturised quantum sources of light \cite{caspani_integrated_2017} on chip-scale photonic integrated circuits developed for quantum computing. Preliminary feasibility studies have also been carried out in partnership with the European Space Agency to mature the supporting technologies needed to implement macroscopic state preparation and interferometry using levitated nanospheres in space~\cite{kaltenbaek_macroscopic_2016}.
\section{Outlook}
In this viewpoint we have reviewed proposals that aim to implement acceleration sensing using spatial superpositions of quantum levitated nanoparticles. Performing macroscopic matter-wave interferometry is now tantalisingly close with the advent of coherent scattering, used to cool a 143\,nm diameter sphere to the ground state of an optical potential \cite{delic_cooling_2020}. A new regime of quantum levitated optomechanics is upon us, 10 years after the first \emph{clamped} human-made object was cooled to the ground state~\cite{oconnell_quantum_2010,aspelmeyer_surf_2010}. The potential advantage of using a macroscopic quantum test-mass has been theoretically predicted using quantum and classical Fisher information analysis; 5-orders of magnitude improved sensitivity is expected over existing cold atom sensors. Although such predictions do not reflect technical feasibility, a potential demonstration of second-generation quantum sensing using optomechanics is certainly on the horizon.
Similar to cold atoms, there are a variety of quantum sensing protocols proposed in levitated optomechanics, whereby the oscillator either remains trapped or undergoes free-fall to enable larger spatial superpositions and averaging times. Many challenges for field-testing free-fall quantum sensors are already being solved by the commercial development of cold atom interferometers, backed by significant funding from governments and industry across the world~\cite{gibney_quantum_2019}. Unique to levitated optomechanics are challenges in reproducible and reliable launch, capture, and characterisation of nanoparticles with size variations, or the fabrication of near-identical nanoparticles.
At the time of writing, the UK, the EU, the US, China, Russia and Canada have or will be committing over £1B each to their respective quantum technology initiatives~\cite{thew_focus_2019}. Although quantum optomechanical sensors may not mature at the same rate as other quantum technologies, progress is undoubtedly linked to the successful commercialisation of existing cold atom sensors or quantum communication devices. These disruptive technologies share common goals, similar routes to market, and interchangeable subcomponents. We call upon the wider scientific community for increased cross pollination of resources and methods, wider engagement with industry, and global collaborations.
\begin{funding} YLL is supported by a Royal Academy of Engineering Intelligence Community Postdoctoral Fellowship Award: ICRF1920\textbackslash3\textbackslash10. MR acknowledges funding from the EPSRC Grant No. EP/S000267/1. JM is supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant Agreement No. 803277), and by EPSRC New Investigator Award EP/S004777/1. \end{funding} \nocite{*}
\printbibliography[category=cited,heading=subbibliography]
\end{document} | arXiv |
\begin{document}
\title{Iwasawa theory for Rankin-Selberg convolution at an Eisenstein prime}
\author{Somnath Jha, Sudhanshu Shekhar, Ravitheja Vangala}
\address{Department of Mathematics and Statistics, IIT Kanpur, 208016, India}\email{[email protected], [email protected], [email protected]}
\thanks{\noindent AMS subject classification: 11F33, 11F67, 11R23.}
\thanks{Keywords: Iwasawa main conjecture, $ p $-adic $ L $-functions, Rankin-Selberg convolutions, Selmer group.}
\begin{abstract} {Let $p$ be an odd prime, $ f$ be a $ p $-ordinary newform of weight $ k $ and $ h $ be a normalized cuspidal $ p $-ordinary Hecke eigenform of weight $ l < k$. In this article, we study the $p$-adic $ L $-function and $ p^{\infty} $-Selmer group of the Rankin-Selberg convolution of $f$ and $h$ under the assumption that $ p $ is an Eisenstein prime for $ h $ i.e. the residual Galois representation of $ h $ at $ p $ is reducible. We show that the $ p $-adic $ L $-function and the characteristic ideal of the $p^\infty$-Selmer group of the Rankin-Selberg convolution of $f, h$ generate the same ideal modulo $ p $ in the Iwasawa algebra i.e. the Rankin-Selberg Iwasawa main conjecture for $f \otimes h$ holds mod $p$. As an application to our results, we explicitly describe a few examples where the above congruences holds.}\end{abstract}
\maketitle
\section*{Introduction}
The study of Rankin-Selberg convolution of two modular forms is of considerable interest in number theory from arithmetic, analytic and automorphic point of view. In this article, we study Iwasawa theory for Rankin-Selberg convolution of two $ p $-ordinary modular forms $f$ and $h$ where the weight of $ f $ is strictly greater than the weight of $ h $ and $ p $ is an Eisenstein prime for $ h $.
Iwasawa main conjecture for elliptic curves and modular forms (at an ordinary prime) has been established (in a large number of cases) by the fundamental works of Kato \cite{kato} and Skinner-Urban \cite{SU}. As a next step, it is natural to study the Iwasawa theory of Rankin-Selberg $L$-function. Let $ f $ and $ h$ be two $ p $-ordinary modular forms.
In this setting, important works have been done by
Lei-Loffler-Zerbes \cite{llz}, Kings-Loffler-Zerbes \cite{klz} and X. Wan \cite{Wan}. In \cite[Theorem 7.5.4]{llz} and \cite[Corollary D]{klz}, one side divisibility in the Iwasawa main conjecture was shown
(i.e. the characteristic ideal of the $ p^\infty $-Selmer group of $ f \otimes h$ divides the $ p $-adic $ L $-function of $ f \otimes h$)
provided $ \rho_f \otimes \rho_h $ is irreducible and assuming additional hypotheses. In \cite[Theorem 1.2]{Wan}, X. Wan proves the other divisibility in the main conjecture
under suitable assumptions which includes $ f $ is a CM form and $ \bar{\rho}_h $ is irreducible. Thus following the works of \cite{llz}, \cite{klz} and \cite{Wan}, the Iwasawa main conjecture is still open in many cases; for example when $ (i) $ $ \rho_f \otimes \rho_h $ is reducible or $(ii)$ $ f $ is a non-CM eigenform.
In our setting, $ p $ is an Eisenstein prime for $ h $ i.e. $ \bar{\rho}_h $ is reducible, so our assumptions are complimentary to those mentioned in \cite{llz}, \cite{klz} and \cite{Wan}. In fact, our result in this article can be interpreted in terms of congruence of modular forms. Indeed, the congruences of modular forms (cf. \cite{Hidaadjoint}) is an important topic of study in the arithmetic of elliptic curves, modular forms and Iwasawa theory for congruent modular forms has been extensively studied following the works of \cite{gv}, \cite{Vatsal}, \cite{epw} etc.
In this article, we combine ideas from these two topics; Iwasawa theory of the Rankin-Selberg convolution and the congruence of modular forms to arrive at our result.
We now briefly recall the terminology needed to introduce our main theorem. The precise definitions are given later at appropriate places. Let $ p $ be an odd prime. Let $ \mathbb{Q}_{\mathrm{cyc}} $ be the cyclotomic $ \mathbb{Z}_{p} $-extension of $ \mathbb{Q} $ and $ \Gamma := \mathrm{Gal}(\mathbb{Q}_{\mathrm{cyc}}/\mathbb{Q}) $. Let $ f(z) = \sum_{n=1}^{\infty} a(n,f) q^n \in S_{k}(\Gamma_{0}(N),\eta)$ be a $ p $-ordinary primitive form
with $ p \nmid N $ and $ h \in S_{k}(\Gamma_{0}(I),\psi)$ be a $ p $-ordinary Hecke eigenform
with $2 \leq l < k$.
Fix a number field $ L $ containing the Fourier coefficients of $ f,h $ and the values of $ \eta , \psi$. Let $ L_\mathfrak{p} $ denote the completion of $ L $ at a prime $ \mathfrak{p} $ lying above $ p $ induced by the embedding $ i_p: \bar{\mathbb{Q}} \rightarrow \bar{\mathbb{Q}}_p $ and $ \pi $ be a uniformizer of the ring of integers $ \mathcal{O} $ of $ L_\mathfrak{p} $. For $ \mathfrak{g} \in \{f,h\} $, let $ \rho_{\mathfrak{g}} : G_{\mathbb{Q}} \rightarrow \mathrm{Aut}(V_{\mathfrak{g}}) $ be the $ p $-adic Galois representation attached to $ \mathfrak{g} $ and $ T_{\mathfrak{g}} $ be an $ \mathcal{O} $ lattice inside $ V_{\mathfrak{g}}$. Set $ A_f := V_f/T_f$ and $ A_f^{-} $ be the unramified quotient of $ A_f $ defined using the $ p $-ordinarity of $ f $. The crucial assumption in this article is the existence of the following exact sequence of $ G_{\mathbb{Q}} $ modules
\begin{small}
\begin{align}\label{intro rho_h}
0 \rightarrow \bar{\xi}_1 \rightarrow T_h/\pi \rightarrow \bar{\xi}_2 \rightarrow 0.
\end{align}
\end{small} As $ h $ is $ p $-ordinary, we may assume $ \xi_{2} $ is unramified. Let $ \chi_p$ and $\omega_{p} $ be the $ p $-adic cyclotomic character and Teichm\"uller character respectively.
Let $ A_j $ be the discrete $ \mathcal{O} $ lattice $ \big( (V_f \otimes V_g)/ (T_f \otimes T_g) \big) \otimes \chi_{p}^{-j} $. Using the $ p $-ordinarity and following \cite{gr1}, we define the $ p^{\infty}$- Selmer groups $ S_{\mathrm{Gr}}(A_f/\mathbb{Q}_\mathrm{cyc})$, $ S_{\mathrm{Gr}}(A_j/\mathbb{Q}_\mathrm{cyc})$ attached to $ A_f $ and $ A_j $ respectively. In this setting, Hida \cite{Hidarankin2} has constructed the $ p $-adic $ L $-function $ L_{p, f \otimes h,j} $ attached to the Rankin-Selberg convolution $ f \otimes h $ (see Theorem~\ref{padic rankin} and \eqref{def: p-adic L-function f x h} for the definition). Let $ L_{p,f,j} $ be the $ p $-adic $ L $-function attached to $ f $ as constructed in \cite{MTT}, \cite{Kitagawa} with an appropriate choice of periods, as explained in Theorem~\ref{period and integral measure}, \ref{c(f) and petterson} and \eqref{def: p-adic L-function f}.
\begin{comment}
Let $ p $ be an odd prime. Let $ \bar{\mathbb{Q}} \subset \mathbb{C}$ denote the algebraic closure of $ \mathbb{Q} $ in $ \mathbb{C} $. Fix an embedding $ i_p: \bar{\mathbb{Q}} \rightarrow \bar{\mathbb{Q}}_p$. Let $ \mathbb{Q}_{\mathrm{cyc}} $ be the cyclotomic $ \mathbb{Z}_{p} $-extension of $ \mathbb{Q} $ and $ \Gamma := \mathrm{Gal}(\mathbb{Q}_{\mathrm{cyc}}/\mathbb{Q}) $. For every place $ v $ of $ \mathbb{Q}_{\mathrm{cyc}} $ let $ I_{\mathrm{cyc},v} $ denote the inertia group at $ v $. For any finite set of primes $ \Sigma $ containing $ p $ and archimedean places, let $ \mathbb{Q}_{\Sigma} $ be the maximal extension of $ \mathbb{Q} $ unramified outside $ \Sigma $.
Let $ f(z) = \sum_{n=1}^{\infty} a(n,f) q^n \in S_{k}(\Gamma_{0}(N),\eta)$ be a newform of level $ N $, weight $ k $ and nebentypus $ \eta $. We assume $ (N,p) = 1 $ and $ f $ is $ p $-ordinary i.e. $ a(p,f) $ is a $ p $-adic unit. Let $ f_0 = \sum_{n=1}^{\infty} a(n,f_0)q^n \in S_k(Np,\eta)$ be the $ p $-stabilization of $ f $ and $ u_f := a(p,f_0) $ be the unique root of the polynomial $ X^2- a(p,f) X + \eta(p)p^{k-1} $ which is a $ p $-adic unit. Let $ h(z) = \sum_{n=1}^{\infty} a(n,h) q^n \in S_{k}(\Gamma_{0}(I),\psi)$ be a Hecke eigenform of level $ I $, weight $2 \leq l < k $ and nebentypus $ \psi $. Fix a number field $ L $ containing $ u_f $, Fourier coefficients of $ f,h $ and values of $ \eta , \psi$. Let $ L_\mathfrak{p} $ denote the completion of $ L $ at a prime $ \mathfrak{p} $ lying above $ p $ induced by the embedding $ \iota_p $ and $ \pi $ be a uniformizer of ring of integers $ \mathcal{O} $ of $ L_\mathfrak{p} $. Let $ \rho_f, \rho_h: G_{\mathbb{Q}} \rightarrow \mathrm{GL}_2(L_\mathfrak{p}) $ denote the Galois representation attached to $ f,h $ respectively. We assume that the residual Galois representation $ \bar{\rho}_h $ of $ h $ is reducible and $ \bar{\rho}_h^{ss} $ the semi simplification of $ \bar{\rho}_h $ is
$\bar{\xi}_{1} \oplus \bar{\xi}_{2}$. Let $ M := \mathrm{cond}(\bar{\rho}_h) $ and $ I := I_0 p^{\alpha} $, $ M := M_0 p^{s} $ with $ (p,I_0) =1 = (p,M_0) $. We now introduce a number $ m $ which plays an important role in this article
\begin{align*}
m := \prod_{\substack{\ell - \mathrm{prime} \\ \ell \mid I_0, \ell \nmid M_0 }} \ell \times \prod_{\substack{\ell - \mathrm{prime} \\ \ell \mid M_0, \ell \mid (I_0/M_0)}} \ell .
\end{align*}
For simplicity, let $ \tilde{f} = f |\iota_m$ (resp. $ \tilde{f}_0 = f_0 |\iota_m $) denote the twist of $ f $ (resp. $ f_0 $) by the trivial character modulo $ m $. Similarly, let $ \tilde{h} = h |\iota_m $. Define the Rankin-Selberg convolution of $ \tilde{f},\tilde{h} $ by
\begin{align*}
D_{IN}(s,\tilde{f},\tilde{h}) := L_{IN}(2s+2-k-l, \psi \eta)
L(s,\tilde{f},\tilde{h}) := L_{IN}(2s+2-k-l, \psi \eta)
\sum_{n=1}^{\infty} a(n,\tilde{f}) a(n,\tilde{h}) n^{-s},
\end{align*}
where $L_{IN}(2s+2-k-l, \psi \eta)$ denotes the
Dirichlet $L$-function of $ \psi \eta $ with the Euler factors at the primes dividing $IN$ omitted from its Euler product. Assume that the surjective homomorphism
$\phi_{\tilde{f}}: h_{k}(\Gamma_{0}(Nm^2p), \eta \iota_{m}^2; L) \rightarrow L $ given by $ T \rightarrow a(1,\tilde{f}_{0}|T) $
induces the decomposition
\begin{align}\label{splitting of hecke algebra intro}
h_{k}(\Gamma_{0}(Nm^2p), \eta\iota_{m}^2; L) = L \oplus A.
\end{align}
Let $\Sigma_{0} = \{v : v \mid m \}$ be a set finitely many places contained in $ \mathbb{Q}_\mathrm{cyc} $. In this setting, Hida \cite{Hidarankin2} has constructed the $ p $-adic $ L $-function associated to the Rankin-Selberg convolution of $ \tilde{f}, \tilde{h} $. For $ 0 \leq j \leq k-l-1 $ and a finite order character $ \phi: (1+p \mathbb{Z}_{p}) \rightarrow \mathbb{C}_p^{\times} $, we have
\begin{align*}
L^{\Sigma_0}_{p, f \otimes h,j} (\phi)
= c(\tilde{f}_0) t p^{\beta l/2} p^{\beta j} p^{(2-k)/2}
a(p,f_0)^{1-\beta} \frac{D_{INp}(l+j,\tilde{f}_{0}, (\tilde{g}^{\rho}|\phi^{-1})|_{l}\tau_{I_0mp^{\beta}})}{(2i)^{k+l+2j} \pi^{l+2j+1} <\tilde{f}_{0}^{\rho}|_{k}\tau_{Nm^2p} , \tilde{f}_{0}>_{Nm^2p}},
\end{align*}
where $\beta $ is the smallest positive integer such that $ \tilde{g}^{\rho}|\phi^{-1}
\in M_{l}(\Gamma_{1}(I_0 m p^{\beta}))$, $ \tau_{I_0 m p^{\beta}} =
\begin{psmallmatrix} 0 & 1 \\ I_0 m p^{\beta} & 0 \end{psmallmatrix} $
and $ t = [N,I_0 m] N^{k/2} (I_0m)^{(l+2j)/2} \Gamma(l+j)\Gamma(j+1) $ with
$ [N,I_0 m] $ is equal to the least common multiple of $ N $ and $I_0 m$. The $ p $-adic $ L $-function is one of the ingredients of Iwasawa main conjecture.
We now describe the other ingredient of the Iwasawa main conjecture namely the characteristic ideal of the dual Selmer group. Assume that $ g $ is also ordinary at $ p $. Let $ V_f $ and $ V_h $ be the two dimensional $ L_{\mathfrak{p}} $-vector space attached to $ \rho_f $ and $ \rho_h $ respectively. Choose a $ G_{\mathbb{Q}} $-invariant $ \mathcal{O} $-lattice $ T_f \subset V_{f} $ and $ T_h \subset V_h $ respectively. Under the assumption $ f $ is $ p $-ordinary, $ V_f $ has a filtration $ 0 \rightarrow V_f^{+} \rightarrow V_f \rightarrow V_f^{-} \rightarrow 0 $ as $ G_p $-module, where $ G_p $ is the decomposition group at $ p $. Set $ T_f^{\pm} = T_f \cap V_f^{\pm} $, $ A_f = V_f/T_f $ and $ A_f^{-} = A_{f}/(V_f^+/T_f^+)$. Let $ \chi_p $ and $ \omega_{p}$ denote the $ p $-adic cyclotomic character and Teichm\"uller character respectively. For every integer $ j $, let $ V_j = V_f \otimes_{\mathbb{Q}_{p}} V_g \otimes_{\mathbb{Q}_{p}} \chi_p^{-j}$, $ T_j = T_f \otimes_{\mathbb{Z}_{p}} T_g \otimes_{\mathbb{Z}_{p}} \chi_p^{-j} $ and $ A_j = V_j/T_j$. Following \cite{gr1}, we define the Greenberg Selmer groups $ S^{\Sigma_{0}}_{\mathrm{Gr}}(A_f/\mathbb{Q}_\mathrm{cyc})$, $ S^{\Sigma_{0}}_{\mathrm{Gr}}(A_j/\mathbb{Q}_\mathrm{cyc})$ attached to $ A_f $ and $ A_j $ respectively. These are discrete $ \mathbb{Z}_{p}[[\Gamma]] $ modules. The Pontryagin dual $ S^{\Sigma_{0}}_{\mathrm{Gr}}(A_f/\mathbb{Q}_\mathrm{cyc})^\vee$ and $ S^{\Sigma_{0}}_{\mathrm{Gr}}(A_j/\mathbb{Q}_\mathrm{cyc})^\vee$ is defined as $ \mathrm{Hom}_{\mathrm{cont}}(S^{\Sigma_{0}}_{\mathrm{Gr}}(A_f/\mathbb{Q}_\mathrm{cyc}), \mathbb{Q}_p/\mathbb{Z}_{p}) $ and $ \mathrm{Hom}_{\mathrm{cont}}(S^{\Sigma_{0}}_{\mathrm{Gr}}(A_j/\mathbb{Q}_\mathrm{cyc}), \mathbb{Q}_p/\mathbb{Z}_{p}) $ respectively. Denote the characteristic ideal of $S^{\Sigma_{0}}_{\mathrm{Gr}}(A_j/\mathbb{Q}_\mathrm{cyc})^\vee$ by $C_{\mathbb{Z}_{p}[[\Gamma]]}\big((S^{\Sigma_{0}}_{\mathrm{Gr}}(A_j)/\mathbb{Q}_{\mathrm{cyc}})^\vee\big)$. Put $ \Sigma = \{ v : v \mid pNI, v \mid \infty \}$. Let $ \xi_{1}, \xi_2 $ be Dirichlet characters such that $ \xi_i \equiv \bar{\xi}_i \mod \pi$ for $ i=1,2 $ as constructed in \S~\ref{section: simplyfying rankin}. Let $ \mu_p $ be the group of $ p^{\mathrm{th}} $-roots of unity.
\end{comment}
For an integer $ m $, let $ \iota_{m} $ be the trivial character modulo $ m $. The following is our main result:
\begin{theorem}\label{congruence main conjecture intro}
Let $ p\geq 5 $. Let $ f \in S_{k}(\Gamma_{0}(N),\eta)$ be a $ p $-ordinary newform with $ p \nmid N $. Let $ h \in S_{k}(\Gamma_{0}(I_0p^\alpha),\psi)$ be a $ p $-ordinary Hecke eigenform with $ 2 \leq l < k$ and $ p \nmid I_0 $. Let $ f_0 $ be the $ p $-stabilisation of $ f $. Assume that the semi-simplification of the residual Galois representation of h, $\bar{\rho}_h^{ss} \cong \bar{\xi}_1 \oplus \bar{\xi}_2$ with $ \bar{\xi}_{2} $ unramified at $ p $. Set $ M_0 $ is the prime to $ p $-part of the conductor of $ \bar{\rho}_h$ and $ \mathcal{M}:=\{\ell \text{ prime}: \ell \mid I_0/M_0 \text{ and } \ell^2\mid M_0\}$. Put $ m := \prod_{\ell \in \mathcal{M}}\ell $ and $ \Sigma_{0} := \{ v \text{ is a prime in } \mathbb{Q}_\mathrm{cyc}: v \mid m \}$. Further assume that
\begin{enumerate}[label=$(\roman*)$]
\item The residual Galois representation $ \bar{\rho}_f $ is irreducible and $ f $ is $ p $-distinguished.
\item $h_{k}(\Gamma_{0}(Nm^2p), \eta\iota_{m}^2; L) $ is a direct summand of $ L $ via the map $ T(n) \mapsto \iota_{m}(n) a(n,f_0) $.
\item $ (N,M_0) =1 $.
\item
$ \psi $ has order coprime to $ p $.
\item If $ (p-1) $ divides any one of the following integers $ l -1, l, \ldots, k-3, k-2 $ and $ H^{0}(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A_{f}^{-}[\pi](\bar{\xi}_2)) \neq 0$, then $ \bar{\rho}_h|I_{\mathrm{cyc},v} \cong \bar{\xi}_1 \oplus \bar{\xi}_2 $ for all $ v \mid p $.
\item The dual Selmer group $S^{\Sigma_0}_{\mathrm{Gr}}( A_f(\xi_i\omega_p^{-j})/\mathbb{Q}_\mathrm{cyc})^\vee$ is a finitely generated $\mathbb{Z}_p$ module for $i=1, 2$.
\end{enumerate}
Then for every $ l-1 \leq j \leq k-2 $, the following congruence of $ p $-adic $ L $-functions holds in $ \mathbb{Z}_{p}[[\Gamma]] $:
\begin{small}
\begin{align}\label{intro congruence of p-adic L-functions}
(L^{\Sigma_{0}}_{p,f \otimes h,j}) \equiv (L^{\Sigma_{0}}_{p,f,\xi_1, j}) (L^{\Sigma_{0}}_{p,f,\xi_2, j}) \mod \pi.
\end{align}
\end{small}
Also the following congruence of characteristic ideals of $ p^\infty $ Selmer groups holds in $ \mathbb{Z}_{p}[[\Gamma]] $:
\begin{small}
\begin{align}\label{intro congruence of char ideals}
C_{\mathbb{Z}_p[[\Gamma]]}\Big(S^{\Sigma_0}_{\mathrm{Gr}}( A_j/\mathbb{Q}_\mathrm{cyc})^\vee\Big) \equiv C_{\mathbb{Z}_p[[\Gamma]]}\Big(S^{\Sigma_0}_{\mathrm{Gr}}( A_f(\xi_1 \omega_{p}^{-j})/\mathbb{Q}_\mathrm{cyc})^\vee\Big) C_{\mathbb{Z}_p[[\Gamma]]}\Big(S^{\Sigma_0}_{\mathrm{Gr}}( A_f(\xi_2 \omega_{p}^{-j})/\mathbb{Q}_\mathrm{cyc})^\vee\Big) \mod \pi.
\end{align}
\end{small}
Furthermore, if the Iwasawa main conjecture holds for $ f|\iota_m \otimes \xi_{1} \omega_p^{-j}$ and $ f|\iota_m \otimes \xi_{2} \omega_p^{-j}$ over $ \mathbb{Q}_{\mathrm{cyc}} $, then for every $ l-1 \leq j \leq k-2 $, we have the following congruence of ideals in the Iwasawa algebra $ \mathbb{Z}_{p}[[\Gamma]] $
\begin{small}
\begin{align}\label{intro IMC}
(L_{p,f\otimes h,j}) \equiv C_{\mathbb{Z}_p[[\Gamma]]}\Big(S_{\mathrm{Gr}}( A_j/\mathbb{Q}_\mathrm{cyc})^\vee\Big) \mod \pi.
\end{align}
\end{small}
In particular, $ L_{p,f\otimes h,j} $ is a unit in $ \mathbb{Z}_{p}[[\Gamma]] $ if and only if $ C_{\mathbb{Z}_p[[\Gamma]]}\Big(S_{\mathrm{Gr}}( A_j/\mathbb{Q}_\mathrm{cyc})^\vee\Big) $ is a unit $ \mathbb{Z}_{p}[[\Gamma]] $.
\end{theorem}
\begin{remark}
\begin{enumerate}[label=$(\roman*)$]
\item The above theorem continues to hold for $ p = 3 $ if $ \Gamma_{0}(N)/\{\pm I\} $ has no non-trivial torsion elements. In fact, the congruence \eqref{intro congruence of char ideals} always holds for $ p = 3 $.
\item Under the hypotheses $(i)-(iii)$ of Theorem~\ref{congruence main conjecture intro}, we prove the congruence \eqref{intro congruence of p-adic L-functions} in Theorem~\ref{analytic final}.
\item On the other hand, we show that hypotheses $(i), (iv), (v)$ and $(vi)$ are needed to arrive at the congruence \eqref{intro congruence of char ideals} in Theorem~\ref{thm:congruence of ideals}.
\item The Iwasawa main conjecture for modular form (as stated in Conjecture~\ref{IWC for modular form}), which is needed to deduce congruence \eqref{intro IMC} from the congruences \eqref{intro congruence of p-adic L-functions} and \eqref{intro congruence of char ideals}, is known for large class of modular forms by the results of \cite{kato} and \cite{SU}. More precisely, we only need the Iwasawa main conjecture for modular form modulo $ \pi $ (i.e. $(L^{\Sigma_{0}}_{p,f,\xi_i, j} ) \equiv C_{\mathbb{Z}_p[[\Gamma]]}\big(S^{\Sigma_0}_{\mathrm{Gr}}( A_f(\xi_i \omega_{p}^{-j})/\mathbb{Q}_\mathrm{cyc})^\vee\big) \mod \pi$ for $ i =1,2 $), to deduce~\eqref{intro IMC} from \eqref{intro congruence of p-adic L-functions} and \eqref{intro congruence of char ideals}.
\item By Lemma~\ref{splitting criteria hecke algebra}, the hypothesis $(ii)$ of Theorem~\ref{congruence main conjecture intro} holds if $ m=1 $. Further by Remark~\ref{rem: CE}, the hypothesis $(ii)$ also holds if $ (m,N) =1 $, $ N $ is cube free and the Tate conjecture \cite[\S 1]{CE} is true.
Note that for a Dirichlet character $ \chi $ of conductor $ m^\delta $, we have $ \rho_{f|\chi} \otimes \rho_{h|\chi^{-1}} \cong \rho_{f} \otimes \rho_{h}$. As explained in the discussion following Remark~\ref{rem: CE}, it is in fact possible to remove the hypothesis $(ii)$ of Theorem~\ref{congruence main conjecture intro}. However, for simplicity, we keep hypothesis $(ii)$ in Theorem~\ref{congruence main conjecture intro}.
\end{enumerate}
\end{remark}
\iffalse
\textcolor{red}{In \cite[Theorem 7.5.4]{llz}, \cite[Corollary D]{klz} it was shown $ (L_{p,f\otimes h,j}) \subseteq C_{\mathbb{Z}_p[[\Gamma]]}\Big(S_{\mathrm{Gr}}( A_j/\mathbb{Q}_\mathrm{cyc})^\vee\Big)$ when $ \rho_f \otimes \rho_h $ is irreducible and some additional hypotheses. In the article \cite{Wan}, X. Wan proves the other containment $ C_{\mathbb{Z}_p[[\Gamma]]}\Big(S_{\mathrm{Gr}}( A_j/\mathbb{Q}_\mathrm{cyc})^\vee\Big) \subseteq (L_{p,f\otimes h,j}) $ under the assumption $ f $ is a CM form and $ \bar{\rho}_h $ is irreducible. The results of \cite{llz}, \cite{klz} and \cite{Wan} don't seem to hold simultaneously, so the Iwasawa main conjecture is not completely established in general. Since $ p $ is an Eisenstein prime for $ h $ i.e. $ \bar{\rho}_h $ is reducible, our assumptions are complimentary to those mentioned in \cite{llz}, \cite{klz} and \cite{Wan}.}
\fi
As an application of Theorem~\ref{congruence main conjecture intro}, we deduce (see Example~\ref{example 1} for more details):
\begin{theorem}\label{thm: IMC example}
Let $ p=11 $, $ \Delta \in S_{12}(\mathrm{SL}_2(\mathbb{Z})) $ be the Ramanujan Delta function, $ h \in S_{2}(\Gamma_{0}(23)) $ be the newform of label LMFDB $23.2.a$ and $ \chi_K $ be the quadratic character of $ \mathbb{Q}(\sqrt{-23})$. Put $ f = \Delta \otimes \chi_K $. Then Iwasawa main conjecture holds for $ f \otimes h $ modulo $ \pi $
that is, the following congruences hold in $ \mathbb{Z}_{p}[[\Gamma]] $:
\begin{small}
\begin{equation*}
(L_{p, f \otimes h, j}) \equiv C_{\mathbb{Z}_{p}[[\Gamma]]}\Big((S_{\mathrm{Gr}}(A_j)/\mathbb{Q}_{\mathrm{cyc}})^\vee\Big) \mod \pi \quad \mathrm{for} ~ 1 \leq j \leq 10.
\end{equation*}
\end{small}
Further, $L_{p, f \otimes h, j}$
and $ C_{\mathbb{Z}_{p}[[\Gamma]]}(S_{\mathrm{Gr}}(A_j)/\mathbb{Q}_{\mathrm{cyc}})^{\vee} $ are units in $\mathbb{Z}_{p}[[\Gamma]]$ for $ j \neq 4,5 $. \qedhere
\end{theorem}
\begin{comment}
The modular form $ f$ (resp. $ h $) in Theorem~\ref{thm: IMC example} is a $ p $-ordinary newform with residual representation at $ p =11 $ is irreducible (resp. reducible). Further, the semi-simplification of $ \bar{\rho}_h$ is $ \omega_{p} \oplus 1 $. For $ 0 \leq j \leq 10 $, we show that the Iwasawa main conjecture holds for $ f \otimes \omega_{p}^{j} $ and the dual of Greenberg Selmer group of $ f \otimes \omega_{p}^{j} $ over $ \mathbb{Q}_{\mathrm{cyc}} $ is finitely generated. Further, for $ j=4,5 $ we deduce that the characteristic polynomial $ C_{\mathbb{Z}_{p}[[\Gamma]]}(S^{\Sigma_{0}}_{\mathrm{Gr}}(A_j)/\mathbb{Q}_{\mathrm{cyc}})^{\vee} = T u(T)$ for some unit $ u(T) \in \mathbb{Z}_p[[T]] $.
\end{comment}
\begin{comment}
We now outline a proof of Theorem~\ref{congruence main conjecture intro}. The congruence in Theorem~\ref{congruence main conjecture intro} obtained by combining the \textquote{analytic} congruence involving $ p $-adic $ L $-functions of $ f \otimes h $ and $ f \otimes \xi_{i} $ for $ i=1,2 $ and \textquote{algebra} congruence involving characteristic ideals of $ f \otimes h $ and $ f \otimes \xi_{i} $ for $ i=1,2 $. These two congruences put together and assuming the Iwasawa main conjecture for $ f \otimes \xi_{i} $ for $ i=1,2 $ completes the proof of Theorem~\ref{congruence main conjecture intro} (see Theorem~\ref{congruence main conjecture }).
\end{comment}
As stated in Theorem~\ref{congruence main conjecture intro} the congruence \eqref{intro IMC} is obtained via the congruences \eqref{intro congruence of p-adic L-functions} and \eqref{intro congruence of char ideals}. We proceed in the analytic side and algebraic side separately to establish the congruences \eqref{intro congruence of p-adic L-functions} and \eqref{intro congruence of char ideals} respectively.
On the analytic side, we start by lifting the characters $ \bar{\xi}_{1}$ and $\bar{\xi}_{2} $ to the Dirichlet characters $ \xi_{1} $ and $ \xi_{2} $. This enables us to define a suitable $ p $-ordinary and primitive Eisenstein series $ g $ such that the residual Galois representation $\bar{\rho}_g \cong \bar{\xi}_{1} \oplus \bar{\xi}_2 $ and $ g|\iota_{m} \equiv h|\iota_{m} \mod \pi$. In fact, the congruence of Fourier coefficients implies the $ p $-adic Rankin-Selberg $ L $-functions of $ \tilde{f} \otimes h|\iota_{m} $ and $ \tilde{f} \otimes g|\iota_{m} $ are congruent. Thus it suffices to show that the congruence \eqref{intro congruence of p-adic L-functions} holds with the $ p $-adic Rankin-Selberg $ L $-function of $ \tilde{f} \otimes h|\iota_{m} $ replaced by the $ p $-adic Rankin-Selberg $ L $-function of $ \tilde{f} \otimes g|\iota_{m} $.
To establish this analogue of congruence \eqref{intro congruence of p-adic L-functions} for the $ p $-adic $ L $-function of $ \tilde{f} \otimes g|\iota_{m} $, we generalise the strategy of \cite{Vatsal} from the setting of elliptic modular forms to our Rankin-Selberg setting. This can be explained in two steps. First, we express the special values of the Rankin-Selberg $ L $-function of $ \tilde{f} \otimes h|\iota_{m} $ as a product of the special values of $ L $-functions of $ \tilde{f}_0 \otimes \xi_1 $ and $ \tilde{f}_0 \otimes \xi_2 $ (see \eqref{eq: rankin as product trivial char} and \eqref{eq: rankin as product nontrivial char}).
From this to arrive at the required congruence of $ p $-adic $ L $-functions involves a delicate choice of periods $ \Omega^{\pm}_{\tilde{f}} $ such that $(i)$ The $p$-adic $ L $-functions $ L^{\Sigma_{0}}_{p,f, \xi_{i},j} $ attached to $ \tilde{f} \times \xi_{i} $ are $ \mathcal{O} $-valued for $ i=1,2 $ and $ 0 \leq j \leq k-2 $. $(ii)$ The $ \mathcal{O} $-valued $ p $-adic $ L $-function $L^{\Sigma_{0}}_{p, f \otimes h, j}$ differs from the product $ L^{\Sigma_{0}}_{p,f, \xi_{1},j} \times L^{\Sigma_{0}}_{p,f, \xi_{2},j} $ by a $ p $-adic unit. In fact, to achieve $(i)$, we choose periods using certain parabolic cohomology groups and then show that with this choice of period, the $ p $-adic $ L $-function becomes $ \mathcal{O} $-valued (see Theorems~\ref{choice of period and petterson innerproduct}, \ref{period and integral measure}). On the other hand, we use results from \cite[Chapter 5]{Hida3} and \cite{Hidaadjoint} on the adjoint $ L $-function of a modular form to show that the condition $(ii)$ is satisfied. Building up on these results, we obtain the congruence of $ p $-adic $ L $-functions in equation \eqref{intro congruence of p-adic L-functions} in Theorems~\ref{analytic final1}, \ref{analytic final}.
On the algebraic side, using the $ p $-ordinarity of $ f,h $, it follows that $S_{\mathrm{Gr}}^{\Sigma_{0}}(B_j/\mathbb{Q}_{\mathrm{cyc}})^\vee $ is a torsion $ \mathbb{Z}_{p}[[\Gamma]] $ module for $ B_j \in \{ A_j, A_f(\xi_{1}\omega_p^{-j}), A_f(\xi_{2}\omega_p^{-j})\}$ (\cite{kato}).
The key ingredient in the congruence \eqref{intro congruence of char ideals} is the following exact sequence, which is proved in Lemma~\ref{lem: dual selmer group sequence}:
\begin{small}
\begin{align}\label{intro exact sequence of dual selmer}
0 \rightarrow S_{\mathrm{Gr}}^{\Sigma_{0}}(A_f(\bar{\xi}_1 \omega_{p}^{-j})[\pi]/\mathbb{Q}_{\mathrm{cyc}})^{\vee} \rightarrow S_{\mathrm{Gr}}^{\Sigma_{0}}(A_j[\pi]/\mathbb{Q}_{\mathrm{cyc}})^{\vee} \rightarrow S_{\mathrm{Gr}}^{\Sigma_{0}}(A_f(\bar{\xi}_2 \omega_{p}^{-j})[\pi]/\mathbb{Q}_{\mathrm{cyc}})^\vee \rightarrow 0.
\end{align} \end{small}
Further with $ B_j $ as above, we show that $S_{\mathrm{Gr}}^{\Sigma_{0}}(B_j/\mathbb{Q}_{\mathrm{cyc}})^\vee $ has no non-zero pseudo-null submodule using a result of \cite{we}. The last fact enables us to show that the base change holds for the characteristic ideals i.e. $C_{\mathbb{Z}_{p}[[\Gamma]]}(S_{\mathrm{Gr}}^{\Sigma_{0}}(B_j/\mathbb{Q}_{\mathrm{cyc}})^{\vee}) \mod \pi = C_{\mathbb{F}_{p}[[\Gamma]]}\big((S_{\mathrm{Gr}}^{\Sigma_{0}}(B_j/\mathbb{Q}_{\mathrm{cyc}})[\pi]\big)^{\vee})$. Note that in this setting, $ S_{\mathrm{Gr}}^{\Sigma_{0}}(B_j[\pi]/\mathbb{Q}_{\mathrm{cyc}}) \cong S_{\mathrm{Gr}}^{\Sigma_{0}}(B_j/\mathbb{Q}_{\mathrm{cyc}})[\pi] $ ({Lemma~\ref{inside-out-lem}}).
Putting all these together, the congruence in \eqref{intro congruence of char ideals} follows (Theorem~\ref{thm:congruence of ideals}).
We now describe the key ideas involved to derive the exact sequence \eqref{intro exact sequence of dual selmer}: Tensoring the exact sequence \eqref{intro rho_h} with $ A_f \otimes \omega_{p}^{-j} $, we obtain
\begin{small}
\begin{align*}
0 \rightarrow A_f(\bar{\xi}_1 \omega_{p}^{-j})[\pi] \rightarrow A_j[\pi] \rightarrow A_f(\bar{\xi}_2 \omega_{p}^{-j})[\pi] \rightarrow 0.
\end{align*}
\end{small}
We determine the Selmer group $ S^{\Sigma_{0}}_{\mathrm{Gr}}(B_j[\pi]/\mathbb{Q}_\mathrm{cyc})$
explicitly under hypotheses $(iii), (iv)$ of Theorem~\ref{congruence main conjecture intro}. With this explicit description, $ S^{\Sigma_{0}}_{\mathrm{Gr}}(B_j[\pi]/\mathbb{Q}_\mathrm{cyc})$ is essentially determined by the corresponding residual representation. Using this description and the
hypotheses $(v), (vi)$ of
Theorem~\ref{congruence main conjecture intro}, we deduce \eqref{intro exact sequence of dual selmer}.
\begin{comment}
Let $\{(\rho_{\ell},V_{\ell}) \}_\ell$ be a family of $ \ell $-adic Galois representation of $ G_{\mathbb{Q}_\ell} $ and assume $ \{ V_{\ell} \} $ are compatible. Further, let $ V_{j} = \{ V_{\ell} \otimes \chi_{\ell}^{-j} \} $ be the $ j^{\mathrm{th}} $-Tate twist of $ \{V_{\ell}\} $. For each $ j $, let $ L_{V_{j}}(s) $ denote the $ L $-function attached to $ \{V_{j}\} $. Let $ (j+1) $ be a critical value of $ L_{V_{0}}(s) $ in the sense of Delinge. Then $ 1 $ is a critical value of $ L_{V_{j}}(s) $. Following \cite{gr1}, one can associate the $p$-adic $ L $-function to $ V_{f}(-j)$ which we denote by $ L_{p,V_{j}}(\cdot) $ and state the main conjecture for $V_{j} $. In this article, we are interested in the families of representation $\{ V_{\ell} \}$ arising from a modular form and the Rankin-Selberg product $ \tilde{f} \otimes h|\iota_{m} $. \end{comment}
In \S\ref{sec: Prelims and setup}, we recall the basics of $ p $-adic modular forms and the $ p $-adic Rankin-Selberg $ L $-function due to Hida \cite{Hidarankin2}. In \S\ref{section: simplyfying rankin}, we define the Eisenstein series $ g $ and show that the $p$-adic $ L $-function of $ \tilde{f} \times g|\iota_{m} $ and $ \tilde{f} \times h|\iota_{m} $ are congruent modulo $ \pi $. We also express the special values of the Rankin-Selberg $ L $-function of $ \tilde{f} \times g|\iota_{m} $ as a product of the special values of the $ L $-functions attached to $ \tilde{f} \otimes \xi_1$ and $ \tilde{f} \otimes \xi_2$. Next in \S\ref{sec: periods and congruence}, we make an appropriate choice of periods $ \Omega_{\tilde{f}}^{\pm} $ and use it to obtain that the ideal generated by the $ p $-adic $ L $-function of $ \tilde{f} \times h|\iota_{m} $ is congruent to the ideal generated by the product of the $ p $-adic $ L $-functions associated to $ \tilde{f} \otimes \xi_1 $ and $ \tilde{f} \otimes \xi_2 $ (Theorem~\ref{analytic final}).
In \S\ref{sec: Selmer groups}, we recall the $ p^\infty $-Selmer groups of the Rankin-Selberg convolution $ f \otimes h $ and the modular form $ f \otimes\xi_{i} $ and further give explicit descriptions of these Selmer groups. In \S\ref{sec: Congruences of char ideals}, we show that the characteristic ideal of the $p^\infty $-Selmer group of $ f \otimes h $ is congruent to the product of the characteristic ideals attached to the $p^\infty $-Selmer groups of $ f \otimes \xi_1 $ and $ f \otimes \xi_2$. In \S\ref{sec: IMC}, we prove our main theorem (Theorem~\ref{congruence main conjecture }) establishing the Iwasawa main conjecture modulo $ \pi $ for $ f \otimes h$. We also discuss a few concrete examples illustrating the result of Theorem~\ref{congruence main conjecture }.
\thanks{{\bf Acknowledgement:} S. Jha acknowledges the support of SERB MTR/2019/000996 grant. A part of this work was done at ICTS-TIFR and we acknowledge the program ICTS/ecl2022/8. We thank Aribam Chandrakant for the help with the computations on $ \mu $-invariants. R. Vangala acknowledges the support of IPDF of IIT Kanpur.}
\section{Preliminaries and Setup}\label{sec: Prelims and setup}
In this section, we begin by recalling $ p $-adic modular forms, measures, Rankin-Selberg $ L $-function and $ p $-adic Rankin-Selberg $ L $-function. Throughout this section $ J $ denotes an arbitrary positive integer.
\subsection{$ p $-adic modular forms}
In this subsection, we briefly recall the definition of the space of $p$-adic modular forms in the sense of Serre.
For more details we refer the reader to \cite{Hidarankin1}, \cite{Hidarankin2}. We fix embeddings $ i_{p}: \bar{\mathbb{Q}} \rightarrow
\bar{\mathbb{Q}}_{p} $ and $ i_{\infty}: \bar{\mathbb{Q}} \hookrightarrow \mathbb{C} $.
Let $ M_{k} (\Gamma_{0}(J), \psi) $
(resp. $ M_{k} (\Gamma_{1}(J)) $) is the space of modular
forms with coefficients in $ \mathbb{C} $, nebentypus $ \psi $ and the congruence subgroup $\Gamma_{0}(J)$ (resp. $ \Gamma_{1}(J) $).
For a subring $ R \subset \bar{\mathbb{Q}} $, let
$M_{k}(\Gamma_{0}(J), \psi ; R)$ (resp.
$M_{k}(\Gamma_{1}(J); R) $) denote the subspace of $ M_{k} (\Gamma_{0}(J), \psi) $
(resp. $ M_{k} (\Gamma_{1}(J)) $) consisting of all modular forms with Fourier coefficients in $ R $.
Also, let $ S_{k} (\Gamma_{0}(J), \psi) $
(resp. $S_{k} (\Gamma_{1}(J)) $) be the space of cusp forms with coefficients in $ \mathbb{C} $, nebentypus $ \psi $ and the congruence subgroup $ \Gamma_{0}(J) $ (resp. $ \Gamma_{1}(J) $). Put
\begin{small}{
\begin{align*}
S_{k}(\Gamma_{0}(J), \psi ; R) :=
M_{k}(\Gamma_{0}(J), \psi ; R)
\cap S_{k} (\Gamma_{0}(J), \psi), \quad
S_{k}(\Gamma_{1}(J); R) := M_{k}(\Gamma_{1}(J); R) \cap
S_{k} (\Gamma_{1}(J)).
\end{align*}}
\end{small}
For every modular form $ f = \sum_{n\geq 0}^{} a(n,f) q^n $ with Fourier coefficients in $\bar{\mathbb{Q}}$, we define a $p$-adic norm $ |f|_{p} $ by $|f|_{p} : = \sup_{n} |a(n,f)|_{p}$.
For a finite extension $ K_0$ of $ \mathbb{Q} $, let
$K$ be the closure of $ K_0 $ in $ \mathbb{C}_{p} $, where $ \mathbb{C}_{p} $ is the completion of $ \bar{\mathbb{Q}}_{p} $ and $ \mathcal{O}_{K} $ be the ring of integers of $ K $. Let
$ M_{k}(\Gamma_{0}(J), \psi ; K) $
(resp. $M_{k}(\Gamma_{1}(J) ; K)$) denote the completion of
$ M_{k}(\Gamma_{0}(J), \psi ; K_0) $
(resp. $M_{k}(\Gamma_{1}(J) ; K_0)$)
with respect to the norm $ |\cdot|_{p} $ in $ K[[q]] $.
Then it is known that (see \cite[Pg 170]{Hidarankin1})
\begin{small}
\begin{align*}
M_{k}(\Gamma_{0}(J), \psi ; K) =
M_{k}(\Gamma_{0}(J), \psi ; K_0) \otimes_{K_{0}}
K, \quad
M_{k}(\Gamma_{1}(J) ; K) = M_{k}(\Gamma_{1}(J) ; K_0)
\otimes_{K_{0}} K.
\end{align*}
\end{small}
Put $M_{k}(\Gamma_{0}(J), \psi ; \mathcal{O}_K) := \mathcal{O}_{K}[[q]] \cap M_{k}(\Gamma_{0}(J), \psi ; K)$ and $M_{k}(\Gamma_{1}(J) ; \mathcal{O}_K) := \mathcal{O}_{K}[[q]] \cap M_{k}(\Gamma_{1}(J) ; K)$. For $ A \in \{ K, \mathcal{O}_{K} \}$, set
\begin{small}
\begin{align*}
M_{k}(J;A) = \cup_{n=0}^{\infty}
M_{k}(\Gamma_{1}(Jp^n);A) \quad \mathrm{and} \quad
M_{k}(J, \psi ;A) = \cup_{n=0}^{\infty}
M_{k}(\Gamma_{0}(Jp^n), \psi;A).
\end{align*}
\end{small}
Let $ \overline{M}_{k}(J, \psi ;A) $ (resp. $\overline{M}_{k}(J;A)$)
denote the completion of $ M_{k}(J, \psi ;A) $
(resp. $M_{k}(J;A)$) under the norm $ | \cdot |_{p} $
in $ K[[q]] $. Any element of the space $\overline{M}_{k}(J;A) $
will be called a $p$-adic modular form. Similarly, one can define the
spaces $ \overline{S}_{k}(J, \psi ;A) $ and $\overline{S}_{k}(J;A)$).
Next, we consider the Hecke
algebras of the space of $p$-adic modular forms. Denote by $ H_{k}(\Gamma_{0}(J), \psi ; R) $
(resp. $H_{k}(\Gamma_{1}(J) ; R)$), the Hecke algebra corresponding to the space of modular forms $ M_{k} (\Gamma_{0}(J), \psi;R) $
(resp. $ M_{k} (\Gamma_{1}(J);R) $). Let $ h_{k}(\Gamma_{0}(J), \psi ; R) $
(resp. $h_{k}(\Gamma_{1}(J) ; R)$) denote the Hecke algebra corresponding to the space of cusp forms
$ S_{k} (\Gamma_{0}(J), \psi;R) $
(resp. $ S_{k} (\Gamma_{1}(J);R) $). Set
\begin{small}
\begin{alignat*}{2}
H_{k}(J, \psi; \mathcal{O}_{K} ) &= \varprojlim_{n}
H_{k}(\Gamma_{0}(Jp^n), \psi; \mathcal{O}_{K} )
\quad \text{ and } \quad
H_{k}(J; \mathcal{O}_{K} ) &&=
\varprojlim_{n} H_{k}(\Gamma_{1}(Jp^n);\mathcal{O}_{K} ), \\
h_{k}(J, \psi; \mathcal{O}_{K} ) &= \varprojlim_{n}
h_{k}(\Gamma_{0}(Jp^n), \psi; \mathcal{O}_{K} )
\quad \text{ and } \quad
h_{k}(J; \mathcal{O}_{K} ) &&=
\varprojlim_{n} h_{k}(\Gamma_{1}(Jp^n);\mathcal{O}_{K} ).
\end{alignat*}
\end{small}
\begin{definition}\label{idempotent defn.}
\textbf{(Idempotent)}(\cite[\S 2]{Hidarankin2})
Let \begin{small}$ U(p) $\end{small} be the $ p^{\text{th}} $-Hecke operator in \begin{small}$H_{k}(\Gamma_{0}(Jp^n), \psi; \mathcal{O}_{K} )$\end{small} and \begin{small}$H_{k}(\Gamma_{1}(Jp^n);\mathcal{O}_{K} )$\end{small}. For every $ n $, let $ e_n $ be the idempotent in
\begin{small}$H_{k}(\Gamma_{0}(Jp^n), \psi; \mathcal{O}_{K} )$\end{small} and \begin{small}$H_{k}(\Gamma_{1}(Jp^n);\mathcal{O}_{K} )$\end{small} defined by \begin{small}$ \lim\limits_{m \rightarrow \infty} U(p)^{m!} $\end{small}. The idempotent operator $e$ in \begin{small}$ H_{k}(J, \psi; \mathcal{O}_{K} ) $\end{small} and \begin{small}$ H_{k}(J; \mathcal{O}_{K} ) $\end{small} is defined as $ \varprojlim_{n} e_{n} $.
\end{definition}
\subsection{Measures and $ p $-adic Rankin-Selberg Convolution}
In this subsection, we briefly recall the construction of $ p $-adic Rankin-Selberg $L$-function due to Hida \cite{Hidarankin2}. For a topological ring $ A $, let $ C(\mathbb{Z}_{p}^{\times}; A ) $ and
$ LC(\mathbb{Z}_{p}^{\times}; A ) $ denote
the space of continuous (resp. locally constant)
functions on $ \mathbb{Z}_{p}^{\times} $ with values in $ A $. Let $ g = \sum_{n=0}^{\infty}
a(n,g) q^n \in M_{l}(\Gamma_{0}(J),\psi;\mathcal{O}_K)$. Then we consider the arithmetic measure (see \cite[Page 36, Example b]{Hidarankin2}) $ \mu_{g} $ of weight $l$, defined by
\begin{small}
\begin{align}\label{Def measure mu star}
\mu_{g}(\phi) = \sum_{n =1}^{\infty} \phi(n)
a(g, n) q^n, \quad \forall ~ \phi \in
C(\mathbb{Z}_{p}^{\times}; \mathcal{O}_K),
\end{align}
\end{small}
where $ \phi \equiv 0 $ on $ \mathbb{Z}_{p} \setminus \mathbb{Z}_{p}^\times$. Let $ J_{0} $ be the prime to $ p $-part of $ J $.
Let $L$ be a positive integer such that $ J_{0} \mid L $. Then we
have a modified arithmetic measure defined as
$ \mu_{g}^{L}(\phi) := \mu_{g}\vert [L/J_{0}] $, where
$ [L/J_{0}] : \bar{S}_{l}(J_{0};\mathcal{O}_{K}) \rightarrow
\bar{S}_{l}(L;\mathcal{O}_{K}) $
is the linear map defined by
$
(f \vert [L/J_{0}])(z) = f(zL/J_{0}),
~ \forall ~ f \in \bar{S}_{l}(N_{0};\mathcal{O}_{K}).
$
Put $ Z_{L} = \mathbb{Z}_{p}^{\times} \times
(\mathbb{Z}/L\mathbb{Z})^{\times} $. For $ z \in Z_{L} $
we denote its component in $ \mathbb{Z}_{p}^{\times} $ by
$ z_{p} $. We consider the action (depending on the weight and the nebentypus of $ g $) of the group $ Z_L $ on $ C(\mathbb{Z}_{p}^{\times}; \mathcal{O}_{K} ) $ by the formula
$ (z \ast \phi)(x) := \psi(z) z_{p}^{l}\phi(z_{p}^2 x)$ for
$ z \in Z_{L} $ and $ \phi \in C(\mathbb{Z}_{p}^{\times}; \mathcal{O}_{K} ) $.
We also consider the arithmetic measure of weight one defined by
\begin{small}
\begin{align*}
2 E(\phi ) = \sum_{\substack{n=1\\ (n,p)=1}}^{\infty}
\sum_{\substack{d \mid n \\ (d,L)=1}} \mathrm{sgn}(d) \phi(d) q^{n}
\in \mathbb{Z}_{p} [[q]].
\end{align*}
\end{small}
For a given integer $ k>l $ and a finite order character
$ \eta: Z_{L} \rightarrow \mathbb{Z}_{p}^{\times} $,
we consider the arithmetic measure
$
(\mu_{g}^{L} \star E)_{\eta,k} : C(\mathbb{Z}_{p}^{\times};
\mathcal{O}_K ) \rightarrow \bar{S}_{k}(L;\mathcal{O}_{K})
$
of weight $k$ and character $ \eta $ defined by convolution of
$ \mu_{g}^{L}$ and $ E $ as follows:
\begin{small}
\begin{align*}
(\mu_{g}^{L} \star E)_{\eta,k} (\phi) :=
\int_{\mathbb{Z}_{p}^{\times} } \int_{Z_{L}}
\eta(z) z_{p}^{k-1} (z^{-1} \ast \phi)(x) dE(z)
d\mu_{g}^{L}(x).
\end{align*}
\end{small}
By \cite[(9.3)]{Hidarankin2} (see also \cite[Section 2]{Bouganis}), for a finite order character $ \phi \in C(Z_{L}; \mathcal{O}_{K}) $,
we have
\begin{small}
\begin{equation}\label{convolution mu and Eisenstien measure}
\begin{aligned}
(\mu_{g}^{L} \star E)_{\eta,k} (x_p^{j}\phi) & =
\int_{\mathbb{Z}_{p}^{\times} } \int_{Z_{L}}
\eta(z) z_{p}^{k-1} (z^{-1} \ast x_p^{j}\phi)(x) dE(z) d\mu_{g}^{L}(x) \\
& = \int_{\mathbb{Z}_{p}^{\times} } \int_{Z_{L}}
\eta(z) z_{p}^{k-1} \psi(z)^{-1} z_{p}^{-l} z_{p}^{-2j} \phi(z_{p}^{-2})
\phi( x) dE(z) d\mu_{g}^{L}(x) \\
& = \mu_{g}^{L}(x_{p}^{j}\phi) \cdot E(\eta \cdot \psi^{-1} \cdot (\phi_{p}^{-2}) \mathfrak{Z}_{p}^{k-l-2j-1}),
\end{aligned}
\end{equation}
\end{small}
where $ \mathfrak{Z}_{p}(z)=z_{p} $ and $ \phi_{p}(z)
= \phi(\mathfrak{Z}_{p}(z)) $.
Let $ f = \sum_{n=1}^{\infty} a(n,f) q^n$ be a normalized Hecke eigenform of weight
$ k > l$, level $ N_f $ and nebentypus $ \eta $. We assume $ f $ is $p$-ordinary i.e. $ | i_p(a(n,f)) |_{p} =1 $. We define the $ p $-stabilization of $ f $ by
\begin{small}
\begin{align}\label{p-stabilization of f}
f_{0}(z) =
\begin{cases}
f(z) &\text{ if } p \mid N_{f},\\
f(z) - \beta_{f} f(pz) & \text{ otherwise},
\end{cases}
\end{align}
\end{small}
where $ \beta_{f} $ is the unique root of
$ X^2 - a(p,f) X + \eta(p) p^{k-1} = 0 $ with $ |\beta_{f} |_{p} < 1 $.
It is well-known that $ N_{f_{0}} = N_{f} $ if $ p \mid N_{f} $ and
$ N_{f_{0}} = pN_{f} $ if $ p \nmid N_{f} $ and nebentypus of $ f_{0} $
is $ \eta $. Note that $u_f:=a(p,f_0)$ is the unique $p$-adic unit root of the Hecke polynomial of $f$ at $p$. For simplicity, by a slight abuse of notation,
we will denote $ N_f $ by $N$. We assume that the Fourier coefficients of $f$ and the values of $ \eta $ lie in $ \mathcal{O}_{K} $. Hence we obtain a surjective homomorphism
$\phi_{f}: h_{k}(\Gamma_{0}(Np), \eta; \mathcal{O}_{K}) \rightarrow \mathcal{O}_{K}$ induced by $ T(n) \rightarrow a(n,f_{0}) $.
Assume that the map $ \phi_{f} $ induces the decomposition
\begin{align}\label{splitting of hecke algebra}
h_{k}(\Gamma_{0}(Np), \eta ; K) = K \oplus A.
\end{align}
We will need the decomposition in \eqref{splitting of hecke algebra} for a twist of certain specific modular form and will be made explicit in \S~\ref{section: simplyfying rankin}. Let $ 1_{f_0} $ be the idempotent attached to the first summand.
We fix a constant $ c(f_0) \in \mathcal{O}_{K}$ such that
$ c(f_0) 1_{f_0} \in
h_{k}(\Gamma_{0}(Np), \eta ; \mathcal{O}_{K})$. The idempotent $ 1_{f_0} $ induces a map
\begin{small}
\[
l_{f_{0}} : e \bar{S}_{k}(N, \eta ; K)
\rightarrow K
\]
\end{small}
defined by $l_{f_0}(ex) = a(1, x \vert e \vert 1_{f_0}) $ where $e \in H_k(\Gamma_{0}(N), \eta ;\mathcal{O}_K)$ is the idempotent operator defined in
Definition~\ref{idempotent defn.} and $ x \in
\bar{S}_{k}(N, \eta ; K) $. It follows from
\cite[Proposition 4.1]{Hidarankin1} that $ e \bar{S}_{k}(N, \eta ; K) \subset S_{k}(\Gamma_{0}(Np), \eta ; K) $ and the map $ l_{f_0} $
is well-defined. Let $L$ be the least common multiple of
$ J_{0} $ and $ N $ where $ J_{0} $ denotes prime to $ p $ part of $ J $. Let $ \omega_p: (\mathbb{Z}/{p}\mathbb{Z})^{\times} \rightarrow \mathbb{Z}_{p}^{\times}$ denote the Teichm\"uller character and $ \chi $ be a Dirichlet character of $ (\mathbb{Z}/Np\mathbb{Z})^{\times} $. For an integer $ r \geq 1 $ and a finite order character $ \epsilon : (1+p\mathbb{Z}_{p}) \rightarrow
\mathcal{O}_{K}^{\times} $ of conductor $p^r$, let
$Tr_{L/N}: M_{k}(\Gamma_{0}(Lp^r), \epsilon \chi ;
\mathcal{O}_K) \rightarrow M_{k}(\Gamma_{0}(Np^r), \epsilon \chi ; \mathcal{O}_K)
$
denote the trace operator considered in \cite[VI.(1.7)]{Hidarankin2}. For $ g \in M_{l}(\Gamma_{0}(J), \psi)$, put $ g^{\rho}(z) = \sum_{n=0}^{\infty} \overline{a(n,g)} q^n \in M_{l}(\Gamma_{0}(J), \psi^{-1})$. Take $ \epsilon \chi = \eta $ and define the measure $ \mu_{f \times g} $ as follows:
\begin{align}\label{Def of convolution}
\mu_{f \times g}(\phi) :=
c(f_0) \circ l_{f_0} \circ Tr_{L/N} \circ
e((\mu_{g^{\rho}}^{L} \star E)_{\eta,k}(\phi^{-1})), ~~ \forall \text{ finite order character }\phi \in C(\mathbb{Z}_{p}^{\times},\mathcal{O}_{K}),
\end{align}
where $\circ $ denotes the composition of maps.
The Rankin-Selberg $ L $-function of $ f \otimes g $ is defined by
\begin{small}
\begin{align*}
D_{JN}(s,f,g) := L_{JN}(2s+2-k-l, \psi \eta)
L(s,f,g) := L_{JN}(2s+2-k-l, \psi \eta)
\sum_{n=1}^{\infty} a(n,f) a(n,g) n^{-s},
\end{align*}
\end{small}
where $L_{JN}(2s+2-k-l, \psi \eta)$ denotes the
Dirichlet $L$-function of $ \psi \eta $ with the Euler factors at the primes dividing $JN$ omitted from its Euler product (see \cite[\S 1]{Hidarankin1}). For $ f, h \in S_k(N,\chi) $, define the Petersson inner product of
$f$ and $h$ by \begin{small}$ \langle f,h \rangle_N := \int_{\mathbb{H}/\Gamma_{0}(N)} f(z)\overline{h(z)} y^{k-2} ~ dz$,\end{small}
where $ y = \text{Im}(z) $. With the notation as above we recall the following theorem:
\begin{theorem}\label{padic rankin}$($\cite[Theorem 5.1 and Section 8]{Hidarankin1}, \cite[Theorem 2.9]{Bouganis}$)$ Let $ f \in S_{k}(\Gamma_{0}(N), \eta) $ be a normalised $ p $-ordinary eigenform with $ p \nmid N $ and $ g \in M_{l}(\Gamma_{0}(Jp^{\alpha}),\psi) $ with $ k>l $ and $ (J,p) =1 $.
Assume that \eqref{splitting of hecke algebra} holds.
For every finite order character $ \phi \in C(\mathbb{Z}_{p}^{\times} ,
\bar{\mathbb{Q}}_p) $ and $ 0 \leq j \leq k-l-1 $, we have
\begin{small}
\begin{align*}
\mu_{f \times g}(x_p^{j} \phi ) = \int_{\mathbb{Z}_{p}^{\times} } x_p^{j} \phi ~ d\mu_{f \times g}
= c(f_0) t p^{\beta l/2} p^{\beta j} p^{(2-k)/2}
a(p,f_0)^{1-\beta} \frac{D_{JNp}(l+j,f_{0},
\mu_{g^{\rho}}(\phi^{-1})|_{l}\tau_{Jp^{\beta}})}{(2i)^{k+l+2j} \pi^{l+2j+1}
{ \langle f_{0}^{\rho}|_{k}\tau_{Np} , f_{0} \rangle_{Np}}},
\end{align*}
\end{small}
where $\beta $ is the smallest positive integer such that $ \mu_{g^{\rho}}(\phi^{-1})
\in M_{l}(\Gamma_{1}(Jp^{\beta}))$, $ \tau_{Jp^{\beta}} =
\begin{psmallmatrix} 0 & 1 \\ Jp^{\beta} & 0 \end{psmallmatrix} $
and $ t = [N,J] N^{k/2} J^{(l+2j)/2} \Gamma(l+j)\Gamma(j+1) $ with
$ [N,J] $ is equal to the least common multiple of $ N $ and $J$.
\end{theorem}
\section{Towards the congruence of the p-adic L-functions}\label{section: simplyfying rankin}
Fix $f(z) = \sum_{n=1}^{\infty} a(n,f) q^n \in S_{k}(\Gamma_0(N), \eta)$ a primitive form. We shall assume through out this
article that $ p \nmid N $ and $ f $ is $p$-ordinary.
Let $ f_0 $ be the $ p $-stabilization of $ f $ as in \eqref{p-stabilization of f}.
For an integer $ J $, let $ J_0 $ denote the prime to $ p $-part of $ J$. For a normalised Hecke eigenform $ \mathfrak{g} $ with nebentypus $\chi$, let
$K_{\mathfrak{g}}$ be the number field generated by the Fourier coefficients of $\mathfrak{g}$ and the values of $\chi$.
We also fix $ h(z) = \sum_{n=1}^{\infty} a(n,h) q^n
\in M_{l}(\Gamma_{0}(I), \psi)$,
a normalized newform of weight
$ 2 \leq l < k$. Assume that $ h $ is $ p $-ordinary. Put $ I = I_{0}p^{\alpha} $
where $ I_0 $ is co-prime to $ p $. Let $ K $ be a number field containing $ K_f$ and $ K_h $. Let $K_{\mathfrak p}$ denote the completion of $ K $ at a prime $\mathfrak{p}$ lying above $p$ induced by the embedding $i_p:\bar{\mathbb{Q}} \rightarrow \bar{\mathbb{Q}}_p$ and $ \mathcal{O}_{K_{\mathfrak{p}}} $ be the ring of integers of $K_{\mathfrak p}$ and $ \pi $ be a uniformizer of $ \mathcal{O}_{K} $. Let $ (\rho_{h},V_{h}) $ be the $ p $-adic Galois
representation attached to the modular form $h$ (see Theorem~\ref{rhof}). Choose a
rank two $ \mathcal{O}_{K} $ submodule
$T_h$ of $V_h$ which is invariant under the action of
the absolute Galois group $ G_{\mathbb{Q}} := \operatorname{Gal} (\bar{\mathbb{Q}}|\mathbb{Q}) $. Let $\bar{\rho}_{h}: G_{\mathbb{Q}} \longrightarrow \mathrm{GL_2}(T_h/\pi)$ be the reduction of $ \rho $ modulo $ \pi $. Assume that the semi-simplification of $ T_{h}/\pi $ is isomorphic to $ \bar{\xi}_{1} \oplus \bar{\xi}_{2}$, i.e.
\begin{align}\label{eq: splitting rho_h}
0 \rightarrow \bar{\xi}_{1} \rightarrow T_{h}/ \pi
\rightarrow \bar{\xi}_{2} \rightarrow 0.
\end{align}
In this section, we use the residual characters $ \bar{\xi}_1 $ and $ \bar{\xi}_{2} $ to construct an Eisenstein series $ g $ congruent to $ h $. We then express the special values of the Rankin-Selberg $ L $-function $ f_0 \otimes g $ as a product of the special values of $ L $-function associated to $ f_0 \otimes \xi_{1} $ and $ f_0 \otimes \xi_{2} $. In the next section, using these results we shall show that the $ p$-adic Rankin-Selberg $ L $-function associated to $ f \otimes g $ is congruent to the product of $ p $-adic $ L $-functions of $ f \otimes \xi_1 $ and $ f \otimes \xi_2 $.
We begin by constructing the Dirichlet characters $ \xi_i $ whose reduction is $ \bar{\xi}_{i} $ for $ i =1,2 $.
\begin{lemma}\label{lemma lifting characters}
Let $ \bar{\xi}: (\mathbb{Z}/v\mathbb{Z})^{\times} \rightarrow \mathbb{F}_{p^r}^{\times} $ be a character.
Then there exists a Dirichlet character $ \xi: (\mathbb{Z}/v \mathbb{Z})^{\times} \rightarrow \mu_{p^a -1}$ such that the reduction of $ \xi $ equals $ \bar{\xi} $, where $\mu_{p^a -1} : = \{ z \in \mathbb{C} : z^{p^a -1} = 1\}$. Further, if the conductor of $ \bar{\xi} $ equals $ v_0 p^a $ with $ p \nmid v_0 $, then the conductor of $ \xi $ is $ v_0 p^{a'} $ with $ a' = \min\{a,1\} $.
\end{lemma}
\begin{proof}
Without any loss of generality, assume that $ \bar{\xi} $ is primitive. Since $ \bar{\mathbb{F}}_{p^r} $ doesn't contain non-trivial $ p^{\mathrm{th}} $ root of unity,
we get that the image of $ p $-Sylow subgroup of
$(\mathbb{Z}/v\mathbb{Z})^{\times}$ is trivial under $ \bar{\xi} $. As $ \bar{\xi} $ is primitive, we obtain $ p^2 \nmid v $ and $ v = v_0 p^a $ with $ a \in \{ 0,1 \} $ and $ p \nmid v_0 $. Choose a finite extension $ L/\mathbb{Q}_p $ such that $ (\mathcal{O}_L/\pi)^{\times} \cong \mathbb{F}_{p^a}^{\times} $. Composing $ \bar{\xi} $ with the Teichm\"uller character $ \omega: (\mathcal{O}_L/\pi)^{\times} \rightarrow \mu_{p^a -1}$ we obtain a lift $ \xi: (\mathbb{Z}/v \mathbb{Z})^{\times} \rightarrow \mu_{p^a -1} \hookrightarrow \mathbb{C}^{\times}$ of $ \bar{\xi} $. If the conductor of $ \xi $ is strictly less than $ v $, then it would imply that the conductor of $ \bar{\xi} $ is strictly less than $ v $. Thus the conductor of $ \xi $ is $ v_0 p^a $. This finishes the proof.
\end{proof}
Since $ h $ is $ p $-ordinary, either $ \bar{\xi}_{1} $ or $ \bar{\xi}_{2} $ is unramified at $ p $ (see Theorem~\ref{rhof}$(ii)$). Without any loss of generality, assume $ \bar{\xi}_{2} $ is unramified and $ \bar{\xi}_{2}(\mathrm{Frob}_p) \neq 0$, where $ \mathrm{Frob}_p $ is the arithmetic Frobenius at $ p $. From Lemma~\ref{lemma lifting characters}, it follows that there exists a Dirichlet character $ \xi_2 $ whose reduction equals $ \bar{\xi}_2 $ and the conductor of $ \xi_{2} $ is $ M_2 $ with $ p \nmid M_2$. Similarly, we can lift $ \bar{\xi}_{1} \bar{\omega}_p^{1-l}$ to a Dirichlet character $ \xi_{1} \omega_p^{1-l} $ such that
the conductor of $ \xi_{1} \omega_p^{1-l} $ is of the form $ M_1 p^{s} $ with $ (M_1, p) = 1$ and $ s \in \{ 0,1\} $.
Since the
conductor of $ (T_h/\pi) $ divides the
conductor of $ T_{h} $, we obtain $ M_1 M_2 \mid I_{0} $. Set $ M := M_1 M_2 p^{s}$. Then $ M_{0} = M_{1} M_{2} $ and $ M_{0} \mid I_{0} $.
Put
$\mathcal{M} = \{ r \text{ is prime} : r \mid (I_0/ M_0) \text{ or } r^2 \mid M_0 \}$ and set
\begin{align}\label{definition m}
m := \prod_{r \in \mathcal{M}} r.
\end{align}
For any Dirichlet character $ \chi $, cond$(\chi)$ denote the conductor of $ \chi $, $ \chi_0 $ denote the primitive character associated to $ \chi $ and $ G(\chi) := \sum_{a =1}^ {\mathrm{cond}(\chi) } \chi_0(a) e^{2 \pi i a/\mathrm{cond}(\chi) }$ denote the Gauss sum of $ \chi $. Further, for a prime $ r $, let $\mathrm{cond}_{r}(\chi) $ denote the $ r $-part of $ \mathrm{cond}(\chi) $.
Let $ \chi_{r} : G_{\mathbb{Q}} \rightarrow \mathbb{Z}_{r}^{\times}$ denote the $ r $-adic cyclotomic character.
\begin{lemma}\label{Eisenstein series Hida}
Let $\theta $ and $ \chi $ be primitive Dirichlet characters modulo $ u $ and $ v $ respectively with $ \theta\chi(-1) = (-1)^{k'} $. Then
\begin{small}{
\begin{align*}
E_{k'}(\theta,\chi)(z) = \delta_1(u) L(0,\chi) +
\delta(v) L(1-k', \theta) +\delta_{2}(u,v) \frac{i}{2 \pi (z-\bar{z})}
+ \sum_{n=1}^{\infty} \sum_{0 < d \mid n} \theta(d)
\xi(n/d) d^{k'-1} q^{n} \in M_{k'}(uv,\theta\chi),
\end{align*}
} \end{small}
\begin{scriptsize}
\begin{align*}
\mathrm{where} ~~ \delta_1(u) = \begin{cases}
2^{-1} & \text{ if } k'= u=1,\\
0 & \text{ otherwise},
\end{cases} \qquad
\delta_2(u,v) = \begin{cases}
2^{-1} & \text{ if } k'=2 \text{ and } u=v=1,\\
0 & \text{ otherwise},
\end{cases} \qquad
\delta(v) &= \begin{cases}
2^{-1} & \text{ if } v=1,\\
0 & \text{ otherwise}.
\end{cases}
\end{align*}
\end{scriptsize}
Further, we have
\begin{enumerate}[label=$(\roman*)$]
\item $(E_{k'}(\theta,\chi)\vert \tau_{uv})(z) =
(uv^{-1})^{k'/2} \chi(-1) G(\chi)/G(\theta^{-1})
E_{k'}(\chi^{-1}, \theta^{-1})(z)$, where $ \tau_{uv}= \begin{psmallmatrix} 0 & -1 \\ uv & 0
\end{psmallmatrix} $.
\item $L(s,E_{k'}(\theta,\chi)) = L(s-k'+1,\theta)L(s,\chi)$.
\item The representation
$ \rho_{E_{k'}(\theta,\chi)}: G_{\mathbb{Q}} \rightarrow \mathrm{GL}_{2}(\bar{\mathbb{Q}}_{p})
$
is isomorphic to $ \theta \chi_{p}^{l-1} \oplus \chi $. Furthermore, the representation $\rho_{E_{k'}(\theta,\chi)}$ is unramified at all primes $r \nmid puv$.
\end{enumerate}
\end{lemma}
\begin{proof}
The main assertion and part $(i)$ are immediate from \cite[Lemma 5.3]{Hidarankin2}. Part $(ii)$ can be proved following the argument in \cite[Theorem 4.7.1]{Miyake}.
The part $(iii)$ follows from \cite[Theorem 9.6.6]{DS05}.
\end{proof}
Note that
$ \bar{\xi}_{1} \bar{\xi}_{2} \bar{\omega}_p^{1-l}(-1)
= \bar{\psi}(-1) = (-1)^{l} $. As $p$ is odd,
$ \xi_{1} \xi_{2} \omega_p^{1-l} (-1) = (-1)^{l} $. Define $ g (z) := E_{l}(\xi_{1} \omega_p^{1-l}, \xi_2)(z) $. From Lemma~\ref{Eisenstein series Hida},
$ g (z) = \sum_{n \geq 0} a(n,g) q^n \in
M_{l}(\Gamma_{0}(M), \xi_{1} \xi_{2}
\omega_p^{1-l}) $, with
$
a(n,g) = \sum_{0 < d \mid n}^{} \xi_1 \omega_p^{1-l}(d) \xi_{2}(n/d)
d^{l-1} ~ \forall ~n \geq 1
$. Further, we have $ \bar{\rho}_g \simeq \bar{\xi}_1 \oplus \bar{\xi}_{2} \simeq (T_h/\pi)^{ss}$. As $ \xi_2(p) \neq 0$, it follows that $ g $ is $ p $-ordinary and $ p $-minimal (see \cite[Line 7, Page 39]{Hidarankin2}). For an integer $ J $, let $ \iota_{J} $ be the trivial character modulo $ J $. For a Dirichlet character $ \chi $ and a modular form $ F = \sum_{n \geq 0} a(n,F) q^n$, let $ (F\vert \chi) (z) := \sum_{n \geq 0} \chi(n) a(n,F) q^n$ denote the twist of $ F $ by $ \chi$.
We now show that $ g|\iota_{mp}$ and $ h|\iota_{mp}$ are congruent to each other modulo $ \pi $.
\begin{lemma}\label{lem: congruence of g,h}
With the notation as above, we have $ h |\iota_{pm} \equiv g |\iota_{pm} \mod \pi$. \end{lemma} \begin{proof}
Let $ r $ be a prime. Since $( T_{h}/\pi)^{ss} $ and $ (T_{g}/\pi)^{ss} $ are the same, we get $ a(r,h|\iota_{pm}) = a(r,h) \equiv a(r,g) = a(r,g|\iota_{pm}) \mod \pi$, $ \forall$ $ r \nmid pmM_{0}$. If $ r =p $ or $ r \mid m $, then $ a(r,h|\iota_{pm}) = 0 = a(r,g|\iota_{pm}) $. Next, consider $ r \mid M_0 $ and $ r \nmid pm $. This forces that $ r \nmid I_0/M_0 $, $ r \| M_{0} $ and $ r \|I_{0} $. Since $ \psi $ and $ \xi_{1} \xi_{2} \omega_p^{1-l} $ have the same reduction modulo $ p $, we also get that $ r $ divides the conductor of $ \psi $. As $ \text{cond}(\psi) \mid I $ and $ r \| I $, we obtain $ r \| \mathrm{cond}(\psi) $. Thus by \cite[Theorem 3.26]{Hida3}, $ a(r, h) = \chi(\text{Frob}_{r}) $, where $ \chi $ is the unique
unramified character appearing in the restriction of $ \rho_{h} $ to the decomposition subgroup at $r$. Since $ r \| M_0 $ and $ M_0 = M_1 M_2$, we get
either $ r \| M_1, r \nmid M_2 $ or $ r \| M_2, r \nmid M_1$.
Suppose we are in the situation where $r \|M_{1}$ and $ r \nmid M_2 $. Then $ \xi_2 $ is
unramified at $ r $ and $ \xi_1 $ is ramified at $ r $. Also $ a(r,g) = \xi_{2}(r) = \xi_{2}(\text{Frob}_{r})$. Since
$ (T_{h}/\pi)^{\mathrm{ss}} = \bar{\xi}_{1} \oplus \bar{\xi}_{2} $ and $ \bar{\xi}_{1} $ is ramified at $ r $, we obtain $ \bar{\chi} = \bar{\xi}_{2} $. Thus $ a(r,h) = \chi(\text{Frob}_{r}) \equiv
\bar{\xi}_{2}(\text{Frob}_{r}) \equiv a(r,g) \mod \pi$. If $ r \| M_{2} $ and $ r \nmid M_1 $, then $ a(r,g) = \xi_{1}(r) \omega_p^{1-l}(r) r^{l-1} \equiv \xi_{1}(r) \mod p$. Again by a similar argument,
we deduce $ a(r,h) = \chi(\text{Frob}_{r}) \equiv
\bar{\xi}_{1}(\text{Frob}_{r}) \equiv a(r,h) \mod \pi$. Thus in either case, we have
$
a(r,h|\iota_{pm}) = a(r,h) \equiv a(r,g) = a(r,g|\iota_{pm}) \mod \pi
$.
This proves the lemma. \end{proof} We note the following observation made while proving in the lemma above: \begin{lemma}\label{lem: exactly divides}
If $ r \mid I_0 $ and $ r \nmid pm $, then $ r \| I_0 $ and $ r \| \mathrm{cond}(\psi) $. \end{lemma}
Set $\tilde{f} := f|\iota_{m} \in M_k(N_{\tilde{f}},\eta \iota_{m}^2)$ and
$\tilde{f}_{0} := f_0|\iota_{m} \in M_k(N_{\tilde{f}}p,\eta \iota_{m}^2) $ and $ \xi = \xi_{1}\xi_{2} \omega_{p}^{1-l} $. Note that $ \tilde{f}_0 $ is an eigenform and induces the map $\phi_{\tilde{f}} : h_{k}(\Gamma_{0}(N_{\tilde{f}}p), \eta\iota_{m} ; \mathcal{O}_K) \rightarrow \mathcal{O}_K$ given by $ T(n) \mapsto \iota_{m}(n) a(n,f_0) $.
\begin{lemma}\label{congruence of measures}
Let $ h\vert \iota_{mp} \equiv g \vert \iota_{mp} \mod \pi $. Assume that $\phi_{\tilde{f}}$ induces the decomposition $h_{k}(\Gamma_{0}(N_{\tilde{f}}p), \eta\iota_{m} ; K) = K \oplus A$ as in \eqref{splitting of hecke algebra}. Then for every finite order character $ \phi \in C(\mathbb{Z}_{p}^{\times} ; \bar{\mathbb{Q}}_p) $ and $ 0 \leq j \leq k-l-1 $, we have
\[
\mu_{\tilde{f} \times g|\iota_{m}} (x_p^j\phi) \equiv
\mu_{\tilde{f} \times h|\iota_{m}} (x_p^j \phi) \mod \pi.
\] \end{lemma} \begin{proof}
To ease the notation, we denote $ g|\iota_{m}$ and $h|\iota_{m} $ by $ \tilde{g} $ and $ \tilde{h} $ respectively. As $ \tilde{h}|\iota_{p} \equiv \tilde{g}|\iota_{p} \mod \pi $ and $ \phi(n) = 0 $ for $ p \mid n $, we obtain $ \mu_{\tilde{g}^{\rho}}(\phi)
\equiv \mu_{\tilde{h}^{\rho}}(\phi) \mod \pi$ from (\ref{Def measure mu star}).
Hence $ \mu^{L}_{\tilde{g}^{\rho}}(\phi) \equiv \mu^{L}_{\tilde{h}^{\rho}}(\phi)
\mod \pi $. Since $ \psi \equiv \xi_{1} \xi_{2} \omega_p^{1-l} \mod \pi$, it follows that
$
E(\eta \cdot \psi^{-1} \cdot
(\phi_{p}^{-2}) \mathfrak{Z}_{p}^{k-l-2j-1})
\equiv E(\eta \cdot \xi^{-1} \cdot (\phi_{p}^{-2}) \mathfrak{Z}_{p}^{k-l-2j-1}) \mod \pi $.
Thus
we deduce from (\ref{convolution mu and Eisenstien measure}) that
$
(\mu_{\tilde{g}^{\rho}}^{L} \star E)_{\eta,k} (x_p^{j} \phi) \equiv
(\mu_{\tilde{h}^{\rho}}^{L} \star E)_{\eta,k} (x_p^{j} \phi) \mod \pi
$. As $ (\mu_{\tilde{g}^{\rho}}^{L} \star E)_{\eta,k} (\phi) $ and
$(\mu_{\tilde{h}^{\rho}}^{L} \star E)_{\eta,k} (\phi)$ both have level $ L $ and observe that $ Tr_{L/N}$ is identity on
$(\mu_{\tilde{g}^{\rho}}^{L} \star E)_{\eta,k} (\phi)$ and on
$(\mu_{\tilde{h}^{\rho}}^{L} \star E)_{\eta,k} (\phi)$.
Since the idempotent $ e$, the trace operator $Tr_{L/N} $ and $ c(f) l_{\tilde{f_{0}}} $ preserve integral structures, the lemma follows from (\ref{Def of convolution}). \end{proof}
\begin{remark}
The result in Lemma \ref{congruence of measures} is a consequence of \cite[Theorem 1.2]{Delbergo} and Lemma~\ref{lem: congruence of g,h}. Using Lemma~\ref{congruence of measures}, we establish our final congruence on $ p $-adic $L$-functions in Theorem~\ref{analytic final}. \end{remark} \begin{comment}
Recall that $ \mathrm{cond}(\xi_{1}\omega_p^{1-l}) = M_{1}p^{s} $ and $ \mathrm{cond}(\xi_{2}) = M_2 $ with $ p \nmid M_{1} M_{2} $ and $ s \leq 1 $. Moreover, we have $ g(z) = \sum_{n=0}^{\infty} a(n,g) q^n \in M_{l}(\Gamma_0(M), \xi_{1} \xi_{2} \omega_p^{1-l}) $ is an Eisenstein series, where $ M = \mathrm{cond}(\xi_{1}\omega_p^{1-l}) \mathrm{cond}(\xi_{2})$. Let $ P_{p}(g,T) $ denote the characteristic polynomial of $ g $ at $ p $. Then
\begin{align}
P_{p}(g,T) = \begin{cases}
\psi(p) p^{l-1} T^2 - a(p,g) T + 1 & \text{ if } p \nmid M, \\
1 - a(p,g) T & \text{ if } p \mid M.
\end{cases}
\end{align}
\begin{lemma}
With notation as above and the assumption
\begin{center}
$(\dagger) \qquad \qquad \qquad$
If $ C(\xi_{1}) $ and $ C(\xi_{2} \omega^{1-l}) $ are divisible by
$ p $ i.e., $ s_1 = s_2 =1 $, then $ p \mid C(\xi_{1}^{-1} \xi_{2} \omega_p^{1-l} )$.
\end{center}
Then $ g $ is $ p $-minimal.
\end{lemma}
\begin{proof}
If $ s_1 = s_2 =0$, then clearly $ g $ is $ p $-minimal. Let $ \phi : \mathbb{Z}_{p}^{\times} \rightarrow \mathbb{C}^{\times}$ be a finite order character. Then for $ p \nmid n $ we have
\[
\phi(n) \sum_{0 < d \mid n} \xi_{1}(d) \xi_{2} \omega^{1-l}(n/d) d^{k-1}
= \sum_{0 < d \mid n} \xi_{1}\phi(d) \xi_{2} \omega_p^{1-l}\phi(n/d) d^{k-1}.
\]
Let $ g' $ be the Eisenstein series obtained from $ \xi_{1}\phi $
and $ \xi_{2} \omega_p^{1-l}\phi $ using the recipe given in Lemma~\ref{Eisenstein series Hida}. Then from above we have
\begin{align}\label{diff g' and twist of g}
g'(z) - g|\phi(z) = \sum_{p \mid n}a(n,g') q^{n} =
\sum_{p \mid n} \sum_{0 < d \mid n} \xi_{1}\phi(d) \xi_{2} \omega^{1-l}\phi(n/d) d^{k-1} q^n.
\end{align}
If $p$ divides $ C(\xi_{1}\phi) $ and $ C(\xi_{2} \omega_p^{1-l}\phi) $,
then $ g' = g|\phi $ and conductor of $ g' $ is equal to
$ C(\xi_{1}\phi) C(\xi_{2} \omega^{1-l}\phi) \geq M$. Assume
$ p \nmid C(\xi_{1}\phi) $ or $ p \nmid C(\xi_{2} \omega_p^{1-l}\phi) $.
If $ s =1 $, then $ p \| C(\xi_{1}\phi)C(\xi_{2} \omega_p^{1-l}\phi)$.
If $ s =2 $, then by assumption $ (\dagger) $ we again get
$ p \| C(\xi_{1}\phi)C(\xi_{2} \omega_p^{1-l}\phi)$. Thus in either case
the left hand side of \eqref{diff g' and twist of g} is non-zero
and $ p\| C(g') $.
Thus by \cite[Lemma 4.6.4(ii)]{Miyake} and \eqref{diff g' and twist of g}
we obtain $ p^2 \mid C(g|\phi) $.
This implies $ C(g|\phi) \geq C(g) $.
\end{proof} \end{comment}
For $ a,b \in \mathbb{Z}$, put $ (a,b) = \gcd $ of $ a , b $. Set $ \mathcal{M}'= \{ r \text{ is a prime}: r \mid m \text{ and } a(r,g) \neq 0 \} $ and $ m' = \prod_{r \in \mathcal{M}'} r $. Also put $ m_1 = (m',M) $ and $ m_2 = m'/m_1 $. From \cite[Lemma 4.6.5]{Miyake}, it follows that $ g^{\rho} | \iota_{m} = g^{\rho} | \iota_{m'} \in S_{l}(\Gamma_{1}(M m_1 m_2^2),\xi^{-1}\iota_{m}^2)$. Set \begin{equation}\label{c(m) expression}
c(m) = (-1)^{\sigma_0(m_1)} a(m_1,g) \xi^{-1}(m_2) m_{1}^{-l/2} m_2^{-1} \in \mathcal{O}_K^\times, \end{equation} where $ \sigma_0(m_1) := $ number of positive divisors of $ m_1 $. Indeed to see that $ c(m) \in \mathcal{O}_K^\times$, it is enough to observe that $a(m_1,g)$ is a $p$-adic unit as $ p \nmid m $ and $ m_1, m_2 \mid m $. For every prime $ r \mid m_1 $, we have $ a(r,g) \neq 0 $ and $ r \mid M_1 M_2$. Thus $ r \mid M_1, r \nmid M_2$ or $r \nmid M_1, r \mid M_2$. Hence $ a(r,g) = \xi_{1}\omega_{p}^{1-l}(r) r^{l-1}$ or $ a(r,g) =\xi_{2}(r)$ is a $ p $-adic unit. As $ m $ is square free and $ g $ is an eigenform, we conclude that $a(m_1,g) $ is a $p$-adic unit.
Let $ P_{p}(g,T) $ denote the characteristic polynomial of $ g $ at $ p $. Then \begin{small}
\begin{align}
P_{p}(g,T) = \begin{cases}
\xi(p) p^{l-1} T^2 - a(p,g) T + 1 & \text{ if } p \nmid M, \\
1 - a(p,g) T & \text{ if } p \mid M.
\end{cases}
\end{align} \end{small} Finally for a Dirichlet character $\chi$, put \begin{small}
\begin{equation}\label{root number of g definition}
W(g^{\rho}|\chi) = (\mathrm{cond}(\xi_2\chi) /\mathrm{cond}(\xi_{1}\omega_{p}^{1-l}\chi))^{-l/2} \xi_{2}(-1) G(\xi_{2}^{-1}\chi^{-1})/G(\xi_{1}\omega_p^{1-l}\chi).
\end{equation} \end{small}
It is easy to see that $ W(g^{\rho}|\iota_{p}) = (\mathrm{cond}(\xi_2) /\mathrm{cond}(\xi_{1}\omega_{p}^{1-l}))^{-l/2} \xi_{2}(-1) G(\xi_{2}^{-1})/G(\xi_{1}\omega_p^{1-l}) = W(g^\rho)$. \begin{comment}
\begin{lemma}
With notation as above and the assumption
\begin{center}
$(\dagger) \qquad \qquad \qquad$
If $ C(\xi_{1}) $ and $ C(\xi_{2} \omega^{1-l}) $ are divisible by
$ p $ i.e., $ s_1 = s_2 =1 $, then $ p \mid C(\xi_{1}^{-1} \xi_{2} \omega_p^{1-l} )$.
\end{center}
Then $ g $ is $ p $-minimal.
\end{lemma}
\begin{proof}
If $ s_1 = s_2 =0$, then clearly $ g $ is $ p $-minimal. Let $ \phi : \mathbb{Z}_{p}^{\times} \rightarrow \mathbb{C}^{\times}$ be a finite order character. Then for $ p \nmid n $ we have
\[
\phi(n) \sum_{0 < d \mid n} \xi_{1}(d) \xi_{2} \omega^{1-l}(n/d) d^{k-1}
= \sum_{0 < d \mid n} \xi_{1}\phi(d) \xi_{2} \omega_p^{1-l}\phi(n/d) d^{k-1}.
\]
Let $ g' $ be the Eisenstein series obtained from $ \xi_{1}\phi $
and $ \xi_{2} \omega_p^{1-l}\phi $ using the recipe given in Lemma~\ref{Eisenstein series Hida}. Then from above we have
\begin{align}\label{diff g' and twist of g}
g'(z) - g|\phi(z) = \sum_{p \mid n}a(n,g') q^{n} =
\sum_{p \mid n} \sum_{0 < d \mid n} \xi_{1}\phi(d) \xi_{2} \omega^{1-l}\phi(n/d) d^{k-1} q^n.
\end{align}
If $p$ divides $ C(\xi_{1}\phi) $ and $ C(\xi_{2} \omega_p^{1-l}\phi) $,
then $ g' = g|\phi $ and conductor of $ g' $ is equal to
$ C(\xi_{1}\phi) C(\xi_{2} \omega^{1-l}\phi) \geq M$. Assume
$ p \nmid C(\xi_{1}\phi) $ or $ p \nmid C(\xi_{2} \omega_p^{1-l}\phi) $.
If $ s =1 $, then $ p \| C(\xi_{1}\phi)C(\xi_{2} \omega_p^{1-l}\phi)$.
If $ s =2 $, then by assumption $ (\dagger) $ we again get
$ p \| C(\xi_{1}\phi)C(\xi_{2} \omega_p^{1-l}\phi)$. Thus in either case
the left hand side of \eqref{diff g' and twist of g} is non-zero
and $ p\| C(g') $.
Thus by \cite[Lemma 4.6.4(ii)]{Miyake} and \eqref{diff g' and twist of g}
we obtain $ p^2 \mid C(g|\phi) $.
This implies $ C(g|\phi) \geq C(g) $.
\end{proof} \end{comment}
\begin{lemma}\label{special value f,g and trivial character}
Let $g = E_{l}(\xi_{1} \omega_p^{1-l}, \xi_2) $ and $ g'= E_{l} (\xi_{2},\xi_{1}{\omega_p}^{1-l})$ be as in Lemma~\ref{Eisenstein series Hida}.
For $ 0 \leq j \leq k-l-1 $, we have
\begin{small}
\begin{align*}
p^{\beta (l+2j)/2} D_{I_{0}Np}(l+j,\tilde{f_{0}},
\mu_{g^\rho|\iota_{m}}(\iota_{p})
|\tau_{M_{0}m_1m_2^2p^{\beta}}) = c(m) p^{s (l+j)/2}
W(g^\rho|\iota_{p}) u_{f}^{\beta-s} P_{p}(g^{\rho},p^ju_{f}^{-1})
D_{I_{0}mNp}(l+j,\tilde{f_{0}}, g').
\end{align*}
\end{small} \end{lemma} \begin{proof}
Since $ g $ is a Hecke eigenform, for $ n \geq 1 $ and a prime $ r $ we have (see \cite[Line -1, Page 46]{Hidarankin2})
\begin{align}\label{iota fourier}
a(n,g^\rho|\iota_{r}) =
\begin{cases}
a(n, g^{\rho}) - a(r,g^{\rho}) a(n,g^{\rho}|[r]) & \text{ if } r \mid M, \\
a(n,g^{\rho}) - a(r,g^{\rho}) a(n,g^{\rho}|[r])+\xi^{-1}(r)r^{l-1}a(n,g^{\rho}|[r^2])
& \text{ if } r \nmid M.
\end{cases}
\end{align}
Iterating, we get
$
a(n, g^{\rho}|\iota_{m'}) = \sum_{0 < d \mid m_1m_2^2} c_{d} a(n, g^{\rho}|[d])
$, where $ c_{m_1m^2_2} = (-1)^{\sigma_0(m_1)} a(m_1,g) \xi^{-1}(m_2) m_2^{l-1}$. Note that $ g^{\rho}|\iota_{m} = g^{\rho}|\iota_{m'} $ and $ c(m) = (m_1 m_2^2)^{-l/2} c_{m_1m^2_2} $ by \eqref{c(m) expression}. For every $ d \mid m_1m_2^2 $ and $ i \geq 0 $, by Lemma~\ref{Eisenstein series Hida}
we obtain
\begin{small}
\begin{equation}\label{Atkin involution and d, g}
\begin{aligned}
g^{\rho}|[dp^i]|\tau_{M_0 m_1m^2_2 p^{\beta}} & = (dp^i)^{-l/2}
g^{\rho}|\begin{psmallmatrix}
dp^i & 0 \\ 0 &1 \end{psmallmatrix}
\begin{psmallmatrix}
0 & -1 \\ M_0 m_1 m^2_2 p^{\beta} &0 \end{psmallmatrix} \\
& =(dp^i)^{-l/2}
(m_1m^2_2 p^{\beta-s-i}/d)^{l/2}
g^{\rho}|\tau_{M_0 p^{s}}| [m_1m^2_2 p^{\beta-s-i}/d ] \\
& = W(g^{\rho})(dp^i)^{-l/2} (m_1m^2_2 p^{\beta-s-i}/d)^{l/2}
g'|[m_1m^2_2 p^{\beta-s-i}/d].
\end{aligned}
\end{equation}
\end{small}
\underline{Case $ p \nmid M$}: In this case, $ g^{\rho} $ is $ p $-minimal and primitive. Thus by \cite[Lemma 5.2]{Hidarankin2}, we have $ g| \iota_{p} \in M_{l}(\Gamma(Mp^2), \xi\iota_{p}^2) $. It follows from \cite[Lemma 4.6.5]{Miyake} that
$ \mu_{g^{\rho} | \iota_{m}}(\iota_{p}) \in S_{l}(\Gamma_{0}(M m_1m^2_2 p^{2}),\xi^{-1}\iota_{mp}^2)$. Therefore $ s =0 $ and $ \beta =2 $. In this case,
$a(n, g^{\rho}|\iota_{m}\iota_{p}) = \sum_{0 < d \mid m_1m^2_2} c_{d} \big(
a(n,g^{\rho}|[d]) - a(p,g^{\rho}) a(n,g^{\rho}|[dp])+ \xi^{-1}(p)p^{l-1} a(n,g^{\rho}|[dp^2]) \big)$.
Thus by \eqref{Atkin involution and d, g}, we have
\begin{small}
\begin{align*}
& D_{I_{0}Np}(l+j,\tilde{f_{0}}, \mu_{g^{\rho}|\iota_{m}}(\iota_{p})
|\tau_{M_{0}m_1m^2_2p^{\beta}})
\\ &~~~ = \sum_{0 < d \mid m_1m^2_2} c_{d} W(g^{\rho})
(m_1m^2_2 /d^2)^{l/2} \times \Big( p^l D_{I_{0}Np}(l+j,\tilde{f_{0}},
g'|[m_1m^2_2 p^{2}/d]) - a(p,g^{\rho}) D_{I_{0}Np}(l+j,\tilde{f_{0}}, g'|[m_1m^2_2 p/d]) \\
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad + p^{-l} p^{1-l} \xi^{-1} (p) D_{I_{0}Np}(l+j,\tilde{f_{0}},
g'|[m_1m^2_2/d]) \Big).
\end{align*}
\end{small}
\begin{comment}
\begin{small}
\begin{align*}
D_{I_{0}Np}(l+j,\tilde{f_{0}}, \mu_{g^{\rho}|\iota_{m}}(\iota_{p})
|\tau_{M_{0}m_1m^2_2p^{\beta}})
& = \sum_{0 < d \mid m_1m^2_2} c_{d} W(g^{\rho})
(m_1m^2_2/d^2)^{l/2} \big[ p^lD_{I_{0}Np}(l+j,\tilde{f_{0}},
g'|[m_1m^2_2 p^{2}/d]) \\ &- D_{I_{0}Np}(l+j,\tilde{f_{0}}, g'|[m_1m^2_2 p/d]) + p^{-l} p^{1-l} \xi(p) D_{I_{0}Np}(l+j,\tilde{f_{0}},
g'|[m_1m^2_2/d])\big].
\end{align*}
\end{small}
\end{comment}
Since $ L(l+j,\tilde{f_{0}}, g'|[d]) =0 $ if
$ (d,m) \neq 1 $ and
$ L(l+j,\tilde{f_{0}}, g'|[p^i]) = p^{-i(l+j)} u_{f}^{i}
L(l+j,\tilde{f_{0}}, g')$, we get
\begin{small}
\begin{align}\label{atkin and trivial char}
D_{I_{0}Np}(l+j,\tilde{f_{0}},
\mu_{g^{\rho}|\iota_{m}}(\iota_{p})
|\tau_{M_{0}m_1m^2_2p^{\beta}})
& = c_{m_1m^2_2} W(g^{\rho}) (m_1m^2_2)^{-l/2} \Big(
p^{l} D_{I_{0}Np}(l+j,\tilde{f_{0}},
g'|[ p^{2}]) \nonumber \\
& \qquad - a(p,g^\rho)
D_{I_{0}Np}(l+j,\tilde{f_{0}}, g'|[ p]) + p^{-l} p^{1-l} \xi^{-1}(p) D_{I_{0}Np}(l+j,\tilde{f_{0}}, g') \Big) \\
&= c_{m_1m^2_2} (m_1m^2_2)^{-l/2} p^{-l -2j} W(g^{\rho}) u_{f}^{2} P_{p}(g^{\rho},p^{j} u_{f}^{-1})
D_{I_{0}Np}(l+j,\tilde{f_{0}}, g') \nonumber .
\end{align}
\end{small}
\underline{Case $ p \mid M $}:
Since $\mathrm{cond}(\xi_1 \omega_{p}^{1-l}) = M_1p^s $ $\mathrm{cond}(\xi_2) = M_2 $ with $ p \nmid M_1M_2 $ and $ s \leq 1 $, we get $ p|| M $. By \cite[Lemma 5.2 (i)]{Hidarankin2}, $ g^{\rho}| \iota_p \in M_{l}(\Gamma_0(M_{0}p^2),
\xi^{-1} \iota_{p}^{2})$. Thus
$ g^{\rho}|\iota_{m} \iota_p \in M_{l}(\Gamma_0(M_{0}m_1m_2^2 p^2),
\xi^{-1} \iota_{m}^{2} \iota_{p}^{2}) $ by \cite[Lemma 4.6.5]{Miyake}.
In this case, $ s =1 $ and $ \beta =2 $ and
$a(n, g^{\rho}|\iota_{m}\iota_{p}) = \sum_{0 < d \mid m_1m^2_2 } c_{d}
\big(a(n,g^{\rho}|[d]) - a(p,g^{\rho}) a(n,g^{\rho}|[dp])\big)$.
Then by \eqref{Atkin involution and d, g}, we get
\begin{small}
\begin{align*}
&D_{I_{0}Np}(l+j,\tilde{f_{0}}, \mu_{g^{\rho}|\iota_{m}}(\iota_{p})
|\tau_{M_{0}m_1m_2^2 p^{\beta}}) = \\
& \quad \sum_{0 < d \mid m_1m_2^2} c_{d} W(g^{\rho})
(m_1m_2^2 /d^2)^{l/2} \Big( D_{I_{0}Np}(l+j,\tilde{f_{0}},
g'|[m_1m_2^2 p/d]) - p^{-l/2}
a(p,g^{\rho}) D_{I_{0}Np}(l+j,\tilde{f_{0}}, g'|[m_1m_2^2 /d]) \Big).
\end{align*}
\end{small}
By a similar argument as in \eqref{atkin and trivial char}, we have
\begin{small}
\begin{align*}
D_{I_{0}Np}(l+j,\tilde{f_{0}}, \mu_{g^{\rho}|\iota_{m}}(\iota_{p})
|\tau_{M_{0}m_1m_2^2p^{\beta}})
& = c_{m_1m_2^2} (m_1m_2^2)^{-l/2} p^{-l/2}W(g^{\rho}) \\
& \qquad \qquad \times \Big(
p^{l} D_{I_{0}Np}(l+j,\tilde{f_{0}},
g'|[p]) - a(p,g^{\rho}) D_{I_{0}Np}(l+j,\tilde{f_{0}}, g')\Big) \\
& = c_{m_1m_2^2} (m_1m_2^2)^{-l/2} p^{-j-l/2} W(g^{\rho}) u_{f} P_{p}(p^j u_f^{-1},g^{\rho})
D_{I_{0}Np}(l+j,\tilde{f_{0}}, g')).
\end{align*}
\end{small}
This finishes the proof.
\begin{comment}
Finally we consider the case $ p \mid M $ and $ a(p,g) = 0 $.
In this case $ g | \iota_{p} = p $. Thus $ \beta = s $. We get
\begin{small}
\begin{align*}
D_{M_{0}mNp}(l,\tilde{f_{0}}, \mu_{g|\iota_{m}}(\iota_{p})
|\tau_{M_{0}m^2p^{\beta}})
& = \sum_{0 < d \mid m^2} c_{d} W(g)d^{-l/2}
(m^2 /d)^{l/2} D_{M_{0}mNp}(l,\tilde{f_{0}}, g'|[m^2 /d]) \\
& = c_{m^2} W(g) m^{-l} D_{M_{0}mNp}(l,\tilde{f_{0}},g') \\
& = W(g) m^{-l} P_{p}(g,u_{f}^{-1}) \xi(m)
D_{M_{0}mNp}(l,\tilde{f_{0}},g').
\end{align*}
\end{small}
Noting that $ P_{p}(g,T) =1 $ in this case finishes the proof of the lemma.
\end{comment} \end{proof}
Recall for a Dirichlet character $ \chi $, $ \chi_0 $ denotes the primitive Dirichlet character associated to $ \chi $. \begin{lemma}\label{special value f,g and non-trivial character}
Let $g = E_{l}(\xi_{1} \omega_p^{1-l}, \xi_2) $,
$ g'= E_{l} (\xi_{2},\xi_{1}{\omega_p}^{1-l})$ be as in Lemma \ref{special value f,g and trivial character} and $ \phi$ be a non-trivial finite order character on $\mathbb{Z}_{p}^{\times}$. If $ ( \xi_{1}\omega_p^{1-l} \phi)_0(p) =0$, then we have $ p^\beta = \mathrm{cond}_{p}(\xi_1 \omega_{p}^{1-l} \phi)\mathrm{cond}_{p}(\xi_2 \phi)$ and
\begin{align*}
D_{I_{0}Np}(l+j,\tilde{f_{0}},
\mu_{g^{\rho}|\iota_{m}}(\bar{\phi})
|\tau_{M_{0}m_1m_2^2p^{\beta}}) =
c(m) W(g^{\rho}\vert\bar{\phi})
D_{I_{0}Np}(l+j,\tilde{f_{0}}, g'\vert \phi),
\quad \text{for } 0 \leq j \leq k-l-1.
\end{align*}
\end{lemma} \begin{proof}
By \cite[Lemma 5.2 (i)]{Hidarankin2} and $ \xi_2 \vert \mathbb{Z}_{p}^{\times} =1$, we have $ p^\beta
= \mathrm{cond}_p(\xi_{2}\phi) \mathrm{cond}_p(\xi_{1}\omega_{p}^{1-l}\phi) $. As observed in Lemma~\ref{special value f,g and trivial character}, we have $ g^{\rho}\vert\iota_{m} \in S_{l}(Mm_1m_2^2,\xi^{-1}\iota_{m}^2)$. Further, note that
\begin{small}
\begin{align*}
a(n, \mu_{g^{\rho}|\iota_{m}}(\bar{\phi})) = \sum_{0 < d \mid m_1m_2^2} c_{d} a(n, g^{\rho}|[d]| \bar{\phi}) = \sum_{0 < d \mid m_1m_2^2} c_{d} a(n, g^{\rho}| \bar{\phi}| [d]),
\end{align*}
\end{small}
where $c_{m_1m^2_2} = (-1)^{\sigma_0(m_1)} a(m_1,g) \xi^{-1}(m_2) m_2^{l-1} = c(m)(m_1m_2^2)^{l/2}$. Also
\begin{align*}
\begin{split}
g^{\rho}\vert\bar{\phi}|[d]|\tau_{M_0 m_1 m_2^2 p^{\beta}} & = d^{-l/2}
g^{\rho}\vert\bar{\phi}|\begin{psmallmatrix}
d & 0 \\ 0 &1 \end{psmallmatrix}
\begin{psmallmatrix}
0 & 1 \\ M_0 m_1m_2^2 p^{\beta} &0 \end{psmallmatrix}
= d^{-l/2} (m_1 m_2^2/d)^{l/2}
g^{\rho}\vert \bar{\phi} |\tau_{M_0 p^{\beta}}| [m_1m_2^2/d ].
\end{split}
\end{align*}
Note that $ L(s,\tilde{f_{0}},g^{\rho}\vert \bar{\phi} |\tau_{M_0 p^{\beta}}| [d ]) =0$ if $ (d,m) \neq 1 $. Thus
\begin{align}\label{non trivial char atkin}
D_{I_{0}Np}(l+j,\tilde{f_{0}}, \mu_{g^{\rho}|\iota_{m}}(\bar{\phi})
|\tau_{M_{0}m_1m_2^2p^{\beta}})
& = \sum_{0 < d \mid m_1m_2^2} c_{d} d^{-l/2}
(m_1m_2^2 /d)^{l/2} D_{I_{0}Np}(l+j,\tilde{f_{0}}, g^{\rho}\vert \bar{\phi} |\tau_{M_0 p^{\beta}}| [m_1m_2^2/d ]) \nonumber \\
& = c_{m_1m_2^2} (m_1m_2^2)^{-l/2} D_{I_{0}Np}(l+j,\tilde{f_{0}},g^{\rho}\vert \bar{\phi} |\tau_{M_0 p^{\beta}}).
\end{align}
Since $ (\xi_{1}\omega_p^{1-l} \phi)_0(p) =0$ and $ p \nmid \mathrm{cond}(\xi_2) $, it follows that $ \xi_{1}\omega_p^{1-l} \phi(p) =0 $ and $ \xi_{2} \phi(p) = 0 $. Also $ \xi_{2}(d) \xi_{1}\omega_p^{1-l}(n/d) \phi(n) = \xi_{2}\phi(d) \xi_{1}\omega_{p}^{1-l} \phi(n/d) $ for $ p \nmid n $. Hence $
g^{\rho}\vert \bar{\phi} = E_{l}(\xi_{1}^{-1}\omega_p^{l-1}\phi^{-1},\xi_{2}^{-1}\phi^{-1})$.
Thus by Lemma~\ref{Eisenstein series Hida} (i),
$g^{\rho}\vert \bar{\phi} |\tau_{M_0 p^{\beta}} = E_{l}(\xi_{1}^{-1}\omega_p^{l-1}\phi^{-1}, \xi_{2}^{-1}\phi^{-1})\vert \tau_{M_0 p^{\beta}} = W(g^{\rho}\vert\bar{\phi}) E_{l} (\xi_{2}\phi, \xi_{1}\omega_p^{1-l}\phi)$. Substituting this in \eqref{non trivial char atkin}, the lemma follows.
\end{proof}
By Lemma~\ref{Eisenstein series Hida} (ii), we already know that $ L(l+j,g'|\phi) =
L(l+j, \xi_{1} \omega_p^{1-l}\phi) L(j+1, \xi_{2} \phi) $.
Thus by Rankin-Selberg method (See \cite[Lemma 1]{Shimura1}), we have
$
D_{I_{0}Np}(l+j,\tilde{f_{0}},g'|\phi)
= L(l+j,\tilde{f_{0}},\xi_{1} \omega_p^{1-l} \phi) L(j+1,\tilde{f_{0}}, \xi_{2} \phi)$. Using this, we obtain the following result:
\begin{comment}
Note that
\[
L(s,\tilde{f_{0}},\phi) = (1-\beta_f \phi(p) p^{-s}) L(s,\tilde{f},\phi) = \left( 1 - \frac{\eta(p) \phi(p) p^{k-1-s}}{u_f} \right) L(s,\tilde{f},\phi).
\]
Using this in \eqref{rankin as product of two l-functions} we get
\begin{align}\label{rankin as product of two l-functions 1}
\begin{split}
D_{M_{0}mNp}(s,\tilde{f_{0}},g'|\bar{\phi}) =
&\left( 1 - \frac{\eta(p) \bar{\xi}_{2}\bar{\phi}(p) p^{k-s+l-2}}{u_f} \right) L(s-l+1,\tilde{f},\xi_{2} \phi) \\ & \qquad \qquad \qquad \qquad \qquad \times
\left( 1 - \frac{\eta(p) \xi_{1}\omega_p^{1-l} \phi(p) p^{k-1-s}}{u_f} \right) L(s,\tilde{f},\xi_{1} \omega_p^{1-l} \phi).
\end{split}
\end{align}
\end{comment}
\begin{corollary}
Let $g = E_{l}(\xi_{1} \omega_p^{1-l}, \xi_2)$, $ 0 \leq j \leq k-l-1 $ and $ \phi $ be a finite order character on $ \mathbb{Z}_p^{\times} $. Further, let $ c(m)$ be the $ p $-adic unit as defined in \eqref{c(m) expression} and $ W(g^\rho|\bar{\phi}) $ as in \eqref{root number of g definition}.
\begin{enumerate}
\item[$(i)$] For trivial character $ \iota_{p} $, we have $ \beta =2 $ and
\begin{align}\label{eq: rankin as product trivial char}
p^{\beta (l+2j)/2} D_{M_{0}mNp}(l+j,\tilde{f_{0}},
\mu_{g^{\rho}|\iota_{m}}(\iota_{p})
|\tau_{M_{0}m^2p^{\beta}}) = p^{s(l+2j)/2}
W(g^{\rho}|\iota_{p}) c(m) u_{f}^{\beta-s} P_{p}(g^{\rho},p^ju_{f}^{-1}) \times \nonumber \\
L(j+1,\tilde{f}_0,\xi_{2})
L(l+j,\tilde{f}_0,\xi_{1} \omega_p^{1-l} ).
\end{align}
\item[$(ii)$] For a non-trivial character $ \phi $ with $ (\xi_{1} \omega_p^{1-l} \phi )_0(p) =0 $, we have $ p^\beta = \mathrm{cond}_p(\xi_{1}\omega_{p}^{1-l}\phi)\mathrm{cond}_p(\xi_{2}\phi)$ and
\begin{equation}\label{eq: rankin as product nontrivial char}
D_{M_{0}mNp}(l+j,\tilde{f_{0}},
\mu_{g^{\rho}|\iota_{m}}(\bar{\phi})
|\tau_{M_{0}m^2p^{\beta}}) =
W(g^{\rho}\vert\bar{\phi})c(m)
L(j+1,\tilde{f}_0,\xi_{2} \phi)
L(l+j,\tilde{f}_0,\xi_{1} \omega_p^{1-l} \phi)
\end{equation}
\end{enumerate}
\end{corollary}
\noindent For every finite order character $ \phi \in C(\mathbb{Z}_{p}^\times, \bar{\mathbb{Q}}_p)$, it follows from \eqref{eq: rankin as product trivial char}, \eqref{eq: rankin as product nontrivial char} and Theorem~\ref{padic rankin} that $ \mu_{\tilde{f} \times g|\iota_{m}}(\phi) $ can be written as a product of the special values of $ L $-function associated to $ \tilde{f}_0 \otimes \xi_{1}\omega_{p}^{1-l} $ and $ \tilde{f}_0 \otimes \xi_{2}$.
\section{Periods and the congruences of p-adic L-functions}\label{sec: periods and congruence}
In this section, we show that the $p $-adic Rankin-Selberg $ L $-function of $ \tilde{f} \otimes h|\iota_{m} $ is congruent to the product of the $p$-adic $ L $-functions of $ \tilde{f_{0}} \otimes \xi_{1} \omega_{p}^{1-l}$ and $ \tilde{f_{0}} \otimes \xi_{2} $. In order to do this, we make an appropriate choice of periods in \S\ref{subsection: relation periods}.
\subsection{Relation between the periods}\label{subsection: relation periods}
We begin by recalling an algebraicity result due to Shimura.
\begin{theorem} $\mathrm{(}$\cite[Theorem 1]{Shimura1}$ \mathrm{ ) } $ Let $ F \in S_{k}(\Gamma_{1}(N))$ be a
normalised Hecke eigenform. There exist complex periods $ \Omega_{F}^{+} $ and $ \Omega_{F}^{-} $ such that for every Dirichlet character $ \chi $,
\begin{small}
\begin{align}\label{eq: special values of f}
\frac{L(j,F,\chi)}{(2 \pi i)^{j} G(\chi)
\Omega_{F}^{\mathrm{sgn}((-1)^j \chi(-1))}} \in \bar{\mathbb{Q}}, \quad \text{for} \quad 1 \leq j \leq k-1.
\end{align}
\end{small}
Further, we have $ \frac{\Omega_{F}^{+}
\Omega_{F}^{-} }
{2 G(\chi) \langle F , F \rangle} \in \bar{\mathbb{Q}}$ and $ \Omega_{F}^{\pm} $ are well-defined up to an element of $ \bar{\mathbb{Q}}^{\times} $.
\end{theorem}
We next recall the $ p $-adic $ L $-function associated to an eigenform. See \cite[\S 14]{MTT} and \cite[\S 3.4.4]{SU}.
\begin{theorem}\label{p-adic l-function of modular form}
Let $ F = \sum a(n,F) q^n \in S_{k}(\Gamma_{0}(N), \chi)$ be a $ p $-ordinary eigenform and $\Omega^{+}_{F}, \Omega^{-}_{F}$ be complex periods satisfying \eqref{eq: special values of f}. Then there exist a bounded measure $ \mu_F $ such that for a finite order Dirichlet character $ \phi $ of conductor $ p^{v}L$ with $ p \nmid L $ and $ 0 \leq j \leq k-2 $, we have
\begin{small}
\begin{align*}
\mu_{F}(x_{p}^{j}\phi ) = \int_{(\mathbb{Z}/L\mathbb{Z})^{\times} \times \mathbb{Z}_{p}^{\times}} x_{p}^{j}\phi ~d\mu_{F} = \frac{e_{p}(u_F, x_p^j \phi )}{u_{F}^{v}}
\frac{ (p^{v}L)^{j+1}}{(-2\pi i)^{j+1} }\frac{j!}{G(\phi)}
\frac{L(j+1,F,\phi)}{
\Omega_{F}^{\mathrm{sgn}((-1)^j \phi(-1))} }.
\end{align*}
\end{small}
Here $ e_{p}(u_F, x_p^j \phi ) = \left( 1 -
\frac{\phi_0(p) \chi(p) p^{k-2-j}}{u_{F}}\right)
\left( 1 - \frac{\bar{\phi}_0(p)p^{j}}{u_{F}}\right) $ is the $ p $-adic multiplier, $ \phi_0 $ is the primitive character associated to $ \phi $ and $ u_F $ is the unique $ p $-adic unit root of $ X^2 - a(p,F)X+\chi(p) p^{k-1} $.
\end{theorem}
Observe that for $ l \leq j \leq k-1 $, we have
$
(-1)^{j} \xi_{2}\phi(-1) (-1)^{j-l+1} \xi_{1} \omega_p^{1-l}\phi(-1) = (-1)^{l-1} \xi_{1}\xi_{2} \omega_p^{1-l} (-1) =(-1)^{l-1}(-1)^{l} = -1$. Hence $ \mathrm{sgn}((-1)^{j}\xi_{2}\phi(-1)) = -\mathrm{sgn}((-1)^{j-l+1} \xi_{1} \omega_p^{1-l}\phi(-1) )$.
Recall that $ f_0 $ is the $ p $-stabilization of the $p $-ordinary newform $ f$ and $\tilde{f}_0 := f_0|\iota_{m}$. Thus, for $ l \leq j \leq k -1$, we have
\begin{small}
\begin{align*}
\frac{L(j,\tilde{f}_0, \xi_{2} \phi)}{(2 \pi i)^{j}G(\xi_{2}\phi)
\Omega_{\tilde{f}_0 }^{\pm} } \cdot \frac{L(j-l+1,\tilde{f}_0, \xi_{1}^{-1} \omega_p^{1-l}\phi)}
{(2 \pi i)^{j-l+1} G(\xi_{1} \omega_p^{1-l}\phi)
\Omega_{\tilde{f}_0 }^{\mp} } \in \bar{\mathbb{Q}}.
\end{align*}
\end{small}
This implies that the period corresponding to the special values on the right hand side of \eqref{eq: rankin as product trivial char} and \eqref{eq: rankin as product nontrivial char} is equal to $ \Omega_{\tilde{f}_{0}}^{+} \Omega_{\tilde{f}_{0}}^{-} $ up to a power of $ \pi $. Further, by Theorem~\ref{padic rankin}, we have the period corresponding to the special value on the left hand side of \eqref{eq: rankin as product trivial char} and \eqref{eq: rankin as product nontrivial char} equals $ c(\tilde{f})/\langle \tilde{f}_0|\tau_{Nm^2p^{\beta}},\tilde{f}_0\rangle $ up to a power of $ \pi $. Next, we want to choose periods $ \Omega_{F}^{\pm} $ such that
$ p $-adic $ L $-function $ L_{p}(F,\cdot) $ associated to $ F $ in Theorem~\ref{p-adic l-function of modular form} has integral power series expansion and $ \langle F\vert \tau_{N} , F \rangle / \Omega_{F}^{+} \Omega_{F}^{-} $ can be explicitly specified up to a $ p $-adic unit.
We now recall relevant cohomology groups. Let $ \Gamma $ be a congruence subgroup. Let $A$ be a ring and $ \mathcal{M} $ be an $ A[\Gamma] $-module.
\begin{comment}\textcolor{red}{Throughout this subsection, fix a Dirichlet character $ \chi $ modulo $ N $. Let $ L(n,\chi;A) = L(n,A) \otimes \chi$ be the left $ \Gamma_{0}(N) $-module with diagonal action.}
\end{comment}
Let $ \widetilde{\mathcal{M}}$ denote the sheaf of locally constant sections of the covering $
\Gamma_0(N) \backslash (\mathbb{H} \times\mathcal{ M}) \rightarrow \Gamma_0(N) \backslash \mathbb{H}
$. If $ \Gamma/\{\pm \mathrm{I}\} $ has no torsion elements or $ A $ is a field of characteristic zero, then by \cite[Proposition 8.1]{Shimura book} there is a canonical isomorphism
$H^{1}(\Gamma,\mathcal{M}) \cong H^{1}(\Gamma \backslash \mathbb{H}, \widetilde{\mathcal{M}})$.
Set $ X_{\Gamma} := \Gamma \backslash \mathbb{H} $
and $ {Y}_{\Gamma} $ be the Borel-Serre compactification of
$ X_{\Gamma} $. Then we have
$ \partial Y_{\Gamma} =
\sqcup_{\Gamma \backslash P^{1}(\mathbb{Q})} S^{1}
$
and
$
H^{i}(\partial Y_{\Gamma} , \widetilde{\mathcal{M}} )
= \bigoplus_{s \in \Gamma_0(N) \backslash P^{1}(\mathbb{Q})} H^{i} ( \Gamma_{s}, \mathcal{M}),
$
where $ \Gamma_{s} $ denotes the stabilizer of $ s $ in $ \Gamma $.
Define the parabolic cohomology
\begin{small}
\begin{align*}
H^{i}_{P}(Y_{\Gamma} , \widetilde{\mathcal{M}}) &= \operatorname{Ker}\Big(\mathrm{res}: H^{i}(Y_{\Gamma} , \widetilde{\mathcal{M}}) \longrightarrow H^{i}(\partial Y_{\Gamma} , \widetilde{\mathcal{M}} )\Big) \\
H^{i}_{P}(\Gamma, \mathcal{M}) &= \operatorname{Ker}\Big(\mathrm{res}: H^{i}(\Gamma, \mathcal{M}) \longrightarrow \bigoplus\limits_{s \in \Gamma \backslash P^{1}(\mathbb{Q})} H^{i} ( \Gamma_s, \mathcal{M})\Big),
\end{align*}
\end{small}
where $\mathrm{res}$ are the corresponding restriction maps. Under the isomorphism $ H^{1}(\Gamma,\mathcal{M}) \cong H^{1}(X_\Gamma, \widetilde{\mathcal{M}}) $, we have $ H^{i}_{P}(Y_{\Gamma} , \widetilde{\mathcal{M}}) \cong H^{i}_{P}(\Gamma, \mathcal{M}) $.
There is a well-defined
action of Hecke algebra on the cohomology groups $H^{i}(Y_{\Gamma} , \widetilde{\mathcal{M}})$ and $H^{i}_{P}(\Gamma, \mathcal{M}) $ (see \cite[\S8.3]{Shimura book} or \cite[\S 6.3]{hida_93}).
We now introduce the modules $ \mathcal{M} $ which are of interest for us. For a non-negative integer $n$, we denote by $L(n,A)$ the symmetric polynomial algebra over $A$ of degree $n$. Thus $L(n,A)$ consists of the homogeneous polynomials of degree $n$ in variables $ X $ and $ Y $, with coefficients in $A$. The semigroup $\operatorname{GL}_2(\mathbb{Q}) \cap \text{M}_2(\mathbb{Z})$ acts on $L(n,A)$ by
\begin{small}
\begin{align*}
(\gamma \cdot P)(X,Y) =
P(aX+bY, cX+dY),
\qquad \forall ~ \gamma = \begin{psmallmatrix} a & b \\ c & d \end{psmallmatrix},
\end{align*}
\end{small}
Note that the above action is a left action and is as
defined in \cite[(8.2.1)]{Shimura book} and \cite[\S 3.3]{Kitagawa}. Note that Hida \cite[Page 165]{hida_93} instead considers the right action of $\operatorname{GL}_2(\mathbb{Q}) \cap \text{M}_2(\mathbb{Z})$ on $ L(n,A) $.
For a Dirichlet character $ \chi $ modulo N, let $ L(n,\chi;A) $ denote the $ \Gamma_{0}(N) $-module $ L(n,A) $ with the action
\begin{small}
\[
(\gamma P)(X,Y) = \chi(\gamma)P(aX+bY, cX+dY) = \chi(d) P(aX+bY, cX+dY),~\forall ~ \gamma = \begin{psmallmatrix} a & b \\ c & d \end{psmallmatrix} \in \Gamma_{0}(N).
\]
\end{small}
Now this action coincides with the action defined on \cite[Page 177]{hida_93}. To ease the notation, we denote $\widetilde{L(n,A)}$ (resp. $\widetilde{L(n,\chi;A)}$) by $ \mathcal{L}(n,A) $ (resp. $ \mathcal{L}(n,\chi;A) $).
We have the inclusion $H^{i}_{P}(\Gamma_0(N), L(n,\chi;A)) \hookrightarrow H^{i}_{P}(\Gamma_1(N), L(n,A)) $.
Let $S_{k}(\Gamma)$ be the space of cusp forms on $ \Gamma $ with coefficients in $ \mathbb{C} $ and
$
\bar{S}_{k}(\Gamma) = \{ \bar{ F} : F \in S_{k}(\Gamma) \}
$ be the space of anti-holomorphic cusp forms on $ \Gamma$.
For every $ F \in S_{k}(\Gamma)$ (resp. $\bar{S}_{k}(\Gamma) $), define an $L(n,\mathbb{C})$-valued differential $1$-form by
$\omega(F)(z) =
F(z)(zX+Y)^{k-2} d z$
(resp. $ \omega(F)(z) = F(z)(\bar{z}X+Y)^{k-2} d \bar{z}$).
With the action as above, it can be checked that $ \gamma^{\ast} \omega(F) = \det(\gamma)^{1-k/2} \gamma \cdot \omega(F\vert \gamma) ~ \forall \gamma \in \operatorname{GL}_2(\mathbb{Q}) \cap \text{M}_2(\mathbb{Z})$ (see \cite[\S 6.2]{hida_93}).
For every $ F \in S_{k}(\Gamma)$ (resp. $ \bar{S}_{k}(\Gamma) $), set $ \delta(F)(\gamma) := \int_{ \infty}^{\gamma \infty} \omega(F) ~\forall \gamma \in \Gamma$.
It can be checked that $ \delta(F) $ is a $ 1 $ cocycle and $\delta: S_{k}(\Gamma) \oplus \bar{S}_{k}(\Gamma) \rightarrow H^{1}(\Gamma, L(k-2,\mathbb{C}) )$ is a homomorphism. A similar construction, defines the map $\delta: S_{k}(\Gamma_0(N), \chi) \oplus \bar{S}_{k}(\Gamma_0(N),\chi^{-1}) \rightarrow H^{1}(\Gamma_0(N), L(k-2,\chi;\mathbb{C}) )$. \begin{theorem} $\mathbf{(}\textbf{Eichler-Shimura} \mathbf{)} ~ \mathrm{(}$\cite[\S 6.2, Theorem 1]{hida_93}$\mathrm{)}$\label{Eichler Shimura}
With the notation as above. The maps $\delta
:S_{k}(\Gamma) \oplus \bar{S}_{k}(\Gamma) \rightarrow H^{1}_{P}(\Gamma, L(k-2,\mathbb{C})) \text{ and }
\delta: S_{k}(\Gamma_0(N), \chi) \oplus \bar{S}_{k}(\Gamma_0(N),\chi^{-1}) \rightarrow
H^{1}_{P}(\Gamma_0(N) , L(k-2,\chi;\mathbb{C}) )$
are Hecke equivariant isomorphisms. \end{theorem} If $ \varepsilon := \begin{psmallmatrix} -1 & 0 \\ 0 & 1 \end{psmallmatrix} $ normalises $ \Gamma $ (resp. $\Gamma _{0}(N)$), then $ \varepsilon $ acts on $ H^{i}_{P}( \Gamma , L(n, A) ) $ (resp. $ H^{i}_{P}( \Gamma_0(N) , L(n, \chi;A) ) $). When $ A = \mathbb{C} $, we obtain $ (\varepsilon \cdot \omega)(z) = \varepsilon\omega(\varepsilon z) = \varepsilon\omega(-\bar{z}) $ at the level of differential forms. If $ 1/2 \in A $, then the map $ \omega \mapsto (\omega + \varepsilon \cdot \omega , \omega - \varepsilon \cdot \omega ) $ induces the decompositions $H^{i}_{P}( \Gamma_1(N), L(n,A) ) = H^{i}_{P}( \Gamma_1(N), L(n,A) )^{+} \oplus H^{i}_{P}(\Gamma_1(N) , L(n,A) )^{-}$ and $H^{i}_{P}( \Gamma_0(N), L(n,\chi;A) ) = H^{i}_{P}( \Gamma_0(N), L(n,\chi;A) )^{+} \oplus H^{i}_{P}(\Gamma_0(N) , L(n,\chi;A) )^{-}$. For a homomorphism $ \lambda: h_{k}(\Gamma_1(N);A) \rightarrow A$, define \begin{small}
\begin{align*}
H^{i}_{P}(\Gamma_1(N) , L(n,A) )^{\pm} [\lambda] &= \{ x \in H^{i}_{P}(\Gamma_1(N) , L(n;A) )^{\pm} : x|T = \lambda(T) x, ~ \forall ~T \in h_{k}(\Gamma_1(N);A) \}, \\
H^{i}_{P}(\Gamma_0(N) , L(n,\chi;A) )^{\pm} [\lambda] &= \{ x \in H^{i}_{P}(\Gamma_0(N) , L(n,\chi;A) )^{\pm} : x|T = \lambda(T) x, ~ \forall ~T \in h_{k}(\Gamma_0(N),\chi;A) \}.
\end{align*} \end{small} \begin{lemma}$\mathrm{(}$\cite[\S 6.3, (11)]{hida_93}, \cite[Theorem 3.2]{Kitagawa}$\mathrm{)}$ \label{multiplicity one of parabolic cohomology}
Let the notation be as above. For a field $ K $ of characteristic zero, we have
$ H^{i}_{P}(\Gamma_0(N) , L(n,\chi;K) )^{\pm} $ $($resp. $H^{i}_{P}(\Gamma_1(N) , L(n,K) )^{\pm})$ is a free module of rank one over $h_k(\Gamma_{0}(N),\chi;K) $ $($resp. $h_{k}(\Gamma_1(N);K))$.
As a consequence, $ H^{i}_{P}(\Gamma_0(N) , L(n,\chi;K) )^{\pm} [\lambda] = H^{i}_{P}(\Gamma_1(N) , L(n,K) )^{\pm})[\lambda]$ is a one dimensional vector space over $K$. \end{lemma}
Suppose $ 1/n! \in A $. Define a pairing $ [ ~ , ~] $ on $ L(n,\chi;A) \times L(n,\chi^{-1};A) \rightarrow A$ by \cite[\S 6.2, (2a)]{hida_93} \begin{small}
\begin{align*}
\Bigg[ \sum_{j=0}^{n} a_{j} X^{j}Y^{n-j}, \sum_{j=0}^{n} b_{j} X^{j}Y^{n-j} \Bigg] = \sum_{j=0}^{n} (-1)^j \binom{n}{j}^{-1} a_j b_{n-j}.
\end{align*} \end{small} Note that $ [(zX+Y)^n, (\bar{z}X+Y)^n] = \sum_{j=0}^{n} (-1)^j \binom{n}{j} z^{n-j} \bar{z}^{j} = (z-\bar{z})^{n}$. The above pairing is perfect as $n!$ is invertible in $A$. Write $Y_N :=Y_{\Gamma_{0}(N)}$ to ease the notation. By Poincar\'e duality, the above paring $ [ ~ , ~ ]$ induces a pairing (\cite[\S 6.2, (3a)]{hida_93}): \begin{small}
\begin{align*}
H^{1}_{c}(Y_{N} , \mathcal{L}(n,\chi;\mathbb{C}) ) \times
H^{1} (Y_{N} , \mathcal{L}(n,\chi^{-1};\mathbb{C})) \xrightarrow[]{\cup} H^{2}_{c}(Y_{N} , \mathcal{L}(n,\chi;\mathbb{C}) \times \mathcal{L}(n,\chi^{-1};\mathbb{C}))
\xrightarrow[]{[ ~ , ~ ]} H^{2}_{c}(Y_{N} , \mathbb{C} )\xrightarrow[]{\simeq} \mathbb{C},
\end{align*} \end{small}
where the first map is the wedge product and the last map is integrating the $2$-form on $ Y_N$. We continue to denote this paring by $[ ~ , ~]$. Further, for $x \in H^{1}(\Gamma_0(N), L(k-2,\chi;\mathbb{C}) ) $, define $ (x|\tau_N) (\gamma) := \tau_N \cdot x(\tau_N \gamma \tau_N^{-1}) $, for all $ \gamma \in \Gamma_0(N) $. This in turn defines an action of $ \tau $ on $ H^{1} (Y_{N} , \mathcal{L}(n,\chi;\mathbb{C})) \cong H^{1}(\Gamma_0(N), L(n,\chi;\mathbb{C})) $. Consider the pairing $\langle x, y \rangle := [x, y|\tau_N] = [x|\tau_N, y] $, where $ \tau_N = \begin{psmallmatrix} 0 & -1 \\ N & 0 \end{psmallmatrix} $. We need the following version of \cite[Theorem 5.16]{Hida3}: \begin{theorem}\label{choice of period and petterson innerproduct}
Let $ F = \sum a(n,F) q^n \in S_{k}(\Gamma_{0}(N),\chi; K)$ be a normalised eigenform and $ \mathcal{O}_{K} $ be ring of integers of $K$.
Fix a generator $ \xi^{\pm} $ of the eigenspace $\{ x\in H^{i}_{P}\big( \Gamma_0(N) , L(k-2, \chi ;\mathcal{O}_{K}) \big)^{\pm}: x|T(n) = a(n,F) x, ~ \forall \, n \in \mathbb{N} \}$. Then there exist complex periods $ \Omega_{F}^{\pm} $ such that
\begin{enumerate}[label=$(\roman*)$]
\item $ \delta(F)^{\pm} : = \delta(F) \pm \varepsilon \cdot \delta(F) = \Omega_{F}^{\pm} \xi^{\pm}$.
\item Further, we have
$2^{k} i^{k+1} N^{k/2-1} \langle F^\rho|\tau_N , F \rangle_N = \Omega_{F}^{+}\Omega_{F}^{-} \langle \xi^{+}, \xi^{-} \rangle$.
\end{enumerate} \end{theorem} \begin{proof}
The first assertion follows from Lemma~\ref{multiplicity one of parabolic cohomology} and the fact that $ H^{1}_{P}\big(\Gamma_0(N) ,L(k-2,\chi;\mathbb{C}) \big) = H^{1}_{P}\big(\Gamma_0(N), L(k-2,\chi;\mathcal{O}_{K}) \big) \otimes \mathbb{C}$.
Note that $ (\delta(F)| \tau_{N})
= (-1)^{k-2} N^{k/2-1} \delta(F| \tau_{N})$
and $ (\varepsilon \cdot \omega(F)) (z) = \varepsilon \omega(F)(-\bar{z}) = - F(-\bar{z}) (\bar{z} X + Y)^{k-2} d \bar{z} = - \bar{F}^{\rho}(z) (\bar{z} X + Y)^{k-2} d \bar{z} $. The assertion $(ii)$ was proved in \cite[Theorem 5.16]{Hida3} when $ F $ is primitive. The proof extends to this case, we omit the details.
\begin{comment}
To prove (2), we proceed as in \cite[Theorem 5.16]{Hida3} where the
above result \textcolor{red}{is stated for} $ F $ is a newform. From the definition
of the pairing, we have
\begin{small}
\begin{align*}
\Omega_{F}^{+}\Omega_{F}^{-} \langle \xi^{+},
\xi^{-} \rangle = \langle \Omega_{F}^{+} \xi^{+},
\Omega_{F}^{-} \xi^{-} \rangle
= \langle \delta(F)^{+}, \delta(F)^{-} \rangle
= \langle \delta(F) + \varepsilon \cdot \delta(F) , \
\delta(F) - \varepsilon \cdot \delta(F) \rangle.
\end{align*}
\end{small}
Note that $ (\delta(F)| \tau_{N})
= (-1)^{k-2} N^{k/2-1} \delta(F| \tau_{N})$
and $ (\varepsilon \cdot \omega(F)) (z) = \varepsilon \omega(F)(-\bar{z}) = - F(-\bar{z}) (\bar{z} X + Y)^{k-2} d \bar{z} = - \bar{F}^{\rho}(z) (\bar{z} X + Y)^{k-2} d \bar{z} $. Thus
\begin{small}
\begin{align*}
\langle \delta(F) + \varepsilon \cdot \delta(F) ,
\delta(F) - \varepsilon \cdot \delta(F) \rangle &=
[ (\delta(F) + \varepsilon \cdot \delta(F))|\tau_N ,
\delta(F) - \varepsilon \cdot \delta(F)] \\
&= [\delta(F)|\tau_N,-\varepsilon \cdot \delta(F)] +
[(\varepsilon \cdot \delta(F))|\tau_N, \delta(F)] \\
& = [ \varepsilon \cdot \delta(F) , \delta(F)|\tau_N] + [ \varepsilon \cdot \delta(F), \delta(F)|\tau_N] = 2 [\varepsilon \cdot \delta(F) , \delta(F)|\tau_N],
\end{align*}
\end{small}
where in the penultimate step we have used $ dz \wedge dz =0$,
$ d\bar{z} \wedge d\bar{z} =0$, and in the last step we have
used $ [x,y] = -[y,x] $ and $ [x|\tau_N, y] = [x,y|\tau_N] $. Using the definition of the pairing, we get
\begin{small}
\begin{align*}
[ \varepsilon \cdot \delta(F) , \delta(F)|\tau_N]
&=-(-1)^{k-2} N^{k/2-1} \int_{\mathbb{H}/\Gamma_{0}(N)} (F|\tau_N)(z) \bar{F}^{\rho}(z) [(\bar{z}X+Y)^{k-2}, (zX+Y)^{k-2}] d\bar{z} \wedge dz \\
& =- 2^{k-1} i^{k-1} N^{k/2-1} \int_{\mathbb{H}/\Gamma_{0}(N)}
(F\vert \tau_N)(z)\bar{F}^{\rho}(z) y^{k-2}dx dy = 2^{k-1} i^{k+1} N^{k/2-1} \langle F^\rho|\tau_N , F \rangle_N.
\end{align*}
\end{small}
This finishes the proof of the theorem.
\end{comment} \end{proof} \begin{remark}
By virtue of Lemma~\ref{multiplicity one of parabolic cohomology}, it follows that the choice of period in Theorem~\ref{choice of period and petterson innerproduct} is the same even if we replace the cohomology group $ H^{1}_{P}\big(\Gamma_0(N), L(k-2,\chi;\mathcal{O}_{K})\big) $ by $H^{1}_{P}\big(\Gamma_1(N), L(k-2;\mathcal{O}_{K}) \big)$. \end{remark} We next show that the periods chosen above have the property that the $ p $-adic $ L $-function $ L_{p}(F, \cdot) $ in Theorem~\ref{p-adic l-function of modular form} has power series expansion with coefficients in $ \mathcal{O}_K $. To do this, it suffices to show that the modular symbol attached to $ F $, when multiplied by $ 1/\Omega_{F}^{\pm} $, is integral (see \cite{MTT}, \cite{Kitagawa}).
Let $ \Delta = $ Div$(P^{1}(\mathbb{Q})) $ denote the group of divisors generated by $ \mathbb{P}^{1}(\mathbb{Q}) $. Let $ \Delta_0 $ denote the subgroup of $ \Delta $ consisting of divisors of degree zero. Recall that $ A $ is a ring with $ 1/2 \in A $ and $ \mathcal{M} $ is an $ A[\Gamma]$-module. For $ \gamma \in \operatorname{GL}_2(\mathbb{Q}) \cap \mathrm{M}_{2}(\mathbb{Z}) $ and $ \{r\} \in \mathbb{P}^{1}(\mathbb{Q}) $, define \begin{small}
\begin{align*}
(\gamma \cdot \Psi) (\{r\}) = \gamma \Psi(\{ \gamma^{-1} r\} ), \quad \forall ~
\Psi \in \mathrm{Hom}_{\mathbb{Z}}(\Delta, \mathcal{M}) \text{ or } \mathrm{Hom}_{\mathbb{Z}}(\Delta_0, \mathcal{M}).
\end{align*} \end{small}
Let Symb$_{\Gamma}(\mathcal{M}) := \text{Hom}_{\Gamma}(\Delta_0,\mathcal{M})$ be the group of modular symbols and BSymb$_{\Gamma}(\mathcal{M}) := \text{Hom}_{\Gamma}(\Delta,\mathcal{M})$ be the group of boundary modular symbols. There is a natural restriction map $\mathrm{res}:$ BSymb$_{\Gamma}(\mathcal{M}) \rightarrow$ Symb$_{\Gamma}(\mathcal{M})$. \begin{comment} For $ F \in S_{k}(\Gamma)$ and $ \{r\},\{s\} \in \mathbb{P}^1(\mathbb{Q}) $, define
\begin{small}
\begin{align*}
\Psi_{F}(\{r\}-\{s\}) := \int_{r}^{s} \omega(F) = \int_{r}^{s} F(z)(zX+Y)^{k-2} dz.
\end{align*}
\end{small}
Similarly, if $ F \in \bar{S}_{k}(\Gamma)$, we define $ \Psi_F $ by
$
\Psi_{F}(\{r\}-\{s\}) := \int_{r}^{s} \omega(F) = \int_{r}^{s} F(z)(\bar{z} X + Y)^{k-2} d\bar{z}
$.
It can be checked that $ \Psi_{F}(\{ \gamma r\} - \{ \gamma s\}) = \gamma \Psi_{F}(\{r\}-\{s\})~\forall ~ \gamma \in \Gamma $ and $ \{r\},\{s\} \in \mathbb{P}^1(\mathbb{Q}) $. Thus $ \Psi_{F} \in \mathrm{Hom}_{\Gamma}(\Delta_0, L(n, \mathbb{C})) = \text{Symb}_{\Gamma}(L(n, \mathbb{C}))$. \end{comment} There is an well-defined action of Hecke algebra on Symb$_{\Gamma}(\mathcal{M})$ (see \cite[\S 4]{MTT}).
\iffalse We next recall a result from \cite{AS} (see also \cite[\S 3.2]{Kitagawa}). \begin{proposition}\label{stevens proposition}
For every integer $ i \geq 0 $, we have following commutative diagram of Hecke modules
\begin{small}
\begin{equation}\label{les cohomology stevens}
\begin{tikzcd}
\cdots \arrow{r} &
H^{i-1}(\Gamma , \mathrm{Hom}_\mathbb{Z}(\Delta_0, L(n,A)) )
\arrow{r} \arrow{d} & H^{i}(\Gamma , L(n,A) ) \arrow{r} \arrow{d} & H^{i}(\Gamma , \mathrm{Hom}_\mathbb{Z}(\Delta, L(n,A)) ) \arrow{r} \arrow{d} & \cdots \\
\cdots \arrow{r} & H^{i}_{c}(Y_{\Gamma} , \mathcal{L}(n,A))
\arrow{r} & H^{i}(Y_{\Gamma} , \mathcal{L}(n,A)) \arrow{r}
& H^{i}(\partial(Y_{\Gamma}) , \mathcal{L}(n,A)) \arrow{r} &
\cdots ,
\end{tikzcd}
\end{equation}
\end{small}
where the vertical arrows are isomorphisms and \textcolor{red}{the map $ H^{i-1}(\Gamma , \mathrm{Hom}_\mathbb{Z}(\Delta_0, L(n,A)) ) \rightarrow H^{i}(\Gamma , L(n,A) ) $ is the connecting homomorphism}. \end{proposition} \begin{comment}
\begin{proof}
We have following commutative diagram of $ \Gamma $-modules \begin{small}
\[
\begin{tikzcd}
0 \arrow{r} & H^{0}(\mathbb{H} , \mathcal{L}(n,A)) \arrow{r} \arrow{d} & H^{0}(\partial \mathbb{H} , \mathcal{L}(n,A)) \arrow{r} \arrow{d} & H^{0}((\mathbb{H},\partial \mathbb{H}) , \mathcal{L}(n,A)) \arrow{r} \arrow{d} & 0\\
0 \arrow{r} & \mathrm{Hom}_{\mathbb{Z}}( \mathbb{Z}, L(n,A)) \arrow{r} & \mathrm{Hom}_{\mathbb{Z}}(\Delta, L(n,A)) \arrow{r} & \mathrm{Hom}_{\mathbb{Z}}(\Delta_0, L(n,A)) \arrow{r} & 0 ,
\end{tikzcd}
\]
\end{small}
where the vertical arrows are isomorphisms. Further, the bottom exact sequence is obtained by dualising the exact sequence $ 0 \rightarrow \Delta_0 \rightarrow \Delta \rightarrow \mathbb{Z}
\rightarrow 0$.
Note that the map $ P(X,Y) \mapsto \{ n \mapsto nP(X,Y) \} $ defines an
$ \Gamma $-equivariant isomorphism between $ L(n,A) $
and $ \mathrm{Hom}_{\mathbb{Z}}( \mathbb{Z}, L(n,A)) $. With this identification, we obtain
\begin{small}
\[
L(n,A) \cong \mathrm{Hom}_{\mathbb{Z}}( \mathbb{Z}, L(n,A)) \rightarrow
\mathrm{Hom}_{\mathbb{Z}}(\Delta, L(n,A))
\]
\end{small}
is given by $ P(X,Y) \mapsto \{ \sum n_{s} \{s\} \mapsto (\sum n_{s})P(X,Y) \} $. Using the Cartan-Leray spectral sequence and applying the functor
$ H^{i}(\Gamma, \cdot ) $, we obtain the proposition.
\end{proof} \end{comment} Taking $ i=1 $ and $ i = 0 $ respectively in \eqref{les cohomology stevens}, we get \begin{small}
\begin{align*}
H^{1}_{c}(Y_{\Gamma} , \mathcal{L}(n,A)) \cong
H^{0}(\Gamma , \mathrm{Hom}_{\mathbb{Z}}(\Delta_0, L(n,A)) )
= \text{Symb}_{\Gamma}(L(n, A)) \\
H^{0}(\partial(Y_{\Gamma}) , \mathcal{L}(n,A)) \cong
H^{0}(\Gamma , \mathrm{Hom}_{\mathbb{Z}}(\Delta, L(n,A)) ) = \text{BSymb}_{\Gamma}(L(n, A)).
\end{align*} \end{small}
Thus by commutative diagram~\eqref{les cohomology stevens}, we have the following exact sequence \begin{small}
\begin{align}\label{les stevens j=1}
0 \rightarrow H^{0}(\Gamma , L(n,A) ) \rightarrow
\text{BSymb}_{\Gamma}(L(n, A)) \xrightarrow{\text{ res }} \text{Symb}_{\Gamma}(L(n, A)) \xrightarrow{\Theta} H^{1}(\Gamma , L(n,A) ).
\end{align} \end{small} From Proposition~\ref{stevens proposition}, \textcolor{red}{we note that the map} $ H^{0}(\Gamma , \mathrm{Hom}_\mathbb{Z}(\Delta_0, L(n,A)))= \text{Symb}_{\Gamma}(L(n, A)) \xrightarrow{\Theta} H^{1}(\Gamma , L(n,A) ) $ is the connecting homomorphism. We now \textcolor{red}{describe} $\Theta$ explicitly. Let $ \Psi \in \text{Symb}_{\Gamma}(L(n, A))$. Then define $ \Phi \in \mathrm{Hom}_{\mathbb{Z}}(\Delta, L(n,A)) $ by $ \Phi(\{ s \}) = \Psi( \{s\} - \{\infty\} ),~ \forall ~ s \in \mathbb{P}^1(\mathbb{Q}). $ Clearly, the image of $\Phi $ under $ \mathrm{Hom}_{\mathbb{Z}}(\Delta, L(n,A)) \rightarrow \mathrm{Hom}_{\mathbb{Z}}(\Delta_0, L(n,A)) $ equals $ \Psi $, i.e., $ \Psi( \{s\} - \{ r \} ) = \Phi(\{ s \}) - \Phi(\{r \}) $. Further \begin{small}
\begin{align*}
(\Phi - \gamma \cdot \Phi)(\{ s \})
= \Psi( \{s\}- \{\infty\} ) - \gamma \Psi(\{ \gamma^{-1} s \} - \{\infty\}) = \Psi( \{s\} - \{\infty\}) - \Psi(\{ s \} - \{\gamma \infty\} ) = \Psi( \{ \gamma \infty\} - \{\infty\} ).
\end{align*} \end{small} Using the commutativity of \eqref{les cohomology stevens} and noting that $ H^{1}_{P}(Y_{\Gamma} , \mathcal{L}(n,A)) $ is the image of $ H^{1}_{c}(Y_{\Gamma}, \mathcal{L}(n,A)) $ in $H^{1}(Y_{\Gamma} , \mathcal{L}(n,A)) $ (see \cite[Page 90]{Kitagawa}), we get $\text{Image} (\Theta ) = H^{1}_{P}(\Gamma, L(n,A)) $. \fi By \cite[Theorem 4.3]{gs} (see also \cite[(9)]{Vatsal}), we have the following exact sequence of Hecke modules \begin{small}
\begin{align}\label{seq:parabolic and symbols}
0 \rightarrow H^{0}(\Gamma, L(n,A)) \rightarrow \text{BSymb}_{\Gamma}(L(n, A))
\xrightarrow{\text{res}} \text{Symb}_{\Gamma}(L(n, A)) \xrightarrow {\Theta}
H^{1}_{P}(\Gamma , L(n,A)) \rightarrow 0.
\end{align} \end{small} \iffalse is exact, where
$ \Theta: \text{Symb}_{\Gamma}(L(n, A)) \rightarrow H^{1}_{P}(\Gamma , L(n,A))$
is given by $ \Theta(\Psi)(\gamma) = \Psi( \{ \gamma \infty\} - \{\infty\} )$ $ \forall$ $\gamma \in \Gamma $ and $ \Psi \in \text{Symb}_{\Gamma}(L(n, A))$. Further, if $ F \in S_k(\Gamma) $ then it follows from above and Theorem~\ref{Eichler Shimura} that \begin{small}
\begin{align}\label{image of modular symbol under beta}
\Theta(\Psi_{F}) = \Psi_{F}(\{\gamma\infty\} - \{\infty\}) = - \int_{\gamma \infty}^{\infty} \omega(F) =-\delta(F).
\end{align} \end{small} Taking invariants (resp. co-invariants) under the action of $ \varepsilon $ in exact sequence \eqref{seq:parabolic and symbols} we get \begin{small}
\begin{align}\label{modular symbols and cohomology exact sequence}
0 \rightarrow H^{0}(\Gamma, L(n,A))^{\pm} \rightarrow \text{BSymb}_{\Gamma}(L(n, A))^{\pm}
\rightarrow \text{Symb}_{\Gamma}(L(n, A))^{\pm} \xrightarrow {\Theta}
H^{1}_{P}(\Gamma , L(n,A))^{\pm} \rightarrow 0.
\end{align} \end{small} \fi \begin{comment}
For every element $ \Psi \in \mathrm{Symb}_{\Gamma}(L(n,A))$
we define map $ \beta_{\Psi}: \Gamma \rightarrow L(n,A)$ given by
$ \beta_{\Psi}(\gamma) = \Psi( \{ \infty \} - \{ \gamma \infty \}) $.
By $ \Gamma $ linearity we have
\begin{align*}
\beta_{\Psi}(\gamma_1 \gamma_2) &= \Psi(\{\infty\} - \{\gamma_1\gamma_2 \infty \}) \\
& = \Psi(\{\infty\} - \{\gamma_1 \infty\}) + \Psi( \{\gamma_1 \infty\} - \{\gamma_1 \gamma_2 \infty\}) \\
& = \beta_{\Psi}(\gamma_1) + \gamma_{1} \beta_{\Psi}(\gamma_2).
\end{align*}
Thus $ \beta_{\Psi} $ defines a 1 co-cycle. We denote the cohomology
class of $ \beta_{\Psi} $ in $H^{1}(\Gamma,L(n,A)) $
by $ [\Psi] $. Thus we obtain
$ \beta: \mathrm{Symb}_{\Gamma}(L(n,A)) \rightarrow H^{1}(\Gamma,L(n,A))$ given by $ \beta(\Psi) := [\Psi] $.
\begin{lemma}\cite{Joel book}\label{Eichler shimura and modular symbols}
We have following \begin{enumerate}[label=(\roman*)]
\item The image of $ \beta $ lies in the parabolic cohomology, i.e., $\beta(\mathrm{Symb}_{\Gamma}(L(n,\mathbb{C})) ) \subset H^{1}_{p}(\Gamma,L(n,\mathbb{C})) $ and is Hecke equivariant.
The kernel of $ \beta $ is $ \mathrm{BSymb}_{\Gamma}(L(n,\mathbb{C})) $
\item We have following commutative diagram
\[
\begin{tikzcd}
S_{k}(\Gamma) \oplus \bar{S}_{k}(\Gamma) \arrow{r}{\delta} \arrow[swap]{dr}{\Psi} & H^{1}_{p}(\Gamma,L(n,\mathbb{C})) \\
& \mathrm{Symb}_{\Gamma}(L(n,\mathbb{C})) \arrow{u}{\beta}
\end{tikzcd}
\]
\end{enumerate}
\end{lemma}
\begin{proof}
The commutativity of diagram follows from the definitions.
To show $ [\Psi] $ lies in $ H^{1}_{p}(Y_{\Gamma},L(n,\mathbb{C}))$ we need to check that $[\Psi] $ lies in the kernel of $ \operatorname{res}: H^{1}(\Gamma,L(n,A))
\rightarrow \oplus_{s \in \Gamma \setminus \mathbb{P}^1(\mathbb{Q}) }
H^{1}(\Gamma_s,L(n,A)) $. Fix $ s \in \Gamma \setminus \mathbb{P}^1(\mathbb{Q}) $
and $ \gamma \in \Gamma_{s} $. Then
\begin{align*}
\beta_{\Psi}(\gamma) & = \Psi_{f}(\{\infty\} -\{ \gamma\infty\}) \\
& = \Psi_{f}(\{\infty\} - \{s\}) + \Psi_{f}(\{s\} -\{\gamma \infty\}) \\
&= \Psi_{f}(\{\infty\} - \{s\}) + \Psi_{f}(\{\gamma s\} - \{\gamma \infty\}) \\
& = (\operatorname{id}-\gamma)\Psi_{f}(\{\infty\} - \{s\}).
\end{align*}
Thus $ \beta_{\Psi}(\gamma) $ represents a boundary class. Hence
$ \beta_{\Psi}(\gamma) $ is zero in $ H^{1}(\Gamma_s,L(n,A)) $.
\end{proof}
By Eichler-Shimura we have $ \delta $ is an isomorphism hence the map $ \beta $
is surjective and the map $ \Psi $ is injective. \end{comment} \iffalse Let $ F(z) = \sum a(n,F)q^n \in S_k(\Gamma_1(N))$ be an eigenform. Let $ \phi_F: h_{k}(\Gamma_1(N); A) \rightarrow A$ be the homomorphism induced by $ T(n) \mapsto a(n,F) $. Set \begin{small}
\begin{align*}
\text{Symb}_{\Gamma}(L(n,A))^\pm[\phi_F] &:= \{ \Psi \in
\text{Symb}_{\Gamma}(L(n,A))^\pm : \Psi|T(n) =a(n,F) \Psi, ~ \forall ~ n \in \mathbb{N}\}
\end{align*} \end{small}
\begin{lemma}\label{lem: isomorphism of eigen spaces}
\textcolor{red}{ Let $ F(z) = \sum a(n,F)q^n \in S_k(\Gamma_1(N))$ be an eigenform with the residual Galois representation at $ p $ is irreducible. Then
$\Theta: \mathrm{Symb}_{\Gamma_{1}(N)}(L(n,A))^\pm [\phi_F] \rightarrow H^{1}_P(\Gamma_{1}(N), L(n,A))^\pm[\phi_F] $ is an isomorphism for $ A = K, \mathcal{O}_K $. Need to say the key idea is due to Vatsal. } \end{lemma} \begin{proof}
To simplify the notation let $ \Gamma = \Gamma_{1}(N) $. First consider the case $ A = K $. By \cite[Proposition 3.3]{Kitagawa} and Lemma~\ref{multiplicity one of parabolic cohomology} we have $\text{Symb}_{\Gamma}(L(n,A))^\pm[\phi_F]$ and $H^{1}_P(\Gamma, L(n,A))^\pm[\phi_F] $ are $ 1 $ dimensional vector spaces over $ K $. Since $ \Theta(\Psi_{F}) = -\delta(F) $, we get $ \Theta \neq 0 $. Hence the lemma follows in the case $ A = K $.
Next consider the case $ A = \mathcal{O}_K $. It follows from the previous case that $ \Theta $ is injective. We need to $ \Theta $ is surjective. Let $ x \in H^{1}_P(\Gamma, L(n,\mathcal{O}_K))^{\pm}[\phi_F]$. From the exact sequence \eqref{modular symbols and cohomology exact sequence} there exists $ y \in \mathrm{Symb}_\Gamma(L(n,\mathcal{O}_K)) $ such that $ \Theta(y) = x $. Again from the previous case there exists $ z \in H^{1}_P(\Gamma, L(n,K))^{\pm}[\phi_F]$ with $ \Theta(z) = x $. Thus $ y - z \in \mathrm{BSymb}_\Gamma (L(n,K)) $. For every prime $ \ell $ consider the integral Hecke operator $\eta_{\ell} := T_{\ell} - 1 - \ell \langle \ell \rangle $ where $ T_\ell, \langle \ell \rangle$ are as in \cite[\S 5.2]{DS05}. By \cite[Lemma 6.9 b]{gs}, we have $ \eta_\ell(y-z) =0 $. Clearly $ \eta_{\ell} y \in \mathrm{Symb}_\Gamma(L(n,\mathcal{O}_K))$ and $ \eta_{\ell} z = (a(\ell,F) - \ell-1) z$. By the assumption $ \bar{\rho}_{F} $ is irreducible there exist a prime $ \ell$ congruent to $ 1 $ modulo $ Np $ such that $ (a(\ell,F) - \ell-1) \in \mathcal{O}_K^\times$. Then for such a prime $ \ell $, we obtain $ z = (a(\ell,F) - \ell-1)^{-1} \eta_{\ell} y \in \mathrm{Symb}_\Gamma(L(n,\mathcal{O}_K)) \cap \mathrm{Symb}_\Gamma(L(n,K))[\phi_{F}]$ and $ \Theta(z) =x $. Hence $ \Theta $ is surjective. This finishes the proof. \end{proof} \fi Following \cite{gs}, \cite{Kitagawa} and \cite{Vatsal} we now show in Theorem~\ref{period and integral measure} that the measure in Theorem~\ref{p-adic l-function of modular form} is $ \mathcal{O}_K $-valued with the choice of periods $ \Omega_{F}^{\pm} $ as specified in Theorem~\ref{choice of period and petterson innerproduct}.
\begin{theorem}\label{period and integral measure}
Let $ F \in S_{k}(\Gamma_{0}(N), \chi; \mathcal{O}_K)$ be a normalised $ p $-ordinary eigenform with the residual Galois representation at $ p $ is irreducible.
Let $ \Omega_{F}^{\pm} $ be the periods as defined in Theorem~\ref{choice of period and petterson innerproduct}. There exists an $ \mathcal{O}_K $-valued measure $ \mu_{F} $ such that for every finite order character $ \phi $ of conductor $ J_0p^v $ with $ p \nmid J_0 $ on $ Z_{L} := \mathbb{Z}_{p}^{\times} \times (\mathbb{Z}/L\mathbb{Z})^\times $ and $ 0 \leq j \leq k-1 $, we have
\begin{small}
\begin{align*}
\mu_{F}(x_p^j \phi) = \int_{Z_L} x_p^j \phi =\frac{1}{u_{F}^{v}} \left( 1 -
\frac{\phi(p) \chi(p) p^{k-2-j}}{u_{F}}\right)
\left( 1 - \frac{\bar{\phi}(p)p^{j}}{u_{F}}\right)
\frac{ \mathrm{cond}(\phi)^{j+1}}{(-2\pi i)^{j+1} } \frac{j!}{G(\phi)}
\frac{L(j+1,F,\phi)}{
\Omega_{F}^{\mathrm{sgn}((-1)^j \phi(-1))} }.
\end{align*}
\end{small}
\end{theorem} \begin{proof}
By \cite[1.6]{Vatsal}, there exist $ \Delta_F^\pm \in \text{Symb}_{\Gamma_{1}(N)}(L(k-2,\mathcal{O}))$ with $\Theta(\Delta^{\pm}(F)) = \delta(F)^{\pm}/\Omega_{F}^\pm$.
Let $ \mu_F $ be the measure attached to the modular symbol $ \Delta_F^{\pm}$ as defined in \cite[(4.16),(4.17)]{gs} (See \cite[\S 4.2]{Kitagawa}). For the interpolation formula, see \cite[(11)]{Vatsal} and \cite[Theorem 4.18]{gs} (See also \cite[Theorem 4.8]{Kitagawa}).
That the measure $ \mu_{F} $ is an $ \mathcal{O}_K $-valued measure follows from \cite[Lemma 4.3]{Kitagawa}.
\begin{comment}
We now consider the case $ m > 1 $. Note that
$ F \vert \iota_{m} = \sum_{0 < d \mid m} c_d F|[d], $
where $ c_d \in \mathcal{O} $. By \cite[Lemmas 4.1- 4.4]{Kitagawa} it is enough to show $ \Psi_{F|\iota_{m}}^{\pm}/ \Omega_{\tilde{F}}^{\pm} $ is an $\mathcal{O}$ -valued modular symbol. Since $ \Psi_{F \vert \iota_{m}} = \sum_{0 < d \mid m} c_d \Psi_{F|[d]} $, it is enough to prove for every $ d \mid m $ that $ \Psi_{F|[d]}^{\pm} / \Omega_{F}^{\pm} $ is an integral modular symbol.
Note that
\begin{align*}
\Psi_{F|[d]} (\{r\} - \{s\}) &= \int_{r}^{s} (F|[d])(z) (zX+Y)^{k-2} dz
= \begin{psmallmatrix}1 & 0 \\ 0 & d \end{psmallmatrix} d^{1-k} \int_{r/d}^{s/d} F(z) (zX+Y)^{k-2} dz \\
& = d^{1-k} \begin{psmallmatrix} 1 & 0 \\ 0 & d \end{psmallmatrix} \Psi_{F}(\{r/d\} - \{s/d\}).
\end{align*}
Thus $ \Psi_{F|[d]}^{\pm} (\{r\} - \{s\}) = d^{1-k} \begin{psmallmatrix} 1 & 0 \\ 0 & d \end{psmallmatrix} \Psi_{F}^{\pm}(\{r/d\} - \{s/d\}) $. Since $ \Psi_{F}^{\pm}/\Omega_{F}^{\pm} \in \text{Symb}_{\Gamma_{0}(N)}(\mathcal{O}) $, we get
$ \Psi_{F|[d]}^{\pm}/ \Omega_{F}^{\pm} $ is also integral.
\end{comment} \end{proof}
\begin{definition}\textbf{(Module of congruences)}
Let $ R $ be a finite, flat and reduced algebra over $ \mathcal{O}_{K} $.
Moreover, assume that we are given
a map $ \lambda: R \rightarrow \mathcal{O}_{K} $ such that it induces a $K$-algebra decomposition
$R \otimes_{ \mathcal{O}_{K}} K \cong K \oplus X$.
Let $ 1_{\lambda} $ be the idempotent {corresponding} to the first summand. Put $\mathfrak{a}$ = Ker($R \rightarrow X) = 1_\lambda R \cap R$, $S$ = Im$(R \rightarrow X)$ and $\mathfrak{b}$ = ker($\lambda$). The module of congruences $ C_0(\lambda) $ is defined by
\begin{align*}
C_0(\lambda) = (R/\mathfrak{a}) \otimes_{R,\lambda} \mathcal{O}_{K} \cong \frac{R}{\mathfrak{a}\oplus \mathfrak{b}} \cong \mathcal{O}_{K}/\lambda(a) \cong
1_{\lambda} R/\mathfrak{a}.
\end{align*} \end{definition}
For $ F \in S_k(N,\chi) $ and $ \Omega_{F}^\pm $ as in Theorem~\ref{period and integral measure}, we now determine $ \langle F^\rho|\tau_{N} , F \rangle_N/\Omega_F^{-}\Omega_F^{-} $ explicitly up to a $ p $-adic unit. Thus $ \Omega_{F}^\pm $ satisfy all the desired properties we mentioned at the beginning of the \S~\ref{subsection: relation periods}. The following theorem is a generalisation of \cite[Theorem 5.20]{Hida3}. \begin{theorem}\label{c(f) and petterson} Let $ p $ be odd and $ k \geq 3 $. Let $ F \in S_{k}(\Gamma_{0}(N),\chi;\mathcal{O}_K) $ be a normalised $ p $-ordinary eigenform. We assume $ p >3 $, if $ \Gamma_{0}(N)/\{\pm1\} $ has non-trivial torsion elements. Let $ \bar{\rho}_{F} $ be the residual representation of $ F $ and $ \xi^{\pm}, \Omega_{F}^{\pm} $ be as in Theorem~\ref{period and integral measure}. We assume that
\begin{enumerate}[label=$(\roman*)$]
\item The homomorphism $ \phi_F: h_{k}(\Gamma_{0}(N), \chi; \mathcal{O}_K) \rightarrow \mathcal{O}_K$
sending $ T(n) \mapsto a(n,F)$ induces the splitting $ h_{k}(\Gamma_{0}(N), \chi; \mathcal{O}_K) \cong \mathcal{O}_K \oplus X$.
\item $\bar{\rho}_{F}$
is absolutely irreducible.
\item $ F $ is $ p $-distinguished i.e. the residual representation $ \bar{\rho}_F $ satisfies $ \bar{\rho}_{F}\vert_ {G_{p}} \cong \begin{psmallmatrix}
\epsilon_{p} & \ast \\ 0 & \delta_{p}
\end{psmallmatrix}$ with $ \epsilon_{p} \neq \delta_{p}$.
\end{enumerate}
\begin{small}
\begin{align*}
\text{Then we have } \quad \frac{c(F)}{\langle \xi^{+}, \xi^{-}\rangle} = c(F) \frac{\Omega_{F}^{+}\Omega_{F}^{-}}{2^{k} i^{k+1}N^{k/2-1} \langle F^\rho|\tau_{N} , F \rangle_N} \in \mathcal{O}_K^{\times}.
\end{align*}
\end{small}
\end{theorem} \begin{proof}
Let $ n =k-2 $. As $ f $ is $ p $-ordinary, by \cite[Theorem 6.25]{Hidaadjoint} we obtain that the pairing $ \langle ~, ~ \rangle $ is perfect on $ 1_F H^{1}_{P}\big(\Gamma_0(N),L(n,\chi;\mathcal{O}_K)\big) $. Thus by \cite[Corollary 6.24]{Hidaadjoint}, we have
\begin{small}
\begin{align*}
|\langle \xi^{+}, \xi^{-}\rangle|_{p}^{-[K:\mathbb{Q}_p]} = \Bigg\lvert \frac{2^{k} (-i)^{k}\Omega_{F}^{+}\Omega_{F}^{-}}{N^{k/2-1} \langle F^\rho|\tau_{N} , F \rangle_N}\Bigg \rvert_{p}^{-[K:\mathbb{Q}_p]} = \Big\lvert \frac{L_{F}}{L^{F}} \Big\rvert.
\end{align*}
\end{small}
Here $ L_{F} $ and $ L^{F} $ are defined as follows: Let $ L = \text{image} ( H^{1}_{P}(\Gamma_{0}(N), L(n, \chi; \mathcal{O}_K ))^{+} \hookrightarrow H^{1}_{P}(\Gamma_{0}(N), L(n, \chi; K ))^{+})$. Then $L_{F} = 1_F L$ and $L^{F}$ is given by the intersection $L_{F} \cap L$ in $ H^{1}_{P}(\Gamma_{0}(N), L(n, \chi; K ))^+ $.
Note that we can remove the assumption $ p \nmid \varphi(N) $ in \cite[Corollary 6.24]{Hidaadjoint} by assuming $ p > 3 $ if $ \Gamma_{0}(N)/\{\pm \mathrm{I}\} $ has non-trivial torsion.
By \cite[Page 229, Lines -7,-4]{Hidaadjoint}, we get
$ L_{F}/L^{F} \cong
C_0(\phi_F)$.
Here, the hypotheses $(ii), (iii)$ are needed to show that the local ring of Hecke algebra $ h_{k}(\Gamma_{0}(N), \chi; \mathcal{O}_K) $ through which $ \phi_F $ factors is reduced. Since $ c(F) $ gives the exact denominator of $ C_0(\phi_F) $, we obtain the theorem. \end{proof}
We end this section by giving a criteria when the map $ \phi_F $ in Theorem~\ref{c(f) and petterson}$(i)$ induces the splitting $h_{k}(\Gamma_{0}(N), \chi; \mathcal{O}_K) \cong \mathcal{O}_K \oplus X$. \begin{lemma}\label{splitting criteria hecke algebra}
Let $ F \in S_{k}(\Gamma_{0}(N),\chi;\mathcal{O}_K) $ be a normalised $ p $-ordinary eigenform. Assume either of the following holds
\begin{enumerate}[label=$(\roman*)$]
\item $ F $ is primitive or $F$ is the $ p $-stabilization of a primitive form.
\item $S_{k}(\Gamma_{0}(N),\chi;K)$ is a semisimple $h_{k}(\Gamma_{0}(N), \chi; K) $ module.
\end{enumerate} Then the homomorphism $ \phi_F: h_{k}(\Gamma_{0}(N), \chi; \mathcal{O}_K) \rightarrow \mathcal{O}_K$ sending $ T(n) \mapsto a(n,F)$ induces the splitting $ h_{k}(\Gamma_{0}(N), \chi; \mathcal{O}_K) \cong \mathcal{O}_K \oplus X$. \end{lemma} \begin{proof}
If $(i)$ holds, then the lemma follows from \cite[Theorem 1.13]{Vatsal}. In case $(ii)$, we get $ h_{k}(\Gamma_{0}(N), \chi; K) \cong \mathrm{Hom}(S_{k}(\Gamma_{0}(N),\chi;K),K) \cong \mathrm{ker}(\phi_{F}) \oplus K$. From this it is easy to deduce the required splitting. \end{proof} \begin{remark}\label{rem: CE}
If $ k=2 $ and $ N $ is cube free, then the condition $(ii)$ of Lemma~\ref{splitting criteria hecke algebra} holds \cite[Theorem 4.2]{CE}, that is $S_{k}(\Gamma_{0}(N),\chi;K)$ is a semisimple. Moreover, if $ k > 2 $ and $ N $ is cube free then the condition $(ii)$ of Lemma~\ref{splitting criteria hecke algebra} holds when the Tate conjecture
\cite[\S 1]{CE} is true \cite[Theorem 4.2]{CE}. \end{remark}
We now sketch an argument to further relax the splitting of the Hecke algebra in \eqref{splitting of hecke algebra}. Let $ f$, $g$ be as in Lemma~\ref{special value f,g and trivial character} and $ h $ be a primitive form as in \S \ref{section: simplyfying rankin}. Assume $ f $ satisfies the hypotheses $(ii)$ and $(iii)$ of Theorem~\ref{c(f) and petterson}. Let $ \chi $ be a Dirichlet character of conductor $ m^\delta $ with $ m $ as defined in \eqref{definition m}. By choosing $ \delta $ sufficiently large, we may assume $ f|\chi , g|\chi^{-1}, h|\chi^{-1}$ are primitive \cite[Theorem 4.1]{Atkin}. Hence $ \mu_{f|\chi \times g|\chi^{-1}}(x_p^{j} \phi ) \equiv \mu_{f|\chi \times h|\chi^{-1}}(x_p^{j} \phi) \mod \pi$ by Lemma~\ref{congruence of measures}. We claim that there exists a $ p $-adic unit $ u $ such that $\mu_{\tilde{f}\times *|\iota_{m}}(x_p^{j} \phi) = u \mu_{f|\chi \times *|\chi^{-1}}(x_p^{j} \phi )$ with $ \ast \in \{g,h\}$. Assume the claim for a moment. Then we have \begin{align*}
\mu_{\tilde{f}\times h|\iota_{m}}(x_p^{j} \phi) \equiv \mu_{\tilde{f}\times g|\iota_{m}}(x_p^{j} \phi) = u \mu_{f|\chi \times g|\chi^{-1}}(x_p^{j} \phi ) \equiv u \mu_{f|\chi \times h|\chi^{-1}}(x_p^{j} \phi) \mod \pi, \end{align*}
for $ u \in \mathcal{O}_K^\times $. As a consequence, we may work with the $p$-adic $ L $-functions $\mu_{f|\chi \times h|\chi^{-1}}(x_p^{j} \phi) $ and $\mu_{f|\chi \times g|\chi^{-1}}(x_p^{j} \phi) $ instead of $\mu_{\tilde{f}\times h|\iota_{m}}$ and $\mu_{\tilde{f}\times g|\iota_{m}}$ and results of \S\ref{section: simplyfying rankin} and \S\ref{sec: Congruences of p-adic L functions} continue to hold. Note that the existence of $ p $-adic $ L $-functions $\mu_{f|\chi \times h|\chi^{-1}}(x_p^{j} \phi) $ and $\mu_{f|\chi \times g|\chi^{-1}}(x_p^{j} \phi) $ do not assume the splitting of Hecke algebra in \eqref{splitting of hecke algebra}. However for simplicity, we keep this assumption $ (ii) $ on the splitting of Hecke algebra in Theorem~\ref{congruence main conjecture intro}.
We now sketch a proof of the claim when $ \ast = g $ and the case $ \ast = h $ is similar. Let $ \phi: \mathbb{Z}_p^\times \rightarrow \mathbb{C}_p $ be a finite order character with $ \xi_1\xi_{2}\omega_{p}^{1-l} \phi (p) =0 $. Thus $ g|\phi^{-1} = E_{l}(\xi_{1}\omega_{p}^{1-l}\phi^{-1}, \xi_{2}\phi^{-1})$ and $ g|\chi^{-1}\phi^{-1} = E_{l}(\xi_{1}\omega_{p}^{1-l}\phi^{-1}\chi^{-1}, \xi_{2}\phi^{-1}\chi^{-1})$. From Lemma~\ref{Eisenstein series Hida}$(i)$, it follows that \begin{small}
\begin{align*}
\mu_{(g|\chi^{-1})^{\rho}}( \phi^{-1})|_{l}\tau_{M_0\mathrm{cond}(\chi)^2p^{\beta}} = W((g|\chi^{-1})^{\rho}|(\phi^{-1})) \mu_{g|\chi^{-1}}(\phi) = \frac{W((g|\chi^{-1})^{\rho}|(\phi^{-1}))}{W(g^{\rho}|(\phi^{-1}))} \mu_{g^{\rho}}(\phi^{-1})|_{l}\tau_{M_0p^{\beta}}|\chi^{-1}.
\end{align*} \end{small}
By \eqref{root number of g definition}, we have $W((g|\chi^{-1})^{\rho}|(\phi^{-1}))/W(g^{\rho}|(\phi^{-1}))$ is a $ p $-adic unit. Let $ \Sigma' $ be the set of primes dividing $ m $. Thus by the Rankin-Selberg method \cite[Lemma 1]{Shimura1}, we have \begin{small} \begin{align}\label{eq: rankin simple general}
D_{M_0\mathrm{cond}(\chi)Np}(l+j,f_{0}|\chi ,
\mu_{(g|\chi^{-1})^{\rho}}(\phi^{-1})|_{l}\tau_{M_0\mathrm{cond}(\chi)^2p^{\beta}})= D^{\Sigma'}_{M_0Np}(l+j,f_{0},
\mu_{g^{\rho}}(\phi^{-1})|_{l}\tau_{M_0p^{\beta}}) \end{align} \end{small}
up to a $ p $-adic unit and here superscript denotes the Euler factors at primes $ \ell \in \Sigma'$ are omitted. Since $ f|\chi $ is primitive, by Lemma~\ref{splitting criteria hecke algebra}, $ f_0|\chi $ satisfies the hypothesis $(i)$ of Theorem~\ref{c(f) and petterson}. Also $ f_0|\chi $ satisfies the hypotheses $ (ii) $ and $ (iii) $ of Theorem~\ref{c(f) and petterson}. Let $\Omega_{f_0}^\pm $ be the periods as in Theorem~\ref{choice of period and petterson innerproduct}. Now it follows from Theorems~\ref{padic rankin}, \ref{c(f) and petterson} and \eqref{eq: rankin simple general} that \begin{small}
\begin{align*}
\mu_{f|\chi \times g|\chi^{-1}}(x_p^{j} \phi ) & = u t p^{\beta j} p^{\beta l /2}
\chi(p)^{1-\beta} a(p,f_0)^{\beta-1} \frac{D_{MNp\mathrm{cond}(\chi)}(l+j,f_{0}|\chi,
\mu_{(g|\chi^{-1})^{\rho}}(\phi^{-1})|_{l}\tau_{Jp^{\beta}})}{(2i)^{k+l+2j} \pi^{l+2j+1} \Omega^{+}_{f_0|\chi}\Omega^{-}_{f_0|\chi}} \\
& = u' t p^{\beta j} p^{\beta l /2}
\chi(p)^{1-\beta} a(p,f_0)^{\beta-1} \frac{D^{\Sigma'}_{M_0Np}(l+j,f_{0},
\mu_{g^{\rho}}(\phi^{-1})|_{l}\tau_{M_0p^{\beta}})}{(2i)^{k+l+2j} \pi^{l+2j+1} \Omega^{+}_{f_0|\chi}\Omega^{-}_{f_0|\chi}}
\end{align*} \end{small}
where $ u,u' $ are $ p $-adic unit. Following the argument of \cite[Lemma 9.6]{SZ}, we can show that $ \Omega^{+}_{f|\chi} \Omega^{-}_{f|\chi}$ differs from $ \Omega^{+}_{f} \Omega^{-}_{f}$ by a $ p $-adic unit. Now the claim follows from \eqref{eq: rankin as product trivial char} and \eqref{eq: rankin as product nontrivial char}.
\begin{comment} \begin{lemma}\label{p-adic rankin special case} Let $ f$, $g$ be as in Lemma~\ref{special value f,g and trivial character}. Assume $ f $ satisfies the hypotheses $(ii)$ and $(iii)$ of Theorem~\ref{c(f) and petterson}. Let $\Omega_{f_0}^\pm $ be the periods as in Theorem~\ref{choice of period and petterson innerproduct}.
Let $ \chi $ be a Dirichlet character, such that $ f|\chi$ and $ g|\chi$ are primitive. Let $ \Sigma $ be the set of primes dividing $ \mathrm{cond}(\chi) $ and $ p \not \in \Sigma $. Then up to a $ p $-adic unit,
\begin{align}\label{eq: rankin simple special}
\mu_{f|\chi \times g|\chi}(x_p^{j} \phi) & = t p^{\beta l/2} p^{\beta j} a(p,f_0)^{1-\beta} \frac{D^{\Sigma}_{M_0Np}(l+j,f_{0},
\mu_{g^{\rho}}(\phi^{-1})|_{l}\tau_{M_0p^{\beta}})}{(2i)^{k+l+2j} \pi^{l+2j+1} \Omega^{+}_{f_0} \Omega^{-}_{f_0}}
\end{align} for every non-trivial finite order character $ \phi: \mathbb{Z}_p^\times \rightarrow \mathbb{C}_p^\times $ with $ \phi \xi_1\xi_2\omega_{p}^{1-l} (p) =0 $ and $ 0 \leq j \leq k-l-1 $. \end{lemma}
\begin{proof}
By Lemma~\ref{splitting criteria hecke algebra}, $ f_0|\chi $ satisfies the hypothesis $(i)$ of Theorem~\ref{c(f) and petterson}. Also $ f_0|\chi $ satisfies the hypotheses $ (ii) $ and $ (iii) $ of Theorem~\ref{c(f) and petterson}. Now it follows from Theorems~\ref{padic rankin}, \ref{c(f) and petterson} that
\begin{small}
\begin{align}\label{eq: rankin simple general}
\mu_{f|\chi \times g|\chi}(x_p^{j} \phi ) = u t p^{\beta j} p^{\beta l /2}
\chi(p)^{1-\beta} a(p,f_0)^{1-\beta} \frac{D_{MNp\mathrm{cond}(\chi)}(l+j,f_{0}|\chi,
\mu_{(g|\chi)^{\rho}}(\phi^{-1})|_{l}\tau_{Jp^{\beta}})}{(2i)^{k+l+2j} \pi^{l+2j+1} \Omega^{+}_{f_0|\chi}\Omega^{-}_{f_0|\chi}},
\end{align} \end{small}
with $ u $ as $ p $-adic unit. Following \cite[Lemma 9.6]{SZ}, we can show $ \Omega^{+}_{f|\chi} \Omega^{-}_{f|\chi}$ differs from $ \Omega^{+}_{f} \Omega^{-}_{f}$ by a $ p $-adic unit. Then from \eqref{eq: rankin simple general}, it suffices to show
\begin{align*}
D_{M_0\mathrm{cond}(\chi)Np}(l+j,f_{0}|\chi ,
\mu_{g^{\rho}}(\chi\phi^{-1})|_{l}\tau_{M_0\mathrm{cond}(\chi)^2p^{\beta}})= D^{\Sigma}_{M_0Np}(l+j,f_{0},
\mu_{g^{\rho}}(\phi^{-1})|_{l}\tau_{M_0p^{\beta}})
\end{align*}
up to a $ p $-adic unit. Since $ g|\chi $ is primitive and $ \xi_1\xi_{2}\omega_{p}^{1-l} \phi (p) =0 $, it follows from Lemma~\ref{Eisenstein series Hida} that \begin{small} \begin{align*}
\mu_{(g|\chi)^{\rho}}(\chi \phi^{-1})|_{l}\tau_{M_0\mathrm{cond}(\chi)^2p^{\beta}} = W((g|\chi)^{\rho}|(\phi^{-1})) \mu_{g|\chi}(\phi) = \frac{W((g|\chi)^{\rho}|(\phi^{-1}))}{W(g^{\rho}|(\phi^{-1}))} \mu_{g^{\rho}}(\phi^{-1})|_{l}\tau_{M_0p^{\beta}}|\chi. \end{align*} \end{small}
By \eqref{root number of g definition}, we have $W((g|\chi)^{\rho}|(\phi^{-1}))/W(g^{\rho}|(\phi^{-1}))$ is $ p $-adic unit. Now \eqref{eq: rankin simple special} follows from Rankin-Selberg method \cite[Lemma 1]{Shimura1}. \end{proof} \end{comment} \iffalse \textcolor{red}{For a finite set of primes $ \Sigma $, define the $ p $-adic $ L $-function we have
\begin{align*}
\mu^{\Sigma}_{F \times G}(x_p^{j} \phi ) = c(f_0) t p^{\beta l/2} p^{\beta j} p^{(2-k)/2}
a(p,F_0)^{1-\beta} \frac{D^{\Sigma}_{JNp}(l+j,F_{0},
\mu_{G^{\rho}}(\phi^{-1})|_{l}\tau_{Jp^{\beta}})}{(2i)^{k+l+2j} \pi^{l+2j+1}
{ \langle F_{0}^{\rho}|_{k}\tau_{Np} , F_{0}\rangle_{Np}}}, \end{align*}}
\begin{lemma}\label{p-adic rankin twist and sigma}
\textcolor{red}{Assume the set up as in Lemma~\ref{p-adic rankin special case} and let $ p \nmid JN $. Let $ \chi $ be a quadratic character such that $ F|\chi \in S_{k}(\Gamma_{0}(N\mathrm{cond}(\chi)^2),\eta\chi^2)$ and $ G|\chi^{-1} \in S_{l}(\Gamma_{0}(J\mathrm{cond}(\chi)^2),\psi\chi^{-2})$ are primitive. Let $ \Sigma $ be the set of primes dividing $ \mathrm{cond}(\chi) $ and $ p \not \in \Sigma $. Then up to $ p $-adic unit, we have
\begin{align*}
\mu^{\Sigma}_{F \times G}(x_p^{j} \phi ) = \mu_{F|\chi \times G|\chi}(x_p^{j} \phi) .
\end{align*} } \end{lemma} \begin{proof}
\textcolor{red}{Note that
\begin{align*}
L(s, F|\chi, G|\chi \phi) = \sum_{n=1}^{\infty} a(n,F|\chi) a(n,G|\chi\phi) = \sum_{n=1}^{\infty} \chi^2(n) a(n,F) a(n,G|\phi) =
L^{\Sigma}(s, F, G| \phi).
\end{align*}
A similar check shows that $ D_{JNp\mathrm{cond}(\chi)}(s, F|\chi, G|\chi \phi) = D_{JNp}^{\Sigma}(s, F, G\phi) $. Since $ p \nmid \mathrm{cond}(\chi) $ we have $ W(G) $ and $ W(G|\chi) $ differ by a $ p $-adic unit. By a similar argument as in \cite[Lemma 9.6]{SZ} we may show that $ \Omega^{+}_{F|\chi} \Omega^{-}_{F|\chi}$ differs from $ \Omega^{+}_{F} \Omega^{-}_{F}$ by a $ p $-adic unit. Now the lemma follows from Lemma~\ref{p-adic rankin special case}.} \end{proof} \fi
\subsection{Congruences of the $p$-adic $L$-functions}\label{sec: Congruences of p-adic L functions}
Throughout \S\ref{sec: Congruences of p-adic L functions}, we choose the periods $ \Omega_{\tilde{f}_0}^{\pm} $ as in Theorem~\ref{choice of period and petterson innerproduct} and the measure $ \mu_F(\tilde{f}_0, \cdot) $ as defined in Theorem~\ref{period and integral measure}. In Theorem~\ref{analytic final}, we prove our main result on the congruences of the $ p $-adic $ L $-functions. We do this first by showing the congruence between the special values in Theorem~\ref{analytic final1}. Then using $ p $-adic Weierstrass preparation theorem, we obtain the desired congruence between the $ p $-adic $ L $-functions.
For $ a,b \in \mathbb{Z} $, write $[a,b] =$ lcm $(a,b)$. Set $t = [Nm^2, M_0m^2/(M_0,m)] (Nm^2)^{-k/2} (M_0m^2/(M_0,m))^{l/2+j} j! (l+j-1)!$ and $ t_0 = t/j!(l+j-1)! $. Recall that $ c(m) $ is the $ p $-adic unit as defined in \eqref{c(m) expression}. \begin{theorem}\label{analytic final1}
Let $ f \in S_{k}(\Gamma_{0}(N), \eta)$ be a $ p $-ordinary and newform with $ p \nmid N $. Let $ h \in S_{l}(\Gamma_{0}(I),\psi) $ be a $p$-ordinary eigenform with $ 2 \leq l < k $ and $ (T_{h}/\pi)^{ss} \cong \bar{\xi}_{1} \oplus \bar{\xi}_{2} $.
Let $ g = E_{l}(\xi_{2} \omega_p^{1-l}, \xi_{1}) $ be as in Lemma~\ref{Eisenstein series Hida} and $ \tilde{f}_0 = f_0 \vert \iota_{m}$. Let $ \mathrm{cond}(\xi_1\omega^{1-l}) = M_1 p^s$ and $ \mathrm{cond}(\xi_2) = M_2 $. Suppose that assumptions in Theorem~\ref{c(f) and petterson} hold for $ F = \tilde{f_{0}} $.
Let $ \phi $ be a finite order character of $ \mathbb{Z}_{p}^{\times} $.
If $ \phi \neq (\xi_{1} \omega_p^{1-l})^{-1}_p$, then for $ 0 \leq j \leq k-l-1 $ we have
\begin{align*}
\mu_{\tilde{f}\times h|\iota_{m}}(x_p^{j} \phi) & \equiv \mu_{\tilde{f} \times g|\iota_{m}}( x_p^{j} \phi)
\equiv (\ast) \mu_{\tilde{f}_0 }( x_p^{j} \xi_1 \phi) \mu_{\tilde{f}_0}(x_p^{j} \xi_2 \phi) \mod \pi.
\end{align*}
Here $ (\ast) $ is the $p$-adic unit given by
$
\frac{t_0 c(m) \xi_2(-1)u_f }{(-1)^{l+1} M_0^{l/2+j}} \frac{p^{(2-k)/2}c(\tilde{f}_0) \Omega_{\tilde{f}_0 }^{+} \Omega_{\tilde{f}_0}^{-}} {2^k i^{k+1}\langle \tilde{f_{0}}^{\rho}\vert\tau_{Nm^2p}, \tilde{f_{0}} \rangle_{Nm^2p}}.
$ \end{theorem} \begin{proof}
As $ p \nmid NM_0m $ and $c(m)$ is a $p$-adic unit, it follows from Theorem~\ref{c(f) and petterson} that $ (\ast) $ is a $ p $-adic unit.
By Lemma~\ref{congruence of measures}, we have
$ \mu_{\tilde{f} \times h|\iota_{m}}(\phi) \equiv \mu_{\tilde{f} \times g|\iota_{m}}(\phi) \mod \pi$.
Note that $ \tilde{f} \in S_{k}(Nm^2, \eta\iota_{m}^2) $
and $ g|\iota_{m} \in M_{l}(M_0m^2p^s/(M_0,m), \xi_{1} \xi_{2}\omega^{1-l}\iota_{m}^2) $.
For $ 0 \leq j \leq k-l-1 $, by Theorem~\ref{padic rankin}, we have
\begin{small}
\begin{align*}
\mu_{\tilde{f} \times g|\iota_{m}}(x_p^j \phi) = c(\tilde{f_{0}}) t p^{\beta (l+2j)/2} p^{(2-k)/2} a(p,\tilde{f_{0}})^{1-\beta} \frac{D_{M_0Nm^2p}(l+j,\tilde{f_{0}}, \mu_{g^{\rho}|\iota_m}(\bar{\phi})\vert\tau_{M_{0}m^2p^{\beta}})}{(2i)^{k+2j+l}\pi^{l+2j+1} \langle \tilde{f_{0}}^{\rho}\vert\tau_{Nm^2p}, \tilde{f_{0}} \rangle_{Nm^2p}}.
\end{align*}
\end{small}
\begin{comment}
By Lemmas~\ref{special value f,g and trivial character} and
\ref{special value f,g and non-trivial character}, we have
\begin{align*}
D_{M_{0}mNp}(l+j,\tilde{f_{0}}, \mu_{g|\iota_{m}}(\bar{\phi})
|\tau_{M_{0}m^2p^{\beta}}) =
\begin{cases}
W(g)m^{-1} \xi(m) p^{(s-\beta)l/2} p^{-\beta j} u_{f}^{\beta-s} P_{p}(g,u_{f}^{-1}) D_{M_{0}mNp}(l+j,\tilde{f_{0}}, g') & \text{ if } \phi = \iota_{p} ,\\
W(g|\bar{\phi}) m^{-1} \xi(m)
D_{M_{0} mNp}(l+j,\tilde{f_{0}}, g'|\phi) &
\text{ otherwise.}
\end{cases}
\end{align*}
Note that $ u_f = a(p, \tilde{f_{0}}) $. Substituting this in above, we get
\begin{align*}
\mu_{\tilde{f} \times g|\iota_{m}}(x_p^{j} \phi) = c(\tilde{f_{0}}) t p^{(2-k)/2} p^{s l/2} W(g) m^{-1} \xi(m) u_{f}^{-s} P_{p}(g,p^j u_{f}^{-1}) \frac{D_{M_{0}mNp}(l+j,\tilde{f_{0}}, g')}{(2i)^{k+l+2j}\pi^{l+2j+1} \langle \tilde{f_{0}}^{\rho}\vert\tau_{Nm^2p}, \tilde{f_{0}} \rangle} ~~\text{ if } \phi = \iota_p,
\end{align*}
and
\begin{align*}
\mu_{\tilde{f} \times g|\iota_{m}}(x_p^j\phi) = c(\tilde{f_{0}}) t p^{(2-k)/2} p^{\beta l/2} p^{\beta j} W(g|\phi) m^{-1} \xi(m) u_{f}^{-\beta} \frac{D_{M_{0}mNp}(l+j,\tilde{f_{0}}, g' |\phi)} {(2i)^{k+l+2j}\pi^{l+2j+1} \langle \tilde{f_{0}}^{\rho}\vert\tau_{Nm^2p}, \tilde{f_{0}} \rangle} ~~\text{ if } \bar{\phi} \neq \iota_p, (\xi_1 \omega^{1-l})_p.
\end{align*}
Taking $ s =l+j $ in \eqref{rankin as product of two l-functions} we
get
\[
D_{M_{0}mNp}(l+j,f, g' |\phi) = L(l+j, \tilde{f_{0}}, \bar{\xi_{1}} \bar{\omega}^{1-l} \phi) L(j+1, \tilde{f_{0}}, \bar{\xi_{2}} \phi).
\]
Substituting this we get
\begin{align*}
\mu_{\tilde{f_{0}} \times g|\iota_{m}}( x_p^{j} \iota_{p}) = c(\tilde{f_{0}}) t p^{s l/2}
p^{(2-k)/2} W(g)m^{-1} \xi(m) u_{f}^{-s} P_{p}(g,p^j u_{f}^{-1}) \frac{L(l+j, \tilde{f_{0}}, \bar{\xi_{1}}\bar{\omega}^{1-l}) L(j+1, \tilde{f_{0}}, \bar{\xi_{2}})}{(2i)^{k+l+2j}\pi^{l+2j+1} \langle \tilde{f_{0}}^{\rho}\vert\tau_{Nm^2p}, \tilde{f_{0}} \rangle} ,
\end{align*}
and if $ \phi \not \in \{ \iota_p, (\xi_1\omega_p^{1-l})_p\}$
\begin{align*}
\mu_{\tilde{f_{0}} \times g|\iota_{m}}(x_p^{j} \phi) = c(\tilde{f_{0}}) t p^{\beta l/2} p^{\beta j} p^{(2-k)/2} W(g|\phi) m^{-1} \xi(m) u_{f}^{-\beta} \frac{L(l+j, \tilde{f_{0}}, \bar{\xi_{1}} \bar{\omega}^{1-l} \bar{\phi}) L(j+1, \tilde{f_{0}}, \bar{\xi_{2}}\bar{\phi})}{(2i)^{k+l+2j}\pi^{l+2j+1} \langle \tilde{f_{0}}^{\rho}\vert\tau_{Nm^2p}, \tilde{f_{0}} \rangle}.
\end{align*}
Since $ a(p,g) = \xi_{2}(p) +\xi_{1}{\omega}^{1-l}(p) p^{l-1}$ we get
\[ P_p(g,p^j u_f^{-1}) = (p^{2j}u_f^{-2} \xi_{1}(p)\xi_{2}{\omega}^{1-l}(p) - a(p,g) p^j u_{f}^{-1}+1 ) = (1 - \xi_{2}(p) p^j u_f^{-1}) (1-\xi_{1}\omega^{1-l}(p)p^{l+j-1}u_f^{-1}).\]
\end{comment}
Note that $P_p(g^{\rho},p^j u_f^{-1}) =
(1 - \overline{\xi_{2}(p)} p^j u_f^{-1}) (1-\overline{\xi_{1}\omega_p^{1-l}(p)}p^{l+j-1}u_f^{-1})
= e_{p}(u_{\tilde{f}_0}, x_p^j \xi_{2}) e_{p}(u_{\tilde{f}_0}, x_p^{l+j-1} \xi_{1}\omega_{p}^{1-l})$, where $ e_{p}(u_{\tilde{f}_0}, \cdot) $ is as defined in Theorem~\ref{p-adic l-function of modular form} and $ u_f = a(p,f_0) = a(p,\tilde{f}_0) $. Thus by \eqref{eq: rankin as product trivial char}, we get
\begin{small}
\begin{align}\label{eq: rankin p-adic as product trivial char}
\begin{split}
\mu_{\tilde{f_{0}} \times g|\iota_{m}}(x_p^{j}\iota_{p}) ~ & {=\joinrel=} ~ c(\tilde{f_{0}}) t p^{s l/2} p^{sj} p^{(2-k)/2} c(m) W(g|\iota_p) u_{f}^{1-s}
e_{p}(u_{\tilde{f}_0}, x_p^j \xi_{2}) e_{p}(u_{\tilde{f}_0}, x_p^{l+j-1} \xi_{1}\omega_{p}^{1-l}) \\
& \qquad \qquad \qquad \times
\frac{ L(j+1,\tilde{f}_0,\xi_{2} \phi) L(l+j, \tilde{f}_0, \xi_{1}\omega_p^{1-l} \phi)} {(2i)^{k+l+2j} \pi^{l+2j+1} \langle \tilde{f_{0}}^{\rho}\vert\tau_{Nm^2p}, \tilde{f_{0}} \rangle_{Nm^2p}} \\
&\stackrel{\eqref{root number of g definition}}{=\joinrel=} ~ \frac{t c(m)\xi_{2}(-1)}{M_0^{l/2+j} } \times u_{\tilde{f}_0}^{1-s} \frac{\mathrm{cond(\xi_{1}\omega_{p}^{1-l})}^{l+j} e_{p}(u_{\tilde{f}_0},x_p^{l+j-1} \xi_{1}\omega_{p}^{1-l}) L(l+j,\tilde{f}_0,\xi_{1}\omega_{p}^{1-l}\phi)}{(2 \pi i)^{l+j}G(\xi_{1}\omega_{p}^{1-l})} \\
& \qquad \qquad \times \frac{p^{(2-k)/2} c(\tilde{f_{0}}) }{2^{k}i^{k+1} \langle \tilde{f_{0}}^{\rho}\vert\tau_{Nm^2p}, \tilde{f_{0}} \rangle_{Nm^2p}}\times \frac{\mathrm{cond(\xi_2)}^{j+1} e_{p}(u_{\tilde{f}_0},x_p^j \xi_{2}) L(j+1,\tilde{f}_0,\xi_{2}\phi )}{(2 \pi i)^{j+1}G(\xi_{2})}.
\end{split}
\end{align}
\end{small}
If $ \phi \neq \iota_p$ and $(\phi\xi_1\omega_p^{1-l})_0(p)=0$, then $ \xi_1 \omega_p^{1-l}\phi(p)= 0= \phi\xi_{2}(p) $. Thus $ e_{p}(u_{\tilde{f}_0},x_p^{l+j-1} \xi_{1}\omega_{p}^{1-l}\phi) =1 = e_{p}(u_{\tilde{f}_0},x_p^j \xi_{2}\phi) $. Let $ p^\beta = \mathrm{cond}_p(\xi_1\omega_p^{1-l}\phi) \mathrm{cond}_p(\xi_2\phi) $. Then it follows from \eqref{eq: rankin as product nontrivial char} that
\begin{small}
\begin{align}\label{eq: rankin p-adic as product non trivial char}
\begin{split}
\mu_{\tilde{f} \times g|\iota_{m}}(x_p^j \phi) & = c(\tilde{f_{0}}) t p^{\beta l/2} p^{\beta j} p^{(2-k)/2} W(g^{\rho}|\bar{\phi}) c(m) u_{f}^{1-\beta} \frac{L(l+j, \tilde{f}_0, \xi_{1} \omega^{1-l} \phi) L(j+1, \tilde{f}_0, \xi_{2}\phi)}{(2i)^{k+l}\pi^{l+2j+1} \langle \tilde{f_{0}}^{\rho}\vert\tau_{Nm^2p}, \tilde{f_0} \rangle_{Nm^2p}} \\
&= \frac{ t c(m)\xi_{2}(-1)}{M_0^{l/2+j}} \times u_{\tilde{f}_0}^{1-\beta} \frac{\mathrm{cond(\xi_{1}\omega_{p}^{1-l}\phi)}^{l+j} e_{p}(u_{\tilde{f}_0},x_p^{l+j-1} \xi_{1}\omega_{p}^{1-l}\phi) L(l+j,\tilde{f}_0,\xi_{1}\omega_{p}^{1-l}\phi)}{(2 \pi i)^{l+j}G(\xi_{1}\omega_{p}^{1-l}\phi)} \\
& \qquad \times \frac{p^{(2-k)/2} c(\tilde{f_{0}}) }{2^k i^{k+1} \langle \tilde{f_{0}}^{\rho}\vert\tau_{Nm^2p}, \tilde{f_{0}} \rangle_{Nm^2p}} \times \frac{\mathrm{cond(\xi_2\phi)}^{j+1} e_{p}(u_{\tilde{f}_0},x_p^j \xi_{2}\phi) L(j+1,\tilde{f}_0,\xi_{2}\phi )}{(2 \pi i)^{j+1}G(\xi_{2}\phi)}
\end{split}
\end{align}
\end{small}
Applying Theorem~\ref{period and integral measure} and choosing $ L = M_0$, we obtain
\begin{small}
\begin{align}\label{eq: product of p-adic L functions of f}
\begin{split}
\mu_{\tilde{f}_0}(x_p^{l+j-1}\xi_{1}\omega_p^{1-l}\phi)
\mu_{\tilde{f}_0}(\tilde{f},x_p^{j} \xi_{2} \phi)
&= \frac{u_{f}^{-(\star)}}{\Omega_{\tilde{f}_0 }^{+} \Omega_{\tilde{f}_0}^{-}}
(l+j-1)!j!
\mathrm{cond}(\xi_{2} \phi)^{j+1} e_{p}(u_{\tilde{f}_0},x_p^{j} \xi_{2}\phi) \frac{L(j+1, \tilde{f}_0, \xi_{2} \phi)}{G({\xi}_{2}{\phi})(-2 \pi i)^{j+1}} \\
&
e_{p}(u_{\tilde{f}_0},x_p^{l+j-1} \xi_{1}\omega_{p}^{1-l}\phi) \mathrm{cond}(\xi_{1} \omega_p^{1-l} \phi)^{l+j} \frac{L(l+j-1,\tilde{f}_0,\xi_{1} \omega_p^{1-l} \phi)} {G(\xi_{1}\omega_p^{1-l}\phi)(-2 \pi i)^{l+j}} ,
\end{split}
\end{align}
\end{small}
where $ (\star) $ is the power of $ p $ dividing and $ \mathrm{cond}(\xi_{1} \omega_p^{1-l} \phi) \mathrm{cond}(\xi_{2} \phi) $. Note that $ (\star) =s $ if $ \phi = \iota_p $ and $ (\star) = \beta $ if $ \phi \neq (\xi_{1} \omega^{1-l})_p^{-1} $.
Substituting \eqref{eq: product of p-adic L functions of f} in \eqref{eq: rankin p-adic as product trivial char} and \eqref{eq: rankin p-adic as product non trivial char}, we get
\begin{small}
\begin{align}\label{eq: last congruence}
\mu_{\tilde{f} \times g|\iota_{m}}(x_p^j \phi) \equiv
\frac{ t_0 c(m)\xi_{2}(-1)}{(-1)^{l+1}M_0^{l/2+j}} \frac{p^{(2-k)/2} c(\tilde{f_{0}}) \Omega_{\tilde{f}_0 }^{+} \Omega_{\tilde{f}_0}^{-}} {2^k i^{k+1}\langle \tilde{f_{0}}^{\rho}\vert\tau_{Nm^2p}, \tilde{f_{0}} \rangle_{Nm^2p}} \mu_{\tilde{f}_0}( x_p^{j} \xi_{1} x_p^{l-1} \omega_p^{1-l} \phi) \mu_{\tilde{f}_0}( x_p^j \xi_{2} \phi) \mod \pi.
\end{align}
\end{small}
Since $ x_p^{l-1} \omega_p^{1-l} \lvert_{\mathbb{Z}_p^{\times}} \equiv 1 \mod p $ and $ \mu_{\tilde{f}_0}(\cdot) $ is $ \mathcal{O}_K $-valued, we obtain $\mu_{\tilde{f}_0}(x_p^{l+j-1} \xi_{1} \omega_p^{1-l} \phi) \equiv \mu_{\tilde{f}_0}(x_p^{j} \xi_{1} \phi) \mod \pi$. Substituting this in \eqref{eq: last congruence}, we deduce the theorem. \end{proof}
\begin{remark}
Taking $ \Omega_{\tilde{f}}^{\pm} = \Omega_{\tilde{f}_0}^{\pm}$ in Theorem~\ref{p-adic l-function of modular form}, we get a bounded measure $ \mu_{\tilde{f}} $. Note that for every Dirichlet character $ \chi $, we have $L(j,\tilde{f_{0}},\chi) = \left( 1 - \frac{\eta(p) \chi(p) p^{k-1-j}}{u_f} \right) L(j,\tilde{f},\chi)$. It follows that $ \mu_{\tilde{f}} = \mu_{\tilde{f}_0} $ with $\mu_{\tilde{f}}, \mu_{\tilde{f}_0}$ as defined in Theorem~\ref{period and integral measure}. So we may replace the measure $ \mu_{\tilde{f}_0} $ in Theorem~\ref{analytic final1} by $ \mu_{\tilde{f}} $. \end{remark}
\begin{comment}
Let $(\rho_{f,\ell},V_{f,\ell})$ be the $ \ell $-adic Galois representation attached to $ f $ and $ V_{f,\ell}(j) = V_{f,\ell} \otimes \chi_{\ell}^{-j}$ be the $ j^\text{th} $ Tate-twist of $ V_{f,\ell} $. Let $ L_{V_f(-j)}(s)$ be the $ L $-function attached to the family $ \{ V_{f,\ell}(-j) \} $. Then for every Dirichlet character $ \chi $, we have $ L_{V_f(-j)}(\chi, s) = L(f,\chi, j+s)$. Thus for every $ 0 \leq j \leq k-2 $, we obtain $ 1 $ is a critical of $ L_{V_{f}(-j)}(s) $. Following \cite{gr1}, one can associate the $p$-adic $ L $-function to $ (V_{f}(-j)) $ which we denote by $ L_{p,f,j}(\cdot) $. \end{comment} \iffalse \textcolor{red}{
We would now like to define power series in $ \mathbb{Z}_{p}[[T]] $ interpolating the special values of the $ \tilde{f} \otimes h $ and $ \tilde{f} \otimes \xi_{i} $ for $ i =1,2 $. Let $ \mathrm{Hom}_{\mathrm{cont}}(1+p\mathbb{Z}_{p}, \mathbb{C}_p^{\times}) $ be the space of continuous $ \mathbb{C}_p^{\times} $-valued homomorphisms on $ (1+p\mathbb{Z}_{p}) $. Since an element of $ \mathbb{Z}_{p}[[T]] $ a priori defines a homomorphism on $ \mathrm{Hom}_{\mathrm{cont}}(1+p\mathbb{Z}_{p}, \mathbb{C}_p^{\times}) $, we can interpolate only one special value corresponding to a given finite order character on $ (1+p\mathbb{Z}_{p}) $. In order to interpolate all the special values we need to define exactly $ r $ many power series in $ \mathbb{Z}_{p}[[T]] $ with $ r = $ number of critical integers for a given $ L $-function. We obtain these power series by restricting the measure on $ \mathbb{Z}_{p}^{\times} $ to an appropriate subset of $ C(\mathbb{Z}_{p}^{\times} ; \mathbb{C}_{p}) $.} \fi Let $\Sigma_{0} $ be the set of all rational primes dividing $ m $. For $ 0 \leq j \leq k-2 $ and Dirichlet character $ \chi $, consider the measure $L^{\Sigma_{0}}_{p,f,\chi, j}$ on $ (1+p\mathbb{Z}_{p}) $ satisfying the interpolation property \begin{equation}\label{def: p-adic L-function f}
L^{\Sigma_{0}}_{p,f,\chi, j}(\phi ) = \mu_{\tilde{f}_0}(x_p^j \chi \phi) = \frac{e_{p}(u_{f},\chi \phi x_p^j)}{u_f^{\mathrm{cond}(\phi)}} \frac{\mathrm{cond}(\chi \phi)^{j+1} j!}{G(\chi \phi)} \frac{L(\tilde{f},\chi \phi,j+1)}{(2 \pi i)^{j+1}\Omega_{\tilde{f}_0}^{\mathrm{sgn}((-1)^j\chi\phi(-1))}} \end{equation} for every finite order character $ \phi: \mathbb{Z}_p^{\times} \rightarrow \mathbb{C}_p^{\times} $ of $ p $-power order.
Let $ \mu_{\tilde{f} \times h|\iota_{m}}$ be the measure as in Theorem~\ref{padic rankin}. For $ l-1 \leq j \leq k-2 $, consider the measure $L^{\Sigma_{0}}_{p,f\otimes h, j}$ on $(1+p\mathbb{Z}_{p})$ satisfying the interpolation property: \begin{align}\label{def: p-adic L-function f x h}
L^{\Sigma_{0}}_{p,f \otimes h,j}(\phi) = \mu_{\tilde{f} \times h|\iota_{m}}( x_p^{j-l+1} \phi ) \end{align} for every finite order character $ \phi: \mathbb{Z}_p^{\times} \rightarrow \mathbb{C}_p^{\times} $ of $ p $-power order. Let $ \mathbb{Q}_{\mathrm{cyc}} $ be the cyclotomic $\mathbb{Z}_p $-extension of $ \mathbb{Q} $ and $ \Gamma = \mathrm{Gal}(\mathbb{Q}_{\mathrm{cyc}}/\mathbb{Q}) \cong \mathbb{Z}_p$. Then it is known that $L^{\Sigma_{0}}_{p,f,\chi, j}$ (resp. $L^{\Sigma_{0}}_{p,f \otimes h,j}$) corresponds to an integral power series in the Iwasawa algebra $\mathbb{Z}_p[[\Gamma]]$ which we continue to denote by $L^{\Sigma_{0}}_{p,f,\chi, j}$ (resp. $L^{\Sigma_{0}}_{p,f \otimes h,j}$). \begin{theorem}\label{analytic final}
Let $ f \in S_{k}(\Gamma_{0}(N), \eta)$ be a $ p $-ordinary newform with $ p \nmid N $. Let $ h \in S_{l}(\Gamma_{0}(I),\psi) $ be a $ p $-ordinary eigenform such that $ 2 \leq l < k $ and $ (T_{h}/\pi)^{ss} \cong \bar{\xi}_{1} \oplus \bar{\xi}_{2} $. Suppose that the assumptions of Theorem~\ref{c(f) and petterson} hold for $F= \tilde{f}_0 $. Then for $l-1 \leq j \leq k-1$, we have the following congruence of ideals in the Iwasawa algebra $ \mathbb{Z}_{p}[[\Gamma]] $
\begin{align*}
(L^{\Sigma_{0}}_{p,f \otimes h,j}) \equiv (L^{\Sigma_{0}}_{p,f,\xi_1, j}) (L^{\Sigma_{0}}_{p,f,\xi_2, j}) \mod \pi.
\end{align*} \end{theorem} \begin{proof}
This follows from Theorem~\ref{analytic final1} and the $ p $-adic Weierstrass preparation theorem (for example see \cite[Theorem 1.10]{Vatsal}). \end{proof} \section{Selmer group of Modular form and Rankin-Selberg product}\label{sec: Selmer groups} In this section we discuss $ p^\infty $-Greenberg Selmer group attached to {a} modular form and {the} Rankin-Selberg product. Under the congruence \eqref{eq: splitting rho_h}, we investigate the relation between $ p^\infty $-Selmer group of $ f \otimes h $ with $ p^\infty $-Selmer groups of $ f \otimes \xi_{1} $ and $ f \otimes \xi_{2} $. \subsection{Background on Selmer groups}\label{sec2} \begin{comment}
We fix an odd prime $p$ and $N \in \mathbb{N}$ with $(N,p)=1$. Let $f = \sum_{n \geq 1} a_n(f)q^n \in S_k(\Gamma_0(N), \eta)$ be a normalized cuspdial Hecke eigenform which is a newform of level $N$. We assume that $f$ is $p$-ordinary i.e. $a_p(f)$ is a $p$-adic unit. Now let $h=\sum_{n \geq 1} a_n(g)q^n \in S_l(\Gamma_0(I), \psi)$ be another normalized cuspidal Hecke eigenform which is also ordinary at $p$. We assume throughout that $k >l \geq 2$. \end{comment}
We recall the following notation from \S\ref{section: simplyfying rankin}: For a normalised eigenform $ F $ with nebentypus $\chi$, $K_F$ denotes the corresponding number field. Let $ L $ be a number field containing $ K_F $ and $\pi_L$ denote a uniformizer of the ring of integers $ \mathcal{O}_{L_{\mathfrak p}} $ of $L_{\mathfrak p}$. Also, recall $\chi_p: G_\mathbb{Q} \longrightarrow \mathbb{Z}_p^\times$ {is} the $p$-adic cyclotomic character. For a discrete $ \mathbb{Z}_{p}[[\Gamma]] $-module $M$, let $M^\vee:=\mathrm{Hom}_{\text{cont}}(M,\frac{\mathbb{Q}_p}{\mathbb{Z}_p})$ be the Pontryagin dual of $ M $. \begin{theorem}\label{rhof}$($Eichler, Shimura, Deligne, Mazur-Wiles, Wiles etc.$)$
Let $F(z) = \sum_{n \geq 1} a(n,F) q^n\in S_k(\Gamma_0(Ap^t),\chi)$ be a $p$-ordinary newform with $k\geq 2$ and $ p \nmid A$. Then there exists a Galois representation,
$\rho_{F}: G_\mathbb{Q} \longrightarrow \mathrm{GL}_2(L_{\mathfrak p})$ such that
\begin{enumerate}[label=$(\roman*)$]
\item \label{unr}
{For} all primes $r \nmid Ap$, $\rho_F$ is unramified and for the $($arithmetic$)$ Frobenius $\mathrm{Frob}_r$ at $r$, we have
$$\mbox{trace}(\rho_{F}(\mathrm{Frob}_r))=a(r, F),\quad \det(\rho_{F}(\mathrm{Frob}_r)) = \chi(r) \chi_p(\mathrm{Frob}_r)^{k-1}
= \chi(r) r^{k-1}.$$
It follows (by the Chebotarev density theorem) that $\det(\rho_{F})=\epsilon \chi_p^{k-1}$.
\item
As $ f $ is $p$-ordinary, let $u_F$ be the unique $p$-adic unit root of $X^2 -a(p,F)X +\epsilon(p) p^{k-1}.$ Let $\lambda_{F}$ be the unramified character with $\lambda_{F}(\mathrm{Frob}_{p})=u_F$. Then \begin{small}
$$ \rho_{F}|_{G_{p}} \sim \begin{pmatrix} \lambda_{F}^{-1} \epsilon \chi_{p}^{k-1} & * \\ 0 & \lambda_{F} \end{pmatrix}.$$
\end{small}
\end{enumerate}
\end{theorem}
Let $V_{\mathfrak{g}} \cong L_{\mathfrak p}^{\oplus 2}$ denote the representation space of $\rho_{\mathfrak{g}}$. By compactness of $G_\mathbb{Q}$, there exists a $ G_{\mathbb{Q}} $ invariant $\mathcal{O}_{L_\mathfrak{p}}$ lattice $T_{\mathfrak{g}}$ of $V_{\mathfrak{g}}$.
Let
$ \bar{\rho}_{\mathfrak{g}}: G_{\mathbb{Q}} \longrightarrow \mathrm{GL_2}(\mathcal{O}_{L_\mathfrak {p}}/{\pi_L})$
be the residual representation of $\rho_{\mathfrak{g}}$.
Recall $f \in S_k(\Gamma_0(N), \eta)$ is a primitive form and $ h\in S_l(\Gamma_0(I_0p^\alpha), \psi) $ are Hecke eigenforms. Further, recall that $ f,h $ are $ p $-ordinary and $ p \nmid N I_0 $. Let $K$ be a number field containing the $K_f, K_h$. To ease the notation, we denote $\mathcal{O}_{K_\mathfrak{p}}$ by $\mathcal{O}$ and $\pi_{K}$ by $ \pi $. Let $\Sigma$ be a finite set of primes of $\mathbb{Q}$ such that $\Sigma \supset \{\infty\} \cup \{p\} \cup \{ \ell : \ell \mid NI\}$ and $\mathbb{Q}_\Sigma$ be the {maximal} algebraic extension of $\mathbb{Q}$ unramified outside $\Sigma$. Set $G_\Sigma(\mathbb{Q}) := \text{Gal}(\mathbb{Q}_{\Sigma}/\mathbb{Q})$. Let us take $\mathfrak{g} \in \{f,h\}$. Let $V_{\mathfrak{g}}=K_\mathfrak p^{\oplus 2}$ be the Galois representation {attached} to $\mathfrak{g}$.
Then we have an induced $G_\mathbb{Q}$ action on the discrete module $A_{\mathfrak{g}}:= V_{\mathfrak{g}}/T_{\mathfrak{g}}$. For a character $ \chi $, let $ V_{\mathfrak{g}} (\chi) := V_{\mathfrak{g}} \otimes_{\mathbb{Q}_p} \chi $ and $ T_{\mathfrak{g}} (\chi) := T_{\mathfrak{g}} \otimes_{\mathbb{Z}_p} \chi $. Further put $ V_{\mathfrak{g}}(j) := V_{\mathfrak{g}} (\chi_{p}^{-j}) $, $ T_{\mathfrak{g}}(j) := T_{\mathfrak{g}} (\chi_{p}^{-j})$ and $ A_{\mathfrak{g}}(j) = V_{\mathfrak{g}} (j)/ T_{\mathfrak{g}}(j)$. We have the canonical maps
$0 \longrightarrow T_{\mathfrak{g}} \longrightarrow V_{\mathfrak{g}} \longrightarrow A_{\mathfrak{g}} \longrightarrow 0$.
As $\mathfrak{g}$ is $p$-ordinary, $T_{\mathfrak{g}}$ has {the} following filtration as a $G_{p}$-module
\begin{small}
\begin{equation}
0 \longrightarrow T_{\mathfrak{g}}^+ \longrightarrow T_{\mathfrak{g}} \longrightarrow T_{\mathfrak{g}}^{-} \longrightarrow 0,
\end{equation}
\end{small}
where both $T_{\mathfrak{g}}^+$ and $T_{\mathfrak{g}}^{-}$ are free $\mathcal{O}$ module of rank $1$ and the action of $G_\mathbb{Q}$ on $T_{\mathfrak{g}}^{-}$ is unramified at $p$. Following \cite{gr1}, for a modular form $\mathfrak{g}$ of weight $k$, define a filtration on $ V_{\mathfrak{g}}(-j)$ with $ 0 \leq j \leq k-2$, by
\begin{small}
\begin{align*}
F^s(V_{\mathfrak{g}}(-j)) =
\begin{cases}
V_{\mathfrak{g}}(-j) & \text{ if } s \leq -j, \\
V_{\mathfrak{g}}^{+}(-j) & \text{ if } -j+1 \leq s \leq k-1-j,\\
0 & \text{ if } s \geq k-j.
\end{cases}
\end{align*}
\end{small}
We have a corresponding filtration on $ T_{\mathfrak{g}}(-j) $. Also, define $A^+_{\mathfrak{g}}(-j) = (V_{\mathfrak{g}}^+(-j)/T_{\mathfrak{g}}^+(-j))$ and
$A_{\mathfrak{g}}^{-}(-j) = A_{\mathfrak{g}}(-j)/A^+_{\mathfrak{g}}(-j)$.
Set $V := V_f \otimes_{K} V_h$ and $T:=T_f\otimes_{\mathcal{O}}T_h$. Then we have an induced filtration on $V$ and $T$ respectively:
\begin{equation}\label{Filtration on T}
\begin{aligned}
&0\subset V_f^+\otimes_KV_h^+ \subset V_f^+\otimes_KV_h \subset V_f^+\otimes_KV_h + V_f\otimes_K V_h^+ \subset V_f\otimes_K V_h, \\
&0\subset T_f^+\otimes_{\mathcal{O}}T_h^+ \subset T_f^+\otimes_{\mathcal{O}}T_h \subset T_f^+\otimes_{\mathcal{O}}T_h + T_f\otimes_{\mathcal{O}} T_h^+ \subset T_f\otimes_{\mathcal{O}} T_h.
\end{aligned}
\end{equation}
For every $l-1 \leq j \leq k-2$, we define $V_j := V \otimes_{\mathbb{Q}_p} \chi_p^{-j} $, $T_j = T \otimes_{\mathbb{Z}_p} \chi_p^{-j}$ and $A_j = \frac{V_j}{T_j} $. Note that $A_j \cong T_f(\chi_p^{-j}) \otimes_{\mathcal{O}} A_h \cong T_h(\chi_p^{-j}) \otimes_{\mathcal{O}} A_f.$
\iffalse
{\color{red} For a modular form $f$ of weight $k$ the filtration on $V=V_f(-j)$ with $0\leq j \leq k-2$, for defining the Greenberg Selmer group (following Greenberg's paper) is given by on $V= V_f(-j)$ for $0 \leq j \leq k-2$.
$F^{s}V =V_f(-j)$ for all $s \leq -j$ and $F^{s}V= V^+_f(-j)$ when $-j +1 \leq s \leq k-1-j$ and $F^{s}V=0 $ for all $s \geq k-j$. Note in Greenberg's notation $F^+V_f(-j) = F^1V_f(-j)= V^+_f(-j)$.
}
\fi
The action of $I_p$ on the successive quotients in our filtration \eqref{Filtration on T} is given by $\chi_p^{k+l-2}, \chi_p^{k-1}, \chi_p^{l-1}, \chi_p^{0}$. In particular, for $l-1 \leq j \leq k-2$, we have the following filtration of $ V_f\otimes V_h(-j)$
\begin{small}
\begin{align*}
F^s(V_f\otimes V_g(-j)) =
\begin{cases}
V_f\otimes V_h(-j) &\text{ if } s \leq -j, \\
V_f^{+} \otimes V_h(-j) + V_f \otimes V_h^+(-j) & \text{ if } -j+1 \leq s \leq l-1-j, \\
V_f^{+} \otimes V_h(-j) & \text{ if } l-j \leq s \leq k-1-j,\\
V_f^{+}\otimes V_h^{+}(-j) & \text{ if } k-j \leq s \leq k+l-2-j, \\
0 & \text{ if } s \geq k+l-1-j.
\end{cases}
\end{align*}
\end{small}
Similarly, define a filtration on $ T_f \otimes T_h(-j)$. Following \cite{gr1}, we have
$F^+(V_j) = V^+_f\otimes V_h(-j)$
and $F^+(T_j) = T^+_f\otimes T_h(-j)$. Finally, we put
\begin{align}\label{def A minus j}
F^+(A_j) :=F^+(V_j)/ {F^+(T_j)} \quad \text{and } \quad A^{-}_j = A_j/{F^+(A_j)}.
\end{align}
Note that $A_j^-=T_h \otimes A_f^-(-j)$, where $ A_f^-(-j) = A_f(-j)/ A_f^{+}(-j)$.
\par Let $ L $ be a number field and $ \Sigma_L $ be the set of primes in $ L $ lying above $ \Sigma $. Put $ G_{\Sigma}(L) = \mathrm{Gal}(L_{\Sigma_{L}}/L) $.
For every prime $ v \in \Sigma_L$ and $ B_j \in \{ A_j, A_f(\xi_{1}\omega_p^{-j}), A_f(\xi_{2}\omega_p^{-j})\} $, let choose a subset $H^1_{\dagger} (L_v,B_j) \subset H^1 (L_v,B_j) := H^1 (G_{L_v},B_j)$.
For this choice, we define $\dagger$-Selmer group $S_\dagger(B_j/L)$ as
\begin{small}
\begin{equation*}
S_{\mathrm{\dagger}}(B_j/L): =\mathrm{Ker}\bigg(H^1(G_\Sigma(L), B_j) \longrightarrow \underset{v \in \Sigma_L}\prod \frac{H^1(L_{v}, B_j)}{H_{\dagger}^1(L_{v}, B_j)}\bigg).
\end{equation*}
\end{small}
For any $r \in \mathbb{N}$, there is a natural inclusion
\begin{equation}\label{ifr}
0 \longrightarrow B_{j}[\pi^r] \stackrel{i_{r}}{\longrightarrow} B_{j}.
\end{equation}
Next we define $\pi^r$ $\dagger$-Selmer group $S_{\dagger}(B_{j}[\pi^r]/L)$ as
\begin{equation*}
S_{\dagger}(B_{j}[\pi^r]/L): =\mathrm{Ker}\Big(H^1(G_\Sigma(L), B_{j}[\pi^r]) \longrightarrow \underset{v \in \Sigma_L}\prod \frac{H^1(L_v, B_{j}[\pi^r])} {H_{\dagger}^1(L_v, B_{j}[\pi^r])}\Big),
\end{equation*}
where $H_\dagger^1(L_v, B_{j}[\pi^r]):= { i_{r}^{*^{-1}}} (H^1_\dagger (L_v,B_{j}))$ for every $v \in \Sigma_L$. Here $i_{r}^{*}: H^1(L_v, B_{j}[\pi^r]) \longrightarrow H^1(L_v, B_{j}) $ is induced from $i_{r}$ in \eqref{ifr}.
Now for $ B_j \in \{ A_j, A_f(\xi_{1}\omega_p^{-j}), A_f(\xi_{2}\omega_p^{-j})\} $ we make a special choice of $H^1_{\dagger} (L_v,B_{j}) $ for every $v \in \Sigma_L$, to define Greenberg
and strict Selmer group. For $\dagger \in \{\mathrm{Gr}, \mathrm{str} \}$, define
\begin{comment}
. Let $\dagger \in \{\mathrm{Gr}, \mathrm{BK} \}$ with $\mathrm{Gr}$ and $\mathrm{BK}$ are defined
\end{comment}
$$
H^1_{\dagger}(L_v, B_{j}) : =
\begin{cases}
\text{Ker}\big(H^1(G_{L_v}, B_{j}) \longrightarrow H^1(I_v, B_{j})\big) & \text{if } q \nmid p \text{ and } \dagger \in \{\mathrm{Gr}, \mathrm{str} \}, \\
\text{Ker}\big(H^1(G_{L_v}, B_{j}) \longrightarrow H^1(I_v, B^-_{j})\big) & \text{if } v \mid p \text{ and } \dagger = \mathrm{Gr}, \\
\text{Ker}\big(H^1(G_{L_v}, B_{j}) \longrightarrow H^1(G_v, B^-_{j})\big) & \text{if } q \mid p \text{ and } \dagger = \mathrm{str}.
\end{cases}
$$
\begin{comment}
For a prime $q \in \mathbb{Q}$, $I_q$ is the inertia group of $\bar{\mathbb{Q}}/\mathbb{Q}$ at $q$ and $p_{j}^*: H^1(\mathbb{Q}_q, V_{j}) \longrightarrow H^1(\mathbb{Q}_q, A_{j})$ is induced from the map $p_{j}$ in \eqref{pif}.
\par Next for a $q \in \Sigma$, set $$
H^1_\mathrm{BK}(\mathbb{Q}_q, A_{j}) : =
\begin{cases}
p_{j}^*(H^1_{\text{unr}}(\mathbb{Q}_q, V_j))& \text{if } q \nmid p \\
p_{j}^*(H^1_{f}(\mathbb{Q}_q, V_j))& \text{if } q \mid p.
\end{cases}
$$
\begin{equation}\label{unr}
\text{where, } \quad H^1_{\text{unr}}(\mathbb{Q}_q, V_{j}) = \mathrm{Ker}\Big(H^1(\mathbb{Q}_q, V_{j}) \longrightarrow H^1(I_q, V_{j})\Big), ~ q \nmid p
\end{equation}
\begin{equation}\label{finite}
\text{and } \quad H^1_{f}(\mathbb{Q}_p, V_{j}) = \mathrm{Ker}\Big(H^1(\mathbb{Q}_q, V_{j}) \longrightarrow H^1(\mathbb{Q}_p, V_{j}\otimes_{\mathbb{Q}_p}B_\mathrm{cris})\Big),
\end{equation} where \textcolor{blue}{$B_\mathrm{cris}$} is as defined by Fontaine in \cite{fon}. This completes the definition of Greenberg and Bloch-Kato Selmer groups.
We also recall the definition of a subgroup $H^1_h(\mathbb{Q}_p,V_{j})$ of $H^1(\mathbb{Q}_p,V_{j})$
\[ H^1_h(\mathbb{Q}_p,V_{j}): = \mathrm{Ker}\big(H^1(\mathbb{Q}_p,V_{j})\longrightarrow H^1(\mathbb{Q}_p,V_{j}\otimes_{\mathbb{Q}_p} B_{dR}\big) \]
where $B_{dR}$ as defined by Fontaine in \cite{fon}.
\end{comment}
Recall that $ \mathbb{Q}_{\mathrm{cyc}} $ is the cyclotomic $ \mathbb{Z}_p $-extension of $ \mathbb{Q} $ and $\Gamma := \mathrm{Gal}(\mathbb{Q}_{\mathrm{cyc}}/\mathbb{Q}) \cong \mathbb{Z}_p$. Set $ \mathbb{Q}_n := \mathbb{Q}_\mathrm{cyc}^{\Gamma^{p^n}} $ for every $ n \geq 1 $. Define
\begin{small}
$$S_\dagger (B_j/\mathbb{Q}_\mathrm{cyc}): = \underset{n}{\varinjlim} ~S_\dagger (B_j/\mathbb{Q}_n) \quad \text{and} \quad S_\dagger(B_j[\pi^r]/\mathbb{Q}_\mathrm{cyc}): = \underset{n}{\varinjlim} ~S_\dagger(B_j[\pi^r]/\mathbb{Q}_n).
$$
\end{small}
\begin{theorem}
The kernel and the cokernel of the map $ S_\dagger(A_j/\mathbb{Q}_n) \longrightarrow S_\dagger(A_j/\mathbb{Q}_\mathrm{cyc})^{\Gamma^{p^n}} $ are finite and uniformly bounded independent of $ n $, for $\dagger \in \{ \mathrm{str}, \mathrm{Gr}\}$.
\end{theorem}
\begin{proof}
Consider the commutative diagram
\begin{comment}
\xymatrix{
0 \ar[r] & S_{\dagger}(A_j/K_n) \ar[r]\ar[d]_{r_{\dagger,n}} & H^1(\mathbb{Q}_\Sigma/K_n, A_j) \ar[r]\ar[d] & \underset{v|q \in \Sigma}\prod \Big( \frac{H^1(K_{n,v}, A_j)}{H_{\dagger}^1(K_{n,v}, A_j)}\Big) \ar[d] \\
0 \ar[r] & S_{\dagger}(A_j/K_{\mathrm{cyc}})^{\Gamma^{p^n}} \ar[r] & H^1(\mathbb{Q}_\Sigma/K_{\mathrm{cyc}}, A_j)^{\Gamma^{p^n}} \ar[r] & \underset{v|q \in \Sigma}\prod \Big( \frac{H^1(K_{\mathrm{cyc},v}, A_j)}{H_{\dagger}^1(K_{\mathrm{cyc},v}, A_j)}\Big)^{\Gamma^{p^n}}. \\
}
\end{comment}
\begin{tiny}
\[
\begin{tikzcd}
0 \arrow[r]
& S_{\dagger}(A_j/\mathbb{Q}_n) \arrow[d, "r_{\dagger,n}"] \arrow[r] & H^1(\mathbb{Q}_\Sigma/\mathbb{Q}_n, A_j) \arrow[d] \arrow[r] & \underset{v \in \Sigma_{\mathbb{Q}_n}}\prod \frac{H^1(\mathbb{Q}_{n,v}, A_j)}{H_{\dagger}^1(\mathbb{Q}_{n,v}, A_j)} \arrow[d]\\
0 \arrow[r]
& S_{\dagger}(A_j/\mathbb{Q}_{\mathrm{cyc}})^{\Gamma^{p^n}} \arrow[r] & H^1(\mathbb{Q}_\Sigma/\mathbb{Q}_{\mathrm{cyc}}, A_j)^{\Gamma^{p^n}} \arrow[r] & \underset{v \in \Sigma_{\mathbb{Q}_n}}\prod \Big( \frac{H^1(\mathbb{Q}_{\mathrm{cyc},v}, A_j)}{H_{\dagger}^1(\mathbb{Q}_{\mathrm{cyc},v}, A_j)}\Big)^{\Gamma^{p^n}}.
\end{tikzcd}
\]
\end{tiny}
To show $\mathrm{ker}(r_{\dagger,n})$ is finite and uniformly bounded for every $n$, by \cite[Theorem 3.5(1)]{o1}, it is enough to show that $H^0(\mathbb{Q}_n, V_j)=0$, for all $n$. In fact, it suffices to show that $H^0(\mathbb{Q}_{n,v}, V_j)=0$, for $ v \mid p$. This will follow if we can establish for $ v \mid p $
\begin{equation}\label{hkjadlk:DK?ALF}
H^0(\mathbb{Q}_{n,v}, V_f^+\otimes V_h(-j))=0 = H^0(\mathbb{Q}_{n,v}, V_f^-\otimes V_h(-j)).
\end{equation}
Further, to establish the second equality in \eqref{hkjadlk:DK?ALF}, it suffices to show, for $ v \mid p $
\begin{equation}\label{hkjadlk:DK?ALF2}
H^0(\mathbb{Q}_{n,v}, V_f^-\otimes V^{+}_h(-j))=0 = H^0(\mathbb{Q}_{n,v}, V_f^-\otimes V^{-}_h(-j)).
\end{equation}
Let $L' = \mathbb{Q}_{n,v}(\mu_{p^\infty}) = \mathbb{Q}_v(\mu_{p^\infty})$. Then $G_{L'}$ acts on $V_f^-\otimes V^+_h(-j)$ and $V_f^-\otimes V^-_h(-j)$) by $\lambda_f\lambda_h^{-1}\phi$ and $\lambda_f\lambda_h$ respectively. Note that as $f,h$ are good at $p$, by Ramanujan–Petersson conjecture, $|\lambda_f(\mathrm{Frob}_{p})|_\mathbb{C} = p_{v}^{\frac{k-1}{2}}$ and $|\lambda_h(\mathrm{Frob}_{p})|_\mathbb{C} =p_{v}^{\frac{l-1}{2}}$, where $p_v$ is the cardinality of the residue field at $v$. Thus $H^0(G_{L'}, V_f^-\otimes V^+_h)=H^0(G_{L'}, V_f^-\otimes V^-_h)=0$ and the vanishing results in \eqref{hkjadlk:DK?ALF2} follows. We can deduce the first vanishing result in \eqref{hkjadlk:DK?ALF} similarly. Putting these together, we get $\text{ker}(r_{\dagger,n})$ is finite and uniformly bounded.
Next we show that $\mathrm{coker}(r_{\mathrm{str},n})$ is finite and uniformly bounded independent of $n$. Let $\tilde{Fr}_{v,n}$ be a fixed lift of $\mathrm{Frob}_v$ in $G_{K_{n,v}}$. Again by \cite[Theorem 3.5(2)]{o1}, it is enough to show for all places $v \mid p$ in $K$, the action of $\tilde{Fr}_{v,n}$ is non-trivial on the successive quotients of the filtration $\frac{F^iV_j}{F^{i+1}V_j}\otimes \chi_p^{-i}$. In our case, the action of $\tilde{Fr}_{v,n}$ on $\frac{F^iV_j}{F^{i+1}V_j}\otimes \chi_p^{-i}$ is given by $\lambda_f^{\pm 1} \lambda_h^{\pm 1} \theta$ where $\theta$ is a finite order character. As before, given that $f,h$ are good at $p$, by Ramanujan–Petersson conjecture, $|\lambda_f(\mathrm{Frob}_p)|_\mathbb{C}=p_v^{\frac{k-1}{2}}$ and $|\lambda_h(\mathrm{Frob}_p)|_\mathbb{C} =p_v^{\frac{l-1}{2}}$, where $p_v$ is the cardinality of the residue field at $v$. Thus the given action of $\tilde{Fr}_{v,n}$ is non-trivial. Hence we deduce $\mathrm{coker}(r_{\mathrm{str},n})$ is finite and uniformly bounded independent of $n$.
Next we will show $\mathrm{coker}(r_{\mathrm{Gr},n})$ is finite and uniformly bounded independent of $n$.
Note that the local condition defining the strict Selmer differs with the Greenberg Selmer only at primes dividing $p$. Let $\phi_{\mathrm{Gr,str}}$ be the natural map $S_{\mathrm{str}}(A_j/\mathbb{Q}_\mathrm{cyc}) \stackrel{\phi_{\mathrm{Gr,str}}}{\longrightarrow} S_{\mathrm{Gr}}(A_j/\mathbb{Q}_\mathrm{cyc})$. From the definitions of Greenberg and strict Selmer group, for every $n$, there is a natural map from $\mathrm{coker}(r_{\mathrm{str},n}) \longrightarrow \mathrm{coker}(r_{\mathrm{Gr},n})$ such that the order of the cokernel is bounded by the order of $\mathrm{coker}(\phi_{\mathrm{Gr,str}})$. Thus it suffices to show $\mathrm{coker}(\phi_{\mathrm{Gr,str}})$ is finite. Now the order of the $\mathrm{coker}(\phi_{\mathrm{Gr,str}})$ is bounded by the order of $\underset{v\mid p}{\oplus}H^1(\mathbb{Q}_{\mathrm{cyc},v}^{unr}/{\mathbb{Q}_{\mathrm{cyc},v}}, A_j^{I_v})$ where $v$ is a prime in $\mathbb{Q}_\mathrm{cyc}$ dividing $p$ and $I_v$ is the inertia subgroup of $\bar{\mathbb{Q}}_p/\mathbb{Q}_{\mathrm{cyc},v}$ at $v$. As $\text{Gal}(\mathbb{Q}_{\mathrm{cyc},v}^{unr}/{\mathbb{Q}_{\mathrm{cyc},v}})$ is topologically cyclic, it suffices to show $H^0(\mathbb{Q}_{\mathrm{cyc},v}^{unr}/{\mathbb{Q}_{\mathrm{cyc},v}}, A_j^{I_v})=A_j^{G_{\mathbb{Q}_{\mathrm{cyc}, v}}}$ is finite for each $v \mid p$. Thus it further reduces to show $H^0(\mathbb{Q}_{\mathrm{cyc}, v}, V_j) =0$. This follows from the proof of \eqref{hkjadlk:DK?ALF}, \eqref{hkjadlk:DK?ALF2} written above.
\end{proof}
For any subset $\Sigma_{0} \subset \Sigma_L$, such that $\{v \text{ prime in } L : v \mid p \} \subset \Sigma_0 \setminus \Sigma$ and $ B_j \in \{ A_j, A_f(\xi_{1}\omega_{p}^{-j}) A_f(\xi_{2}\omega_{p}^{-j})\} $ we define
\begin{small}
\begin{align*}
S^{\Sigma_0}_{\mathrm{Gr}}(B_j[\pi]/L):=\mathrm{Ker}\bigg(H^1(\mathbb{Q}_\Sigma/L, B_j[\pi]) \longrightarrow \underset{v \in \Sigma_L \setminus \Sigma_0}\prod \frac{H^1(L_v, B_j[\pi])}{H_\dagger^1(L_v, B_j[\pi])}\bigg).
\end{align*}
\end{small}
\begin{comment}
We similarly define $$S^{\Sigma_0}_{\mathrm{Gr}}(A_j[\pi]/K):=\mathrm{Ker}\Big(H^1(\mathbb{Q}_\Sigma/K, A_j[\pi]) \longrightarrow \underset{v \in \Sigma \setminus \Sigma_0}\prod \frac{H^1(K_v, A_j[\pi])}{H_\dagger^1(K_v, A_j[\pi])}\Big).$$
\end{comment}
Note that if $H^0(G_L,A_j[\pi])=0$, then the natural map $H^1(\mathbb{Q}_\Sigma/L, A_j[\pi]) \longrightarrow H^1(\mathbb{Q}_\Sigma/L, A_j)[\pi]$ is an isomorphism. The following lemma is immediate from the above definition and the snake lemma.
\begin{lemma}\label{inside-out-lem}
Assume $H^0(G_L, A_j)=0$.
Then the natural map $S^{\Sigma_0}_\mathrm{Gr}(A_j[\pi]/L) \longrightarrow S^{\Sigma_0}_\mathrm{Gr}(A_j/L)[\pi]$ is an isomorphism.
\end{lemma}
\subsection{Explicit description of Selmer Groups}\label{subsec: explicit selmer}
Recall that $ f \in S_{k}(\Gamma_{0}(N), \eta)$, $ h \in S_{l}(\Gamma_{0}(I), \psi)$ and let $ m $ be as in \eqref{definition m}. From now on, we choose and fix $ \Sigma := \{ \ell : \ell \mid pNI\} \cup \{ \ell : \ell \mid \infty\} $. Also recall $ \Sigma_0 := \{ \ell : \ell \mid m\} $. Let $ \Sigma^{\infty} $ (resp. $ \Sigma_{0}^\infty $) be the set of primes in $ \mathbb{Q}_{\mathrm{cyc}} $ lying above $ \Sigma $ (resp. $ \Sigma_{0}$).
In this subsection, we give a more explicit description of $ H^{1}_{\mathrm{Gr}}(\mathbb{Q}_{\mathrm{cyc},v},A_j[\pi]) $ and $ H^{1}_{\mathrm{Gr}}(\mathbb{Q}_{\mathrm{cyc},v},A_f(\xi_i\omega_p^{-j})[\pi]) $ for $ v \in \Sigma^{\infty} \setminus \Sigma^{\infty}_{0} $.
\begin{proposition}\label{prop:Greenberg at p}
Let $ v \in \Sigma^\infty$ and $v \mid p $. Assume the order of $\psi |I_{\mathrm{cyc},v}$ is co-prime to $p$.
Then we have
\begin{align*}
H^{1}_{\mathrm{Gr}} (\mathbb{Q}_{\mathrm{cyc},v},A_j[\pi]) =
\mathrm{Ker}\big(H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A_{j}[\pi]) \longrightarrow H^1(I_{\mathrm{cyc},v}, A^{-}_{j}[\pi])\big).
\end{align*}
\end{proposition}
\begin{proof}
First note that by definition of $ H^1_{\text{Gr}}(\mathbb{Q}_{\mathrm{cyc},v},A_j[\pi]) $, we have the following exact sequence
\begin{equation*}
0\longrightarrow H^1_{\text{Gr}}(\mathbb{Q}_{\mathrm{cyc},v},A_j[\pi]) \longrightarrow H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}},A_j[\pi]) \stackrel{\varphi}{\longrightarrow} H^1(I_{\mathrm{cyc},v},A^-_j).
\end{equation*}
Consider the commutative diagram
\begin{small}
\begin{equation}\label{dqGYaidhkAdoaU}
\begin{tikzcd}
& & H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}},A_j[\pi]) \ar[r,"\nu"] \ar[d,"\cong"] & H^1(I_{\mathrm{cyc},v},A^-_j[\pi]) \ar[d,"\epsilon"] \\
0 \ar[r] & H^1_{\text{Gr}}(G_{\mathbb{Q}_{\mathrm{cyc},v}},A_j[\pi]) \ar[r] & H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}},A_j[\pi]) \ar[r,"\varphi"] & H^1(I_{\mathrm{cyc},v},A^-_j) .
\end{tikzcd}
\end{equation}
\end{small}
From the diagram \eqref{dqGYaidhkAdoaU}, we see that $H^1_{\text{Gr}}(\mathbb{Q}_{\mathrm{cyc},v},A_j[\pi]) = \text{ker}(\varphi) = \nu^{-1}(\text{ker}(\epsilon)).$ Thus it suffices to show $\text{ker}(\epsilon) =0 $.
As $T_f^-$ is unramified at $ v $, we have $H^1(I_{\mathrm{cyc},v}, A_j^-[\pi]) \cong H^1(I_{\mathrm{cyc},v}, A_h(j)[\pi]) \otimes T_f^-$ and $H^1(I_{\mathrm{cyc},v}, A_j^-) \cong H^1(I_{\mathrm{cyc},v}, A_h(j)) \otimes T_f^-$. Let $\epsilon'$ be the natural map $H^1(I_{\mathrm{cyc},v}, A_h(j)[\pi]) \longrightarrow H^1(I_{\mathrm{cyc},v}, A_h(j))$. Further, it suffices to show ker $ (\epsilon') =0 $.
As $ I_{\mathrm{cyc}, v} $ has $ p $-cohomological dimension $ 1 $, $ H^{2}(I_{\mathrm{cyc},v}, A_h^{-}(j)[\pi]) =0 = H^{2}(I_{\mathrm{cyc},v}, A_h^{-}(j)) $ and we have
\begin{comment}
Now we assume $\rho_{h}|_{I_{\mathrm{cyc},v}} \sim \begin{pmatrix} \psi_h & 0 \\ 0 &1 \end{pmatrix}$, where $\psi_h$ is the nebentypus of $h$. We further assume order of $\psi_g$ is coprime to $p$. Then by our choice of lattice, $\bar{\rho}_{h}|_{I_{\mathrm{cyc},v}} \sim \begin{pmatrix} \bar{\psi}_h & 0 \\ 0 &1 \end{pma(j)trix}$.
By splitting of $\rho_{h}|_{I_{\mathrm{cyc},v}}$ and $\bar{\rho}_{h}|_{I_{\mathrm{cyc},v}}$ as above, we have the following commutative diagram of split exact sequences
\end{comment}
\begin{small}
\[
\begin{tikzcd}
H^0(I_{\mathrm{cyc},v}, A_h^-(j)[\pi]) \ar[r] \ar[d,"\cong"] & H^1(I_{\mathrm{cyc},v}, A_h^+(j)[\pi]) \ar[r]\ar[d,"\epsilon_1"] & H^1(I_{\mathrm{cyc},v} , A_h(j)[\pi]) \ar[r]\ar[d,"\epsilon' "] & H^1(I_{\mathrm{cyc},v}, A_h^-(j)[\pi]) \ar[d, "\epsilon_2"] \ar[r]& 0 \\
H^0(I_{\mathrm{cyc},v}, A_h^-(j))[\pi] \ar[r]& H^1(I_{\mathrm{cyc},v}, A_h^+(j))[\pi] \ar[r] & H^1(I_{\mathrm{cyc},v},A_h(j))[\pi] \ar[r] & H^1(I_{\mathrm{cyc},v}, A_h^-(j))[\pi] \ar[r]& 0 .
\end{tikzcd}
\]
\end{small}
Note as $A_h^{-}$ is unramified, $I_{\mathrm{cyc},v}$ acts on the $p$-primary group $A_h^-(j)[\pi]$ by the character $\omega_p^j$ which has order prime to $p$. Thus either $(A_h^{-}(j)[\pi])^{I_{\mathrm{cyc},v}} =0$ or we have $(A_h^{-}(j))^{I_{\mathrm{cyc},v}}$ is divisible. In either case, it follows that $\frac{(A_h^{-}(j))^{I_{\mathrm{cyc},v}}}{\pi} =0$ and hence $\epsilon_2$ is an isomorphism. As $\psi_h \chi_{p}^{l-1}|I_{\mathrm{cyc},v}$ has order prime to $p$, a similar argument shows that $\epsilon_1$ is also an isomorphism. By the Five lemma, it follows that
ker$(\epsilon')$=0.
\end{proof}
\begin{comment}
Case 2: We assume that $\psi_g|_{I_{\mathrm{cyc},v}}=1$ and $* \neq 0$ but $* = 0 ~(mod~ \pi)$. We consider the natural connecting map ${A_g^{-}}^{I_{\mathrm{cyc},v}} \stackrel{\tau}{\longrightarrow} H^1(I_{\mathrm{cyc},v}, A_g^+) $. Note that Im$(\tau)$ is divisible of corank =1. Indeed, this follows from the fact that $* \neq 0$ and ${A_g^{-}}^{I_{\mathrm{cyc},v}} = {A_g^{-}}$ is divisible of corank 1. We define $I':= \mathrm{ker}(\rho_g|_{I_{\mathrm{cyc},v}})$. {\color{red} IN THIS PARTICULAR CASE, $I'$ DEPENDS ON $g$ AND HENCE THE REST OF THE PROOF IS NOT CORRECT} Then the natural map $H^1(I', A_g^+) \rightarrow H^1(I', A_g) $ is injective. So we have the following commutative diagram
\xymatrix{
& {A_g^{-}}^{I_{\mathrm{cyc},v}} \ar[r]^\tau \ar[d] & H^1(I_{\mathrm{cyc},v}, A_g^+) \ar[r]\ar[d]^{f_1} & H^1(I_{\mathrm{cyc},v}, , A_g) \ar[r]\ar[d]^{f_2} & H^1(I_{\mathrm{cyc},v}, A_g^-) \ar[r] \ar[d]^{f_3} & 0 \\
& 0 \ar[r]& H^1(I', A_g^+) \ar[r] & H^1(I',A_g) \ar[r] & H^1(I', A_g^-) \ar[r] & 0 \\
}
Note that $f_1$ is surjective as $I_{\mathrm{cyc},v}/{I'} \cong \mathbb{Z}_p$. Thus using a snake lemma in the above diagram, we get
$$0\longrightarrow \text{Im}(\tau) \longrightarrow \text{ker}(f_1) $$
As $ \text{ker}(f_1) \cong H^1(I_{\mathrm{cyc},v}/{I'}, A_g^+) $, it is divisible of corank 1 and hence the above map $\text{Im}(\tau) \longrightarrow \text{ker}(f_1)$ is an isomorphism.
Now as $* = 0 $ (mod $p$), the commutativity of the following diagram follows:
\xymatrix{
0 \ar[r] & H^1(I_{\mathrm{cyc},v}, A_g^+[\pi]) \ar[r]\ar[d]^{\bar{\epsilon}_1} & H^1(I_{\mathrm{cyc},v}, , A_g[\pi]) \ar[r]\ar[d]^{\epsilon'} & H^1(I_{\mathrm{cyc},v}, A_g^-[\pi]) \ar[r] \ar[d]^{\epsilon_2} & 0 \\
0 \ar[r] & \frac{H^1(I_{\mathrm{cyc},v}, A_g^+)}{\text{Im}(\tau)}[\pi] \ar[r] & H^1(I_{\mathrm{cyc},v},A_g)[\pi] \ar[r] & H^1(I_{\mathrm{cyc},v}, A_g^-)[\pi] & \\
}
Here $\bar{\epsilon}_1$ is induced by the map ${\epsilon}_1: H^1(I_{\mathrm{cyc},v}, A_g^+[\pi]) \longrightarrow H^1(I_{\mathrm{cyc},v}, A_g^+)$. Note as Im$(\tau)$ is divisible, we can write $\frac{H^1(I_{\mathrm{cyc},v}, A_g^+)}{\text{Im}(\tau)}[\pi] = \frac{H^1(I_{\mathrm{cyc},v}, A_g^+)[\pi]}{\text{Im}(\tau)[\pi]}$. Now as $\epsilon_2$ is injective, $\text{ker}(\bar{\epsilon}_1) =\text{ker}(\epsilon')$. Now $\text{ker}(\bar{\epsilon}_1) = {\epsilon}_1^{-1}(\text{Im}(\tau)) \cong {\epsilon}_1^{-1}( H^1(I_{\mathrm{cyc},v}/{I'}, A_g^+) ) \cong H^1(I_{\mathrm{cyc},v}/{I'}, A_g^+[\pi]).$ The last isomorphism follows from the following commutative diagram where both the vertical maps are isomorphisms.
\xymatrix{
0 \ar[r] & H^1(I_{\mathrm{cyc},v}/{I'}, A_g^+[\pi]) \ar[r]\ar[d] & H^1(I_{\mathrm{cyc},v}, , A^+_g[\pi]) \ar[d]\\
0 \ar[r] & H^1(I_{\mathrm{cyc},v}/{I'}, A_g^+)[\pi]\ar[r] & H^1(I_{\mathrm{cyc},v}, , A^+_g)[\pi] \\
}
Thus ker$(\epsilon')=$ ker$(\epsilon_1)$ depends only on $A_g^+[\pi]$. Also $A_g^+[\pi]$ is determined by $A_g[\pi]$ and $A_g^-[\pi]$. Moreover by remark \ref{189}, $A_g^-[\pi]$ is determined by $A_g[\pi]$ for $l > p$. Thus ker$(\epsilon')=$ is determined by $A_g[\pi]$ for $l>p$. Hence for $k>l >p$, $$\text{ker}(\epsilon) \cong \text{ker}(\epsilon')\otimes T_f^- \cong \text{ker}(\epsilon')\otimes T_f^-/{\pi T_f^-} \cong \text{ker}(\epsilon')\otimes A_f^-[\pi] $$ is determined by $A_f[\pi]$ and $A_g[\pi]$.
Case 3: $\psi_g|_{I_{\mathrm{cyc},v}} = 1$ and $* = 0$. This case would also be ok.
Case 4: $\psi_g|_{I_{\mathrm{cyc},v}} \neq 1$.
In the general case, when $\mu_p \not\in K$, let us denote $\Delta= \text{Gal}(K(\mu_{p})_v/K_v)$. Then as $(\# \Delta, p)=1$, using Hoscschild-Serre spectral sequence, we can show that $\text{ker}(\epsilon)=H^1(I_{\mathrm{cyc},v}, A_j^-[\pi])^\Delta$.
\end{comment}
\begin{comment}
Let $ \Sigma_0 =\{v \text{ prime in } K: v \mid p\} $. Now we would like to show that for $\dagger \in \{Str, Gr\}$, $S^{\Sigma_0}_\dagger(A_j[\pi]/K_\mathrm{cyc})$ depends only on the residual representation $A_j[\pi]$. Clearly, it is enough to show that for each $v \in \Sigma_0$, $H^1_\dagger(K_{\mathrm{cyc},v},A_j[\pi])$ depends only on $A_j[\pi]$. First we consider the case $\dagger =Str$. First note that by definition of $ H^1_{Str}(K_{\mathrm{cyc},v},A_j[\pi]) $, we can write, $H^1_{Str}(K_{\mathrm{cyc},v},A_j[\pi]) = \text{Ker}(\phi)$ where
\begin{equation}\label{sjadggqGD}
0\longrightarrow H^1_{Str}(K_{\mathrm{cyc},v},A_j[\pi]) \longrightarrow H^1(K_{\mathrm{cyc},v},A_j[\pi]) \stackrel{\phi}{\longrightarrow} H^1(K_{\mathrm{cyc},v},A^-_j).
\end{equation}
Consider the commutative diagram
\begin{equation}\label{dqGYaidhkAdoaU}
\xymatrix{
& & H^1(K_{\mathrm{cyc},v},A_j[\pi]) \ar[r]^{\psi}\ar[d]^= & H^1(K_{\mathrm{cyc},v},A^-_j[\pi]) \ar[r] \ar[d]^{\theta} & 0 \\
0 \ar[r] & H^1_{Str}(K_{\mathrm{cyc},v},A_j[\pi]) \ar[r] & H^1(K_{\mathrm{cyc},v},A_j[\pi]) \ar[r]^{\phi} & H^1(K_{\mathrm{cyc},v},A^-_j) \\
}
\end{equation}
{\color{red} We know that the map $\psi$ in diagram \eqref{dqGYaidhkAdoaU} is surjective as $p$-cohomological dimension of $G_{K_{\mathrm{cyc},v}}$ =1 for prime $v \mid p$. (reference ?)} From the diagram \eqref{dqGYaidhkAdoaU}, we see that $H^1_{Str}(K_{\mathrm{cyc},v},A_j[\pi]) = \text{ker}(\phi) = \psi^{-1}(\text{ker}(\theta)).$ Thus it suffices to show $\text{ker}(\theta)$ is determined by $A_j[\pi]$. We first show that $\text{ker}(\theta)$ is determined by $A_j^-[\pi]$. We will then go on to show $A_j^-[\pi]$ is determined by $A_j[\pi]$.
We will show that $\text{ker}(\theta) $ is determined by $A^-_j[\pi]$. Let $I_{\mathrm{cyc}, v}$ and $<\text{Fr}_{\mathrm{cyc},v}>$ denote the inertia subgroup and Frobenius quotient of $\bar{\mathbb{Q}}_p/K_{\mathrm{cyc},v}$. We first assume $K$ contains $\mu_p$ i.e. $K_\mathrm{cyc}= K(\mu_{p^\infty})$.
\xymatrix{
0 \ar[r] & H^1(\text{Fr}_{\mathrm{cyc},v}, A_j^-[\pi]) \ar[r]\ar[d] & H^1(K_{\mathrm{cyc},v}, A_j^-[\pi]) \ar[r]\ar[d]^{\theta} & H^1(I_{\mathrm{cyc},v}, A_j^-[\pi]) \ar[r] \ar[d]^{\epsilon} & 0 \\
0 \ar[r] & H^1(\text{Fr}_{\mathrm{cyc},v}, A_j^-) \ar[r] & H^1(K_{\mathrm{cyc},v},A_j^-) \ar[r] & H^1(I_{\mathrm{cyc},v}, A^-_j) \ar[r] & 0 \\ }
Since $ A_j^-$ is divisible, $H^1(\text{Fr}_{\mathrm{cyc},v}, A_j^-)$ is divisible. On the other hand, from the proof of \eqref{hkjadlk:DK?ALF} and \eqref{hkjadlk:DK?ALF2} we get that $H^0(\text{Fr}_{\mathrm{cyc},v}, A_j^-) $ is finite. As $<\text{Fr}_{\mathrm{cyc},v}>$ is topologically cyclic, $H^1(\text{Fr}_{\mathrm{cyc},v}, A_j^-)$ is also finite. Thus $H^1(\text{Fr}_{\mathrm{cyc},v}, A_j^-)=0$. This shows that we have an exact sequence \begin{equation}\label{181}
0\longrightarrow H^1(\text{Fr}_{\mathrm{cyc},v}, A_j^-[\pi])\longrightarrow \text{ker}(\theta) \longrightarrow \text{ker}(\epsilon) \longrightarrow 0. \end{equation} {\color{red} Question: Why is it enough from \eqref{181} to show ker$(\epsilon)$ and $H^1(\text{Fr}_{\mathrm{cyc},v}, A_j^-[\pi]) $ depends on residual representation to conclude the same about $\text{ker}(\theta)$ ?} Now note that $\mu_p \subset K$ and $A_f^-$ is unramified together implies that $H^1(I_{\mathrm{cyc},v}, A_j^-[\pi]) \cong H^1(I_{\mathrm{cyc},v}, A_g[\pi]) $ and $H^1(I_{\mathrm{cyc},v}, A_j^-) \cong H^1(I_{\mathrm{cyc},v}, A_g) $. Thus from \eqref{181}, it suffices to show ker$(\epsilon)$ is determined by $A_g[\pi]$ where $\epsilon$ is the natural map $H^1(I_{\mathrm{cyc},v}, A_g[\pi]) \longrightarrow H^1(I_{\mathrm{cyc},v}, A_g)$.
Now $\rho_{g}|_{I_{\mathrm{cyc},v}} \sim \begin{pmatrix} \psi_g & * \\ 0 &1 \end{pmatrix}$, where $\psi_g$ is the nebentypus of $g$.
\noindent Case 1: We assume that $\psi_g|_{I_{\mathrm{cyc},v}}=1$ and $* \neq 0 ~(mod~ \pi)$.\\
By the given condition, we can check that $(A_g^+)^{I_{\mathrm{cyc},v}} = A_g^{I_{\mathrm{cyc},v}}$ and $(A_g^+[\pi])^{I_{\mathrm{cyc},v}} = A_g^{I_{\mathrm{cyc},v}}[\pi]$. Thus we have the following commutative diagram
\xymatrix{
0 \ar[r] & (A_g^-[\pi])^{I_{\mathrm{cyc},v}} \ar[r] \ar[d]^{\epsilon_0} & H^1(I_{\mathrm{cyc},v}, A_g^+[\pi]) \ar[r]\ar[d]^{\epsilon_1} & H^1(I_{\mathrm{cyc},v}, , A_g[\pi]) \ar[r]\ar[d]^{\epsilon} & H^1(I_{\mathrm{cyc},v}, A_g^-[\pi]) \ar[r] \ar[d]^{\epsilon_2} & 0 \\
0 \ar[r] &{(A_g^-)}^{I_{\mathrm{cyc},v}}[\pi] \ar[r]& H^1(I_{\mathrm{cyc},v}, A_g^+)[\pi] \ar[r] & H^1(I_{\mathrm{cyc},v},A_g)[\pi] \ar[r] & H^1(I_{\mathrm{cyc},v}, A_g^-)[\pi] \ar[r] & 0 \\ } As $\epsilon_i$ is an isomorphism for $0 \leq i \leq 2$, it follows that $\epsilon $ is also an isomorphism and ker$(\epsilon)$=0. Hence from \eqref{181}, ker$(\theta) \cong H^1(\text{Fr}_{\mathrm{cyc},v}, A_j^-[\pi]).$
Case 2: We assume that $\psi_g|_{I_{\mathrm{cyc},v}}=1$ and $* \neq 0$ but $* = 0 ~(mod~ \pi)$. We consider the natural connecting map ${A_g^{-}}^{I_{\mathrm{cyc},v}} \stackrel{\tau}{\longrightarrow} H^1(I_{\mathrm{cyc},v}, A_g^+) $. Note that Im$(\tau)$ is divisible of corank =1. Indeed, this follows from the fact that $* \neq 0$ and ${A_g^{-}}^{I_{\mathrm{cyc},v}} = {A_g^{-}}$ is divisible of corank 1. We define $I':= \mathrm{ker}(\rho_g|_{I_{\mathrm{cyc},v}})$. Then the natural map $H^1(I', A_g^+) \rightarrow H^1(I', A_g) $ is injective. So we have the following commutative diagram
\xymatrix{
& {A_g^{-}}^{I_{\mathrm{cyc},v}} \ar[r]^\tau \ar[d] & H^1(I_{\mathrm{cyc},v}, A_g^+) \ar[r]\ar[d]^{f_1} & H^1(I_{\mathrm{cyc},v}, , A_g) \ar[r]\ar[d]^{f_2} & H^1(I_{\mathrm{cyc},v}, A_g^-) \ar[r] \ar[d]^{f_3} & 0 \\
& 0 \ar[r]& H^1(I', A_g^+) \ar[r] & H^1(I',A_g) \ar[r] & H^1(I', A_g^-) \ar[r] & 0 \\ } Note that $f_1$ is surjective as $I_{\mathrm{cyc},v}/{I'} \cong \mathbb{Z}_p$. Thus using a snake lemma in the above diagram, we get
$$0\longrightarrow \text{Im}(\tau) \longrightarrow \text{ker}(f_1) $$
As $ \text{ker}(f_1) \cong H^1(I_{\mathrm{cyc},v}/{I'}, A_g^+) $, it is divisible of corank 1 and hence the above map $\text{Im}(\tau) \longrightarrow \text{ker}(f_1)$ is an isomorphism.
Now as $* = 0 $ (mod $p$), the commutativity of the following diagram follows:
\xymatrix{
0 \ar[r] & H^1(I_{\mathrm{cyc},v}, A_g^+[\pi]) \ar[r]\ar[d]^{\bar{\epsilon}_1} & H^1(I_{\mathrm{cyc},v}, , A_g[\pi]) \ar[r]\ar[d]^{\epsilon} & H^1(I_{\mathrm{cyc},v}, A_g^-[\pi]) \ar[r] \ar[d]^{\epsilon_2} & 0 \\
0 \ar[r] & \frac{H^1(I_{\mathrm{cyc},v}, A_g^+)}{\text{Im}(\tau)}[\pi] \ar[r] & H^1(I_{\mathrm{cyc},v},A_g)[\pi] \ar[r] & H^1(I_{\mathrm{cyc},v}, A_g^-)[\pi] & \\ }
Here $\bar{\epsilon}_1$ is induced by the map ${\epsilon}_1: H^1(I_{\mathrm{cyc},v}, A_g^+[\pi]) \longrightarrow H^1(I_{\mathrm{cyc},v}, A_g^+)$. Note as Im$(\tau)$ is divisible, we can write $\frac{H^1(I_{\mathrm{cyc},v}, A_g^+)}{\text{Im}(\tau)}[\pi] = \frac{H^1(I_{\mathrm{cyc},v}, A_g^+)[\pi]}{\text{Im}(\tau)[\pi]}$. Now as $\epsilon_2$ is injective, $\text{ker}(\bar{\epsilon}_1) =\text{ker}(\epsilon)$. Now $\text{ker}(\bar{\epsilon}_1) = {\epsilon}_1^{-1}(\text{Im}(\tau)) \cong {\epsilon}_1^{-1}( H^1(I_{\mathrm{cyc},v}/{I'}, A_g^+) ) \cong H^1(I_{\mathrm{cyc},v}/{I'}, A_g^+[\pi]).$ The last isomorphism follows from the following commutative diagram where both the vertical maps are isomorphisms.
\xymatrix{
0 \ar[r] & H^1(I_{\mathrm{cyc},v}/{I'}, A_g^+[\pi]) \ar[r]\ar[d] & H^1(I_{\mathrm{cyc},v}, , A^+_g[\pi]) \ar[d]\\
0 \ar[r] & H^1(I_{\mathrm{cyc},v}/{I'}, A_g^+)[\pi]\ar[r] & H^1(I_{\mathrm{cyc},v}, , A^+_g)[\pi] \\ }
Thus ker$(\epsilon)=$ ker$(\epsilon_1)$ depends only on $A_g^+[\pi]$. Now look at \eqref{181}.
\begin{equation}\label{182}
\begin{tiny}{
0 \rightarrow (A_g^+)^{I_{\mathrm{cyc},v}} \rightarrow A_g^{I_{\mathrm{cyc},v}} \rightarrow {A_g^{-}}^{I_{\mathrm{cyc},v}} \rightarrow H^1(I_{\mathrm{cyc},v}, A_g^+) \rightarrow H^1(I_{\mathrm{cyc},v}, A_g) \rightarrow H^1(I_{\mathrm{cyc},v}, A_g^{-}) \rightarrow 0
}\end{tiny} \end{equation}
Case 3: $\psi_g|_{I_{\mathrm{cyc},v}} \neq 1$.
In the general case, when $\mu_p \not\in K$, let us denote $\Delta= \text{Gal}(K(\mu_{p})_v/K_v)$. Then as $(\# \Delta, p)=1$, using Hoscschild-Serre spectral sequence, we can show that $\text{ker}(\theta)=H^1(\text{Fr}_{\mathrm{cyc},v}, A_j^-[\pi])^\Delta$.
We will now show $A_j^-[\pi]$ is determined by $A_j[\pi]$ to complete the proof. \end{comment}
\begin{rem}\label{189} Assume $p >k$. Then note that $A_j^-[\pi] = (A_f^-\otimes T_h)(j) [\pi] \cong (A_f^-[\pi]\otimes T_h)(j) \cong (A_f^-[\pi]\otimes_{\mathcal{O}/\pi} \frac{T_h}{\pi})(j) \cong ((A_f[\pi])_{I_p}\otimes_{\mathcal{O}/\pi} A_h[\pi] )(j)$. The last equality is true as $p>k$ by \cite[(3.5), (3.6)]{p-selmer}. This shows that if $p>k$ then $A_j^-[\pi]$ is determined by $A_f[\pi] $ and $A_h[\pi]$.
\end{rem}
\begin{proposition}\label{prop:Greenberg away from N} Let $ v \in \Sigma^\infty$, $ v \nmid N pm $ and $ v \mid I $. Assume $ \psi \lvert I_{\mathrm{cyc},v} $ has order prime to $ p $. Then we have \begin{align*}
H^{1}_{\mathrm{Gr}} (\mathbb{Q}_{\mathrm{cyc},v},A_j[\pi]) =
\mathrm{Ker}\big(H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A_{j}[\pi]) \longrightarrow H^1(I_{\mathrm{cyc},v}, A_{j}[\pi])\big). \end{align*} \end{proposition} \begin{proof}
From Lemma~\ref{lem: exactly divides} and \cite[Theorem 3.26 (iii)(a)]{Hida3}, we have
$\rho_{h}|_{I_{\mathrm{cyc},v}} \sim \begin{psmallmatrix} \psi_h & 0 \\ 0 &1 \end{psmallmatrix}$.
By the definition of $ H^1_{\text{Gr}}(G_{\mathbb{Q}_{\mathrm{cyc},v}},A_j[\pi]) $, we have the following exact sequence \begin{equation*}
0\longrightarrow H^1_{\text{Gr}}(\mathbb{Q}_{\mathrm{cyc},v},A_j[\pi]) \longrightarrow H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}},A_j[\pi]) \longrightarrow H^1(I_{\mathrm{cyc},v},A_j). \end{equation*} Rest of the argument is similar to the proof of Proposition~\ref{prop:Greenberg at p}. \end{proof}
\begin{proposition}\label{prop:Greenberg dividing N} Recall that $ M_0 $ is the prime to $ p $-part of $ \mathrm{cond}(\bar{\xi}_1\bar{\omega}_p^{1-l}) \mathrm{cond}(\bar{\xi_2})$. Let $ v \in \Sigma^\infty $, $ v \mid N $ and $ v \nmid pm $. Assume $ (N,M_0) =1 $. Then we have \begin{small}
\begin{align*}
H^{1}_{\mathrm{Gr}} (\mathbb{Q}_{\mathrm{cyc},v},A_j[\pi]) =
\mathrm{Ker}\Big(H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A_{j}[\pi]) \longrightarrow \Big(H^1(I_{\mathrm{cyc},v}, A_{f}(j))[\pi] \otimes \frac{T_h}{\pi}\Big)^{\frac{G_{\mathbb{Q}_{\mathrm{cyc},v}}}{I_{\mathrm{cyc},v}}}\Big).
\end{align*} \end{small}
\end{proposition} \begin{proof} Let $ \Delta_v = G_{\mathbb{Q}_{\mathrm{cyc},v}}/I_{\mathrm{cyc},v} = \mathrm{Gal}(\mathbb{Q}_{\mathrm{cyc},v}^{\text{unr}}/\mathbb{Q}_{\mathrm{cyc},v}) $. Using inflation-restriction sequence, we obtain the image of $ H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}},A_j) \rightarrow H^1(I_{\mathrm{cyc},v},A_j) $ lies in $H^1(I_{\mathrm{cyc},v},A_j) ^{\Delta_v}$. From the definition of $ H^1_{\text{Gr}}(\mathbb{Q}_{\mathrm{cyc},v},A_j[\pi]) $, we have the following exact sequence \begin{small}
\begin{equation*}
0\longrightarrow H^1_{\text{Gr}}(G_{\mathbb{Q}_{\mathrm{cyc},v}},A_j[\pi]) \longrightarrow H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}},A_j[\pi]) \xrightarrow{\varphi} H^1(I_{\mathrm{cyc},v},A_j)^{\Delta_v}.
\end{equation*} \end{small} Since the image of $ \varphi $ is $ \pi $-torsion, we get \begin{small}
\begin{equation}\label{eq:Greenberg away from N}
0\longrightarrow H^1_{\text{Gr}}(G_{\mathbb{Q}_{\mathrm{cyc},v}},A_j[\pi]) \longrightarrow H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}},A_j[\pi]) \stackrel{\varphi}{\longrightarrow} H^1(I_{\mathrm{cyc},v},A_j)^{\Delta_v}[\pi] = \big(H^1(I_{\mathrm{cyc},v},A_j)[\pi] \big)^{\Delta_v} .
\end{equation} \end{small}
As $ v \nmid M_0 pm $, we have $ v \nmid I $.
Since $ A_j = A_{f} \otimes T_h (j)$ and $ T_h $ are unramified at $ v $, we obtain $ H^1(I_{\mathrm{cyc},v},A_j)[\pi] = \big(H^1(I_{\mathrm{cyc},v},A_f(j)) \otimes T_{h} \big) [\pi]= H^1(I_{\mathrm{cyc},v},A_f(j))[\pi] \otimes T_h = H^1(I_{\mathrm{cyc},v},A_f(j))[\pi] \otimes \frac{T_h}{\pi} $. Substituting this in \eqref{eq:Greenberg away from N}, we get the required description of $H^{1}_{\mathrm{Gr}} (\mathbb{Q}_{\mathrm{cyc},v},A_j[\pi])$. \end{proof} \begin{comment} \begin{corollary}\label{cor:selmer A_j}
Let $ (N,M_0) =1 $ and order of $ \bar{\psi}_h \lvert _{I_{\mathrm{cyc},v}} $ is co-prime to $ p $ for $ v \mid I_0 p $. Then
\begin{small}
\begin{equation*}
\begin{split}
S^{\Sigma_0}_{\mathrm{Gr}}(A_j[\pi]/K_{\mathrm{cyc}}) =
\mathrm{Ker}\Big(H^1(\mathbb{Q}_\Sigma/K_{\mathrm{cyc}}, A_j[\pi]) \longrightarrow & \underset{v \mid p} \prod H^1(I_{\mathrm{cyc},v}, A^{-}_j[\pi])\Big) \times \underset{v \nmid Nmp, v \mid I} \prod H^1(I_{\mathrm{cyc},v}, A_j[\pi]) \\ & \qquad \times \underset{v \mid N, v \nmid m} \prod \big( H^{1}(I_{\mathrm{cyc},v}, A_f[\pi]) \otimes_{\mathcal{O}/\pi} T_h/\pi \big)^{\Delta'}\Big).
\end{split}
\end{equation*}
\end{small} \end{corollary} \end{comment} Recall that $ \xi_1 $ and $ \xi_{2} $ are Dirichlet characters whose reductions are $ \bar{\xi}_1 $ and $ \bar{\xi}_2 $ respectively. We now prove an analogous result for $ f \otimes \xi_{1} $ and $ f \otimes \xi_{2} $. \begin{proposition}\label{prop:Greenberg for modular form} Let $ v \in \Sigma^\infty \setminus \Sigma^\infty_{0}$. Then \begin{enumerate}[label=$(\roman*)$]
\item
If $ v \mid p $, then for $ i=1,2 $,
\begin{align*}
H^{1}_{\mathrm{Gr}} (\mathbb{Q}_{\mathrm{cyc},v},A_f(\xi_{i} \omega^{-j})[\pi]) =
\mathrm{Ker}\big(H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A_f(\xi_{i} \omega^{-j})[\pi]) \longrightarrow H^1(I_{\mathrm{cyc},v}, A_{f}^{-}(\xi_{i} \omega^{-j})[\pi])\big).
\end{align*}
\begin{comment}
\begin{align*}
H^{1}_{\mathrm{Gr}} &(K_{\mathrm{cyc},v},A_f(\xi_{1} \omega^{-j})[\pi]) \\
&= \begin{cases}
\mathrm{Ker}\big(H^1(G_{K_{\mathrm{cyc},v}}, A_f(\xi_{1} \omega^{-j})[\pi]) \longrightarrow H^1(I_{\mathrm{cyc},v}, A_{f}^{-}(\xi_{1} \omega^{-j})[\pi])\big)
\text{ if } (\xi_{1} \omega^{-j})|_{I_p} \neq 1 \text{ or } j=0, \\
\mathrm{Ker}\big(H^1(G_{K_{\mathrm{cyc},v}}, A_f(\xi_{1} \omega^{-j})[\pi]) \longrightarrow H^1(I'_{\mathrm{cyc},v}, A_{f}^{-}(\xi_{1} \omega^{-j})[\pi]) \big) \text{ otherwise. }
\end{cases}
\end{align*}
\end{comment}
\item If $ v \nmid Nm $ and $ v \mid I_0 $, then for $ i=1,2 $,
\begin{align*}
H^{1}_{\mathrm{Gr}} (\mathbb{Q}_{\mathrm{cyc},v},A_f(\xi_{i} \omega^{-j})[\pi]) = \mathrm{Ker}\big(H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A_f(\xi_{i} \omega^{-j})[\pi]) \longrightarrow H^1(I_{\mathrm{cyc},v}, A_f(\xi_{i} \omega^{-j})[\pi])\big).
\end{align*}
\item Assume $ (N,M_0) =1 $. If $ v \mid N $ and $ v \nmid m $, then for $ i=1,2 $,
\begin{small}
\begin{align*}
H^{1}_{\mathrm{Gr}} (\mathbb{Q}_{\mathrm{cyc},v},A_f(\xi_{i} \omega^{-j})[\pi]) =
\mathrm{Ker}\Big(H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A_f(\xi_{i} \omega^{-j})[\pi]) \longrightarrow (H^1(I_{\mathrm{cyc},v}, A_{f}(j))[\pi] \otimes \bar{\xi}_{i})^{\frac{G_{\mathbb{Q}_{\mathrm{cyc},v}}}{I_{\mathrm{cyc},v}}}\Big).
\end{align*}
\end{small} \end{enumerate} \end{proposition} \begin{proof} For $ i = 1,2$ and $v \in \Sigma^{\infty} \setminus \Sigma^\infty_{0} $, we have the following commutative diagram \begin{small}
\[
\begin{tikzcd}
H^{1}(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A_f(\xi_i \omega_{p}^{-j})[\pi]) \arrow[r, "\nu"] \arrow[d, "i^{\ast}_1"]
& H^{1}(I_{\mathrm{cyc},v}, A_{f,v}(\xi_i \omega_{p}^{-j})[\pi]) \arrow[d, "\epsilon"] \\
H^{1}(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A_{f}(\xi_i \omega_{p}^{-j})) \arrow[r, "\varphi"]
& H^{1}(I_{\mathrm{cyc},v}, A_{f,v}(\xi_i \omega_{p}^{-j})),
\end{tikzcd}
\] \end{small} where $ A_{f,v} = A_f^{-}$ if $ v \mid p $ and $ A_{f,v} = A_f $ if $ v \nmid p $. By definition, we have $ H^{1}_{\mathrm{Gr}}(G_{\mathbb{Q}_{\mathrm{cyc},v}},A_j[\pi]) = \mathrm{ker}(\psi \circ i_1^{\ast}) = \mathrm{ker}(\epsilon \circ \nu) $. It suffices to show $\mathrm{ker}(\epsilon) = 0$. Furthermore, we have $ \mathrm{ker}(\epsilon) = A_{f,v}(\xi_i \omega_{p}^{-j})^{I_{\mathrm{cyc},v}}/\pi A_{f,v}(\xi_i \omega_{p}^{-j})^{I_{\mathrm{cyc},v}}$. \begin{enumerate}[label=(\roman*)]
\item Since $ \xi_{2} $ is unramified at $ v $ and $ \omega_p $ is ramified at $ v $, we get $ \xi_{2} \omega_p^{-j}|I_p =1 \Longleftrightarrow (p-1) \mid j $. Thus $ A_{f,v}(\xi_2 \omega_{p}^{-j})^{I_{\mathrm{cyc},v}} = A_{f,v}(\xi_2 \omega_{p}^{-j}) $ if $ (p-1) \mid j $ and $ A_{f,v}(\xi_2 \omega_{p}^{-j})^{I_{\mathrm{cyc},v}} = 0$ if $ (p-1) \nmid j $. So in either case, we have $A_{f,v}(\xi_2 \omega_{p}^{-j})^{I_{\mathrm{cyc},v}}$ is $ \pi $-divisible and hence $ \mathrm{ker}(\epsilon) =0 $. Thus the assertion $(i)$ for $ \xi_2 $ follows from the above commutative diagram. Since $ \bar{\xi}_{1}$ has order prime to $ p $, by construction (see the proof of Lemma~\ref{lemma lifting characters}) we have $\xi_{1}$ has order prime to $ p $. Thus $ \bar{\xi}_{1} \bar{\omega}_p^{1-l}|_{I_{\mathrm{cyc},v}} = 1$ if and only if $ \xi_{1} \omega_p^{1-l}|_{I_{\mathrm{cyc},v}} = 1 \mod \pi$. A similar argument as above shows that $A_{f,v}(\xi_1 \omega_{p}^{-j})^{I_{\mathrm{cyc},v}}$ is $ \pi $-divisible and $ \mathrm{ker}(\epsilon)=0 $.
\item Let $ r $ be prime in $ \mathbb{Z} $ lying below $ v $. Then $ r \nmid N p$. Since $ r \nmid m$ and $ r \mid I_0 $, we must have $ r \nmid I_0/M_0 $ and $ r \mid M_0 $. Since $ \mathrm{cond}_{r}(\omega_{p}^{-j}\xi_i) = \mathrm{cond}_{r}(\omega_{p}^{-j}\bar{\xi}_i) $ for $ i =1,2 $, we get $ \mathrm{cond}_{r}(\rho_{f \otimes \xi_{i}\omega_{p}^{-j}}) = \mathrm{cond}_{r}(\xi_{i}\omega_{p}^{-j}) = \mathrm{cond}_{r}(\bar{\xi}_i\omega_{p}^{-j}) = \mathrm{cond}_{r}(\bar{\rho}_{f \otimes \xi_{i}\omega_{p}^{-j}}) $. Thus $ A_{f}(\xi_{i}\omega_p^{-j})^{I_v} $ is $ \pi$-divisible by \cite[Lemma 4.1.2]{epw}. So $ \mathrm{ker}(\epsilon) $ in the above commutative diagram is zero. Now part $(ii)$ follows.
\item Note that $ \xi_{i} $ is unramified is at $ v $. The proof is similar to Proposition~\ref{prop:Greenberg dividing N}. \qedhere \end{enumerate} \end{proof} \begin{comment} \begin{corollary}\label{cor:Selmer of modular form}
Let the assumptions be as in Proposition~\ref{prop:Greenberg for modular form}.
Then
\begin{small}
\begin{equation*}
\begin{split}
S^{\Sigma_0}_{\mathrm{Gr}}(A_f(\xi_i\omega^j)[\pi]/K_{\mathrm{cyc}}) =
\mathrm{Ker}\Big(H^1(\mathbb{Q}_\Sigma/K_{\mathrm{cyc}}, A_f(\xi_i\omega_p^{-j}[\pi]) \longrightarrow & \underset{v \mid p} \prod H^1(I_{\mathrm{cyc},v}, A^{-}_f(\xi_i\omega_p^{-j}[\pi])\Big) \times \underset{v \nmid Nmp, v \mid I} \prod H^1(I_{\mathrm{cyc},v}, A_f(\xi_i\omega_p^{-j}[\pi]) \\ & \qquad \times \underset{v \mid N, v \nmid m} \prod \big( H^{1}(I_{\mathrm{cyc},v}, A_f(\xi_i\omega_p^{-j})[\pi] \big)^{\Delta'}\Big).
\end{split}
\end{equation*}
\end{small} \end{corollary} \end{comment} \section{Congruence of the characteristic ideals}\label{sec: Congruences of char ideals} In this section, we obtain the congruence between the characteristic ideal associated to the Selmer group of $ f \otimes h $ and the product of characteristic ideals associated to the Selmer group of $ f \otimes \xi_1 $, $ f \otimes \xi_2$. \begin{lemma}\label{191} Let $ F$ and $G $ be two normalised $ p $-ordinary cuspidal eigenforms. Further, let $ A_F(\chi_{p}^{-j}), A_G(\chi_{p}^{-j})$ and $A_j$ be the discrete lattices attached to $ F $, $ G $ and $ F \otimes G$ respectively as in \S\ref{sec: Selmer groups}. Assume any one of the following holds: \begin{enumerate}[label=$(\roman*)$]
\item $A_F[\pi]$ is an irreducible $G_\mathbb{Q}$ module and $A_G[\pi]$ is not isomorphic to the dual representation $\mathrm{Hom}(A_G[\pi], \mathcal{O}_K/\pi)$ as $G_\mathbb{Q}$ modules.
\item $A_F[\pi]$ is a reducible $G_\mathbb{Q}$ module and $A_G[\pi]$ is an irreducible $G_\mathbb{Q}$ module.
\end{enumerate} Then $ \mathrm{Hom}(A_j(-1)[\pi], \mathcal{O}_K/\pi)^{G_\mathbb{Q}}=0 $ for every $j$. \end{lemma} \begin{proof} We first consider the case $A_F[\pi]$ is a irreducible $G_\mathbb{Q}$ module and $A_G[\pi]$ is an reducible $G_\mathbb{Q}$ module. Then we have the following exact sequence \begin{equation}\label{192}
0 \longrightarrow \varphi_{1} \otimes A_F[\pi] \longrightarrow A_G[\pi] \otimes A_F[\pi] \longrightarrow \varphi_{2} \otimes A_F[\pi] \longrightarrow 0, \end{equation} where $\varphi_1$ is a one dimensional $G_\mathbb{Q}$ sub-representation of $A_G[\pi] $ and the corresponding quotient is $ \varphi_2$. Since $ A_F[\pi] $ is irreducible, we have $(\varphi_{i}\chi_p^{j} \otimes A_F[\pi])$ is also irreducible for $i=1,2$. The lemma in this case follows first by tensoring $ \chi_{p}^{j} $ and then applying $\mathrm{Hom}$ functor to the sequence \eqref{192}. The case $A_F[\pi]$ is a reducible $G_\mathbb{Q}$ module and $A_G[\pi]$ is an irreducible $G_\mathbb{Q}$ module can be treated similarly. Finally, if $A_F[\pi], A_G[\pi] $ are irreducible and $A_F[\pi]$ is not isomorphic to the dual representation $\mathrm{Hom}(A_G[\pi], \mathcal{O}_K/\pi)$, then we obtain that $A_F[\pi] \otimes A_G[\pi]$ is an irreducible $G_\mathbb{Q}$ module. Now the lemma follows. \iffalse Further, \eqref{192} gives rise to the exact sequence \begin{equation}\label{193}
\resizebox{.9\textwidth}{!}{$
0 \rightarrow \text{Hom}(\varphi_2 \otimes A_f[\pi], \mathcal{O}_K/\pi) \rightarrow \text{Hom}(A_h[\pi] \otimes A_f[\pi], \mathcal{O}_K/\pi)\rightarrow \text{Hom}(\varphi_1 \otimes A_f[\pi], \mathcal{O}_K/\pi) \rightarrow 0,$
} \end{equation} Note that $A_j(-1)[\pi] \cong (A_h[\pi] \otimes A_f[\pi])(\chi_p^{-j-1})$. Now from \eqref{193}, it suffices to show that for every $s \in \mathbb{Z}$, \begin{equation}\label{194}
\text{Hom}((\varphi_{1} \otimes A_f[\pi])(\chi^s_p), \mathcal{O}_K/\pi)^{G_\mathbb{Q}} =0 =\text{Hom}((\varphi_2 \otimes A_f[\pi])(\chi^s_p), \mathcal{O}_K/\pi)^{G_\mathbb{Q}}. \end{equation} As $A_f[\pi]$ is irreducible, both $(\varphi_1 \otimes A_f[\pi])(\chi^s_p)$ and $(\varphi_2 \otimes A_f[\pi])(\chi^s_p)$ are irreducible $G_\mathbb{Q}$ module for any $s \in \mathbb{Z}$. This in turn shows that Hom$((\varphi_1 \otimes A_f[\pi])(\chi^s_p), \mathcal{O}_K/\pi)$ and Hom$((\varphi_2 \otimes A_f[\pi])(\chi^s_p), \mathcal{O}_K/\pi)$ are also irreducible. Thus \eqref{194} holds and the lemma follows in case $1$. In case $3$, it follows that by the given hypothesis that $A_h[\pi] \otimes A_f[\pi]$ is an irreducible $G_\mathbb{Q}$ module and again by a similar argument as in case $1$, we deduce the lemma. \fi \end{proof}
\begin{theorem}\label{no-finite-part-result1} Let $ B_{j} \in \{ A_{f}(\xi_1\omega_p^{-j}), A_{f}(\xi_2\omega_p^{-j}), A_j\}$. Assume that $ A_f[\pi]$ is irreducible and $S_{\mathrm{Gr}}(B_j/\mathbb{Q}_\mathrm{cyc})^\vee$ is a torsion $\mathbb{Z}_p[[\Gamma]]$ module. Then
$S_{\mathrm{Gr}}(B_j/\mathbb{Q}_\mathrm{cyc})^\vee$ has no non-zero pseudo-null $\mathbb{Z}_p[[\Gamma]]$ submodule. \end{theorem} \begin{proof}
We apply \cite[Proposition 1.8]{we} to deduce $S_{\mathrm{Gr}}(B_j/\mathbb{Q}_\mathrm{cyc})^\vee$ has no non-zero pseudo-null $\mathbb{Z}_p[[\Gamma]]$ submodule. Following \cite{we}, define $\mathrm{Sel}_{\mathrm{cr}}(\mathbb{Q}_{\mathrm{cyc}},B_j) := \mathrm{Ker} \big(H^{1} (G_{\Sigma}(\mathbb{Q}_{\mathrm{cyc}}), B_j) \rightarrow \oplus_{v \in \Sigma^\infty} Loc_v \big)$, where $ Loc_v = H^{1}({\mathbb{Q}_{\mathrm{cyc},v}}, B_j)$ if $ v \nmid p $ and $ Loc_v = (\mathrm{image}( H^{1}({\mathbb{Q}_{\mathrm{cyc},v}}, B_j) \rightarrow H^{1}(I_{\mathrm{cyc},v}, B_j^{-}) ))$ if $ v \mid p $. For $v \nmid p$, $G_{\mathbb{Q}_{\mathrm{cyc},v}}/I_{\mathrm{cyc},v}$ has profinite order prime to $ p $, thus $H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}}/I_{\mathrm{cyc},v}, B_{j}^{I_{\mathrm{cyc},v}}) =0$. With this, it is easy to see that $\mathrm{Sel}_{\mathrm{cr}}(\mathbb{Q}_{\mathrm{cyc}},B_j) = S_{\mathrm{Gr}}(B_j/\mathbb{Q}_\mathrm{cyc})$.
As $ A_{f}[\pi] $ is irreducible and $ A_{h}[\pi] $ is reducible, we have $\text{Hom}(B_j(-1)[\pi], \mathcal{O}_K/\pi)^{G_\mathbb{Q}}= 0$ by Lemma~\ref{191}. Since $\big(\mathrm{Hom}(B_j, \frac{K_\mathfrak p}{\mathcal{O}}(1))\otimes \frac{K_\mathfrak p}{\mathcal{O}}\big) [\pi] \cong \text{Hom}(B_j(-1)[\pi], \frac{\mathcal{O}_K}{\pi})$, we get $ (\mathrm{Hom}(B_j, \frac{K_\mathfrak p}{\mathcal{O}}(1))\otimes \frac{K_\mathfrak p}{\mathcal{O}} )^{G_\mathbb{Q}} [\pi] = 0$. Note that $\big(\mathrm{Hom}(B_j, \frac{K_\mathfrak p}{\mathcal{O}}(1))\otimes \frac{K_\mathfrak p}{\mathcal{O}}\big)^{G_\mathbb{Q}}=0$ implies that $H^0(G_{\mathbb{Q}_\mathrm{cyc}}, \mathrm{Hom}(B_j, \frac{K_\mathfrak p}{\mathcal{O}}(1))\otimes \frac{K_\mathfrak p}{\mathcal{O}})=0$ by Nakayama's lemma. As we have assumed that $S_{\mathrm{Gr}}(B_j/\mathbb{Q}_\mathrm{cyc})^\vee$ is a torsion $\mathbb{Z}_p[[\Gamma]]$ module, so the hypotheses of \cite[Proposition 1.8]{we} hold for $ B_j $. Now the proof of the theorem follows. \end{proof} \begin{remark} Let be $ F,G $ be cusp forms as in Lemma~\ref{191} and assume that $S_{\mathrm{Gr}}(A_j/\mathbb{Q}_\mathrm{cyc})^\vee$ is a torsion $\mathbb{Z}_p[[\Gamma]]$ module. If either assumption $(i)$ or assumption $(ii)$ of Lemma~\ref{191} holds, then $S_{\mathrm{Gr}}(A_j/\mathbb{Q}_\mathrm{cyc})^\vee$ has no non-zero pseudo-null submodules. \iffalse \cite{we} defines the Selmer group as $ \mathrm{Ker} \big(H^{1} (G_{\Sigma}(\mathbb{Q}_{\mathrm{cyc}}), B_j) \rightarrow \oplus_{v \in \Sigma} Loc_v \big)$.
As $G_{\mathbb{Q}_{\mathrm{cyc},v}}/I_{\mathrm{cyc},v}$ is a profinite group with (profinite) order prime to $ p $ for $v\nmid p$, $H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}}/I_{\mathrm{cyc},v}, B_{j}^{I_{\mathrm{cyc},v}}) =0$. Using this, it is easy to see that the definition of Selmer group in \cite{we} matches with our definition of Greenberg Selmer group $S_{\mathrm{Gr}}(B_j/\mathbb{Q}_\mathrm{cyc})$. \fi \end{remark} From the exact sequence \eqref{eq: splitting rho_h}, it follows that we have the following exact sequence of $G_\mathbb{Q}$ modules \begin{equation}\label{fund-resb} 0 \longrightarrow \frac{\mathcal{O}_K}{\pi}(\bar{\xi}_1) \longrightarrow A_h[\pi] \longrightarrow \frac{\mathcal{O}_K}{\pi}(\bar{\xi}_2) \longrightarrow 0. \end{equation} Tensoring with $-\otimes_{\mathcal{O}_K} T_f{(-j)} $ and noting that $ T_f{(-j)} $ is a free $ \mathcal{O}_K $-module, we get \begin{equation}\label{fund-reds2} 0 \longrightarrow A_{f}[\pi](\bar{\xi}_1\bar{\omega}_p^{-j}) \longrightarrow A_j[\pi] \longrightarrow A_{f}[\pi](\bar{\xi}_2\bar{\omega}_p^{-j}) \longrightarrow 0. \end{equation}
\begin{lemma}\label{lem: cohomology exact sequence of Q_cyc} Assume $H^2(\mathbb{Q}_\Sigma/\mathbb{Q}_{\mathrm{cyc}}, A_f[\pi](\bar{\xi}_1\omega_p^{-j})) =0$, $A_f[\pi]$ is an irreducible $G_\mathbb{Q}$ module and $ (T_h/\pi)^{ss} \cong \bar{\xi}_1 \oplus \bar{\xi}_2$. Then the following sequence is exact \begin{small}
\begin{equation*}
0 \rightarrow H^1(\mathbb{Q}_\Sigma/\mathbb{Q}_\mathrm{cyc}, A_f[\pi](\bar{\xi}_1\omega_p^{-j})) \rightarrow H^1(\mathbb{Q}_\Sigma/\mathbb{Q}_\mathrm{cyc}, A_j[\pi]) \rightarrow H^1(\mathbb{Q}_\Sigma/\mathbb{Q}_\mathrm{cyc}, A_f[\pi](\bar{\xi}_2\omega_p^{-j})) \rightarrow 0.
\end{equation*} \end{small} \end{lemma} \begin{proof} The exact sequence \eqref{fund-reds2} induces the following exact sequence of cohomology groups \begin{small}
\[
0 \longrightarrow H^1(\mathbb{Q}_\Sigma/\mathbb{Q}_\mathrm{cyc}, A_f[\pi](\bar{\xi}_1\omega_p^{-j})) \longrightarrow H^1(\mathbb{Q}_\Sigma/\mathbb{Q}_\mathrm{cyc}, A_j[\pi]) \longrightarrow H^1(\mathbb{Q}_\Sigma/\mathbb{Q}_\mathrm{cyc}, A_f[\pi](\bar{\xi}_2\omega_p^{-j})).
\] \end{small} By the assumption $H^2(\mathbb{Q}_\Sigma/\mathbb{Q}_{\mathrm{cyc}}, A_f[\pi](\bar{\xi}_1\omega_p^{-j})) =0$, we obtain the right most map is surjective. Note that by our assumption that $A_f[\pi]$ is irreducible $G_\mathbb{Q}$ module, it follows that $A_f[\pi](\bar{\xi}_1\omega_p^{-j})$ is also an irreducible $G_\mathbb{Q}$ module, hence $H^0(\mathbb{Q},A_f[\pi](\bar{\xi}_1\omega_p^{-j}))=0$. Thus by Nakayama lemma, $H^0(\mathbb{Q}_\mathrm{cyc},A_f[\pi](\bar{\xi}_1\omega_p^{-j}))=0$ as well. Hence the left exactness follows in the above exact sequence. \end{proof}
\begin{rem}\label{remark mu in variant and H2} Note that if $S_{\mathrm{Gr}}(A_f(\xi_1\omega_p^{-j})/\mathbb{Q}_\mathrm{cyc})^\vee$ has $\mu$-invariant equal to $0$, i.e. $S_{\mathrm{Gr}}(A_f(\xi_1\omega_p^{-j})/\mathbb{Q}_\mathrm{cyc})^\vee$ is a finitely generated $\mathbb{Z}_p$ module, then $H^2(\mathbb{Q}_\Sigma/\mathbb{Q}_\mathrm{cyc}, A_f(\xi_1\omega_p^{-j})[\pi])$ vanishes. In fact, the vanishing of $H^2(\mathbb{Q}_\Sigma/\mathbb{Q}_{\mathrm{cyc}}, A_f(\xi_1\omega_p^{-j}) [\pi])$ is equivalent to the weaker assumption that $ \mu ( R(A_{f}(\xi_1\omega_p^{-j}))/\mathbb{Q}_{\mathrm{cyc}})^\vee) =0$, where $R(A_{f}(\xi_1\omega_p^{-j}))/\mathbb{Q}_{\mathrm{cyc}})= \mathrm{Ker}(H^1(\mathbb{Q}_{\Sigma}/\mathbb{Q}_{\mathrm{cyc}}, A_{f}(\bar{\xi}_1\omega_p^{-j})[\pi]) \rightarrow \oplus_{v \in \Sigma} H^1(\mathbb{Q}_{\mathrm{cyc},v}, A_{f}(\bar{\xi}_1\omega_p^{-j})[\pi]) )$ is the \textit{fine Selmer group} defined and studied by Coates-Sujatha (cf. \cite{ms}). \end{rem}
We now state a technical assumption, which we will need later. \begin{equation}\label{assumption}
\text{ For a prime } v \mid p \text{ in } \mathbb{Q}_{\mathrm{cyc}} , ~ \bar{\rho}_h |I_{\mathrm{cyc},v} = \bar{\xi}_1 \oplus \bar{\xi}_2 \text{ whenever } H^0(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A^-_f[\pi](\bar{\xi}_2)) \neq 0 \tag{red$_v$}. \end{equation}
Recall that $ \Sigma = \{ \ell: \ell \mid pNI\infty \}$, $ \Sigma_{0} = \{ \ell : \ell \mid m \}$ and $ \Sigma^\infty , \Sigma_0^\infty$ are the corresponding primes of $ \mathbb{Q}_{\mathrm{cyc}} $.
\begin{proposition}\label{prop:exact sequence of inertia} Let $ (N,M_0) =1$ and $ j $ be an integer. Assume that $ A_f[\pi] $ is an irreducible $ G_{\mathbb{Q}} $ module, $ (T_h/\pi)^{ss} \cong \bar{\xi}_1 \oplus \bar{\xi}_2$ and $ \psi \vert_{I_{\mathrm{cyc},v}} $ has order prime to $ p $ for $ v \mid pI_0$. If $ (p-1) \mid j $, then we further assume that \eqref{assumption} holds for all $ v \mid p $.
Then for every $ v \in \Sigma^\infty \setminus \Sigma^\infty_{0} $, we have the following exact sequence \begin{equation}\label{split-local-c2}
0 \rightarrow \frac{H^1(\mathbb{Q}_{\mathrm{cyc},v}, A_f[\pi](\bar{\xi}_1\bar{\omega}_p^{-j}))}{H_{\mathrm{Gr}}^1(\mathbb{Q}_{\mathrm{cyc},v},A_f[\pi](\bar{\xi}_1\bar{\omega}_p^{-j}))} \rightarrow \frac{H^1(\mathbb{Q}_{\mathrm{cyc},v}, A_j[\pi])}{H^1_{\mathrm{Gr}}(\mathbb{Q}_{\mathrm{cyc},v}, A_j[\pi])} \rightarrow \frac{H^1(\mathbb{Q}_{\mathrm{cyc},v}, A_f[\pi](\bar{\xi}_2\bar{\omega}_p^{-j}))}{H^1_{\mathrm{Gr}}(\mathbb{Q}_{\mathrm{cyc},v},A_f[\pi](\bar{\xi}_2\bar{\omega}_p^{-j}))} . \end{equation} \end{proposition} \begin{proof}
As $ p $-cohomological dimension of $ G_{\mathbb{Q}_{\mathrm{cyc},v}} $ is $ 1 $, from \eqref{fund-reds2} we get the exact sequence \begin{align}\label{195}
H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A_f[\pi](\bar{\xi}_1\bar{\omega}_p^{-j})) \rightarrow H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A_j[\pi]) \rightarrow H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A_f[\pi](\bar{\xi}_2\bar{\omega}_p^{-j})) \rightarrow 0. \end{align}
We denote $ G_{\mathbb{Q}_{\mathrm{cyc},v}}/I_{\mathrm{cyc},v} $ by $ \Delta_v $.
\underline{Case $v \mid p$}: As $ p $-cohomological dimension of $ G_{\mathbb{Q}_{\mathrm{cyc},v}} $ is $ 1 $, it follows from \eqref{def A minus j} that $ H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A_j[\pi]) \rightarrow H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A^{-}_j[\pi])$ is surjective. By the inflation restriction sequence, we have the $ H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A^{-}_j[\pi]) \rightarrow H^1(I_{\mathrm{cyc},v}, A^{-}_j[\pi])^{\Delta_v}$ is surjective. Hence the image of the following composition \begin{align*}
H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A_j[\pi]) \rightarrow H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A^{-}_j[\pi])) \rightarrow H^1(I_{\mathrm{cyc},v}, A^{-}_j[\pi])) \end{align*} is equal to $H^1(I_{\mathrm{cyc},v}, A^{-}_j[\pi])^{\Delta_v}$. Similarly image$\Big( H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A_f[\pi](\xi_{i}\omega_{p}^{-j}) \rightarrow H^1(I_{\mathrm{cyc},v}, A^{-}_f[\pi](\xi_{i}\omega_{p}^{-j})\Big) = (H^1(I_{\mathrm{cyc},v}, A^{-}_f[\pi](\xi_{i}\omega_{p}^{-j}))^{\Delta_v}$ for $ i=1,2 $. Thus it follows from \eqref{fund-resb} and Propositions \ref{prop:Greenberg at p}, \ref{prop:Greenberg for modular form} we get the following commutative digram \begin{small}
\begin{equation}\label{commutative diagram}
\begin{tikzcd}
0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\
H^1_{\mathrm{Gr}}(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A_f[\pi](\bar{\xi}_1\bar{\omega}_p^{-j})) \ar[r] \ar[d] & H_{\mathrm{Gr}}^1(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A_j[\pi]) \ar[r] \arrow{d}
& H_{\mathrm{Gr}}^1(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A_f[\pi](\bar{\xi}_2\bar{\omega}_p^{-j}))\ar[d] \ar[r] & 0\\
H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A_f[\pi](\bar{\xi}_1\bar{\omega}_p^{-j})) \ar[r] \ar[d] & H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A_j[\pi]) \ar[r] \ar[d] & H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A_f[\pi](\bar{\xi}_2\bar{\omega}_p^{-j}))\ar[d] \ar[r]& 0 \\
H^{1}(I_{\mathrm{cyc},v}, A^{-}_{f}[\pi](\bar{\xi}_{1}\bar{\omega}_p^{-j}))^{\Delta_v} \ar[r,"\nu"] \ar[d] & H^{1}(I_{\mathrm{cyc},v}, A^{-}_{j}[\pi])^{\Delta_v} \ar[r] \ar[d] & H^{1}(I_{\mathrm{cyc},v}, A^{-}_{f}[\pi](\bar{\xi}_{2}\bar{\omega}_p^{-j}))^{\Delta_v}. \ar[d] & \\
0 & 0 & 0
\end{tikzcd}
\end{equation} \end{small} So to prove the proposition, it suffices to show $ \mathrm{ker}(\nu) = 0$.
If $(p-1) \nmid j$, then $H^{0}(I_{\mathrm{cyc},v},A_{f}^{-}[\pi](\bar{\xi}_{2}\omega_{p}^{-j})) = 0 $. Tensoring \eqref{fund-resb} with $ - \otimes A_{f}^{-} $, we obtain \begin{align*}
0 \rightarrow H^1(I_{\mathrm{cyc},v}, A^{-}_f[\pi](\xi_{1}\omega_{p}^{-j})) \rightarrow
H^1(I_{\mathrm{cyc},v}, A^{-}_j[\pi]) \rightarrow H^1(I_{\mathrm{cyc},v}, A^{-}_f[\pi](\xi_{2}\omega_{p}^{-j})) \rightarrow 0. \end{align*} Taking $ \Delta_v $ invariants, we obtain $ \mathrm{ker}(\nu) =0 $.
Next consider the case $ (p-1) \mid j$ and $H^0(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A^{-}_f[\pi](\xi_{2})) \neq 0 $. Then by the assumption
\eqref{assumption}, we have $ A_{h}[\pi] \cong (T_h/\pi)|I_{\mathrm{cyc},v} \cong \bar{\xi}_{1} \oplus \bar{\xi}_{2} $. Thus we have $H^1(I_{\mathrm{cyc},v}, A^{-}_j[\pi]) = H^1(I_{\mathrm{cyc},v}, A^{-}_f[\pi](\bar{\xi}_{1}\bar{\omega}_{p}^{-j})) \oplus H^1(I_{\mathrm{cyc},v}, A^{-}_f[\pi](\bar{\xi}_{2}\bar{\omega}_{p}^{-j}))$. Again taking $ \Delta_v $ invariants, we get $ \mathrm{ker}(\nu) =0 $.
Finally consider the case $ (p-1) \mid j $ and $ H^0(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A^{-}_f[\pi](\xi_{2})) = 0 $. Then $ H^{0}(\Delta_v,A_{f}^{-}[\pi](\xi_{2})^{I_{\mathrm{cyc},v}}) =0 $. As $ \Delta_v = G_{\mathbb{Q}_{\mathrm{cyc},v}}/I_{\mathrm{cyc},v} $ is procyclic, we get $ H^{1}(\Delta_v,A_{f}^{-}[\pi](\xi_{2})^{I_{\mathrm{cyc},v}}) =0 $. Thus by the inflation restriction sequence and \eqref{fund-resb}, in this case we have the following commutative diagram \begin{small}
\[
\begin{tikzcd}
0 \ar[r] & H^1(\Delta_v, A^{-}_f[\pi](\bar{\xi}_1\bar{\omega}_p^{-j})^{I_{\mathrm{cyc},v}}) \ar[r] \ar[d] & H^1(\Delta_v, A^{-}_j[\pi]^{I_{\mathrm{cyc},v}}) \ar[r] \ar[d] & H^1(\Delta_v, A^{-}_f[\pi](\bar{\xi}_2\bar{\omega}_p^{-j})^{I_{\mathrm{cyc},v}}) =0 \ar[d] & \\
& H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A^{-}_f[\pi](\bar{\xi}_1\bar{\omega}_p^{-j})) \ar[r] & H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A^{-}_j[\pi]) \ar[r] & H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}}, A^{-}_f[\pi](\bar{\xi}_2\bar{\omega}_p^{-j})) \ar[r] & 0 .
\end{tikzcd}
\] \end{small} where all the vertical maps are injective. Now the proposition follows from the snake lemma and the inflation restriction sequence.
\iffalse This is contradiction to the fact that $\begin{pmatrix} \bar{\xi}_1 \omega_p^{-j} & * \\ 0 &\bar{\xi}_2 \bar{\omega}_p^{-j} \end{pmatrix} \overset{ss}{\sim} \begin{pmatrix} \bar{\psi}_h \bar{\omega}_p^{-j} & 0 \\ 0 & \bar{\omega}_p^{-j} \end{pmatrix} $ with $* \neq 0$. Thus the left exactness in \eqref{split-local-c2} follows. This proves part (1). \fi
\underline{Case $v \mid I$, $ v \nmid Npm $}:
Let $ v \mid \ell $ in $ \mathbb{Z} $. Since $ \ell \nmid m$ and $ \ell \mid I_0 $, it forces that $ \ell \nmid I_0/M_0 $ and $ \ell | M = \mathrm{cond}(\xi_1)\mathrm{cond}(\xi_2)$. By Lemma~\ref{lem: exactly divides} and \cite[Theorem 3.26]{Hida3}, we have $ \bar{\rho}_h | G_{\mathbb{Q}_{\mathrm{cyc},v}} \cong \bar{\xi}_1 \oplus \bar{\xi}_2$. Tensoring with $ -\otimes_\mathcal{O} A_f(-j) $ induces following exact sequence of $ \Delta_v $ modules \[ 0 \rightarrow H^1(I_{\mathrm{cyc},v}, A_f[\pi](\bar{\xi}_1\bar{\omega}_p^{-j})) \longrightarrow H^1(I_{\mathrm{cyc},v}, A_j[\pi]) \longrightarrow H^1(I_{\mathrm{cyc},v}, A_f[\pi](\bar{\xi}_2\bar{\omega}_p^{-j})) \rightarrow 0. \] Note that we have an analogue of commutative diagram \eqref{commutative diagram} even in this case. So the assertion follows from Proposition~\ref{prop:Greenberg away from N} and Proposition \ref{prop:Greenberg for modular form}$(ii)$ by a similar argument as in the case $ v \mid p $.
\underline{Case $v \mid N$ and $ v \nmid Im $}: We claim that $ (T_h/\pi) \cong \bar{\xi}_1 \oplus \bar{\xi}_2$ as $ G_{\mathbb{Q}_{\mathrm{cyc},v}} $ modules. Since $ (N, M_0) =1$ and $ v \mid N $, we get $ v \nmid M_0$. Again using $ v \nmid m $, we obtain $ v \nmid I_0 $. Thus $ T_h/\pi $ is unramified at $ v $ and action of $ G_{\mathbb{Q}_{\mathrm{cyc}, v}} $ factors through $ \Delta_v$. As the order of $ \bar{\xi}_1 , \bar{\xi}_2 $ are co-prime to $ p $, the action of $ G_{\mathbb{Q}_{\mathrm{cyc}, v}} $ on $ T_{h}/\pi $ further factors through $ G_{\mathbb{Q}_{\mathrm{cyc}, v}}/N $ such that $ N $ is a finite index normal subgroup of $ G_{\mathbb{Q}_{\mathrm{cyc}, v}} $ containing $ I_{\mathrm{cyc},v} $ and $ G_{\mathbb{Q}_{\mathrm{cyc}, v} } /N $ has order co-prime to $ p $. Thus $ T_{h}/\pi $ is a semi-simple $ G_{\mathbb{Q}_{\mathrm{cyc}, v} } /N $ module. Hence $ T_{h}/\pi \cong \bar{\xi}_1 \oplus \bar{\xi}_2 $ as $ G_{\mathbb{Q}_{\mathrm{cyc}, v} } $ module. Now the proposition in this case follows from Propositions~\ref{prop:Greenberg dividing N}, \ref{prop:Greenberg for modular form} using similar arguments as in previous two cases. \end{proof}
\begin{remark} We will apply Proposition~\ref{prop:exact sequence of inertia} in Theorem~\ref{thm:congruence of ideals}, Theorem~\ref{congruence main conjecture } for $ l-1 \leq j \leq k-2 $. In that case, the assumption \eqref{assumption} in Proposition~\ref{prop:exact sequence of inertia} is required whenever $ \omega_{p}^{j} =1$, for some $ l-1 \leq j \leq k-2 $. In particular, if $ p > k-1 $, then the assumption \eqref{assumption} is not required. \end{remark}
\begin{comment} For $ B_{j} \in \{ A_f(\xi_{1}\omega^{-j}), A_{j}, A_f(\xi_{2}\omega^{-j}) \} $ define \begin{align*}
H^{1}_{\ast}(I_{\mathrm{cyc},v}, B_{j}[\pi]) =
\begin{cases}
H^{1}(I_{\mathrm{cyc},v}, B^{-}_{j}[\pi]) &\text{ if } v \mid p, \\
H^{1}(I_{\mathrm{cyc},v}, B_{j}[\pi]) & \text{ if } v \mid I, v \nmid Nm, \\
\big(H^{1}(I_{\mathrm{cyc},v}, B_{j})[\pi]\big)^{\Delta'} &\text{ if } v \mid N, v \nmid m,
\end{cases} \end{align*} where $ \Delta' = \mathrm{Gal}(K_{v,\mathrm{cyc}}^{\mathrm{unr}}/K_{v,\mathrm{cyc}}) $. \end{comment}
\begin{proposition}\label{prop: selmer exact sequence} Let $ (N,M_0) =1$ and $ \psi \vert_{I_{\mathrm{cyc},v}} $ has order co-prime to $ p $ for all $ v \mid pI_0$. Assume that $A_f[\pi]$ is an irreducible $ G_\mathbb{Q} $ module, $(T_h/\pi)^{ss} \cong \bar{\xi}_1 \oplus \bar{\xi}_2$ and $S^{\Sigma_0}_{\mathrm{Gr}}( A_f(\xi_1\omega_p^{-j})/\mathbb{Q}_\mathrm{cyc})^\vee$ is a finitely generated $\mathbb{Z}_p$ module. If $ (p-1) \mid j $, then assume that the hypothesis \eqref{assumption} holds for all $ v \mid p $. Then we have the following exact sequence \begin{equation}\label{split-selmer-01}
0 \rightarrow S^{\Sigma_0}_{\mathrm{Gr}}( A_f[\pi](\bar{\xi}_1\bar{\omega}_p^{-j})/\mathbb{Q}_\mathrm{cyc}) \rightarrow S^{\Sigma_0}_{\mathrm{Gr}}( A_j[\pi]/\mathbb{Q}_\mathrm{cyc}) \rightarrow S^{\Sigma_0}_{\mathrm{Gr}}( A_f[\pi](\bar{\xi}_2\bar{\omega}_p^{-j})/\mathbb{Q}_\mathrm{cyc}) \longrightarrow 0. \end{equation} \end{proposition} \begin{proof} By Lemma~\ref{lem: cohomology exact sequence of Q_cyc} and Proposition~\ref{prop:exact sequence of inertia}, we have the following commutative diagram \begin{tiny}{
\[
\begin{tikzcd}
0 \arrow{r} & H^1(\mathbb{Q}_\Sigma/\mathbb{Q}_\mathrm{cyc}, A_f[\pi](\bar{\xi}_1\bar{\omega}_p^{-j})) \arrow{r} \arrow{d}{\epsilon_1} & H^1(\mathbb{Q}_\Sigma/\mathbb{Q}_\mathrm{cyc}, A_j[\pi]) \arrow{r} \arrow{d}{\epsilon} & H^1(\mathbb{Q}_\Sigma/\mathbb{Q}_\mathrm{cyc}, A_f[\pi](\bar{\xi}_2\bar{\omega}_p^{-j}))\arrow{r} \arrow{d}{\epsilon_2} & 0\\
0 \arrow{r} & \prod\limits_{v \in \Sigma^\infty \smallsetminus \Sigma^\infty_{0}} \frac{H^{1}(\mathbb{Q}_{\mathrm{cyc},v}, A_{f}(\bar{\xi}_{1}\bar{\omega}_p^{-j})[\pi])}{ H^{1}_{\mathrm{Gr}}(\mathbb{Q}_{\mathrm{cyc},v}, A_{f}(\bar{\xi}_{1}\bar{\omega}_p^{-j})[\pi])} \arrow{r} & \prod\limits_{v \in \Sigma^\infty \smallsetminus \Sigma^\infty_{0}} \frac{H^{1}(\mathbb{Q}_{\mathrm{cyc},v}, A_{j}[\pi])}{H_{\mathrm{Gr}}^{1}(\mathbb{Q}_{\mathrm{cyc},v}, A_{j}[\pi])} \arrow{r} & \prod\limits_{v \in \Sigma^\infty \smallsetminus \Sigma^\infty_{0}} \frac{H^{1}(\mathbb{Q}_{\mathrm{cyc},v}, A_{f}(\bar{\xi}_{2}\bar{\omega}_{p}^{-j})[\pi])}{H_{\mathrm{Gr}}^{1}(\mathbb{Q}_{\mathrm{cyc},v}, A_{f}(\bar{\xi}_{2}\bar{\omega}_{p}^{-j})[\pi])}.
\end{tikzcd}
\] }\end{tiny} Applying the snake lemma to the above commutative diagram, we get the following exact sequence \begin{equation*}
0 \rightarrow S^{\Sigma_0}_{\mathrm{Gr}}( A_f[\pi](\bar{\xi}_1\bar{\omega}_p^{-j})/\mathbb{Q}_\mathrm{cyc}) \rightarrow S^{\Sigma_0}_{\mathrm{Gr}}( A_j[\pi]/\mathbb{Q}_\mathrm{cyc}) \rightarrow S^{\Sigma_0}_{\mathrm{Gr}}( A_f[\pi](\bar{\xi}_2\bar{\omega}_p^{-j})/\mathbb{Q}_\mathrm{cyc}) \rightarrow \mathrm{coker}(\epsilon_1). \end{equation*} To prove the lemma, we need to show coker($\epsilon_1$) is zero.
In this setting, the following global to local map \begin{small}{
\begin{equation*}\label{surj-f-sel3}
H^1(\mathbb{Q}_\Sigma/\mathbb{Q}_\mathrm{cyc}, A_f[\pi](\bar{\xi}_1\bar{\omega}_p^{-j})) \longrightarrow \underset{v \mid p}{\bigoplus}H^1(\mathbb{Q}_{\mathrm{cyc},v}, A^-_f[\pi](\bar{\xi_1}\bar{\omega}_p^{-j})) \underset{v \in \Sigma\setminus \Sigma^\infty_{0} , ~v \nmid p}{\bigoplus} H^1(\mathbb{Q}_{\mathrm{cyc},v},A_f[\pi](\bar{\xi_1}\bar{\omega}_p^{-j}))
\end{equation*} }\end{small} is surjective (see \cite[Theorem 5.2]{ms}). Using the inflation restriction sequence and $ \Delta_v $ is topologically cyclic, we obtain the following surjective map \begin{small}{
\begin{equation*}
H^1(\mathbb{Q}_\Sigma/\mathbb{Q}_\mathrm{cyc}, A_f[\pi](\bar{\xi}_1\bar{\omega}_p^{-j})) \longrightarrow \underset{v \mid p}{\bigoplus}H^1(I_{\mathrm{cyc},v}, A^-_f[\pi](\bar{\xi_1}\bar{\omega}_p^{-j}))^{\Delta_v} \bigoplus_ {v \in \Sigma^\infty\setminus \Sigma^\infty_{0}, v \nmid p} H^1(\mathbb{Q}_{\mathrm{cyc},v},A_f[\pi](\bar{\xi_1}\bar{\omega}_p^{-j}))
\end{equation*} }\end{small} Since $\frac{H^1(\mathbb{Q}_{\mathrm{cyc},v}, A_f[\pi](\bar{\xi}_1\bar{\omega}_p^{-j}))}{H^1_{\mathrm{Gr}}(\mathbb{Q}_{\mathrm{cyc},v}, A_f[\pi](\bar{\xi_1}\bar{\omega}_p^{-j}))} = H^1(I_{\mathrm{cyc},v}, A^-_f[\pi](\bar{\xi_1}\bar{\omega}_p^{-j}))^{\Delta_v} $ for $ v \mid p $ (see Diagram \eqref{commutative diagram}), it follows that \begin{small}{
\begin{equation*}\label{surj-f-sel2}
H^1(\mathbb{Q}_\Sigma/\mathbb{Q}_\mathrm{cyc}, A_f[\pi](\bar{\xi}_1\bar{\omega}_p^{-j})) \longrightarrow \underset{v \mid p}{\bigoplus}\frac{H^1(\mathbb{Q}_{\mathrm{cyc},v}, A^-_f[\pi](\bar{\xi}_1\bar{\omega}_p^{-j}))}{H^1_{\mathrm{Gr}}(\mathbb{Q}_{\mathrm{cyc},v}, A_f[\pi](\bar{\xi_1}\bar{\omega}_p^{-j}))} \underset{\substack{v \in \Sigma^\infty \setminus \Sigma^\infty_{0} \\ v \nmid p}}{\bigoplus} \frac{H^1(\mathbb{Q}_{\mathrm{cyc},v},A_f[\pi](\bar{\xi}_1\bar{\omega}_p^{-j}))}{H^1_{\mathrm{Gr}}(\mathbb{Q}_{\mathrm{cyc},v},A_f[\pi](\bar{\xi}_1\bar{\omega}_p^{-j}))}.
\end{equation*} }\end{small} is also surjective. This shows $ \epsilon_1 $ is surjective. Hence the right exactness follows in \eqref{split-selmer-01}. \begin{comment}
Let $$ S^{\Sigma_0}_{Gr}( A_f[\pi](\bar{\xi}_i\omega_p^{-j})/\mathbb{Q}_\mathrm{cyc})_{p} = \mathrm{Ker}( H^1(\mathbb{Q}_\Sigma/\mathbb{Q}_\mathrm{cyc}, A_f[\pi](\bar{\xi}_i\omega_p^{-j})) \longrightarrow \underset{v \mid p}{\oplus}H^1(I_{\mathrm{cyc},v}, A^-_f[\pi](\bar{\xi}_i\omega_p^{-j})) .$$
Now we claim the following (only left) exact sequence for $i =1,2$
\begin{equation}\label{surj-f-sel1}
0 \rightarrow S^{\Sigma_0}_{Gr}( A_f[\pi](\bar{\xi}_i\omega_p^{-j})/\mathbb{Q}_\mathrm{cyc}) \longrightarrow H^1(\mathbb{Q}_\Sigma/\mathbb{Q}_\mathrm{cyc}, A_f[\pi](\bar{\xi}_i\omega_p^{-j})) \longrightarrow \underset{v \mid p}{\oplus}H^1(I_{\mathrm{cyc},v}, A^-_f[\pi](\bar{\xi}_i\omega_p^{-j}))
\end{equation}
To prove this claim, we consider a diagram similar to \eqref{dqGYaidhkAdoaU} replacing $A_j$ with $A_f(\xi_i\omega_p^{-j})$ and $A^-_j$ with $A^-_f(\xi_i\omega_p^{-j})$. As in diagram \eqref{dqGYaidhkAdoaU}, it is enough to show that $\text{ker}(\epsilon) =0$ and $H^1_{\text{Gr}}(K_{\mathrm{cyc},v}, A_f(\xi_i\omega_p^{-j})[\pi]) = \text{ker}(\psi) $. As $\bar{\psi}_g\mid_{I_\mathrm{cyc}}$ has order prime to $p$, and
$$
\bar{\rho}_{h}|_{I_{\mathrm{cyc},v}} \sim \begin{pmatrix} \bar{\psi}_h & 0 \\ 0 &1 \end{pmatrix} \sim \begin{pmatrix} \bar{\xi}_1 & * \\ 0 &\bar{\xi}_2 \end{pmatrix},
$$
it follows that order of $\bar{\xi}_1$ and $\bar{\xi}_2$ are prime to $p$. It follows that $\frac{(A^-_f(\bar{\xi}_i\omega_p^{-j}))^{I_{\mathrm{cyc},v}}}{\pi} =0$. Hence $\text{ker}(\epsilon) =0$ and \eqref{surj-f-sel1} follows.
Now using \eqref{surj-f-sel1} in \eqref{surj-f-sel2} we have another exact sequence:
\begin{equation}\label{surj-f-sel3}
0 \rightarrow S^{\Sigma_0}_{Gr}( A_f[\pi](\bar{\xi}_i\omega_p^{-j})/\mathbb{Q}_\mathrm{cyc}) \longrightarrow H^1(\mathbb{Q}_\Sigma/\mathbb{Q}_\mathrm{cyc}, A_f[\pi](\bar{\xi}_1\omega_p^{-j})) \longrightarrow \underset{v \mid p}{\bigoplus}\frac{H^1(K_{\mathrm{cyc},v}, A^-_f[\pi](\bar{\xi_1}\omega_p^{-j}))}{H^1(<Fr_{\mathrm{cyc},v}>, (A^-_f[\pi](\bar{\xi_1}\omega_p^{-j}))^{I_{\mathrm{cyc},v}})} \rightarrow 0
\end{equation}
From \eqref{surj-f-sel3}, the right exactness in \eqref{split-selmer-01} follows. \end{comment} \end{proof}
\begin{lemma}\label{lem: dual selmer group sequence} We keep the setting and hypotheses as in Proposition~\ref{prop: selmer exact sequence}. In addition assume that $S^{\Sigma_0}_{\mathrm{Gr}}( A_f(\xi_2\omega_p^{-j})/\mathbb{Q}_\mathrm{cyc})^\vee$ is a finitely generated $\mathbb{Z}_p$ module. Then we have the following exact sequence \begin{small}
\begin{equation}\label{split-selmer-03}
0 \rightarrow \frac{S^{\Sigma_0}_{\mathrm{Gr}}( A_f(\xi_2\omega_p^{-j})/\mathbb{Q}_\mathrm{cyc})^\vee}{\pi} \rightarrow \frac{S^{\Sigma_0}_{\mathrm{Gr}}( A_j/\mathbb{Q}_\mathrm{cyc})^\vee}{\pi} \rightarrow \frac{S^{\Sigma_0}_{\mathrm{Gr}}(A_f(\xi_1\omega_p^{-j})/\mathbb{Q}_\mathrm{cyc})^\vee}{\pi} \rightarrow 0.
\end{equation} \end{small} \end{lemma} \begin{proof}
We claim that $H^0(G_{\mathbb{Q}_\mathrm{cyc}}, A_f(\xi_i \omega_p^{-j}))=0$, for $i=1,2$ and $H^0(G_{\mathbb{Q}_\mathrm{cyc}}, A_j)=0$. From the assumption $A_f[\pi]$ (and hence $A_f[\pi](\bar{\xi}_i\bar{\omega}_p^{-j})$) is an irreducible $G_\mathbb{Q}$ module, we have $H^0(G_{\mathbb{Q}}, A_f[\pi](\bar{\xi}_i\bar{\omega}_p^{-j}))=0$, for $i=1,2$. The result over $\mathbb{Q}_\mathrm{cyc}$ follows from Nakayama's Lemma.
From the exact sequence \eqref{fund-reds2}, it follows that $H^0(G_{\mathbb{Q}_\mathrm{cyc}}, A_j[\pi])=0$.
Now using Lemma \ref{inside-out-lem}, we deduce from \eqref{split-selmer-01} that \begin{equation}\label{split-selmer-02}
0 \rightarrow S^{\Sigma_0}_{\mathrm{Gr}}( A_f(\xi_1\omega_p^{-j})/\mathbb{Q}_\mathrm{cyc})[\pi] \rightarrow S^{\Sigma_0}_{\mathrm{Gr}}( A_j/\mathbb{Q}_\mathrm{cyc})[\pi] \rightarrow S^{\Sigma_0}_{\mathrm{Gr}}( A_f(\xi_2\omega_p^{-j})/\mathbb{Q}_\mathrm{cyc})[\pi] \rightarrow 0. \end{equation} is exact. Applying the Pontryagin dual, we obtain the lemma.
\end{proof}
We next prove that all three modules in \eqref{split-selmer-03} have no non-zero pseudo-null $\mathbb{Z}_{p}[[\Gamma]]$ submodules. \begin{lemma}\label{lem:psuedonullity} Let the hypotheses be as in Lemma~\ref{lem: dual selmer group sequence}. Then the modules $ S^{\Sigma_0}_{\mathrm{Gr}}( A_f(\xi_i\omega_p^{-j})/\mathbb{Q}_\mathrm{cyc})^\vee$ for $ i=1,2 $ and $S^{\Sigma_0}_{\mathrm{Gr}}( A_j/\mathbb{Q}_\mathrm{cyc})^\vee$ have no non-zero pseudo-null $ \mathbb{Z}_{p}[[\Gamma]] $ submodules. \end{lemma} \begin{proof} Let $B \in \{A_j, A_f(\xi_1\omega_p^{-j}), A_f(\xi_2\omega_p^{-j})\}$. We have the following exact sequence \begin{equation}\label{last-alg1}
0 \longrightarrow S_{\mathrm{Gr}}( B/\mathbb{Q}_\mathrm{cyc}) \longrightarrow S^{\Sigma_0}_{\mathrm{Gr}}( B/\mathbb{Q}_\mathrm{cyc}) \longrightarrow \underset{v \in \Sigma^\infty_0}{\prod}H^1(\mathbb{Q}_{\mathrm{cyc},v}, B) \longrightarrow 0. \end{equation} For every $ v \in \Sigma_0$, we have $H^1(G_{\mathbb{Q}_{\mathrm{cyc},v}}/I_{\mathrm{cyc},v}, B^{I_{\mathrm{cyc},v}}) =0$ as the group $G_{K_{\mathrm{cyc},v}}/I_{\mathrm{cyc},v}$ being a pro-$\ell$ group with $\ell\neq p$. We note that, $\prod_{v \in \Sigma^\infty_0} H^1(\mathbb{Q}_{\mathrm{cyc},v}, B)^\vee$ has no non-zero pseudo-null submodule as $B$ is divisible and $G_{\mathbb{Q}_{\mathrm{cyc},v}}$ has $p$-cohomological dimension 1. So it is enough to prove $ S_{\mathrm{Gr}}( B/\mathbb{Q}_\mathrm{cyc})^\vee $ has no non-zero pseudo-null submodules. In order to do this, we first show that $ S_{\mathrm{Gr}}(B/\mathbb{Q}_\mathrm{cyc})^\vee $ is finitely generated over $ \mathbb{Z}_p $. Since $S^{\Sigma_0}_{\mathrm{Gr}}( A_f(\xi_i\omega_p^{-j})/\mathbb{Q}_\mathrm{cyc})^\vee$ are finitely generated $\mathbb{Z}_p$ modules for $i=1,2$, it follows that $S^{\Sigma_0}_{\mathrm{Gr}}( A_j/\mathbb{Q}_\mathrm{cyc})^\vee$ is also a finitely generated $\mathbb{Z}_p$ module by exact sequence \eqref{split-selmer-03}. In particular, $S^{\Sigma_0}_{\mathrm{Gr}}( B/\mathbb{Q}_\mathrm{cyc})^\vee$ is $\mathbb{Z}_p[[\Gamma]]$ torsion. Thus by exact sequence \eqref{last-alg1}, $S_{\mathrm{Gr}}( B/\mathbb{Q}_\mathrm{cyc})^\vee$ is a $\mathbb{Z}_p[[\Gamma]]$ torsion. It then follows from Lemma \ref{191} and Theorem \ref{no-finite-part-result1} that $S_{\mathrm{Gr}}( A_j/\mathbb{Q}_\mathrm{cyc})^\vee$ has no non-zero pseudo-null submodule. Also note that for $i=1,2$, we have $ \text{Hom}(A_f(\xi_i\omega_p^{-j-1})[\pi], \mathcal{O}_K/\pi)^{G_\mathbb{Q}}=0$, as $A_f[\pi]$ is an irreducible $G_\mathbb{Q}$ module. Hence again applying
Theorem \ref{no-finite-part-result1}, we get $S_{\mathrm{Gr}}( A_f(\xi_i\omega_p^{-j})/\mathbb{Q}_\mathrm{cyc})^\vee$ has no non-zero pseudo-null submodule.
\end{proof}
For a finitely generated torsion $\mathbb{Z}_p[[\Gamma]]$ (resp. $\mathbb F_p[[\Gamma]]$) module $\mathcal{M}$, we denote the characteristic element of $ \mathcal{M} $ over $\mathbb{Z}_p[[\Gamma]]$ (resp. $\mathbb F_p[[\Gamma]]$) by $C_{\mathbb{Z}_p[[\Gamma]]}(\mathcal{M})$ (resp. $C_{ \mathbb F_p[[\Gamma]]}(\mathcal{M})$). From Lemma~\ref{lem:psuedonullity} and \cite[Corollary 3.8 (i), Corollary 3.21 (iii)]{SU} we have \begin{equation} C_{\mathbb{Z}_p[[\Gamma]]}(B_j) ~(\text{mod }\pi) = C_{ \mathbb F_p[[\Gamma]]}(B_j/{\pi \mathcal{M}}) \end{equation} for $B_j \in \{S^{\Sigma_0}_{\mathrm{Gr}}( A_j/\mathbb{Q}_\mathrm{cyc})^\vee, S^{\Sigma_0}_{\mathrm{Gr}}( A_f(\xi_1\omega_p^{-j})/\mathbb{Q}_\mathrm{cyc})^\vee, S^{\Sigma_0}_{\mathrm{Gr}}( A_f(\xi_2\omega_p^{-j})/\mathbb{Q}_\mathrm{cyc})^\vee \}$. Using this in \eqref{split-selmer-03}, we deduce the following theorem:
\begin{theorem}\label{thm:congruence of ideals} Let $ (N,M_0) =1$ and $\psi \lvert I_{\mathrm{cyc},v}$ has order prime to $p$ for $ v \mid pI_0$, where $\psi$ is the nebentypus of $h$. Assume $\bar{\rho}_{h}\vert I_{\mathrm{cyc},v} \cong \bar{\xi}_1 \oplus \xi_{2}$, $\bar{\rho}_{h}^{ss} \cong \xi_{1} \oplus \xi_{2}$ and $ A_{f}[\pi] $ is an irreducible $ G_{\mathbb{Q}} $ module and $S^{\Sigma_0}_{\mathrm{Gr}}( A_f(\xi_i\omega_p^{-j})/\mathbb{Q}_\mathrm{cyc})^\vee$ are finitely generated $\mathbb{Z}_p$ modules for $i=1,2$. If $ (p-1) \mid j $, then assume that \eqref{assumption} holds for all $ v \mid p $. Then for $ l-1 \leq j \leq k-2 $, we have \begin{small}{
\begin{equation}\label{split-selmer-04}
C_{\mathbb{Z}_p[[\Gamma]]}\Big(S^{\Sigma_0}_{\mathrm{Gr}}( A_j/\mathbb{Q}_\mathrm{cyc})^\vee\Big) = C_{\mathbb{Z}_p[[\Gamma]]}\Big(S^{\Sigma_0}_{\mathrm{Gr}}( A_f(\xi_1\omega_p^{-j})/\mathbb{Q}_\mathrm{cyc})^\vee\Big) C_{\mathbb{Z}_p[[\Gamma]]}\Big(S^{\Sigma_0}_{\mathrm{Gr}}( A_f(\xi_2\omega_p^{-j})/\mathbb{Q}_\mathrm{cyc})^\vee\Big)~ \text{mod } \pi.
\end{equation} }\end{small} \end{theorem} \begin{remark} As $ f $ is $ p $-ordinary and $ \bar{\rho}_f $ is irreducible, it follows the $C_{\mathbb{Z}_p[[\Gamma]]}\Big(S^{\Sigma_0}_{\mathrm{Gr}}( A_f(\xi_i\omega_p^{-j})/\mathbb{Q}_\mathrm{cyc})^\vee\Big)$ is independent of the choice of the lattice $ T_f $ (See \cite[Page 34]{SU}). Thus by Theorem~\ref{thm:congruence of ideals}, it follows that $C_{\mathbb{Z}_p[[\Gamma]]}\Big(S^{\Sigma_0}_{\mathrm{Gr}}( A_j/\mathbb{Q}_\mathrm{cyc})^\vee\Big) $ is also independent of the choice of the lattice $ T = T_f \otimes T_g $. \end{remark}
\section{Iwasawa main conjecture modulo $\pi$}\label{sec: IMC} Recall that $L^{\Sigma_{0}}_{p,f, \chi,j}$ and $L^{\Sigma_{0}}_{p,f\otimes h,j}$ are the $ p $-adic $ L $-functions attached to $ f \otimes \chi $ and $f\otimes h,$ (see \eqref{def: p-adic L-function f}, \eqref{def: p-adic L-function f x h}). We now recall the Iwasawa Main Conjecture for modular forms due to Greenberg \cite{gr1}. \begin{conjecture}\label{IWC for modular form}$($\textbf{Greenberg Iwasawa Main Conjecture for modular forms}$)$ Let $ f \in S_{k}(\Gamma_{0}(N),\eta) $ be a good $ p $-ordinary eigenform. For every $ 0 \leq j \leq k-1 $, we have \begin{align*}
(L^{\Sigma_{0}}_{p,f, \chi,j}) = C_{\mathbb{Z}_p[[\Gamma]]}\Big(S^{\Sigma_0}_{\mathrm{Gr}}( A_F(\chi \chi_p^{-j})/\mathbb{Q}_\mathrm{cyc})^\vee\Big). \end{align*} \end{conjecture} \begin{remark} Conjecture~\ref{IWC for modular form} is known for a large class of modular forms by the results of Kato \cite{kato} and Skinner-Urban \cite{SU}.
\end{remark} We now state our main result on the Iwasawa Main Conjecture for Rankin-Selberg $ L $-function mod $ \pi $. \begin{theorem}\label{congruence main conjecture } We assume that Conjecture~\ref{IWC for modular form} holds for all $ j $ and $ \xi_1 , \xi_{2}$. Let $ f \in S_{k}(\Gamma_{0}(N),\eta), h \in S_{l}(\Gamma_{0}(I),\psi) $ be $ p $-ordinary eigenforms, $f$ is a newform and the keep the assumptions as in Theorem~\ref{analytic final} and Theorem~\ref{thm:congruence of ideals}. For every $ l-1 \leq j \leq k-2 $, we have \begin{align*}
(L_{p,f\otimes h,j}) \equiv C_{\mathbb{Z}_p[[\Gamma]]}\Big(S_{\mathrm{Gr}}( A_j/\mathbb{Q}_\mathrm{cyc})^\vee\Big) \mod \pi. \end{align*} \end{theorem}
\begin{proof} By Theorem~\ref{analytic final}, we have $ (L^{\Sigma_{0}}_{p,f \otimes h,j}) \equiv (L^{\Sigma_{0}}_{p,f,\xi_1,j}) (L^{\Sigma_{0}}_{p,f, \xi_2,j}) \mod \pi $ for every $ l-1 \leq j \leq k-2 $. By Conjecture~\ref{IWC for modular form} we have \begin{small}
\begin{align*}
(L^{\Sigma_{0}}_{p,f, \xi_i, j} )= C_{\mathbb{Z}_p[[\Gamma]]}\Big(S^{\Sigma_0}_{\mathrm{Gr}}( A_f(\xi_i \chi_p^{-j})/\mathbb{Q}_\mathrm{cyc})^\vee\Big) = C_{\mathbb{Z}_p[[\Gamma]]}\Big(S^{\Sigma_0}_{\mathrm{Gr}}( A_f(\xi_i \omega_p^{-j})/\mathbb{Q}_\mathrm{cyc})^\vee\Big),
\end{align*} \end{small} for $ i=1,2 $. Now it follows from Theorem~\ref{thm:congruence of ideals} that \begin{equation}\label{Sigma congruence} (L^{\Sigma_{0}}_{p,f\otimes h,j}) \equiv C_{\mathbb{Z}_p[[\Gamma]]}\Big(S^{\Sigma_0}_{\mathrm{Gr}}( A_j/\mathbb{Q}_\mathrm{cyc})^\vee\Big) \mod \pi. \end{equation}
Arguing as in \cite[Corollary 2.3, Proposition 2.4]{gv} we have
\begin{small}
\begin{align*}
C_{\mathbb{Z}_p[[\Gamma]]}\Big(S^{\Sigma_0}_{\mathrm{Gr}}( A_j/\mathbb{Q}_\mathrm{cyc})^\vee\Big)
=C_{\mathbb{Z}_p[[\Gamma]]}\Big(S_{\mathrm{Gr}}( A_j/\mathbb{Q}_\mathrm{cyc})^\vee\Big) \prod_{\ell \in \Sigma_{0}} P_{T,\ell}(\ell^{-j-1}\gamma) , \quad
\forall ~~ l-1 \leq j \leq k-2
\end{align*} \end{small} where $ P_{T,\ell}(X) = \det(I -X \mathrm{Frob}_{\ell}\big\vert{V^{I_{\ell}}} ) $ with $ \gamma $ is a topological generator of $ \Gamma = \mathrm{Gal}(\mathbb{Q}_{\mathrm{cyc}}/\mathbb{Q}) $. Note that $ P_{T,\ell}(X)$ is the Euler factor at $ \ell $ for the Rankin-Selberg $ L $-function. Multiplying the appropriate Euler factors on either side of the congruence \eqref{Sigma congruence} we obtain the desired congruence modulo $ \pi $. \end{proof}
\begin{corollary}\label{cor: unit implies unit} Let the setting be as in Theorem~\ref{congruence main conjecture }. Then $ L_{p,f\otimes h,j} $ is a unit in the Iwasawa algebra $ \mathbb{Z}_p[[\Gamma]] $ if and only if $ C_{\mathbb{Z}_p[[\Gamma]]}\Big(S_{\mathrm{Gr}}( A_j/\mathbb{Q}_\mathrm{cyc})^\vee\Big) $ is a unit in the Iwasawa algebra $ \mathbb{Z}_p[[\Gamma]] $. \end{corollary}
\subsection{Examples}\label{section: examples}
\begin{example}\label{example 1} Let $ \Delta = \sum_{n=1}^{\infty} \tau(n) q^n $ be the Ramanujan Delta function. We have $ \Delta \in S_{12}(\mathrm{SL}_{2}(\mathbb{Z})) $ and $ \tau(11) $ is $ 11 $-adic unit. Further $ \bar{\rho}_{\Delta}\vert_{G_{\mathbb{Q}}} \mod 11$ is an irreducible $ G_{\mathbb{Q}} $ module. Consider the elliptic curve $E := X_0(11)$ of conductor $11$ over $\mathbb{Q}$ (LMFDB label 11.2.a.a) defined by $$ y^2 + y = x^3 - x^2 -10x - 20. $$ Let $ f_E $ be the modular form corresponding to $ E $ under modularity theorem. There is a $ \Lambda $-adic form whose specialisation at $ p=11 $ and $ k=2 $ (resp. $ k=12 $) is $ f_E $ (resp. $ \Delta $). Hence the residual Galois representation of $ E $ and $ \Delta $ at $ p=11 $ are isomorphic.
Consider the Eisenstein series $g' = E_2(z) - 23E_2(23z)$, where $ E_{2}(z) = \frac{1}{24} + \sum_{n=1}^{\infty} \sigma(n) q^n $. Then by a result of Mazur \cite[Proposition 5.12(ii)]{Mazur}, there exists a cusp form $h \in S_2(\Gamma_{0}(23))$ (LMFDB label 23.2.a) such that $ a(n,h) \equiv a(n,g') \mod \mathfrak{p}, $ where $ \mathfrak{p} = (11, 4 - \sqrt{5}) \subset \mathbb{Z}[(1 + \sqrt{5})/2]$. By [DS, Theorem 9.6.6], we have $ \rho_{h} \simeq \omega_{p} \oplus 1 $. Thus $\bar{\xi}_{1}=\bar{\omega}_{p}$ and $\bar{\xi}_{2}=1$. By our construction, $\xi_{1}= \omega_{p} $ and $\xi_{2} = 1$. Thus $g=E (1_p ,1)= E_2(z) - pE_2(pz)$, where $1_p$ is the trivial character modulo $p = 11$. As $ \mathrm{cond}(\rho_{h}) =23 $ and $ \mathrm{cond}(\bar{\rho}_{h}) = 11 $, we obtain $ m = 23 $ by \eqref{definition m}.
\begin{comment}
Consider the Eisenstein series $ g' = E_4(z)$, where $ E_4(z) = \frac{1}{240} + \sum_{n=1}^{\infty} \sigma_3(n) d^n $ be the weight $4$ Eisenstein series. Then by result of Dummigan, applied with $ k =4 $, $ p = 43 $ and $ l = 11 $ there exists an eigen form form $ h = \sum_{n=1}^{\infty} a(n,h) q^n \in S_{4}(43) $ such that
\[
a(\ell,h) \equiv 1 + \ell^3 \mod \mathfrak{p},
\]
where $ \mathfrak{p} $ is a prime lying above $ 11 $ in $ \mathbb{Q} (\{a(n,h)\})$. Using Sage one can check that for h given by LMDB label 43.4.a.a satisfies the congruence
\begin{align*}
a(\ell,h) \equiv 1 + \ell^3 \mod \mathfrak{p}, \quad
1 \leq l \leq 13
\end{align*}
where $ \mathfrak{p} $ is a prime lying above $ 11 $ in $ \mathbb{Q}(\lbrace a(n,h) \rbrace)$. By we have Sturm bound equal to $ \frac{4}{12} \cdot 43 \cdot (1 + \frac{1}{43}) \leq 15$. By \cite[Theorem 9.6.6]{DS05} we have
\[
\rho_{g} \simeq \omega^2_{p} \oplus 1.
\]
Thus $ \bar{\xi}_{2} =1 $ and $ \bar{\xi}_{1} = \bar{\omega}_p^2 $. Thus by our construction we have $ \xi_{2} = 1 $ and $ \xi_{1} = \omega $. Thus $ g = E_2(1_p, 1 ) = E_{2}(z) - p E_{2}(pz)$ where $ 1_p $ denotes the trivial character modulo $ p=11 $ and $ m=43 $. \end{comment} Consider the quadratic character $ \chi_{K} = (\frac{-23}{\cdot})$ associated to the field $ K = \mathbb{Q}(\sqrt{-23}) $. Let $ f = \Delta \otimes \chi_{K} $. Then $ f \in S_{12}(\Gamma_{0}(23^2), \chi_{K}^2)$ is a primitive form. Clearly $ f_0\vert \iota_m = f_0 $ is primitive, $ \bar{\rho}_{f_0} $ is $ p $-distinguished and $ \bar{\rho}_{f_0} $ is unramified outside $ 23 \cdot 11 $. Hence, $ f_0 $ satisfies the hypotheses of Theorem~\ref{c(f) and petterson}.
We now verify the hypotheses of Theorem~\ref{thm:congruence of ideals}. We have $ N_f= 23^2 $, $ M_0 =1 $, $ (N_f,M_0) =1 $ and the nebentypus of $ h $ is trivial. Also $ A_{f}[\pi] = A_{\Delta}[\pi] \otimes \chi_K $ is an irreducible $ G_\mathbb{Q} $-module. Note that $ \mathrm{Frob}_{p} $ acts on $ A_{f}^{-}(\xi_{2}) $ by the scalar $ a(p,f) \xi_{2}(p) = \chi_{K}(p) a(p,\Delta) = - \tau(p) = -\tau(11) \neq 1 \mod \pi $. Similarly, it can be checked that $ \mathrm{Frob}_{p} $ acts on $ A_{f}^{-}(\xi_{1}) $ as multiplication by $ 0 $. We next show that $ \mu(S^{\Sigma_0}_{\mathrm{Gr}}(A_{f}(\xi_{1}\omega_{p}^{j})/\mathbb{Q}_{\mathrm{cyc}})^\vee) =0$. Since $ f $ and $ f_{E} \otimes \chi_K $ lie in the same branch of Hida family, it is enough to know that $ \mu (S^{\Sigma_0}_{\mathrm{Gr}}(f_{E} \otimes \chi_K \omega_{p}^j/\mathbb{Q}_{\mathrm{cyc}})^\vee) =0$ (cf. \cite[Theorem 4.3.3]{epw}). By a deep result of Kato \cite{kato}, it reduces to show that the analytic $ \mu $-invariant vanishes i.e. the $ p $-adic $ L $-function $ L_{p}(f_{E} \otimes \chi_K, \omega_{p}^j)$ is not divisible by $ p $. We now compute $ L_{p}(f_{E} \otimes \chi_K, \omega_{p}^j) ( 0) $. From \cite{MTT} (cf., \cite[Section 5.1]{C}) we get \begin{small}
\begin{align}\label{eq: modular symbol and L-value}
L_{p}(f_{E}\otimes \chi_{K}, \omega_{p}^{i})(0) = \begin{cases} \frac{1}{\alpha}
\sum_{b=1}^{p-1} \bar{\omega}_{p}^{i}(b) \text{x}^{sign(\omega_{p}^i)}(b/p) &\text{if} ~
\omega_{p}^{i} \neq 1, \\
(1-\alpha^{-2}) \frac{L(f_{E}\otimes \chi_{K},1)}{\Omega^{+}}
&\text{otherwise},
\end{cases}
\end{align} \end{small}
where $ \text{x}^{\pm} $ are the $ \pm $ modular symbols associated to $ f_{E} \otimes \chi_K $, as defined by Manin and $ \alpha $ is a root of $ p^{\mathrm{th}} $-Hecke polynomial of $ f_{E}\otimes \chi_{K} $ with $ |\alpha|_p =1$. Using SAGE, we compute $ \text{x}^{\pm} $. The values are \begin{small}
\begin{center}
\begin{tabular}{| c | c | c | c | c | c | c | c | c | c | c |}
\hline
$\frac{b}{11}$ & $1/11$ & $ 2/11 $ & $ 3/11 $ & $ 4/11 $ & $ 5/11 $ & $ 6/11 $ & $ 7/11 $ & $ 8/11 $ & $ 9/11 $ & $ 10/11 $ \\
\hline
$ \text{x}^{+} $ & 2 & 0 & 5 & 5 & 0 & 0 & 5 & 5 & 0 & 2 \\
\hline
$ \text{x}^{-} $ & 0 & 0 & -5 & 5 & 0 & 0 & -5 & 5 & 0 & 0 \\
\hline
\end{tabular}
\end{center} \end{small} Let $\zeta_{10}$ be a primitive root of unity and $ \omega_{p}(2) = \zeta_{10} $ and $ \omega_{p}(-1) = -1 $. Then it follows $ \omega_{p}(3) = \omega_{p}(2)^3 \omega_{p}(-1) = - \zeta_{10}^3 $, $\omega_{p}(4) = \zeta_{10}^2 $ and $\omega_{p}(5) = \omega_{p}(2)^4 = \zeta_{10}^4 = \zeta_{10}^3 - \zeta_{10}^2 +\zeta_{10} -1 $. Thus from \eqref{eq: modular symbol and L-value} \begin{small}
\begin{align*}
L_{p}(f_{E}\otimes \chi_{K}, \omega_{p}^{i}) (0) \sim \begin{cases}
4 (4 - \omega_p^{i}(2) - \omega_p^{i}(3) - \omega_p^{i}(4)- \omega_p^{i}(5) ), &\text{if } i ~ \text{is even} \\
10(-\omega_p^{i}(2) + \omega^{i}(3) - \omega_p^{i}(4)- \omega_p^{i}(5) ) , &\text{if } i ~ \text{is odd},
\end{cases}
\end{align*} \end{small} here $ \sim $ denotes up to multiplication by a $p$-adic unit. Taking $ i = 1, 2,\ldots, 9 $ we obtain \begin{small}
\begin{alignat*}{3}
L_{p}(f_{E}\otimes \chi_{K}, \omega_{p}^{1})(0)& \sim -10( \zeta_{10}^3 + \zeta_{10}^2)
& L_{p}(f_{E}\otimes \chi_{K}, \omega_{p}^{2})(0) & \sim 10 \zeta_{10}^3 - 10 \zeta_{10}^2 - 6 \\
L_{p}(f_{E}\otimes \chi_{K}, \omega_{p}^{3})(0)&\sim 10( \zeta_{10}^3 - \zeta_{10}^2 + 2\zeta_{10} - 1)
&L_{p}(f_{E}\otimes \chi_{K}, \omega_{p}^{4})(0) &\sim -10 \zeta_{10}^3 + 10 \zeta_{10}^2 + 4 \\
L_{p}(f_{E}\otimes \chi_{K}, \omega_{p}^{5})(0) & = 0 \qquad \qquad
& L_{p}(f_{E}\otimes \chi_{K}, \omega_{p}^{6})(0) &\sim 10 \zeta_{10}^3 - 10 \zeta_{10}^2 + 4 \\
L_{p}(f_{E}\otimes \chi_{K}, \omega_{p}^{7})(0) & \sim 10(- \zeta_{10}^3 + \zeta_{10}^2 - 2 \zeta_{10} + 1) \quad
& L_{p}(f_{E}\otimes \chi_{K}, \omega_{p}^{8})(0) &\sim 10 \zeta_{10}^3 - 10 \zeta_{10}^2 - 6 \\
L_{p}(f_{E}\otimes \chi_{K}, \omega_{p}^{9})(0) &\sim 10( \zeta_{10}^3 + \zeta_{10}^2)
& L_{p}(f_{E}\otimes \chi_{K}, \omega_{p}^{0})(0) & \sim 24
\end{alignat*} \end{small} We have $ \prod_{i \neq 5} L_{p}(f_{E}\otimes \chi_{K}, \omega_{p}^{i}, \mathbb{Q})(0) \sim 18451200000 $, which is a $ 11 $-adic unit. For $ i=5 $, we have $ L_{p}(f_{E}\otimes \chi_{K}, \omega_{p}^{i}, \mathbb{Q}) = T u(T) $, for some $ u(T) \in \mathbb{Z}_p[[T]]^{\times} $ using SAGE. {This shows that $\prod_{i=0}^{9} L_{p}(f_{E}\otimes \chi_{K}, \omega_{p}^{i}, \mathbb{Q})= Tu(T)$ for some unit $ u \in \mathbb{Z}_p[[T]] $.} Since $ \xi_{1} = \omega_{p} $, we {obtain} $ \mu(S^{\Sigma_0}_{\mathrm{Gr}}(A_{f}(\xi_{1}\omega_p^{j})/\mathbb{Q}_{\mathrm{cyc}})^\vee) = 0$, for all $ j $. This checks all the hypotheses of Theorem~\ref{thm:congruence of ideals} and the Greenberg Iwasawa main conjecture holds for $ f_{E} \otimes \chi_{K} $.
For $ f = \Delta \otimes \chi_{K} $ and $h \in S_{2}(\Gamma_0(23)) $ (LMFDB label 23.2.a) the hypotheses of Theorem~\ref{congruence main conjecture } are met. Therefore by Theorem~\ref{congruence main conjecture }, we conclude that the Iwasawa main conjecture holds $ f \otimes h $ modulo $ \pi $, i.e., \begin{equation}
(L_{p, f \otimes h, j}) = (C_{\mathbb{Z}_{p}[[\Gamma]]}(S_{\mathrm{Gr}}(A_j)/\mathbb{Q}_{\mathrm{cyc}})^\vee) \mod \pi \quad \mathrm{for} ~ 1 \leq j \leq 10. \end{equation}
As $ \mu(S_{\mathrm{Gr}}(A_{f}(\omega_p^{j})/\mathbb{Q}_{\mathrm{cyc}})^\vee) = 0 $ and $ T \nmid L_{p}(f_E\otimes \chi_{K}, \omega_{p}^{i}, \mathbb{Q}) $ for $ i \neq 5 $, it follows that $ L_{p}(f_E\otimes \chi_{K}, \omega_{p}^{i}, \mathbb{Q}) $ is unit whenever $ i \neq 5$. Hence, $L_{p}(f\otimes \chi_{K}, \omega_{p}^{i}, \mathbb{Q}) $ is unit whenever $ i \neq 5$. Applying Theorem~\ref{analytic final}, we obtain $ L_{p, f \otimes h, j} $ whenever $ j \neq 4, 5 $. Thus, $ C_{\mathbb{Z}_{p}[[\Gamma]]}(S_{\mathrm{Gr}} (A_j/\mathbb{Q}_{\mathrm{cyc}})^\vee) $ is a unit in $ \mathbb{Z}_p[[\Gamma]] $ for $ j \neq 4,5 $. Further, if $ j =4,5 $ we have $ C_{\mathbb{Z}_{p}[[\Gamma]]}(S_{\mathrm{Gr}} (A_j/\mathbb{Q}_{\mathrm{cyc}})^\vee) = T u(T)$, for some unit $ u(T) \in \mathbb{Z}_p[[T]] $. \end{example}
\begin{example} Let $ \Delta = \sum_{n=1}^{\infty} \tau(n) q^n $ be Ramanujan Delta function and $ p =691 $. Consider the Hida family passing through $ \Delta $. Then it known that $ E_{12}(z) = -\frac{B_{12}}{2} + \sum_{n=1}^{\infty} \sigma_{11}(n) q^n$ is a member of the family. By specialising appropriately at $ (k, \mu_{691^n}) $, one obtains cusp form of weight $ k $ and level $ 691^{n+1} $ for every $ n \geq 0$. Let $ h $ be a cusp form of weight $ 14 $ and level $ 691 $. Since the residual representation of $ E_{12} $ is reducible, the residual representation of $ g $ is also reducible. Further, $ \mathrm{cond}(\bar{\rho}_h) $ is also a power of $ 691 $ and $ \xi_{1} = \omega_{p}^{-11} $ and $ \xi_2 = 1 $. Hence $ m=1 $ and $ I_0 = M_0 =1 $ in this case. Let $ f = \Delta_{16} $ (resp. $ \Delta_{18} $, $\Delta_{20}$, $ \Delta_{22} $ etc.) be the cusp form of weight $ 16 $ and level $ 1 $. Then $ f $ is $ p $-ordinary and $ \bar{\rho}_{f} $ is irreducible $ \mathrm{G}_{\mathbb{Q}} $ module. Thus $ f $ satisfies hypotheses of Theorem~\ref{c(f) and petterson} and so the hypotheses of Theorem~\ref{analytic final} also hold. Since $ \bar{\rho}_g $ is reducible and nebentypus of $ \psi_h $ is a power of $ \omega_p $ we also get hypotheses of Proposition~\ref{prop:exact sequence of inertia}. It remains to check the hypotheses of Theorem~\ref{thm:congruence of ideals} i.e. Conjecture~\ref{IWC for modular form} holds for $ A_{f}(\xi_{i} \omega_{p}^{-j}) $ for $ 13 \leq j \leq 14$. It is easy to see this hypothesis are met if $ \mu(S^{\Sigma_0}_{\mathrm{Gr}}(A_{f}( \omega_{p}^{-j})/\mathbb{Q}_{\mathrm{cyc}})^\vee) =0 $ for $ j=13, 14 $. It is known that $ \bar{\rho}_f $ is irreducible. As $ f $ is good and ordinary at $ p=691 $, following \cite[Conjecture 1.11]{gr}, it is expected that the $ \mu $-invariant, $ \mu(S^{\Sigma_0}_{\mathrm{Gr}}(A_{f}( \omega_{p}^{-j})/\mathbb{Q}_{\mathrm{cyc}})^\vee) =0 $ for $ j=13 $ and $j= 14 $.
Thus we have shown if $ \mu(S^{\Sigma_0}_{\mathrm{Gr}}(A_{f}( \omega_{p}^{-j})/\mathbb{Q}_{\mathrm{cyc}})^\vee) =0 $ for $ j=13 $ and $j= 14 $, then all the hypotheses of Theorem~\ref{congruence main conjecture } hold for $ f = \Delta_{16} $ and $ h $. Hence the Iwasawa main conjecture holds $ f \otimes h $ modulo $ \pi $, i.e. \begin{equation*}
(L_{p, f \otimes h, j}) = (C_{\mathbb{Z}_{p}[[\Gamma]]}(S_{\mathrm{Gr}}(A_j)/\mathbb{Q}_{\mathrm{cyc}})^\vee) \mod \pi \quad \mathrm{for} ~ 13 \leq j \leq 14. \end{equation*}
\end{example}
\begin{comment} Let $g$ and $h$ be two modular forms such that residual representation $A_g[\pi]$ and $A_h[\pi]$ are isomorphic. Then $S^{\Sigma_0}_{\text{Gr}}(A^1_j[\pi]/K_\mathrm{cyc}) \cong S^{\Sigma_0}_{\text{Gr}}(A^2_j[\pi]/K_\mathrm{cyc})$ are isomorphic. \end{comment}
\end{document} | arXiv |
Comment. The trace of an endomorphism $\alpha$ of a finite-dimensional vector space $V$ over the field $k$ may be defined as the trace of any matrix representing it... To understand total variation we first must find the trace of a square matrix. A square matrix is a matrix that has an equal number of columns and rows. Important examples of square matrices include the variance-covariance and correlation matrices.
4/09/2014 · Trace of a matrix and it's properties explained. To ask your doubts on this topic and much more, click here: http://www.techtud.com/video-lecture/lecture-trace how to get from papeete to bora bora To understand total variation we first must find the trace of a square matrix. A square matrix is a matrix that has an equal number of columns and rows. Important examples of square matrices include the variance-covariance and correlation matrices.
To understand total variation we first must find the trace of a square matrix. A square matrix is a matrix that has an equal number of columns and rows. Important examples of square matrices include the variance-covariance and correlation matrices. | CommonCrawl |
\begin{document}
\title[Exponential decay for BV coefficients]{Exponential time-decay for a one dimensional wave equation with coefficients of bounded variation}
\author{Kiril Datchev} \address{Department of Mathematics, Purdue University, West Lafayette, IN, 47907-2067, USA}
\email{[email protected]} \author{Jacob Shapiro} \address{Department of Mathematics, University of Dayton, Dayton, OH 45469-2316, USA} \email{[email protected]} \thanks{Second author is the corresponding author}
\keywords{resolvent estimate, Schr\"odinger operator, wave decay}
\maketitle
\begin{abstract} We consider the initial-value problem for a one-dimensional wave equation with coefficients that are positive, constant outside of an interval, and have bounded variation (BV). Under the assumption of compact support of the initial data, we prove that the local energy decays exponentially fast in time, and provide the explicit constant to which the solution converges. The key ingredient of the proof is a high frequency resolvent estimate for an associated Helmholtz operator with a BV potential.
\end{abstract}
\section{Introduction and statement of results} \label{introduction}
This paper establishes exponential local energy decay for the solution of the following one dimensional wave equation, with compactly supported initial data: \begin{equation} \label{wave eqn} \begin{cases} \beta(x) \partial_t^2 w(x,t) - \partial_x (\alpha(x) \partial_x w(x,t)) = 0, \qquad (x,t) \in \mathbb{R} \times (0, \infty), \\ w(x,0) = w_0(x), \\ \partial_t w(x,0) = w_1(x), \\ \supp w_0, \, \supp w_1 \subseteq (-R,R), \qquad R > 0. \end{cases} \end{equation} Here, the coefficients $\alpha, \beta : \mathbb{R} \to (0,\infty)$ have bounded variation (BV). We suppose also \begin{equation} \label{infs positive}
\inf_\mathbb{R} \alpha, \, \inf_\mathbb{R} \beta > 0, \end{equation} and that there exist $R_0, \, \alpha_0, \, \beta_0 > 0$, so that \begin{equation} \label{perturbations}
\alpha(x) = \alpha_0, \, \beta(x) = \beta_0, \qquad |x| \ge R_0.
\end{equation}
To begin, we address the well-posedness of \eqref{wave eqn} via the spectral theorem for self-adjoint operators. Let $\mathcal H$ be the Hilbert space $L^2(\mathbb R; \beta(x)dx)$ equipped with the inner product \[\langle u,v \rangle_{\mathcal H} \defeq \int_{\mathbb{R}}\overline{u}(x) v(x) \beta(x) dx.\] (Note that $L^2(\mathbb R; \beta(x)dx) = L^2(\mathbb R; dx)$ as sets, and their respective norms generate the same topology, since $\beta$ has positive upper and lower bounds.) Define the symmetric, nonnegative differential operator \begin{equation} \label{H}
H u \defeq - \beta^{-1} \partial_x (\alpha \partial_x u), \end{equation}
with domain $\mathcal D(H) \defeq \{u \in L^2(\mathbb{R}) : u, \partial_x u \in L^2(\mathbb{R}) \cap L^\infty(\mathbb{R}), \text{ and } \partial_x(\alpha \partial_xu) \in L^2(\mathbb{R}) \}$. We will see from Lemma \ref{self adjointness lemma} in Section \ref{wtd resolv est section} that $H$ is self-adjoint with respect to $\mathcal{D}(H)$. It is also conveniently the case that $D(H^{1/2})$ coincides with the Sobolev space $H^1(\mathbb{R})$ \cite{re22b}. For completeness, we prove this fact in Appendix \ref{H1 appendix}.
Thus, for initial conditions $w_0 \in \mathcal{D}(H)$, $w_1 \in \mathcal{D}(H^{1/2})$,
\begin{equation} \label{soln spectral thm} w(t) = w(\cdot, t) = \cos(t H^{1/2}) w_0 + \frac{\sin(tH^{1/2})}{H^{1/2}} w_1. \end{equation}
is the unique function $w \in C^2((0,\infty), \mathcal{H})$ with $w(0) = w_0$, $\partial_tw(0) = w_1$, and for all $t > 0$, $w(t) \in \mathcal{D}(H)$ and $\partial^2_t w(t) +Hw(t)=0$.
\begin{theorem} \label{LED thm}
Let $\alpha, \, \beta : \mathbb{R} \to (0,\infty)$ have bounded variation and satisfy \eqref{infs positive} and \eqref{perturbations}. Suppose $w_0 \in \mathcal{D}(H), \, w_1 \in \mathcal{D}(H^{1/2})$, and $\supp w_0,\, \supp w_1 \subseteq (-R, R)$ for some $R> 0$. Let $w(t)$ be given by \eqref{soln spectral thm}. For any $R_1 > 0$, there exist $C, c > 0$ so that \begin{equation} \label{LED} \begin{gathered}
\| w(\cdot, t) - w_\infty \|_{H^1(-R_1,R_1)} +
\|\partial_t w(\cdot, t) \|_{L^2(-R_1,R_1)} \\
\le C e^{-c t} (\| w_0\|_{H^1(\mathbb{R})} + \|w_1\|_{L^2(\mathbb{R})}), \qquad t > 0,
\end{gathered} \end{equation} where
\begin{equation} \label{w infty}
w_\infty \defeq \frac{1}{2(\alpha_0 \beta_0)^{1/2}}\int_{\mathbb{R}} w_1(x) \beta(x) dx.
\end{equation} \end{theorem}
Theorem \ref{LED thm} is motivated by the recent article \cite{agpp22}. There, the authors prove \eqref{LED}, with an explicit constant $c$ depending on $\alpha$ and $\beta$, provided that $\alpha$ and $\beta$ are Lipschitz continuous, bounded from above and below by positive constants, and satisfy \eqref{perturbations}. Our result includes natural examples such as cases where $\alpha$ and $\beta$ are piecewise constant and it is easy to see that the exponential decay rate in \eqref{LED} cannot in general be improved to any superexponential rate. See \cite{biz16} for dispersive and Strichartz estimates for one dimensional wave equations with BV coefficients.
To prove Theorem \ref{LED thm}, it suffices to show \eqref{LED} and \eqref{w infty} in the special case
\begin{equation} \label{perturbations of identity}
\alpha(x) = \beta(x) = 1, \qquad |x| \ge R_0.
\end{equation}
Indeed, if $w(x,t)$ solves \eqref{wave eqn} for initial conditions $w_0, \, w_1$ and general $\alpha$ and $\beta$, then the function $u(x,t) \defeq w(\sqrt{\alpha_0}) x, \sqrt{\beta_0}t)$ solves $ (\beta(\sqrt{\alpha_0}x)/\beta_0)\partial_t^2 u - \partial_x ((\alpha(\sqrt{\alpha_0}x)/\alpha_0) \partial_x u) = 0$ with initial conditions $u(x,0) = w_0(\sqrt{\alpha_0}x), \, \partial_t u(x,0) = \sqrt{\beta}_0 w_1(\sqrt{\alpha_0}x)$. Then \eqref{perturbations of identity} applies, giving that $u$ decays according to \eqref{LED} and \eqref{w infty}. The asserted decay for $w$ follows by a change of variables.
For the wave equation with \textit{constant} coefficients and compactly supported initial conditions, it follows readily from D'Alembert's formula that solution to \eqref{wave eqn} converges to $w_\infty$ in finite time. However, for variable coefficients, exponential decay is a typical scenario. This occurs in the setting of reflection and transmission, e.g., when $\alpha \equiv 1$ and $\beta$ assumes precisely two values.
In dimensions two and higher, the recent works \cite{chik20, sh18} treat local energy decay for wave equations with Lipschitz coefficients. Though in higher dimensions, logarithmic, rather than exponential decay, is optimal in general. The study of energy decay more broadly has a long history, going back to the foundational work of Morawetz, Lax--Phillips, and Vainberg \cite{mor, lmp, lp, vai}, which we will not attempt to review here. The reader may consult \cite{bu98, hizw17,sh18,dz} for more historical background and references.
We prove Theorem \ref{LED thm} by analyzing $H$ as a \textit{black box Hamiltonian} in the sense of Sj\"ostrand and Zworski \cite{sz91}. In particular, \eqref{perturbations of identity} implies that for any $\chi \in C_0^\infty(\mathbb R; [0,1])$ that is identically one near $[-R_0, R_0]$, the cutoff resolvent \begin{equation} \label{continued resolv}
\chi R(\lambda) \chi \defeq \chi (H- \lambda^2)^{-1} \chi : \mathcal{H} \to \mathcal{D}(H) \end{equation}
continues meromorphically from $\imag \lambda > 0$ to the complex plane. (Here, we equip $\mathcal{D}(H)$ with the graph norm $u \mapsto (\|u\|^2_{\mathcal{H}} + \|Hu\|^2_{\mathcal{H}})^{1/2}$.) In particular, we establish the following high frequency bound.
\begin{theorem} \label{unif resolv est thm}
Suppose $\alpha, \beta : \mathbb{R} \to (0, \infty)$ have bounded variation and obey \eqref{infs positive} and \eqref{perturbations of identity}. For any $\chi \in C_0^\infty(\mathbb R; [0,1])$ that is identically one near $[-R_0, R_0]$, there exists $C, \, \lambda_0, \, \varepsilon_0 > 0$ so that
\begin{equation} \label{unif resolv est}
\|\chi R(\lambda) \chi\|_{\mathcal{H} \to \mathcal{H}} \le C|\real \lambda |^{-1},
\end{equation}
whenever $|\real \lambda| \ge \lambda_0$, and $|\imag \lambda| \le \varepsilon_0$. \end{theorem}
In Section \ref{high energy bound section}, we achieve \eqref{unif resolv est} by rescaling $H - \lambda^2$ semiclassically, see \eqref{A}, and apply a resolvent estimate for a Schr\"odinger operator with a BV potential, namely Theorem \ref{nontrap thm oned} in Section \ref{wtd resolv est section}. The proof of Theorem \ref{nontrap thm oned} uses a positive commutator argument that relies on some basic calculus facts for BV functions. We collect these facts in Section \ref{bv review section}, and prove them in Appendix \ref{BV appendix}. Finally, in Section \ref{wave decay section}, we prove \eqref{LED} by combining \eqref{unif resolv est} with an argument involving Plancherel's theorem and contour deformation. A similar strategy appears in \cite[Section 3]{vo99}.
Our methods should apply directly to some more general operators, such as the wave operator $\beta(x) \partial_t^2 - \partial_x (\alpha(x) \partial_x) + V(x)$, where $V$ is real-valued, compactly supported, and has BV. In that case, however, the residual $w_\infty$ in \eqref{LED} may be more complicated, as there may or may not be a resonance at zero, and there may also be discrete negative spectrum. See \cite[Theorem 2.9]{dz} for instance, which treats the case $V \not\equiv 0$ and $\alpha, \beta \equiv 1$.
\section{Review of BV} \label{bv review section}
To keep the notation concise, for the rest of the article, we use ``prime" notation to denote differentiation with respect to $x$, e.g., $u' \defeq \partial_x u$.
Let $f : \mathbb{R} \to \mathbb{C}$ be a function of locally bounded variation. For all $x \in \mathbb{R}$, put \begin{equation} \label{LRA}
f^L(x) \defeq \lim_{\delta \to 0^+}f(x-\delta), \qquad f^R(x) \defeq \lim_{\delta \to 0^+}f(x+\delta), \qquad f^A(x) \defeq (f^L(x) + f^R(x))/2, \end{equation} where the limits exist because both the real and imaginary parts of $f$ are a difference of two increasing functions. Recall that $f$ is differentiable Lebesgue almost everywhere, so $f (x)= f^L(x) = f^R(x) = f^A(x)$ for almost all $x \in \mathbb{R}$.
We may decompose $f$ as \begin{equation} \label{decompose f} f = f_{r, +} - f_{r, -} + i( f_{i,+} - f_{i,-}), \end{equation} where the $f_{\sigma,\pm}$, $\sigma \in \{r, i\}$, are increasing functions on $\mathbb{R}$. Each $f^R_{\sigma,\pm}$ uniquely determines a regular Borel measure $\mu_{\sigma,\pm}$ on $\mathbb{R}$ satisfying $\mu_{\sigma, \pm}(x_1, x_2] = f^R_{\sigma, \pm}(x_2) - f^R_{\sigma, \pm}(x_1)$, see \cite[Theorem 1.16]{fo}. We put \begin{equation} \label{df}
df \defeq \mu_{r, +} - \mu_{r, -} + i( \mu_{i,+} - \mu_{i,-}), \end{equation}
which is a complex measure when restricted to any bounded Borel subset. For any $a < b$,
\begin{equation} \label{ftc}
\begin{gathered}
\int_{(a,b]}df = f^R(b) - f^R(a),\\
\int_{(a,b)}df = f^L(b) - f^R(a).
\end{gathered}
\end{equation}
We collect several properties of functions of bounded variation, which are well known, and which we use to prove Theorem \ref{nontrap thm oned} in Section \ref{wtd resolv est section}. Their proofs are deferred to the appendix.
\begin{proposition}[integration by parts] \label{ibp bv prop} Let $f: \mathbb{R} \to \mathbb{C}$ have locally bounded variation. For any $a < b$, and any continuous $\varphi$, with $\varphi'$ piecewise continuous and $\varphi(a) = \varphi(b) = 0$, \begin{equation} \label{Folland IBP}
\int_{(a,b]} \varphi df = -\int_{(a,b]} \varphi' fdx. \end{equation}
\end{proposition}
\begin{proposition}[product rule] \label{prod rule bv prop} Let $f, \, g : \mathbb{R} \to \mathbb{C}$ be functions of locally bounded variation. Then \begin{equation}\label{e:prod}
d(fg) = f^A dg + g^A df \end{equation} as measures on a bounded Borel subset of $\mathbb{R}$. \end{proposition}
\noindent \textbf{Remark:} We note that if $f$ is continuous, then inductively applying \eqref{e:prod} yields $df^n = nf^{n-1} df$.
\begin{proposition}[chain rules] \label{chain rule bv prop} Let $f : \mathbb{R} \to \mathbb{R}$ be continuous and have locally bounded variation. Then, as measures on a bounded Borel set of $\mathbb{R}$, \begin{equation} \label{chain rule continuous}
d(e^f) = e^{f} df. \end{equation} On the other hand, let $x_1, \dots x_N, r_0, r_1 \dots, r_N \in \mathbb{R}$, and consider the function \begin{equation*}
g(x) = r_0 \mathbf{1}_{(-\infty, x_1]} + \sum_{j=1}^{N-1} r_j \mathbf{1}_{(x_j, x_{j+1}]} + r_N \mathbf{1}_{(x_N, \infty)}. \end{equation*} Then \begin{equation} \label{chain rule jumps}
d(e^{g}) = \sum_{j=1}^N(e^{r_j} - e^{r_{j-1}}) \delta_{x_j}, \end{equation} where $\delta_{x_j}$ denotes the dirac measure at $x_j$. \end{proposition} The need to treat separately the case of jump discontinuities in Proposition \ref{chain rule bv prop} was brought to the authors' attention by \cite{pi22, re22a}.
\section{Weighted resolvent estimate} \label{wtd resolv est section} The purpose of this Section is to prove a weighted resolvent estimate for the semiclassical Schr\"odinger operator \begin{equation} \label{defn P}
P = P(h) \defeq -h \partial_x (\alpha(x) h \partial_x ) + V(x) - E : L^2(\mathbb{R}) \to L^2(\mathbb{R}), \qquad E, \, h > 0, \end{equation} which is the key ingredient in the proof of Theorem \ref{unif resolv est thm} in Section \ref{high energy bound section}. We suppose $\alpha$ and $V$ are real-valued functions of bounded variation on $\mathbb{R}$, and \begin{equation} \label{inf alpha positive} \inf_{\mathbb{R}} \alpha > 0. \end{equation} Specifically, we show \begin{lemma} \label{self adjointness lemma} The operator $P : L^2(\mathbb{R}) \to L^2(\mathbb{R})$ is self adjoint with respect to the domain \begin{equation} \label{D}
\mathcal{D} \defeq \{u \in L^2(\mathbb{R}) : u, u' \in L^2(\mathbb{R}) \cap L^\infty(\mathbb{R}), \text{ and } Pu \in L^2(\mathbb{R}) \}, \end{equation} \end{lemma} \noindent and prove the following resolvent bound, for $h$ small, and uniformly down to $[E_{\min}, E_{\max}] \subseteq (0, \infty)$.
\begin{theorem} \label{nontrap thm oned} Fix $[E_{\min}, E_{\max}] \subseteq (0, \infty)$ and $\delta > 0$. Assume $\alpha, V : \mathbb{R} \to \mathbb{R}$ have bounded variation, $\alpha$ obeys \eqref{inf alpha positive}, and \begin{equation} \label{E grtr V}
\sup_{\mathbb{R}} V < E_{\min}. \end{equation} Then there exist $C, h_0 > 0$, so that for all $ E \in [E_{\min}, E_{\max}],$ $h \in (0,h_0]$, and $\varepsilon > 0$, \begin{equation} \label{nontrap est oned}
\|(|x| + 1)^{-\frac{1+\delta}{2}} (P(h) - i\varepsilon)^{-1} (|x| + 1)^{-\frac{1+\delta}{2}}\|_{L^2(\mathbb R) \to L^2(\mathbb R)} \le Ch^{-1}. \end{equation} \end{theorem}
Since $V$ has limited regularity, we have replaced a more typical nontrapping condition, concerning the escape of trajectories $\dot{x} = 2\xi$, $\dot{\xi} = -\partial_x V$ that obey $|\xi|^2 + V(x) = E$, with the simpler condition \eqref{E grtr V}. Indeed, as $\alpha$ and $V$ have only bounded variation, the bicharacteristic flow is not necessarily well defined. Moreover, in Section \ref{high energy bound section}, we shall see that \eqref{E grtr V} is a natural assumption, given that the coefficients of the operator $H$ obey \eqref{infs positive}.
To prove Theorem \ref{nontrap thm oned}, we employ a positive commutator-style argument in the context of the spherical energy method. This strategy has long been used to prove semiclassical resolvent estimates \cite{cv, da, kv, dash20, gash22}. In fact, as we are in one dimension, we just use the pointwise energy \begin{equation} \label{defn F}
F(x) = F[u](x) \defeq \alpha(x)|h \partial_x u(x)|^2 + (E- V(x))|u(x)|^2, \qquad u \in \mathcal{D}. \end{equation}
The goal is to construct a suitable weight function $w(x)$ so that the derivative of $wF$, in the sense of distributions, has a favorable sign. From \eqref{deriv wF with alpha} below, we see that $w$ ought to be designed so that $(w(E - V))'$ has a positive lower bound. If $V$ only has bounded variation, this derivative must be interpreted as a measure, and extra care is needed to control the point masses arising from the discontinuities of $V$ (see \eqref{lower bd first meas}).
We first give our attention to Lemma \ref{self adjointness lemma}, which is essentially well known. Our present proof is adapted from \cite[Section 2]{dash20}.
\begin{proof}[Proof of Lemma \ref{self adjointness lemma}] Let \begin{equation*}
\mathcal D_{\max} \defeq \{u \in L^2(\mathbb{R}) : u, \, \alpha u' \text{ are locally absolutely continuous and } Pu \in L^2(\mathbb{R})\}, \end{equation*} By, \cite[Lemma 10.3.1]{ze}, $\mathcal D_{\max}$ is dense in $L^2(\mathbb{R})$. We begin by proving \begin{equation} \label{Dmax is D} \mathcal D_\textrm{max} = \mathcal D. \end{equation}
Indeed, for any $a>0$ and $u \in \mathcal D_\textrm{max}$, by integration by parts and Cauchy--Schwarz, \begin{equation*} \begin{gathered}
\inf \alpha \int_{-a}^a|u'|^2 \le \int_{-a}^a \alpha u'\bar u' = \alpha u' \overline{u} \rvert_{-a}^a + h^{-2} \int_{-a}^a Pu \bar u - h^{-2} \int_{-a}^a Vu \bar u \\
\le 2 \sup \alpha \sup_{[-a,a]}|u'|\sup_{[-a,a]} |u| + h^{-2}\sup|V| \|u\|_{L^2}^2 + h^{-2}\|Pu\|_{L^2}\|u\|_{L^2} ,\\
\sup_{[-a,a]}|u|^2 =\sup_{x \in [-a,a]} \left(|u(0)|^2 + 2 \real \int_0^x u'\bar u\right) \le |u(0)|^2 + 2\left(\int_{-a}^a|u'|^2\right)^{1/2}\|u\|_{L^2},\\
(\inf \alpha)^2 \sup_{[-a,a]} |u'|^2 \le \sup_{[-a,a]} |\alpha u'|^2 = \sup_{x \in [-a,a]} \left( |(\alpha u')(0)|^2 + 2 \real \int_0^x (\alpha u')'\alpha \overline{u}' \right) \\
\le |(\alpha u')(0)|^2 + 2 h^{-2} ( \sup (\alpha |V|) \|u\|_{L^2} + \sup \alpha \|Pu\|_{L^2})\left(\int_{-a}^a|u'|^2\right)^{1/2}. \end{gathered} \end{equation*} This is a system of inequalities of the form $x^2 \le A + Byz$, $y^2 \le C + Dx$, $z^2 \le E + Fx$. Thus, for any $\gamma > 0$, \begin{equation} \label{combine ineq system} \begin{split}
x^2 &\le A + \frac{B}{2\gamma} + \gamma (yz)^2 \le A + \frac{B}{2\gamma} + \gamma (C + Dx)(E+ Fx) \\
&\le A + \frac{B}{2\gamma} + \gamma CE + \gamma \frac{ (CF)^2 + (DE)^2}{2} + (\gamma^2 + \gamma DF) x^2. \end{split} \end{equation} Choosing $\gamma$ small enough allows one to absorb all the terms involving $x^2$ on the right side of \eqref{combine ineq system}, into the left side. Hence $x$, $y$ and $z$ are all bounded independently of $a$. Letting $a \to \infty$, we conclude that $u' \in L^2(\mathbb{R})$ and $u, u' \in L^\infty(\mathbb{R})$. Hence $\mathcal D_{\max} \subseteq \mathcal D$. The inclusion $\mathcal{D} \subseteq \mathcal D_{\max}$ follows because $Pu \in L^2(\mathbb{R})$ implies $(\alpha u')' \in L^2(\mathbb{R})$, which in turns gives that $\alpha u'$ is locally absolutely continuous.
Equip $P$ with the domain $\mathcal D_\textrm{max} = \mathcal D \subseteq L^2(\mathbb{R})$. By integration by parts, $P \subseteq P^*$. But, by Sturm--Liouville theory, $P^* \subseteq P$; see \cite[Equation 10.3.2]{ze}. Hence $P=P^*$.
\end{proof}
We now prove Theorem \ref{nontrap thm oned}, with the argument proceeding in two steps. First, as described above, we build a weight $w$ so that, $d(wF)$ has a desirable lower bound in the sense of measures--see \eqref{deriv wF with alpha}. This yields the Carleman estimate \eqref{penult est dim one}, which implies the resolvent estimate \eqref{final estimate one dim}.
\begin{proof}[Proof of Theorem \ref{nontrap thm oned}] Decompose \begin{equation*} \begin{gathered} dV = dV^d + dV^c, \\ d\alpha = d\alpha^d + d\alpha^c, \\ \end{gathered} \end{equation*} into their discrete and continuous parts. Let $J_V$, respectively $J_\alpha$ be the sets of ``positive jumps" of $V$, $\alpha$ respectively. That is $J_V$ is the set of $x$-values such that $(V^R - V^L)(x) > 0$, and similarly for $J_\alpha$. Since $V$ and $\alpha$ have bounded variation, both $J_V$ and $J_\alpha$ are at most countable. We denote by $\{x_j\}_j$ an enumeration of $J_V \cup J_\alpha$. Additionally, let \begin{equation*} \begin{gathered} dV^c = dV^c_+ - dV^c_-,\\ d\alpha^c = d\alpha^c_+ - d\alpha^c_-, \\ \end{gathered} \end{equation*} be Jordan decompositions for $dV^c$, $d\alpha^c$ respectively.
For each $N \in \mathbb{N}$, let $x_{1,N}, x_{2,N}, \dots, x_{N,N}$ be the elements of $\{x_j\}_{j=1}^N$ relabeled in increasing order. Define the function $q_{1,N}$ by \begin{equation} q_{1,N}(x) \defeq r_{0,N} \mathbf{1}_{(-\infty, x_{1,N}]} + \sum^{N-1}_{j = 1} r_{j,N} \mathbf{1}_{(x_{j,N}, x_{j+1,N}]} + r_{N,N} \mathbf{1}_{(x_{N,N}, \infty)}, \label{q1} \end{equation} where the numbers $\{r_{j,N}\}_{j=0}^N$ are defined recursively as follows: \begin{gather}
r_{0, N} = 0, \qquad r_{j,N} = r_{j-1,N} + \log \max \Big\{1 + \frac{2A_{j,N}}{1 - A_{j,N}}, 1 + \frac{2B_{j,N}}{1 - B_{j,N}} \Big\}, \label{rj} \\
A_{j,N} \defeq \frac{(V^R- V^L)(x_{j,N})}{2(E - V)^A(x_{j,N})} \in [0, 1), \qquad B_{j,N} \defeq \frac{(\alpha^R- \alpha^L)(x_{j,N}) }{2\alpha^A(x_{j,N})} \in [0, 1). \label{Aj and Bj} \end{gather}
When $N = 1$, we omit the summation from \eqref{q1}. Moreover, if $\{x_j\}_j$ is a finite set, we work only with a single function $q_{1,N_1}$, where $x_1 < \cdots < x_{N_1}$ is the ordering of $J_V \cup J_\alpha$.
Since $V$ and $\alpha$ have bounded variation, \begin{equation} \label{apply BV} \sum_j \max\{ (V^R - V^L)(x_j), (\alpha^R - \alpha^L)(x_j)\} < \infty. \end{equation} Thus $\max q_{1,N} = r_{N,N}$ is bounded uniformly in $N$, by \begin{equation} \label{rNN}
\begin{split}
r_{N,N} &= \sum_{j =1}^N r_{j,N} - r_{j-1,N} \\
&= \sum_{j =1}^N \log \max \Big\{1 + \frac{2A_{j,N}}{1 - A_{j,N}}, 1 + \frac{2B_{j,N}}{1 - B_{j,N}} \Big\} \\
&\le \sum_{j=1}^N \max \Big\{\frac{2A_{j,N}}{1 - A_{j,N}},\frac{2B_{j,N}}{1 - B_{j,N}} \Big\} \\
&\le \sum_{j=1}^N \max \Big\{ \frac{(V^R - V^L)(x_{j,N})}{(E - V)^A(x_{j,N}) - \tfrac{1}{2}(V^R-V^L)(x_{j,N})}, \frac{(\alpha^R - \alpha^L)(x_{j,N})}{\alpha^A(x_{j,N}) - \tfrac{1}{2}(\alpha^R - \alpha^L)(x_{j,N})} \Big \} < \infty.
\end{split} \end{equation}
Next, we put \begin{equation} \label{q2}
q_{2}(x) \defeq \int_{-\infty}^x \big[ k dV^c_{+} + \tfrac{2}{\inf \alpha} d\alpha^c_{+} + (|x'|+1)^{-1-\delta}dx' \big], \end{equation} where $k > 0$ is chosen large enough so that \begin{equation} \label{cond k}
k \left(E_{\min} - \sup_{\mathbb{R}} V\right) \ge 1. \end{equation} To implement the energy method outlined in Section \ref{introduction}, we will in fact use a family of weight functions depending on $N$, \begin{equation} \label{wN} w(x) = w_N(x) = e^{q_{1,N}(x) + q_2(x)}, \qquad N \in \mathbb{N}. \end{equation} According to \eqref{chain rule continuous} and \eqref{chain rule jumps}, \begin{equation} \label{dw}
dw(x) = \sum_{j=1}^N e^{q_2} (e^{r_{j,N}} - e^{r_{j-1,N}}) \delta_{x_{j,N}} + w^A(\tfrac{2}{\inf \alpha} d\alpha^c_+ + k dV^c_+ + (|x| + 1)^{-1 -\delta}). \end{equation}
We now establish lower bounds on the measures $d(w(E-V))$ and $dw - (\alpha^A)^{-1}w^A d\alpha$, which we need in the estimate \eqref{deriv wF with alpha} below. For $d(w(E-V))$, we have, by \eqref{e:prod}, \eqref{cond k} and \eqref{dw}, \begin{equation} \label{lower bd first meas} \begin{split}
d(w&(E-V)) \\
&\ge (E-V)^Adw - w^A (dV^d + dV^c_+) \\
&\ge \sum_{j =1}^N e^{q_2} \Big((E-V)^A(e^{r_{j,N}} - e^{r_{j-1,N}}) -(V^R - V^L)(\tfrac{1}{2} e^{r_{j,N}} + \tfrac{1}{2} e^{r_{j-1,N}} ) \Big)\delta_{x_{j,N}} \\
&- \sum_{x \in J_{V} \setminus \{x_{j,N}\}_{j=1}^N} w^A (V^R - V^L) \delta_{x}\\
&+w^A (k(E_{\min}-V)^A -1)dV^c_+ + w^A(E-V)^A(|x| + 1)^{-1 -\delta}. \end{split} \end{equation} with the inequalities holding in the sense of measures. As for $dw - (\alpha^A)^{-1}w^A d\alpha$, \begin{equation} \label{lower bd second meas} \begin{split}
dw& - (\alpha^A)^{-1}w^A d\alpha \\
&\ge dw - (\alpha^A)^{-1}w^A (d\alpha^d + d\alpha^c_+) \\
&\ge \sum_{j =1}^N e^{q_2} \Big((e^{r_{j,N}} - e^{r_{j-1,N}}) -\frac{(\alpha^R - \alpha^L)}{\alpha^A}(\tfrac{1}{2} e^{r_{j,N}} + \tfrac{1}{2} e^{r_{j-1.N}} ) \Big)\delta_{x_{j,N}} \\
&- \sum_{x \in J_{\alpha} \setminus \{x_{j,N}\}_{j=1}^N} (\alpha^A)^{-1} w^A (\alpha^R - \alpha^L) \delta_{x}\\
&+ w^A (\tfrac{2}{ \inf \alpha} - \tfrac{1}{\alpha^A} )d\alpha^c_+ + w^A(|x| + 1)^{-1 -\delta}. \end{split} \end{equation} The first term in line five of \eqref{lower bd first meas} is nonnegative by \eqref{cond k}; the first term of line four of \eqref{lower bd second meas} is nonnegative since $\inf \alpha < 2\alpha^A$. Furthermore, the third line of \eqref{lower bd first meas} and the third line of \eqref{lower bd second meas}, are nonnegative by \eqref{rj} and \eqref{Aj and Bj}.
Thus we conclude \begin{equation} \label{final lwr bds} \begin{gathered}
d(w(E-V)) \ge w^A(E_{\min}-V)^A(|x| + 1)^{-1 -\delta} - \sum_{x \in J_{V} \setminus \{x_{j,N}\}_{j=1}^N} w^A (V^R - V^L) \delta_{x}, \\
dw - (\alpha^A)^{-1}w^A d\alpha \ge w^A(|x| + 1)^{-1 -\delta} - \sum_{x \in J_{\alpha} \setminus \{x_{j,N}\}_{j=1}^N} (\alpha^A)^{-1} w^A (\alpha^R - \alpha^L) \delta_{x},
\end{gathered} \end{equation} which are the lower bounds we shall employ in \eqref{deriv wF with alpha}.
Next, define the pointwise energy \begin{equation} \label{defn F with alpha}
F(x) = F[u](x) \defeq \alpha(x)|hu'(x)|^2 + (E- V(x))|u(x)|^2, \qquad x \in \mathbb{R}, \end{equation} with \begin{equation} \label{defn u}
u = (P(h)- i\varepsilon)^{-1}(|x| + 1)^{-\frac{1+\delta}{2}} f \in \mathcal{D}, \qquad \varepsilon >0,\, f \in L^2(\mathbb{R}). \end{equation} By \eqref{D}, $u, \, u' \in L^2(\mathbb{R}) \cap L^\infty(\mathbb{R}),$ and $(\alpha u')' \in L^2(\mathbb{R})$. Moreover, in the calculations to follow, we work with fixed representatives of $u$ and $u'$, such that both $u$ and $\alpha u'$ are locally absolutely continuous. This is justified by \eqref{Dmax is D}.
From \eqref{e:prod}, we see that $dF$ is given by \begin{equation*}
dF = h^2(\alpha u') d(\overline{u}') + h^2(\overline{u}')^A (\alpha u')' -|u|^2dV + 2(E - V)^A \real\left( u \overline{u}' \right). \end{equation*} Using \begin{equation*}
(\alpha u')' = (u')^A d\alpha + \alpha^A d(u') \implies d(u') = \frac{(\alpha u')'}{\alpha^A} -\frac{(u')^A}{\alpha^A} d\alpha, \end{equation*} we arrive at \begin{equation} \label{F prime with alpha}
dF = \tfrac{h^2}{\alpha^A} (\alpha u') (\alpha \overline{u}')' + h^2(\overline{u}')^A (\alpha u')' - \tfrac{h^2}{\alpha^A}(\alpha u')(\overline{u}')^A d\alpha -|u|^2dV + 2(E - V)^A \real\left( u \overline{u}' \right). \end{equation}
We now multiply \eqref{defn F with alpha} by $w$ and compute $d(wF)$: \begin{equation} \label{deriv wF with alpha} \begin{split}
d(wF) &= F^Adw + w^AdF \\
&= h^2 (\alpha u')(\overline{u}')^A dw + (E - V)^A|u|^2dw \\
&+ \tfrac{h^2}{\alpha^A} w^A (\alpha u') (\alpha \overline{u}')' + h^2 w^A (\overline{u}')^A (\alpha u')' - \tfrac{h^2}{\alpha^A}w^A(\alpha u')(\overline{u}')^A d\alpha \\
&-w^A|u|^2dV + 2w^A(E - V)^A \real\left( u \overline{u}' \right). \\
&= -w^A \left(-\tfrac{h^2}{\alpha^A} (\alpha u') (\alpha \overline{u}')' - h^2 (\overline{u}')^A (\alpha u')' + 2(V - E)^A \real(u \overline{u}')- 2\real (i\varepsilon u \overline{u}') \right) \\
&+ 2\varepsilon w^A \imag \left(u \overline{u}'\right) + |u|^2d(w(E-V)) + h^2 (\alpha u')(\overline{u}')^A \Big(dw - w^A\tfrac{d\alpha}{\alpha^A}\Big) \\
&\ge -w^A \left(-\tfrac{h^2}{\alpha^A} (\alpha u') (\alpha \overline{u}')' - h^2 (\overline{u}')^A (\alpha u')' + 2(V - E)^A \real(u \overline{u}')- 2\real (i\varepsilon u \overline{u}') \right) \\
&+2\varepsilon w^A \imag \left(u \overline{u}'\right)+(|x| + 1)^{-1-\delta}( (E_{\min} - \sup_\mathbb{R} V)|u|^2 + h^2 (\alpha u')(\overline{u}')^A) \\
&- \sum_{x \in J_{V} \setminus \{x_{j,N}\}_{j=1}^N} w^A |u|^2 (V^R - V^L) \delta_{x} - \sum_{x \in J_{\alpha} \setminus \{x_{j,N}\}_{j=1}^N} (\alpha^A)^{-1} h^2 w^A (\alpha u')(\overline{u}')^A (\alpha^R - \alpha^L) \delta_{x}.
\end{split} \end{equation} To get lines seven and eight we plugged in \eqref{final lwr bds} and used $w^A \ge 1$.
We now integrate both sides of \eqref{deriv wF with alpha} over all of $\mathbb{R}$. Since $F \in L^1(\mathbb{R})$ and is continuous off of a countable set, $F(x)$ tends to zero along a sequence of $x$-values tending to $+\infty$, and at which $F(x) = F^R(x) = F^L(x)$. Similarly, $F(x) = F^R(x) = F^L(x) \to 0$ along a sequence of $x$-values tending to $-\infty$. Thus \eqref{ftc} gives $\int_\mathbb{R} d(wF) = 0$. Since the average values of functions that appear are equal to the functions themselves Lebesgue almost-everywhere, for each $N$, we arrive at,
\begin{equation} \begin{split} \label{pre penult est before limit}
(1/&\max w)\int (|x| + 1)^{-1-\delta}\big((E_{\min} - \sup_\mathbb{R} V)|u|^2 + \inf \alpha |hu'|^2\big) \\ &\le \int 2|(P(h) - i\varepsilon)u) \overline{u}'| + 2 \varepsilon |u u'|\\
&+ \sum_{x \in J_{V} \setminus \{x_{j,N}\}_{j=1}^N} |u|^2 (V^R - V^L) \delta_{x} + \sum_{x \in J_{\alpha} \setminus \{x_{j,N}\}_{j=1}^N} (\alpha^A)^{-1} h^2 (\alpha u')(\overline{u}')^A (\alpha^R - \alpha^L) \delta_{x}.
\end{split} \end{equation} Sending $N \to \infty$, recalling \eqref{apply BV} (which gives $\sup_N (\max w) < \infty$ via \eqref{rNN}), \eqref{defn u}, and\\ $u, u' \in L^\infty(\mathbb{R})$, and using Young's inequality, we find \begin{equation} \label{pre penult est} \begin{split}
\int (|x| &+ 1)^{-1-\delta}\big(|u|^2 + |hu'|^2\big)\\
& \le C \int \frac{1}{\gamma h^2} |f|^2 + \gamma (|x| + 1)^{-1-\delta}|hu'|^2 + 2 \varepsilon |u u'| \qquad h, \, \gamma > 0. \end{split} \end{equation} Here and below, $C>0$ is a constant that may change from line to line, but it is always independent of $u$, $\varepsilon$, and $h$.
The second term on the right side of \eqref{pre penult est} can be absorbed into the left side by selecting $\gamma$ small enough. As for the term involving $\varepsilon$, by Young's inequality, \begin{equation*}
\int |u \overline{u}'| \le \frac{1}{2 h \inf \alpha} \int |u|^2 + \frac{1}{2h} \int \alpha |h u'|^2, \qquad h > 0. \end{equation*} Then \begin{equation*}
\begin{split}
\int \alpha |hu'|^2 &= \real \int -h^2(\alpha u')' \overline{u} \\
&= \real \int \left((P(h) - i\varepsilon) - V + E\right)u \overline{u} \\
&\le\frac{1}{2} \int |(|x| + 1)^{-\frac{1+ \delta}{2}}f|^2 + \left(\frac{1}{2} + \|E_{\max} - V\|_{L^\infty}\right)\int |u|^2.
\end{split} \end{equation*} Substituting these observations and calculations into \eqref{pre penult est} gives, for $\varepsilon, h > 0$, \begin{equation} \label{penult est dim one}
\int (|x| + 1)^{-1-\delta}( |u|^2 + |hu'|^2) \le \frac{C}{h^2} \int |f|^2 + \frac{C \varepsilon}{h} \int |u|^2. \end{equation}
To finish, we rewrite $\varepsilon \int |u|^2$ and estimate, for any $\gamma > 0$, \begin{equation} \label{gamma eqn dim one}
\begin{split}
\varepsilon \int |u|^2 &= - \imag \int (P(h) - i\varepsilon)u\overline{u} \\
&\le \frac{1}{\gamma } \int |f|^2 + \gamma \int (|x| + 1)^{-1-\delta}|u|^2.
\end{split} \end{equation}
If we now take $\gamma$ sufficiently small (depending on $C$ and $h$), we may absorb the integral of \\ $(|x| + 1)^{-1-\delta}|u|^2$ in \eqref{gamma eqn dim one} into the left side of \eqref{penult est dim one} to achieve \begin{equation} \label{final estimate one dim}
\int (|x| + 1)^{-1-\delta}(|u|^2 + |hu'|^2 ) \le \frac{C}{h^2} \int |f|^2, \qquad \varepsilon >0, \, h \in (0,1]. \end{equation} This completes the proof of \eqref{nontrap est oned}. \\ \end{proof}
\section{High frequency bound on the cutoff resolvent} \label{high energy bound section}
In this Section, we prove Theorem \ref{unif resolv est thm} as an application of Theorem \ref{nontrap thm oned}. We return to working with the operator $H : \mathcal{D}(H) \to \mathcal{H}$ as defined by \eqref{H}, where $\alpha, \beta : \mathbb{R} \to (0,\infty)$ are BV functions obeying \eqref{infs positive} and \eqref{perturbations of identity}.
In that situation, $H$ is a \textit{black box Hamiltonian} in the sense of Sj\"ostrand and Zworski \cite{sz91}, as defined in \cite[Definition 4.1]{dz}. More precisely, in our setting this means the following. First, if $u \in \mathcal{D}(H)$, then $u|_{\mathbb R\setminus [-R_0,R_0]} \in H^2(\mathbb R\setminus [-R_0,R_0])$. Second, for any $u \in \mathcal{D}(H)$, we have $(Hu)|_{\mathbb R\setminus [-R_0,R_0]} = - u''|_{\mathbb R\setminus [-R_0,R_0]}$. Third, any $u \in H^2(\mathbb R)$ which vanishes on a neighborhood of $[-R_0,R_0]$ is also in $\mathcal{D}(H)$. Fourth, $\textbf{1}_{[-R_0,R_0]} (H+i)^{-1}$ is compact on $\mathcal H$; this last condition follows from the fact that $\mathcal{D}(H) \subseteq H^1(\mathbb{R})$.
Then, by \cite[Theorem 4.4]{dz}, for any $\chi \in C_0^\infty(\mathbb R; [0,1])$ that is identically one near $[-R_0, R_0]$, the cutoff resolvent \eqref{continued resolv} continues meromorphically $\mathcal{H} \to \mathcal{D}(H)$ from $\imag \lambda > 0$ to the complex plane. The poles of this continuation are precisely at those values $\lambda$ for which there is a solution $u$ to $Hu=\lambda^2 u$ having $u, u', Hu \in L^2_{\text{loc}}(\mathbb{R})$ in the sense of distributions, and which is outgoing, i.e. obeys \begin{equation}\label{e:uoutgoing} \pm x \ge R_0 \qquad \Longrightarrow \qquad u(x) = c_{\pm} e^{\pm i \lambda x}, \end{equation} for some nonzero constants $c_{\pm}$.
Observe that $\lambda = 0$ is such a pole because we may take $u(x) = 1$ for all $x$. Observe further that this is the only pole in the closed half plane $\imag \lambda \ge 0$. Indeed, if $u$ satisfying \eqref{e:uoutgoing} solves $Hu = \lambda^2u$ with $\imag \lambda >0$, then $u \in \mathcal D(H)$ and we have $\lambda^2 \|u\|^2_{\mathcal H} = \langle H u , u \rangle_{\mathcal H} = \int_{\mathbb R} \alpha |u'|^2\ge 0$, which implies $\|u\|_{\mathcal H} = 0$ since $\lambda^2 \ge 0$ is impossible when $\imag \lambda > 0$. For $\lambda \in \mathbb R \setminus \{0\}$ this follows as in the proof of \cite[(2.2.12)]{dz}.
\begin{proof}[Proof of Theorem \ref{unif resolv est thm}] Set $V_{\beta} \defeq 1- \beta$ and $ \mathcal{O} \defeq \{ \lambda \in \mathbb{C} : \real \lambda \neq 0, \text{ } \imag \lambda > 0 \}$. Note that $\supp V_\beta \subseteq [-R_0, R_0]$. Define on $\mathcal{O}$ the following families of operators $\mathcal{H} \to \mathcal{H}$ with domain $\mathcal{D}(H)$, \begin{gather} \begin{split} A(\lambda) &\defeq (\real \lambda)^{-2} \beta (H - \lambda^2) \\ &=-(\real \lambda)^{-2} \partial_x \alpha \partial_x + V_\beta + (\imag \lambda)^2(\real \lambda)^{-2} \beta - i2 \imag \lambda(\real \lambda)^{-1}\beta - 1, \end{split} \label{A} \\ B(\lambda) \defeq -(\real \lambda)^{-2} \partial_x \alpha \partial_x + V_\beta + (\imag \lambda)^2(\real \lambda)^{-2} - 1 - i2 \imag \lambda(\real \lambda)^{-1}, \nonumber \end{gather} Furthermore, define on $\mathcal{O}$ the family $\mathcal{H} \to \mathcal{H}$, \begin{equation*} D(\lambda) \defeq (\imag \lambda)^2(\real \lambda)^{-2}V_\beta - i2 \imag \lambda(\real \lambda)^{-1}V_\beta. \end{equation*}
We have, \begin{equation*} B(\lambda) - A(\lambda) = D(\lambda). \end{equation*} Composing with inverses gives \begin{equation*} A(\lambda)^{-1} - B(\lambda)^{-1} = B(\lambda)^{-1} D(\lambda) A(\lambda)^{-1} \implies( I - B(\lambda)^{-1}D(\lambda))A(\lambda)^{-1} = B(\lambda)^{-1}, \end{equation*} Multiplying on the left and right by $\chi$ and noticing that $D(\lambda) = \chi D(\lambda) \chi$, we arrive at \begin{equation} \label{prelim resolv id A and B} (I - \chi B(\lambda)^{-1} \chi D(\lambda)) \chi A(\lambda)^{-1} \chi = \chi B(\lambda)^{-1} \chi, \qquad \lambda \in \mathcal{O}. \end{equation}
Next, choose $\lambda_0, \, \varepsilon_0 > 0$ so that $\sup_{\mathbb{R}}V_\beta < 1 - \varepsilon_0^2 \lambda_0^{-2}$. Identifying $E_{\min} \defeq 1 - \varepsilon_0^2 \lambda_0^{-2}$, $E_{\max} = 1$, and $h \defeq|\real \lambda|^{-1}$, we see that Theorem \ref{nontrap thm oned} applies to $B(\lambda)^{-1}$. So for some $C > 0$ and a possibly larger $\lambda_0$, we have \begin{equation}\label{est for B}
\| \chi B(\lambda)^{-1} \chi \|_{\mathcal{H} \to \mathcal{H}} \le C|\real \lambda|, \qquad |\real \lambda| \ge \lambda_0, \, 0 < \imag \lambda \le \varepsilon_0 . \end{equation} Moreover, \begin{equation} \label{est for D}
\|D(\lambda)\|_{\mathcal{H} \to \mathcal{H}} \le \varepsilon_0 \|V_\beta\|_{L^\infty} \big( \frac{ 1}{\lambda^2_0} + \frac{ 2}{\lambda_0} \big), \qquad |\real \lambda| \ge \lambda_0, \, 0 < \imag \lambda \le \varepsilon_0. \end{equation}
Thus, increasing $\lambda_0$ again if needed, we can invert $(I - \chi B(\lambda)^{-1} \chi D(\lambda))$ by a Neumann series when $|\real \lambda| \ge \lambda_0$, $0 < \imag \lambda < \varepsilon_0$. From \eqref{prelim resolv id A and B}, \eqref{est for B}, and \eqref{est for D}, we find \begin{equation} \label{Neumann A and B}
\chi A(\lambda)^{-1} \chi = \left(\sum_{k=0}^\infty (\chi B(\lambda)^{-1} \chi D(\lambda))^k \right) \chi B(\lambda)^{-1} \chi, \quad |\real \lambda | \ge \lambda_0, \, 0 < \imag \lambda \le \varepsilon_0. \end{equation}
Since \begin{equation*} \chi R(\lambda) \chi = (\real \lambda)^{-2} \chi A(\lambda)^{-1} \chi \beta, \qquad \lambda \in \mathcal{O}, \end{equation*}
\eqref{unif resolv est} follows from \eqref{est for B}, \eqref{est for D}, and \eqref{Neumann A and B}, at least when $|\real \lambda | \ge \lambda_0, \, 0 \le \imag \lambda \le \varepsilon_0$. To get \eqref{unif resolv est} for $|\real \lambda | \ge \lambda_0, \, |\imag \lambda| \le \varepsilon_0$, we appeal to a resolvent identity argument due to Vodev \cite[Theorem 1.5]{vo14}, which was adapted to the non-semiclassical (see, for instance, \cite[Lemma 5.1]{sh18}). It yields, for possibly smaller $\varepsilon_0$, holomorphicity of $\chi R(\lambda) \chi$ in $|\real \lambda | \ge \lambda_0, \, - \varepsilon_0 \le \imag \lambda \le 0$, along with a bound of the form \eqref{unif resolv est} there. \\ \end{proof}
To conclude this section, we consider the two by two matrix operator \begin{equation*}
G \defeq -i \begin{pmatrix} 0 & 1 \\ -H & 0 \end{pmatrix} : \mathcal{D}(H) \oplus \mathcal{H} \to \mathcal{H} \oplus \mathcal{H}, \end{equation*} which arises naturally from rewriting \eqref{wave eqn} as a first order system. A short computation yields, \begin{equation} \label{inv G plus lambda} (G + \lambda)^{-1} = \begin{pmatrix} -\lambda R(\lambda) & -iR(\lambda) \\ i \lambda^2 R(\lambda) + i & -\lambda R(\lambda) \end{pmatrix}, \qquad \imag \lambda > 0. \end{equation}
The following Corollary of Theorem \ref{unif resolv est thm} is essentially well-known, and is an important input to the proof of Theorem \ref{LED thm} in Section \ref{wave decay section}. We give the proof by recalling several results from \cite{bu03, vo14, dz}. \begin{corollary} \label{matrix op cor} Let $\chi \in C^\infty_0(\mathbb{R}; [0,1])$ be identically one near $[-R_0, R_0]$. The operator \begin{equation} \label{matrix op cutoffs}
\chi (G + \lambda)^{-1} \chi \defeq \begin{pmatrix} -\lambda \chi R(\lambda) \chi & -i\chi R(\lambda)\chi \\ i \lambda^2 \chi R(\lambda) \chi + i \chi^2 & -\lambda \chi R(\lambda) \chi \end{pmatrix} : H^1(\mathbb{R}) \oplus L^2(\mathbb{R}) \to H^1(\mathbb{R}) \oplus L^2(\mathbb{R}) \end{equation} continues meromorphically from $\imag \lambda > 0$ to $\mathbb{C}$. It has no poles on $\mathbb R \setminus \{0\}$ and at $\lambda = 0$ it has a simple pole: more precisely, if $w_0 \in H^1(\mathbb R)$ and $w_1 \in L^2(\mathbb R)$, then \begin{equation}\label{e:residuecomp} \lim_{\lambda \to 0} \lambda \chi (G+\lambda)^{-1} \chi \begin{pmatrix} w_0 \\ w_1 \end{pmatrix} = \begin{pmatrix} -i \lim_{\lambda \to 0} \lambda \chi R(\lambda) \chi w_1\\ 0 \end{pmatrix} = \begin{pmatrix} \frac 12\langle \chi, w_1 \rangle_{\mathcal H} \chi \\ 0 \end{pmatrix}. \end{equation}
Furthermore, there exist $C,\, \lambda_0, \, \varepsilon_0 > 0$ so that
\begin{equation} \label{matrix op unif bd}
\|\chi (G + \lambda)^{-1} \chi \|_{H^1(\mathbb{R}) \oplus L^2(\mathbb{R}) \to H^1(\mathbb{R}) \oplus L^2(\mathbb{R})} \le C,
\end{equation}
whenever $|\real \lambda| \ge \lambda_0$, and $|\imag \lambda| \le \varepsilon_0$. \end{corollary}
\begin{proof} As described above, by \cite[Theorem 4.4]{dz} and the proof of \cite[(2.2.12)]{dz}, the operator $\chi R(\lambda) \chi : L^2(\mathbb{R}) \to \mathcal{D}(H)$ continues meromorphically from $\imag \lambda >0$ to $\mathbb{C}$, and has no poles in $\mathbb{R} \setminus \{0\}$. This implies that each entry of \eqref{matrix op cutoffs} continues meromorphically as an operator between the appropriate spaces, again without poles in $\mathbb{R} \setminus \{0\}$.
Next, as in the proof of \cite[Theorem 2.7]{dz}, \eqref{perturbations of identity} implies that near $\lambda = 0$, \begin{equation} \label{resolv near zero} \chi R(\lambda) \chi w_1 = \frac i {2 \lambda} \langle \chi, w_1 \rangle_{\mathcal H} \chi + A(\lambda)w_1, \end{equation} where $A(\lambda) : \mathcal{H} \to \mathcal{D}(H)$ is holomorphic near zero, and hence we have \eqref{e:residuecomp}.
With \eqref{unif resolv est} already in hand, to establish \eqref{matrix op unif bd}, it suffices to supply $\lambda_0, \varepsilon_0 > 0$ so that \begin{gather} \lambda^2 \chi R(\lambda) \chi + \chi^2 = \chi H R(\lambda) \chi: H^1(\mathbb{R}) \to L^2(\mathbb{R}) , \label{two one entry} \\ \lambda \chi R(\lambda) \chi : H^1(\mathbb{R}) \to H^1(\mathbb{R}), \label{one one entry} \end{gather}
are uniformly bounded for $ |\real \lambda| \ge \lambda_0$ and $|\imag \lambda| \le \varepsilon_0$. When $ |\real \lambda| \ge \lambda_0$ and $0 < \imag \lambda \le \varepsilon_0$ this follows from the proof of \cite[Proposition 2.4]{bu03}, see in particular \cite[(2.14), (2.17), and (2.19)]{bu03}. To extend these bounds to strips below the real axis, we use once more Vodev's resolvent identity (\cite[Theorem 1.5]{vo14} and \cite[Lemma 5.1]{sh18}).
\end{proof}
\section{Wave decay} \label{wave decay section}
\begin{proof}[Proof of Theorem \ref{LED thm}]
This section follows part of Section 3 of \cite{vo99}.
Recall that we use $w(t)$ to denote the solution \eqref{soln spectral thm} to \eqref{wave eqn}, with initial data $w_0 \in \mathcal{D}(H)$ and $w_1 \in \mathcal{D}(H^{1/2})$. We have $\supp w_0, \, \supp w_1 \subseteq (-R,R)$, and the coefficients of \eqref{wave eqn} obey \eqref{infs positive} and \eqref{perturbations of identity}. We want to show that the local energy $\| w(\cdot, t) - w_\infty \|_{H^1(-R_1,R_1)} + \|\partial_t w(\cdot, t) \|_{L^2(-R_1,R_1)}$ decays exponentially, for a suitable constant $w_\infty$.
Choose $\chi \in C_0^\infty(\mathbb R; [0,1])$ such that $\chi =1$ near $[-R_1, R_1] \cup [-R, R] \cup [R_0, R_0]$ ($R_0$ given as in \eqref{perturbations of identity}). Recall from Corollary \ref{matrix op cor} that there exist $C$, $\lambda_0$, $\varepsilon_0 > 0$ such that \begin{equation*}
\|\chi(G + \lambda)^{-1}\chi f\| \le C\|f\|, \end{equation*}
whenever $|\real \lambda | \ge \lambda_0$ and $|\imag \lambda| \le \varepsilon_0$, where here and for the rest of this section all norms are $H^1(\mathbb{R}) \oplus L^2(\mathbb{R})$ unless otherwise specified.
We have \[\begin{split} w(t) &= \cos(t H^{1/2}) w_0 + \sin(tH^{1/2})H^{-1/2} w_1,\\ \partial_t w(t) &= -\sin(tH^{1/2}) H^{1/2} w_0 + \cos(tH^{1/2})w_1, \\ \partial_t^2 w(t) &= - H w(t). \end{split}\] Consequently, after defining \[
f \defeq \left( \begin{array}{c}w_0\\w_1 \end{array} \right), \qquad U(t) f \defeq \left( \begin{array}{c} w(t) \\ \partial_t w(t)\end{array}\right), \] we have \begin{equation}\label{e:utfbound}
\|U(t) f \| \le C\|f\|, \qquad \partial_t U(t) f =i G U(t) f, \qquad U(t)U(s) f = U(t+s)f, \end{equation} for all real $t$ and $s$, for some $C>0$ independent of $t$ and $f$. (Note that $U(t)f$ is still defined even if only $w_0 \in \mathcal{D}(H^{1/2})$, $w_1 \in \mathcal{H}$.)
Take $\varphi \in C^\infty(\mathbb R; [0,1])$ which is $0$ on $(-\infty,1]$ and $1$ on $[2,\infty)$ and put \[
W(t) f \defeq \varphi(t) U(t) f = \int_{\imag \lambda = \varepsilon} e^{-it\lambda} \check W(\lambda) \, d \lambda, \qquad \check W(\lambda) \defeq \frac 1 {2\pi} \int_{\mathbb{R}} e^{is\lambda}W(s)fds. \] Since $\partial_t W(t)f = \varphi'(t) U(t)f + i G W(t)f$ we get \[
W(t) f = \int_{\imag \lambda = \varepsilon} e^{-it\lambda} (G + \lambda)^{-1}(i\varphi' U f)\check{~}(\lambda)\,d \lambda. \]
Since $\supp w_0, \, \supp w_1 \subseteq (-R,R)$, by finite speed of propagation for the wave equation, and increasing $R > 0$ if necessary, we have that, $x \mapsto U(t)f$ is supported in $(-R,R)$ for all $t \in [0,2]$. By continuity of integration, the same is true of $x \mapsto (i\varphi' U f)\check{~}(\lambda)$ for every $\lambda$. Hence $ \lambda \mapsto (i\varphi' U f)\check{~}(\lambda)$ is entire and rapidly decaying as $|\real \lambda| \to \infty$ with $|\imag \lambda|$ remaining bounded and further $(i\varphi' U f)\check{~}(\lambda) = \chi(i\varphi' U f)\check{~}(\lambda)$ . Take $\varepsilon \in(0,\varepsilon_0)$ small enough that $\lambda = 0$ is the only pole of $\chi R(\lambda)\chi $ (and hence also of $\chi (G+\lambda)^{-1}\chi$ by \eqref{inv G plus lambda}) in the half plane $\imag \lambda \ge -\varepsilon$. By deformation of contour, \[ \chi W(t)f = \lim_{\lambda \to 0} \lambda \chi (G+\lambda)^{-1} \chi \int_{\mathbb{R}} \varphi'(s) U(s) f\,ds + \int_{\imag \lambda = -\varepsilon} e^{-it\lambda} \chi (G + \lambda)^{-1}\chi (i\varphi' U f)\check{~}(\lambda) d \lambda. \] To simplify this, use \eqref{e:residuecomp} and put \[
W_1(t)f := \int_{-\infty}^\infty e^{-it\lambda} (G + \lambda - i \varepsilon)^{-1} (i\varphi' U f)\check{~}(\lambda - i \varepsilon)\, d \lambda, \] to obtain \[ \chi W(t) f = \begin{pmatrix} \frac 12 \chi \int_{\mathbb{R}} \int_0^2 \beta(x) \chi(x) \varphi'(s) \partial_sw(s,x)ds dx \\ 0 \end{pmatrix} + e^{-\varepsilon t} \chi W_1(t)f. \] To simplify the first term, we integrate by parts in $s$, using $\varphi' = -(1-\varphi)'$, to obtain \[ \int_{\mathbb{R}} \int_0^2 \beta(x) \chi(x) \varphi'(s) \partial_sw(s,x)\,ds\,dx =\int_{\mathbb{R}}\beta \chi w_1 +\int_{\mathbb{R}} \int_0^2 \beta(x) \chi(x) (1-\varphi(s)) \partial_s^2w(s,x)\,ds\,dx. \] Now observe that $\partial_s^2 w = - H w$ and $\langle \chi, H w(s) \rangle_{\mathcal{H}} =0$ for $s \in [0,2]$ (the latter fact following from $\chi = 1$ near $[-R, R]$ and $\supp w(s) \subseteq (-R, R)$ for $s \in [0,2]$). Thus \[ \chi W(t) f = \frac 12 \begin{pmatrix} \langle \chi, w_1 \rangle_{\mathcal{H}} \chi \\ 0 \end{pmatrix}+ e^{-\varepsilon t} \chi W_1(t)f. \]
It now suffices to show that \[
\|\chi W_1(t)f\| \le C e^{\varepsilon t/2}\|f\|. \]
To prove this, we first use Plancherel's theorem, along with the fact that by \eqref{e:utfbound}, the operator norm $\|U(t)\|_{H^1(\mathbb{R}) \oplus L^2(\mathbb{R}) \to H^1(\mathbb{R}) \oplus L^2(\mathbb{R})}$ is uniformly bounded for all $t \in \mathbb R$, as well as the fact that by Corollary \ref{matrix op cor}, for any $\varepsilon>0$ small enough, the operator norm $\|\chi (G+\lambda-i\varepsilon)^{-1} \chi\|_{H^1(\mathbb{R}) \oplus L^2(\mathbb{R}) \to H^1(\mathbb{R}) \oplus L^2(\mathbb{R})}$ is uniformly bounded for all $\lambda \in \mathbb R$, to obtain \begin{equation}\label{e:planch}\begin{split}
\int \|\chi W_1(t)f\|^2 \, dt &= C \int \|\chi (G + \lambda - i \varepsilon)^{-1} (\varphi' U f)\check{~}(\lambda - i \varepsilon)\|^2 \, d \lambda \\
&\le C \int \|(\varphi' U f)\check{~}(\lambda - i \varepsilon)\|^2 \, d \lambda \\ &= C \int e^{2\varepsilon t} \|\varphi'(t) U(t) f\|^2 \, d t \le C \|f\|^2. \end{split}\end{equation} Next, compute \[
(\partial_t - i G) \chi W_1(t)f = - i [G,\chi] W_1(t)f + \varepsilon \chi W_1(t)f - i \chi \int e^{-it\lambda }(i\varphi' U f)\check{~}(\lambda - i \varepsilon)\, d \lambda \qefed \widetilde W_1(t)f. \] Integrating both sides of $\partial_s( U(t-s) \chi W_1(s)f) = U(t-s)\widetilde W_1(s)f$ from $s=0$ to $s=t$ gives \[
\chi W_1(t)f = U(t)\chi W_1(0)f + U(t)\int_0^t U(-s)\widetilde W_1(s)f\,ds. \] Thus \[
\|\chi W_1(t) f\| \le C \big(\|f\| + \int_0^t \|\widetilde W_1(s)f\| ds \big) \le C\big( \|f\| + t^{1/2} \Big(\int_0^t \|\widetilde W_1(s)f\|^2 ds\Big)^{1/2} \big). \]
Now check that, since $\|[G,\chi] W_1(t)f\| \le C\|W_1(t)f\|$, calculating as in \eqref{e:planch}, we obtain \\ $\int \|\widetilde W_1(s)f\|^2\,ds \le C \|f\|^2$, and hence \[
\|\chi W_1(t)f\| \le C (1 + t^{1/2}) \|f\| \] as desired.\\ \end{proof}
\section{Acknowledgments}
K.D. gratefully acknowledges support under NSF grant DMS 1708511. J.S. gratefully acknowledges support from four sources: ARC grant DP180100589, NSF grant DMS 1440140 while in residence at the Mathematical Sciences Research institute in Berkeley, CA, NSF grant DMS 2204322, and a University of Dayton Catholic Intellectual Tradition grant. Thanks also to the anonymous referee for helpful comments and corrections.
\appendix
\section{Characterization of $\mathcal{D}(H^{1/2})$} \label{H1 appendix}
In this Appendix we show
\begin{lemma}[{\cite{re22b}}] \label{square root lemma}
It holds that $\mathcal{D}(H^{1/2}) = H^1(\mathbb{R})$, and that\\ $u \mapsto \|u\|_{H^1}$, $u \mapsto (\|u\|^2_{\mathcal{H}} + \|H^{1/2} u\|^2_{\mathcal{H}})^{1/2}$ are equivalent norms. \end{lemma}
\begin{proof}
First, recall the well-known fact that $\mathcal{D}(H^{1/2})$ equals the form domain associated to $H$, namely, $\mathcal{D}(H^{1/2})$ is the completion of $\mathcal{D}(H)$ with respect to the norm $\| u\|^2_{+1} \defeq \langle Hu,u \rangle_{L^2} + \langle u,u \rangle_{L^2}$. On $\mathcal{D}(H)$, it's clear that there exist $C, c> 0$ so that $ c\| \cdot \|^2_{H^1} \le \| \cdot \|^2_{+1} \le C\| \cdot \|^2_{H^1}$.
If $u \in \mathcal{D}(H^{1/2})$, then there exists a $\|\cdot\|_{+1}$-Cauchy sequence $u_j \in \mathcal{D}(H) $ converging to $u$ in $\mathcal{H}$ (or, equivalently, converging to $u \in L^2(\mathbb{R})$). Because $\|\cdot\|_{+1}$ and $\| \cdot \|_{H^1}$ are equivalent on $\mathcal{D}(H)$, we get that the $u_j$ are also $\|\cdot\|_{H^1(\mathbb{R})}$-Cauchy. By completeness of $H^1(\mathbb{R})$, we conclude $u \in H^1(\mathbb{R})$. We also have \begin{equation*}
\|H^{1/2}u \|^2_{\mathcal{H}} = \lim_{j \to \infty}\|H^{1/2}u_j \|^2_{\mathcal{H}} = \lim_{j \to \infty} \langle Hu_j, u_j \rangle_{\mathcal{H}} \le C \lim_{j \to \infty} \|u_j \|^2_{H^1} = C \| u\|^2_{H^1}, \end{equation*} where the first equals sign follows since $H^{1/2}$ is a closed operator.
To show $H^1(\mathbb{R}) \subseteq \mathcal{D}(H^{1/2})$, first suppose $u \in H^1(\mathbb{R})$ has compact support. Approximate $\alpha u'$ in $L^2(\mathbb{R})$ by $\tilde{v}_j \in C^\infty_0(\mathbb{R})$ which have support in a fixed compact set. Choose $\varphi_0 \in C^\infty_0(\mathbb{R})$ with $\int \varphi_0/ \alpha = 1$, and put \begin{equation*}
v_j \defeq \tilde{v}_j - \big(\int \tilde{v}_j/ \alpha \big) \varphi_0. \end{equation*}
Then $\int v_j/\alpha = 0$ and the $v_j/\alpha \to u'$ in $L^2(\mathbb{R})$ since $\int u' = 0$. We clearly have \\$u_j \defeq \int_{-\infty}^x v_j/\alpha \in \mathcal{D}(H)$. Moreover, because $\int_{-\infty}^x v_j/\alpha \to \int_{-\infty}^x u' = u(x)$ locally uniformly in $x$, it follows that $u_j \to u$ in $H^1(\mathbb{R})$, and that the $u_j$ are $\| \cdot \|_{+1}$-Cauchy. Hence $u \in \mathcal{D}(H^{1/2})$.
For general $u \in H^1(\mathbb{R})$, choose a sequence $\tilde{u}_j$ of compactly supported functions with \\ $\|\tilde{u}_j - u\|_{H^1} \le 2^{-j-1}$. For each $j$, use the construction of the previous paragraph to find $u_j \in \mathcal{D}(H)$ with $\|u_j - \tilde{u}_j\|_{H^1} \le 2^{-j-1}$. Then the $u_j \to u$ in $H^1(\mathbb{R})$ and \begin{equation*}
\| u_{j} - u_{k} \|^2_{+1} \le C\|u_j - u_k\|_{H^1} \to 0 \qquad \text{as $j,\, k \to \infty$.} \end{equation*} Thus $u \in \mathcal{D}(H^{1/2})$ and \begin{equation*}
c\|u\|^2_{H^1} = c\lim_{j \to \infty}\|u_j\|^2_{H^1} \le \lim_{j \to \infty} \| u_j\|^2_{+1} = \lim_{j \to \infty} \big( \| H^{1/2} u_j\|^2_{\mathcal{H}} + \| u_j\|^2_{\mathcal{H}} \big) = \| H^{1/2} u\|^2_{\mathcal{H}} + \| u\|^2_{\mathcal{H}}. \end{equation*} \end{proof}
\section{Elementary properties of BV functions} \label{BV appendix}
This appendix collects some facts about functions of bounded variation which can be found in \cite{vh} and \cite{afp}. The main results are the integration by parts formula \eqref{Folland IBP}, the product rule \eqref{e:prod}, and the chain rules \eqref{chain rule continuous} and \eqref{chain rule jumps}. The books \cite{vh} and \cite{afp} are mostly concerned with higher dimensional problems, so we present proofs for the much simpler one dimensional case here.
We continue to use the notation \eqref{LRA} and \eqref{df} from Section \ref{bv review section}. For $\psi \in L^1(\mathbb R)$ compactly supported and satisfying $\int \psi = 1$, and for $\varepsilon >0 $, let \begin{equation}\label{e:reg}
f_\varepsilon(x) = \int f(x-\varepsilon y)\psi(y)dy = \varepsilon^{-1} \int f(y) \psi(\varepsilon^{-1}(x-y))dy. \end{equation} Then, accordingly as $\psi$ is supported in $[0,\infty)$ or supported in $(-\infty,0]$ or even, we have \begin{equation}\label{e:limconv}
\lim_{\varepsilon \to 0^+} f_\varepsilon = f^L \ \textrm{or } f^R \ \textrm{or } f^A, \qquad \text{pointwise on $\mathbb{R}$.} \end{equation} Indeed, use the dominated convergence theorem in the first two cases and average them to get the third case.
\begin{proof}[Proof of Proposition \ref{ibp bv prop}] The integration by parts formula \eqref{Folland IBP} follows as in the proof of \cite[Theorem 3.36]{fo}. Indeed, let $\Omega = \{(x,y) \in \mathbb{R}^2: a < x \le y \le b \}$. Since $\varphi$ is continuous and $\varphi'$ is piecewise continuous, it holds that $d\varphi = \varphi' dx$. Using Fubini's theorem, we evaluate the product measure $df \times d\varphi$ two different ways, \begin{equation*}
\begin{gathered}
\int_{(a,b] \times (a,b]} \mathbf{1}_\Omega (x,y) df(x) \times d\varphi(y)
= \int_{(a,b]} \int_{(a,y]} df(x) d\varphi(y) \\
= \int_{(a,b]} (f^R(y) - f^R(a)) \varphi'(y)dy = \int_{(a,b]} f(y) \varphi'(y) dy,
\end{gathered} \end{equation*} where we used that $f^R = f$ Lebesgue almost everyone, and that the boundary terms vanish since $\varphi(a) = \varphi(b) = 0.$ Similarly, \begin{equation*} \begin{gathered}
\int_{(a,b] \times (a,b]} \mathbf{1}_\Omega (x,y) d_f(x) \times d\varphi(y)
= \int_{(a,b]} \int_{[x,b]} d\varphi(y) df(x)
= -\int_{(a,b]} \varphi(x)df(x).
\end{gathered}
\end{equation*} \end{proof}
\iffalse \begin{proof}[Proof of Proposition \ref{ftc bv prop}] Let $\varphi_n$ be the following sequence of `trapezoid functions': \[
\varphi_n(x) = \begin{cases}
n(x-a), \qquad &a \le x \le a + \frac 1n, \\
1, \qquad &a + \frac 1n \le x \le b - \frac 1n, \\
n(b-x), \qquad &b - \frac 1n \le x \le b.
\end{cases} \] From the proof of \cite[Theorem 3.36]{fo}, we have \begin{equation} \label{Folland IBP}
\int_{(a,b]} \varphi df = -\int_{(a,b]} \varphi' fdx, \end{equation} for any $a < b$, and any continuous $\varphi$, with $\varphi'$ piecewise continuous and $\varphi(a) = \varphi(b) = 0$. Thus \begin{equation*} \begin{split}
\int_{(a,b)} \varphi_n df &= \int_{(a,b]} \varphi_n df \\
& = n \int_{b-\frac 1n}^bfdx - n \int_a^{a+\frac 1n}fdx\\
&= \int f(b - n^{-1} (\cdot))\textbf{1}_{(0,1)} - f(a - n^{-1} (\cdot))\textbf{1}_{(-1,0)}dx,
\end{split}
\end{equation*} where $\mathbf{1}_E$ denotes the characteristic function of $E$. Then let $n \to \infty$, and use the dominated convergence theorem on the left hand side and \eqref{e:limconv} on the right hand side.\\ \end{proof} \fi
\begin{proof}[Proof of Proposition \ref{prod rule bv prop}] Let $\psi \in C_0^\infty(\mathbb R)$ be an even function satisfying $\int \psi =1$. For any $\varepsilon>0$, define $f_\varepsilon$ by \eqref{e:reg}, and for any, $\eta>0$ define $g_\eta$ similarly. Then \begin{equation}\label{e:prodreg}
(f_\varepsilon g_\eta)' = f_\varepsilon (g_\eta)' + g_\eta (f_\varepsilon)'. \end{equation} We now show that taking $\eta \to 0^+$ and then $\varepsilon \to 0^+$ in \eqref{e:prodreg} gives \eqref{e:prod}. Let $\varphi \in C_0^\infty(\mathbb R)$. First, by integration by parts, \[
\lim_{\varepsilon \to 0^+} \lim_{\eta \to 0^+} \int \varphi (f_\varepsilon g_\eta)'dx = - \lim_{\varepsilon \to 0^+} \lim_{\eta \to 0^+} \int \varphi' f_\varepsilon g_\eta dx.
\]
Then, we observe that $\int \varphi' f_\varepsilon g_\eta dx \to \int \varphi' f_\varepsilon g dx$ by the dominated convergence theorem. Indeed, \\ $g_\eta \to g^A \stackrel{a.e.}{=} g $ by \eqref{e:limconv}, and $|\varphi' f_\varepsilon g_\eta|$ is uniformly bounded for $\varepsilon$ fixed and $\eta$ small. Similarly, \\ $\int \varphi' f_\varepsilon g dx \to \int \varphi' f g dx.$ Finally, $-\int \varphi' f g dx = \int \varphi d(f g)$ by \eqref{Folland IBP}.
Next \begin{equation*} \begin{split}
\lim_{\varepsilon \to 0^+} \lim_{\eta \to 0^+} \int \varphi f_\varepsilon g_\eta' dx &= -\lim_{\varepsilon \to 0^+} \lim_{\eta \to 0^+} \int (\varphi f_\varepsilon)' g_\eta dx \\
& = -\lim_{\varepsilon \to 0^+} \int (\varphi f_\varepsilon)' g dx \\
&= \lim_{\varepsilon \to 0^+} \int \varphi f_\varepsilon dg \\
&= \int \varphi f^A dg. \end{split} \end{equation*} For the first equal sign, we integrate by parts; for the second, we use the dominated convergence theorem, as in the previous paragraph. The third equal sign follows from \eqref{Folland IBP}, and the fourth from another application of the dominated convergence theorem (and \eqref{e:limconv}).
Continuing, by \eqref{e:reg}, \eqref{Folland IBP} and Fubini's theorem, \begin{equation} \label{exchange ep} \begin{split}
\int \varphi g^A(f_\varepsilon)'dx &= \int \varphi(x) g^A(x) \varepsilon^{-2} \left[\int \psi'(\varepsilon^{-1}(x-y))f(y) dy \right]dx \\ &= \int \varphi(x) g^A(x) \varepsilon^{-1} \left[\int \psi(\varepsilon^{-1}(x-y))df(y) \right]dx\\
&= \varepsilon^{-1}\int\left[\int \varphi(x) g^A(x) \psi(\varepsilon^{-1}(y-x))dx \right]df(y) = \int (\varphi g^A)_\varepsilon df, \end{split} \end{equation} where for the third equal sign we used that $\psi$ is even. Since $\varphi$ and $\psi$ have compact support, the integrals against $df$ make sense, and the application of Fubini's theorem is justified (even though $df$ may be finite only after it is restricted to a bounded Borel set). Finally, \[
\lim_{\varepsilon \to 0^+} \lim_{\eta \to 0^+} \int \varphi g_\eta (f_\varepsilon)'dx = \lim_{\varepsilon \to 0^+} \int \varphi g^A(f_\varepsilon)'dx =\lim_{\varepsilon \to 0^+} \int (\varphi g^A)_\varepsilon df= \int \varphi g^A df, \] by the dominated convergence theorem, \eqref{e:limconv}, and \eqref{exchange ep}.\\ \end{proof}
\begin{proof}[Proof of Proposition \ref{chain rule bv prop}] Using the decomposition \eqref{decompose f}, we see that $e^f = e^{f_{r,+}} e^{-f_{r,-}}$ has locally bounded variation, as it is a product of functions of locally bounded variation.
Let $\varphi, \psi \in C_0^\infty(\mathbb R)$, with $\psi$ even and satisfying $\int \psi =1$. To show \eqref{chain rule continuous}:
\begin{equation*} \begin{split}
\int \varphi d(e^f) &= - \int e^{f} \varphi' dx\\
&= -\int \lim_{N \to \infty} \sum_{n = 0}^N \frac{f^n}{n!} \varphi' dx \\
&= - \lim_{N \to \infty} \sum_{n = 0}^N \int \frac{(f^n)}{n!} \varphi' dx \\
&= \lim_{N \to \infty} \Big( \sum_{n=1}^N \int \frac{\varphi}{n!} df^n - \sum_{n=0}^N \int d (\varphi \frac{f^n}{n!}) \Big) \\
&= \lim_{N \to \infty} \sum_{n=1}^N \int \frac{\varphi}{(n-1)!} f^{n-1} df \\
&= \int \varphi e^{f} df.\\ \end{split} \end{equation*}
The first equal sign follows from \eqref{Folland IBP}. The third and sixth equal signs use the dominated convergence theorem; the fourth follows by \eqref{e:prod}, and the fifth by \eqref{ftc} and the Remark after \eqref{e:prod}.
For \eqref{chain rule jumps}, we first note that, because $g$ has locally bounded variation, so does $e^g$. We compute, \begin{equation*} \begin{split}
\int \varphi d(e^g) &= - \int e^{g} \varphi' dx\\
&= - \int_{-\infty}^{x_1} e^{r_0} \varphi' dx - \sum_{j=1}^{N-1} \int_{x_{j}}^{x_{j+1}} e^{r_j} \varphi' dx - \int_{x_N}^\infty e^{r_N} \varphi' dx \\
&=\sum_{j=1}^N (e^{r_j} - e^{r_{j-1}})\varphi(x_j). \end{split} \end{equation*} \\ \end{proof}
\end{document} | arXiv |
GPAW
Features and algorithms
Introductory tutorials
Specialized tutorials
Calculating band gap using the GLLB-sc functional
Electronic Band Structure Unfolding for Supercell Calculations
Spin-orbit coupling
Dipole-layer corrections in GPAW
DFT+U theory
Jellium
Bare Coulomb potential for hydrogen
Muon Site
Calculating the formation energies of charged defects
Tutorial: STM images - Al(111)
Bader Analysis
Getting the all-electron density
Obtaining all-electron wave functions and electrostatic potential
NEB calculations parallelized over images
PBE0 calculations for bulk silicon
XAS theory
Linear dielectric response of an extended system
The Quantum Electrostatic Heterostructure (QEH) model
Quasi-particle spectrum in the GW approximation: tutorial
The Bethe-Salpeter equation and Excitons
Calculating RPA correlation energies
Correlation energies from TDDFT
Correlation energies within the range-separated RPA
Continuum Solvent Model (CSM)
Orbital-free Density Functional Theory (OFDFT)
Theoretical introduction
Running the OFDFT GPAW module
Berry phase calculations
Interface to Wannier90
Solvated Jellium Method (SJM)
Atomic PAW Setups
Bugs!
Orbital-free Density Functional Theory (OFDFT)¶
This page introduces the orbital-free DFT method in a comprehensive way. If you are already familiar with the theory and want to learn how to use the orbital-free GPAW module you can skip the introduction and go directly to the section running the code.
Theoretical introduction¶
Orbital-based (Kohn-Sham) density functional theory¶
Density functional theory (DFT) has become possibly the most popular method for electronic structure calculations. This is due to its balance between accuracy and computational cost. However, the success of DFT mostly relies on the introduction of the Kohn-Sham single-particle ansatz. 1 DFT, as formulated by Hohenberg and Kohn in their seminal paper, 2 is an exact theory. In principle all the properties of a system of interacting electrons in an external potential (for example that determined by the charged atomic nuclei) can be derived from the knowledge of the electronic density \(n\) and the universal energy functional \(E[n]\) , where the electronic density can be obtained variationally as the density that minimizes \(E[n]\) . The general form of this functional is
\[E[n] = \langle \Psi | \hat{T} | \Psi \rangle + \langle \Psi | \hat{V} | \Psi \rangle,\]
where \(\hat{T}\) and \(\hat{V}\) , are the kinetic and potential energy operators, respectively, and \(|\Psi \rangle\) is the many-body wave function. The exact kinetic energy functional is then
\[T[n] = \langle \Psi | \hat{T} | \Psi \rangle.\]
In practice, the form of the universal density functional is unknown and we must rely on approximations. As we have already mentioned, introducing the Kohn-Sham single-particle ansatz is the most popular strategy to tackle this problem. Kohn and Sham proposed that the electronic density can be expressed as a sum of the density of a set of \(N\) non-interacting single-particle wave functions, also called orbitals:
\[n(\textbf{r}) = \sum_{i=1}^N | \psi_i (\textbf{r})|^2.\]
The Kohn-Sham energy functional (in atomic units) now becomes
\[E_\text{KS} [n] = -\frac{1}{2} \sum_{i=1}^{N} \langle \psi_i | \nabla_i^2 | \psi_i \rangle + \int \text{d}\textbf{r} \, V_\text{ext} (\textbf{r}) \, n(\textbf{r}) + \frac{1}{2} \int \int \text{d}\textbf{r} \, \text{d}\textbf{r}' \, \frac{n (\textbf{r}) \, n (\textbf{r}')}{|\textbf{r} - \textbf{r}'|} + E_\text{xc} [n(\textbf{r})].\]
The first term, denoted Kohn-Sham kinetic energy functional \(T_\text{s}[n]\) , now depends explicitly on the orbitals. All the other terms, including the exchange-correlation term \(E_\text{xc} [n]\) , depend only implicitly on the orbitals, because the density is calculated from them. Applying a variational principle to the expression for the total Kohn-Sham energy (e.g. that it is minimal with respect to changes in the wave functions), this formulation in turns leads to \(N\) Kohn-Sham Schrödinger-like equations (one per orbital):
\[\hat{H}_\text{KS} \, \psi_i (\textbf{r}) = \epsilon_i \, \psi_i (\textbf{r}),\]
that need to be solved in order to obtain the orbitals. The aim of orbital-free DFT is to avoid the need to solve the \(N\) equations by removing the explicit dependence of the kinetic energy term on the orbitals, effectively obtaining a kinetic energy functional \(T[n]\) that depends explicitly only on the density. The motivation for this objective is straightforward: by reducing the complexity of the problem from \(N\) particles to one "particle" the computational cost is greatly reduced. In particular, the scaling law for the time cost versus system size is reduced from cubic (Kohn-Sham DFT) to linear (orbital-free DFT). The question that follows is a no-brainer: if orbital-free DFT is so wonderful why is it not the standard implementation of DFT?
Orbital-free density functional theory¶
An orbital-free formulation of DFT is more in line with the original spirit of the Hohenberg-Kohn theorems, [#hohenberg-kohn] whereby the universal energy functional can be cast in terms of the electronic density alone. The success of the Kohn-Sham method relies on the fact that it provides an accurate description of the kinetic energy, which is the leading term in the total energy. All the many-body effects neglected by the Kohn-Sham independent-particle formulation are "pushed" into the exchange-correlation energy functional, which is then estimated by e.g. local-density, generalized-gradient or hybrid-functional (which typically include Hartree-Fock exchange) approximations. Therefore the accuracy that can be achieved within the realm of orbital-free DFT calculations heavily depends on the quality of approximated orbital-free kinetic energy functionals. As a historically important development and to illustrate how critical the quality of the kinetic energy functional is, consider the kinetic energy functional of the non-interacting homogeneous electron gas, also known as the Thomas-Fermi kinetic functional:
\[T_\text{TF} = \frac{3}{10} (3 \pi^2)^{2/3} \int \text{d}\textbf{r} \, [ n (\textbf{r})]^{5/3}.\]
When the Thomas-Fermi functional is used to represent the kinetic energy of electrons in matter, one obtains results that are quantitatively quite far from reality but, more importantly, are also qualitatively incorrect. For instance, DFT calculations based on the Thomas-Fermi functional fail to reproduce molecular bonding of simple diatomic molecules, such as H2, N2, O2, CO, etc. 3 On the other hand, calculations based on local-density approximations (LDAs) for the exchange-correlation functional (i.e. at the same level of approximation as the TF functional) used in combination with Kohn-Sham kinetic energies have been quite successful at describing qualitative and quantitative properties of matter, such as shell structure, molecular bonding, phase diagrams, elastic and structural properties, and so on. It becomes clear at this point that the prospects of orbital-free DFT becoming a successful electronic structure method rely of refining the approximation of the kinetic energy functional as an explicit functional of the density alone. We shall come back to this issue later on.
Orbital-free implementation in GPAW: reusing a Kohn-Sham calculator¶
Many years of development and popularization of DFT have left us with a variety of efficient codes to solve the Kohn-Sham equations and an active community hungry for new functionals. It would then be a great advantage if OFDFT calculations could be carried out reusing the computational tools already available. Levy et al. 4 showed that it is possible to reformulate the orbital-free problem in such a convenient way. The total orbital-free (i.e. explicitly density-dependent) energy functional can be expressed as
\[E_\text{OF} [n] = \underbrace{\int \text{d}\textbf{r} \, n^{1/2} (\textbf{r}) \left( - \frac{1}{2} \nabla^2 \right) \, n^{1/2} (\textbf{r})}_{T_\text{W} [n]} + J[n] + V[n] + E_\text{xc} [n] + T_\text{s} [n] - T_\text{W} [n],\]
where the first and last terms, known as the Weizsäcker functional, are just subtracting each other. \(J[n]\) and \(V[n]\) are the classical electrostatic energies due to electron-electron and electron-nuclei interactions, respectively, and \(E_\text{xc}[n]\) is the exchange-correlation energy functional, whose approximate form can correspond to any of the usual LDAs or GGAs developed for Kohn-Sham DFT available for GPAW. The kinetic energy functional \(T_\text{s} [n]\) is the non-interacting Kohn-Sham kinetic energy, and the last two terms combined are known as the Pauli functional,
\[T_\theta [n] = T_\text{s}[n] - T_\text{W} [n].\]
Levy et al. showed that a Kohn-Sham-like equation, derived variationally from the equation above, holds for the square root of the density:
\[\left( - \frac{1}{2} \nabla^2 + V_\text{eff}(\textbf{r}) \right) \, n^{1/2} (\textbf{r}) = \mu \, n^{1/2} (\textbf{r}),\]
where \(\mu\) is the negative of the ionization energy. By making the equivalence between a single orbital and the square root of the density,
\[\psi_0 (\textbf{r}) = n^{1/2} (\textbf{r})\]
, with the condition that \(\psi_0 (\textbf{r})\) renormalizes to the total number of electrons in the system, i.e.
\[\int \text{d} \textbf{r} \, |\psi_0 (\textbf{r})|^2 = N\]
, we can rewrite Levy's expression in terms of this orbital,
\[\left( - \frac{1}{2} \nabla^2 + V_\text{eff}(\textbf{r}) \right) \, \psi_0 (\textbf{r}) = \mu \, \psi_0 (\textbf{r}),\]
and use GPAW's Kohn-Sham solver truncated to a single orbital with its occupancy set to the total number of electrons. 5 This effectively orbital-free equation can be solved self-consistently using GPAW's iterative algorithms originally designed to solve the Kohn-Sham equations. The development of accurate orbital-free kinetic functionals will focus on obtaining a close approximation to \(T_\text{s} [n].\) Historically, proposed orbital-free kinetic functionals incorporate only a fraction of the von Weizsäcker term, parametrized by \(\lambda\) and the full Thomas-Fermi contribution, or the other way around, where the Thomas-Fermi part is considered to be the correction to the Weizsäcker term. This dichotomy is known as the \(\text{"} \lambda and \gamma\) controversy". 9 Both OF approximations to \(T_\text{s} [n]\) derive from the more general form
\[T_\text{s} [n] \approx \gamma T_\text{TF} [n] + \lambda T_\text{W} [n]\]
, and the corresponding Pauli functional is
\[T_\theta [n] \approx \gamma T_\text{TF} [n] + (\lambda - 1) T_\text{W} [n].\]
Since one could choose to construct a kinetic functional which does not explicitly include the Thomas-Fermi part, Thomas-Fermi is only one among possible OF kinetic functionals, we can express in a more general form \(T_\text{s} [n]\) as
\[T_\text{s} [n] \approx T_\text{r} [n] + \lambda T_\text{W} [n],\]
where r stands for "rest", referring to the approximation to the total kinetic functional minus the included fraction of Weizsäcker. In practice, the "rest" term will be included in the code as part of the definition of the exchange-correlation functional, and the Weizsäcker contribution will be included via an additional parameter (see the next subsection on "λ scaling"). This more general form leads to the Pauli functional expressed as
\[T_\theta [n] \approx T_\text{r} [n] + (\lambda - 1) T_\text{W} [n].\]
\(\lambda\) scaling
When using only a fraction of the Weizsäcker term the orbital-free equation needs to be rearranged in the following way due to convergence issues and practicalities of the implementation: 5
\[\left( - \frac{1}{2} \nabla^2 + \frac{1}{\lambda} V_\text{eff}' (\textbf{r}) \right) \, \psi_0 (\textbf{r}) = \frac{\mu}{\lambda} \, \psi_0 (\textbf{r}).\]
The new modified effective potential has the form:
\[V_\text{eff}' (\textbf{r}) = \frac{\delta}{\delta n} \left( T_\text{r} [n] + J[n] + V[n] + E_\text{xc} [n] \right).\]
Because of practical considerations, the term \(T_\text{r} [n]\) is included as part of a parametrized exchange-correlation energy functional when running GPAW's OFDFT module, as explained in detail in the section on running the code. Read through the next section to learn how the kinetic functional is defined in terms of how the present GPAW OFDFT implementation works. Construction and suitability of orbital-free kinetic energy functionals Although in principle any kinetic energy functional available from LibXC can be used to run OFDFT calculations in GPAW, we have only tested extensively a parametrized combination of Thomas-Fermi and von Weizsäcker, in combination with LDA exchange and correlation. On the list of ongoing research is the derivation of more accurate kinetic energy functionals. The recurrent (parametrized) form of the kinetic energy functional used in the examples below is
\[E_\text{OF} [n; \lambda , \gamma] = \lambda T_\text{W}[n] + \gamma T_\text{TF}[n] + J[n] + V[n] + E_\text{xc}^\text{PW}[n],\]
where the fractions of Thomas-Fermi and von Weizsäcker are given by \(\gamma\) and \(\lambda\) , respectively, and the exchange-correlation energy functional is the Perdew-Wang LDA (although we could have chosen any other LDA of GGA functional). An extensive and detailed study on the performance of this parametrized functional for atoms has been presented in the paper by Espinosa Leal et al. 6 An important thing to note is that because of how the implementation is done in GPAW, your kinetic energy functional must always contain a fraction of Weizsäcker, \(\lambda T_\text{W} [n]\) , where \(\lambda\) is set by the use of the keyword tw_coeff. The definition of the remainder of the kinetic functional, \(T_\text{r} [n] = T[n] - \lambda T_\text{W} [n]\) , is done through the definition of the XC functional choosing a kinetic functional from those available in LibXC and prepending a number for the corresponding fraction to be incorporated into \(T[n]\) . For instance, in the example above, \(T[n] - \lambda T_\text{W} [n] = \gamma T_\text{TF} [n]\) . When defining this kinetic functional in GPAW, say for \(\lambda = 0.2 , \, \gamma = 0.8\) , we would do:
lambda = 0.2
gamma = '0.8'
# Fraction of Weizsacker introduced through eigensolver definition
eigensolver = CG(tw_coeff=lambda)
# Fraction of Thomas-Fermi included in the definition of the XC functional
xcname = gamma + '_LDA_K_TF+1.0_LDA_X+1.0_LDA_C_PW'
A note on convergence¶
Convergence problems have been one of the historical obstacles to the development and spreading of OFDFT. Convergence instabilities of the self-consistency cycle have been attributed to the quality of the kinetic energy functional. 7 As a general rule, the more inaccurate the approximated orbital-free kinetic functional the more severe convergence problems will be. If you experience convergence problems, chances are that you are using an unreasonable approximation for the kinetic functional of your system.
Running the OFDFT GPAW module¶
Running GPAW's OFDFT module consists of two steps. The first thing to do is to generate the OFDFT PAW setups for each element and functional of interest. This needs to be done only once. The second step is to run the calculation itself. Both steps are described in detail below.
Setup generation¶
Before an orbital-free calculation can be carried out the PAW setups need to be generated. Currently, only a 1s projector can be used for setup generation, but the plan is to extend this capability in the future to be able to use a more flexible basis. Below we give a code example to generate the setup for a N atom with \(\lambda = 1, \, \gamma = 1\) and Perdew-Wang LDA exchange-correlation. The code includes the optimum cutoff distances for the augmentation spheres for all the atoms in the first three rows of the periodic table. N and the different functional options can be replaced by the desired values. Note that the definition of the functional is done separately for the Weizsäcker part (through the tw_coeff keyword) and the rest (Thomas-Fermi in the present case) which is done through the definition of the exchange-correlation functional. The orbital-free mode is enabled through the option orbital_free=True. Also note that the parametrized exchange-correlation functionality allows to use a linear combination of the different exchange-correlation functionals available from LibXC by changing the number prepended. For instance, xcname='1.0_LDA_X+0.5_LDA_C_PW+0.5_LDA_C_PZ' would combine half Perdew-Wang with half Perdew-Zunger LDA exchange-correlation functionals.
from gpaw.atom.generator import Generator
# List of elements for which setups will be generated
elements = ['N']
# Fraction of Weizsacker
lambda_coeff = 1.0
# Fraction of Thomas-Fermi
gamma_coeff = 1.0
# Select optimum cutoff and grid
for symbol in elements:
gpernode = 800
if symbol == 'H':
rcut = 0.9
elif symbol in ['He' or 'Li']:
elif symbol in ['Be', 'B', 'C', 'N', 'O', 'F', 'Ne']:
elif symbol in ['Na', 'Mg', 'Al', 'Si', 'P', 'S', 'Cl', 'Ar']:
# If the lambda scaling is used change name to differentiate the setup
name = 'lambda_{0}'.format(lambda_coeff)
# Use of Kinetic functional (minus the Tw contribution) inside the
# xc definition
pauliname = '{0}_LDA_K_TF+1.0_LDA_X+1.0_LDA_C_PW'.format(gamma_coeff)
# Calculate OFDFT density
g = Generator(symbol, xcname=pauliname, scalarrel=False,
orbital_free=True, tw_coeff=lambda_coeff,
gpernode=gpernode)
g.run(exx=False, name=name, use_restart_file=False,
rcut=rcut,
write_xml=True)
Running a simple OFDFT calculation¶
Once the needed setups have been generated, an OFDFT calculation can be run similarly to any standard Kohn-Sham GPAW calculation. Remember to make the path where you saved your OFDFT setups available to GPAW via the setup_paths function, as in the example below, where we run a PAW calculation for a N atom. Also remember the name of your XC functional needs to match the name of the corresponding setup you generated. GPAW will recognize the setup as an OFDFT setup and the orbital-free mode will be automatically enabled.
from ase import Atoms
from ase.parallel import paropen
from gpaw import GPAW
from gpaw.mixer import Mixer
from gpaw.eigensolvers import CG
from gpaw.poisson import PoissonSolver
from gpaw import setup_paths
setup_paths.insert(0, '.')
# Usual GPAW definitions
h = 0.18
a = 12.00
c = a/2
# XC functional + kinetic functional (minus the Tw contribution) to be used
xcname = '1.0_LDA_K_TF+1.0_LDA_X+1.0_LDA_C_PW'
# Fraction of Tw
filename = 'atoms_'+name+'.dat'
f = paropen(filename,'w')
mixer = Mixer()
eigensolver = CG(tw_coeff=lambda_coeff)
poissonsolver=PoissonSolver()
molecule = Atoms(symbol,
positions=[(c,c,c)] ,
cell=(a,a,a))
calc = GPAW(h=h,
xc=xcname,
maxiter=240,
eigensolver=eigensolver,
mixer=mixer,
setups=name,
poissonsolver=poissonsolver)
molecule.set_calculator(calc)
E = molecule.get_total_energy()
f.write('{0}\t{1}\n'.format(symbol,E))
Another example calculation¶
Here you will learn how to run a GPAW OFDFT calculation for the binding energy of an N2 molecule. Any other GPAW method, as explained in the different GPAW tutorials, can also be used with OFDFT by employing the definition of the GPAW calculator detailed here. In the present example, our kinetic energy functional will be \(T [n] = T_\text{W} + T_\text{TF}\) , corresponding to \(\lambda = 1, \, \gamma = 1\) , and our XC functional will be Perdew-Wang LDA. Below you will find the steps you need to follow in this tutorial.
Generate the setups Follow the instructions given in the section on running OFDFT for PAW setup generation for N. This should generate a file called "N.lambda_1.0.1.0_LDA_K_TF+1.0_LDA_X+1.0_LDA_C_PW" in your current directory, which contains the OFDFT PAW setup information for N generated with the desired energy functional.
Run the grid calculations We will now run grid calculations for atomic N and molecular N2 using the setup generated in the preceding step. For the atomic calculation use the code given in the second part of the running OFDFT section. The energy calculated by GPAW on the grid is given with respect to the total energy of the atomic calculation done during setup generation, and should be close to zero. For the N2 molecule, we first need to optimize the bond length. In order to do so, plot the system's energy as a function of interatomic distance and look for the minimum, for instance by adding the following loop to your script (since the experimental bond length is about 1.098 Å we will start searching in that region):
for d in [0.9, 1.0, 1.1, 1.2, 1.3, 1.4]:
molecule = Atoms('N2',
positions=([c - d/2, c,c], [c+d/2,c,c]),
We plot the output, which looks like this:
Since errors with OFDFT can be quite large, the initial range was very wide (between 0.9 Å and 1.4 Å), and a 4th-order polynomial is required to fit the data. The analysis reveals that for this particular energy functional the equilibrium bond length of N2 is close to 1.2 Å, which allows us to refine the search by adding further data points around that value:
Refining the range allows us to establish the interatomic distance in N2 at approximately 1.229 Å for this functional, about +12% error compared to the experimental value. The energy calculation for N2 is thus performed for this value. The results are summarized in the table below: System Energy (eV) Bond length (Å) N 0.00723 n/a N2 -13.25142 1.229 The binding energy, ` 2 E(text{N}) - E(text{N$2$}) is 13.266 eV. For reference, the experimental value is about 9.79 eV.
Accuracy¶
For PAW calculations, as well as for pseudopotentials, the formalism itself will introduce (hopefully small) errors compared to full potential calculations, often referred to as "all-electron calculations" in the context of the frozen-core formalism. These errors also affect usual Kohn-Sham calculations, not only OFDFT. In the case of GPAW, the main sources of error will be the cutoff of the augmentation spheres and the grid spacing, both in the radial grid for atomic setup generation and the regular grid for PAW calculations. For N2 there is a full potential OFDFT calculation that we can use for reference, by Chan et al. 8 Since this reference calculation did not include correlation, in order to compare our result to Chan's we need to perform the same calculation as above removing the 1.0_LDA_C_PW from the definition of the XC functional, which yields 12.602 eV for our PAW calculation. The reference full-potential binding energy from Chan is 12.599 eV (0.463 Hartree) giving a deviation of only 0.004 eV. As is the case for Kohn-Sham PAW calculations in GPAW, one would also need to check the effect of varying the cutoff radius of the setups and the grid spacing on the results. For the comparison with Chan's reference binding energy of N2, the table below summarizes the effect on the PAW calculation of changing the cutoff during setup generation while keeping the other parameters unchanged: Rcut (Bohr) N2 binding energy deviation (eV) 0.9 -0.091 1.0 -0.075 1.1 -0.031 1.2 -0.004 Note that as the cutoff is reduced the potential becomes harder (less smooth) and a finer grid would be required to keep the error small in the PAW calculation (this roughly corresponds to increasing the size of the basis in plane-waves calculations). The "default" values given in the scripts of the present tutorial correspond to our own optimization of these values, but depending on your own requirements for accuracy and your specific system under study you might have to consider optimizing these values yourself. A much larger source of error than the technical parameters discussed above, and even more so for OFDFT calculations, is the choice of functional. For a test on the performance of different possible kinetic functionals, you can vary the values of \(\lambda\) and \(\gamma\) and repeat the calculation, in order to check how the choice of OF functional affects the values of bond length and binding energy.
Citation information¶
If you use GPAW's OFDFT module for the compilation of published work, remember to add (in addition to the general GPAW and PAW references) a citation to the implementation paper 5 .
Kohn and L. J. Sham, Self-Consistent Equations Including Exchange and Correlation Effects, Phys. Rev. 140, A1133 (1965).
Hohenberg and W. Kohn, Inhomogeneous Electron Gas, Phys. Rev. 136, B864 (1964).
Teller, On the Stability of Molecules in the Thomas-Fermi Theory, Rev. Mod. Phys. 34, 627 (1962).
Levy, J. P. Perdew, and V. Sahni, Exact differential equation for the density and ionization energy of a many-particle system, Phys. Rev. A 30, 2745 (1984).
Lehtomäki, I. Makkonen, M. A. Caro, A. Harju, and O. Lopez-Acevedo, Orbital-free density functional theory implementation with the projector augmented-wave method, J. Chem. Phys. 141, 234102 (2014).
Espinosa Leal, A. Karpenko, M. A. Caro, and O. Lopez-Acevedo, Optimizing a parametrized Thomas-Fermi-Dirac-Weizsäcker density functional for atoms, Phys. Chem. Chem. Phys. , (2015) DOI: 10.1039/C5CP01211B.
Karasiev, S. B. Trickey, Issues and challenges in orbital-free density functional calculations, Comp. Phys. Comm. 183, 2519 (2012).
K.-L. Chan, A. J. Cohen and N. C. Handy, Thomas-Fermi-Dirac-von Weizsäcker models in finite systems, J. Chem. Phys. 114, 631 (2001).
Ludeña and V. V. Karasiev, Kinetic energy functionals: history, challenges and prospects, in Reviews of Modern Quantum Chemistry Vol. 1, pp612 (World Scientific, Singapore, 2002).
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\begin{document}
\title{Weight Changing Operators for Automorphic Forms on Grassmannians and Differential Properties of Certain Theta Lifts}
\author{Shaul Zemel\thanks{This work was carried out while I was working at the Technical University of Darmstadt, Germany. The initial part of this work was supported by the Minerva Fellowship (Max-Planck-Gesellschaft).}}
\maketitle
\section*{Introduction}
The classical Shimura--Maa\ss\ operators $\frac{\partial}{\partial_{\tau}}+\frac{k}{2iy}$ and $y^{2}\frac{\partial}{\partial_{\overline{\tau}}}$ are well-known for taking (elliptic, real-analytic) modular forms of weight $k$ to modular forms of weight $k+2$ and $k-2$ respectively. In addition, \cite{[Ma1]} and \cite{[Ma2]} consider differential operators which have a similar effect on Siegel modular forms, a work which was generalized in \cite{[Sh2]}. The following paper \cite{[Sh3]} concerns differential operators on functions on unitary groups which have related properties. All these operators have number-theoretic as well as representation-theoretic (or Lie-algebraic) interpretations, and are therefore the subject of many research papers (see, e.g., the reference \cite{[Sh1]}, which is strongly related to the case considered in this paper, as well as \cite{[Sh4]} for some generalizations of the results of the previously mentioned references or the investigation of invariant differential operators appearing in \cite{[Sh5]}, for example).
Our first goal is to define similar operators for modular (or automorphic) forms on another type of Shimura varieties, namely quotients of Grassmannians of vector spaces of signature $(2,b_{-})$. These are obtained by interpolating the square of the Shimura--Maa\ss\ operators from the case $b_{-}=1$, the multiple Shimura--Maa\ss\ operators obtained in the case $b_{-}=2$, and the operators for Siegel modular forms appearing in the case $b_{-}=3$. One may use Lie-theoretic considerations in order to establish the existence of such operators, but obtaining their explicit formula in this way is very tedious, because of the change of coordinates between the tube domain model and the transitive free action of an appropriate parabolic subgroup of $SO^{+}(V)$. We also remark that \cite{[Sh1]} also considers differential operators on automorphic forms on orthogonal groups. However, the operators defined in that reference take automorphic forms of some weight (i.e., a representation of the maximal compact subgroup) $\rho$ to automorphic forms having weight $\rho\otimes\eta$ for some $b_{-}$-dimensional representation $\eta$, hence in particular take scalar-valued automorphic forms on Grassmannians to vector-valued functions. Moreover, since that reference works with the coordinates arising from the bounded model while we consider the tube domain model (since the explicit formulae for the theta functions are more neatly presented in this model), an appropriate change of coordinates must be employed. It is true that after this change of coordinates, using the natural bilinear form on the tangent space of the Grassmannian in the tube domain model we may indeed obtain differential operators which remain in the scalar-valued realm. Indeed, after some additional normalization we obtain the operators defined in this paper using this method. However, the calculations involved are very delicate, laborious, long, an unenlightening, for which reason we have chosen to state and prove the formulae for the operators directly.
The second goal of this paper is to present two applications of these weight changing operators, in the theory of theta lifts. We recall the generalization, defined in \cite{[B]}, for the Doi--Naganuma lifting first introduced in \cite{[DN]} and \cite{[Ng]}. This map is given in \cite{[B]} in terms of a singular theta lift, and takes weakly holomorphic elliptic modular forms to meromorphic modular forms on Grassmannians. On the other hand, \cite{[Ze2]} defines a similar theta lift, using the same theta functions with polynomials. The first result of this paper states that in the case of an even dimension, a power of our weight raising operator sends the theta lift from \cite{[Ze2]} to the generalized Doi--Naganuma lifting of \cite{[B]}.
In addition, recall that the theta lift from Section 13 of \cite{[B]} (which is also studied extensively in \cite{[Bru]} and others) is a \emph{real} function. No automorphic forms of non-zero weight can be real. As a second application for our operators we define a notion of $m$-real automorphic forms of positive weight $m$, and show that in case one applies the theta lift from Section 14 of \cite{[B]} (or from \cite{[Ze2]}) to a modular form with real Fourier coefficients, then resulting theta lift is $m$-real.
The first half of the paper contains numerous statements whose proofs are delayed to later sections. We choose this way of presentation since most of the proofs consist of direct calculations, which may divert the reader's attention from the main ideas. Specifically, the paper is divided into 4 sections. In Section \ref{Operators} we define the weight raising and weight lowering operators and state their properties. Section \ref{Lifts} presents the images of certain functions under the weight raising operators, and proves the main theorem. Section \ref{Proofswcop} presents the proofs for the assertions of Section \ref{Operators}, while Section \ref{Proofsact} contains the missing proofs of Section \ref{Lifts}.
I would like to thank J. Bruinier for numerous suggestions and intriguing discussions regarding the results of this paper.
\section{Weight Changing Operators for Automorphic Forms on Orthogonal Groups \label{Operators}}
In this Section we present automorphic forms on complex manifolds arising as orthogonal Shimura varieties of signature $(2,b_{-})$, introduce the weight raising and weight lowering operators on such forms, and give some of their properties. The proofs of most assertions are postponed to Section \ref{Proofswcop}.
Let $V$ be a real vector space with a non-degenerate bilinear form of signature $(b_{+},b_{-})$. The pairing of $x$ and $y$ in $V$ is written $(x,y)$, and $x^{2}$ stands for the norm $(x,x)$ of $x$. For $S \subseteq V$, $S^{\perp}$ denotes the subspace of $V$ which is perpendicular to $S$. The
\emph{Grassmannian} $G(V)$ of $V$ is defined to be the set of all decompositions of $V$ into the orthogonal direct sum of a positive definite space $v_{+}$ and a negative definite space $v_{-}$. In the case $b_{+}=2$ (which is the only case we consider in this paper), it is shown in Section 13 of \cite{[B]}, Sections 3.2 and 3.3 of \cite{[Bru]}, or Subsection 2.2 of \cite{[Ze2]} (among others), that $G(V)$ carries a complex structure and has several equivalent models, which we now briefly present. Let \[P=\big\{Z_{V}=X_{V}+iY_{V} \in V_{\mathbb{C}}=V\otimes_{\mathbb{R}}\mathbb{C}\big|Z_{V}^{2}=0,\ (Z_{V},\overline{Z_{V}})>0\big\}.\] $Z_{V} \in V_{\mathbb{C}}$ lies in $P$ if and only if $X_{V}$ and $Y_{V}$ are orthogonal and have the same positive norm. $P$ has two connected components (which are interchanged by complex conjugation), and let $P^{+}$ be one component. The map \[P^{+} \to G(V),\quad Z_{V}\mapsto\mathbb{R}X_{V}\oplus\mathbb{R}Y_{V}\] is surjective, and $C^{*}$ acts freely and transitively on each fiber of this map by multiplication. This realizes $G(V)$ as the image of $P^{+}$ in the projective space $\mathbb{P}(L_{\mathbb{C}})$, which is an analytically open subset of the (algebraic) quadric $Z_{V}^{2}=0$, yielding a complex structure on $G(V)$. This is the \emph{projective model} of $G(V)$.
Let $z$ be a non-zero vector in $V$ which is \emph{isotropic}, i.e., $z^{2}=0$. The vector space $K_{\mathbb{R}}=z^{\perp}/\mathbb{R}z$ is non-degenerate and Lorentzian of signature $(1,b_{-}-1)$. Choosing some $\zeta \in V$ with $(z,\zeta)=1$ and restricting the projection $z^{\perp} \to K_{\mathbb{R}}$ to $\{z,\zeta\}^{\perp}$ gives an isomorphism. We thus write $V$ as $K_{\mathbb{R}}\times\mathbb{R}\times\mathbb{R}$, in which \[(\alpha,a,b)=a\zeta+bz+\big(\alpha\in\{z,\zeta\}^{\perp} \cong K_{\mathbb{R}}\big),\quad(\alpha,a,b)^{2}=\alpha^{2}+2ab+a^{2}\zeta^{2}.\] A (holomorphic) section $s:G(V) \to P^{+}$ is defined by the pairing with $z$ being 1. Subtracting $\zeta$ from any $s$-image and taking the $K_{\mathbb{C}}$-image of the result yields a biholomorphism between $G(V) \cong s\big(G(V)\big)$ and the \emph{tube domain} $K_{\mathbb{R}}+iC$, where $C$ is a cone of positive norm vectors in the Lorentzian space $K_{\mathbb{R}}$. $C$ is called the \emph{positive cone}, and it is determined by the choice of $z$ and the connected component $P^{+}$. The inverse biholomorphism takes $Z=X+iY \in K_{\mathbb{C}}$ to \[Z_{V,Z}=\bigg(Z,1,\frac{-Z^{2}-\zeta^{2}}{2}\bigg)=\bigg(X,1,\frac{Y^{2}-X^{2} -\zeta^{2}}{2}\bigg)+i\big(Y,0,-(X,Y)\big),\] with the real and imaginary parts denoted $X_{V,Z}$ and $Y_{V,Z}$ respectively. They are orthogonal and have norm $Y^{2}>0$. This identifies $G(V)$ with the \emph{tube domain model} $K_{\mathbb{R}}+iC$. Taking the other connected component of $P$ corresponds to taking the other cone $-C$ to be the positive cone, and to the conjugate complex structure.
The subgroup $O^{+}(V)$ consisting of elements of $O(V)$ preserving the orientation on the positive definite part acts on $P^{+}$ and $G(V)$, respecting the projection. Elements of $O(V) \setminus O^{+}(V)$ interchange the connected components of $P$. The action of $O^{+}(V)$ (and also of the connected component $SO^{+}(V)$) on $G(V)$ is transitive, with the stabilizer $K$ (or $SK \leq SO^{+}(V)$) of a point being isomorphic to $SO(2) \times O(n)$ (resp. $SO(2) \times SO(n)$). Therefore $G(V)$ is isomrphic to $O^{+}(V)/K$ and to $SO^{+}(V)/SK$. Given an isotropic $z$ as above, the action of $O^{+}(V)$ transfers to $K_{\mathbb{R}}+iC$, and for $M \in O^{+}(V)$ and $Z \in K_{\mathbb{R}}+iC$ we have \[MZ_{V,Z}=J(M,Z)Z_{V,MZ},\quad\mathrm{with}\quad J(M,Z)=(MZ_{V,Z},z)\in\mathbb{C}^{*}.\] $J$ is a \emph{factor of automorphy}, namely the equality \[J(MN,Z)=J(M,NZ)J(N,Z)\] holds for all $Z \in K_{\mathbb{R}}+iC$ and $M$ and $N$ in $O^{+}(V)$. For such $M$ we define the \emph{slash operator} of weight $m$, and more generally of weight $(m,n)$, by \[\Phi[M]_{m,n}(Z)=J(M,Z)^{-m}\overline{J(M,Z)}^{-n}\Phi(MZ),\qquad[M]_{m}=[M]_{ m,0}.\] The fact that $(Z_{V},\overline{Z_{V}})=2Y^{2}$ and the definition of $J(M,Z)$ yield the equalities \begin{equation}
\big(\Im(MZ)\big)^{2}=\frac{Y^{2}}{|J(M,Z)|^{2}}\quad\mathrm{and}\quad\big(F(Y^{ 2})^{t}\big)[M]_{m,n}=F[M]_{m+t,n+t}(Y^{2})^{t} \label{Y2mod} \end{equation} the latter holding for every $m$, $n$, $t$, and function $F$ on $K_{\mathbb{R}}+iC$ (see Lemma 3.20 of \cite{[Bru]} for the first equality in Equation \eqref{Y2mod}, and the second one follows immediately).
The invariant measure on $K_{\mathbb{R}}+iC$ is $\frac{dXdY}{(Y^{2})^{b_{-}}}$ (see Section 4.1 of \cite{[Bru]}, but one can also prove this directly, using the generators of $O^{+}(V)$ considered in Section \ref{Proofswcop} below). Note that this measure depends on the choice of a basis for $K_{\mathbb{R}}+iC$, but changing the basis only multiplies this measure by a positive global scalar. Let $\Gamma$ be a discrete subgroup $\Gamma$ of $O^{+}(V)$ of cofinite volume. In most of the interesting cases $\Gamma$ will be either the $O^{+}$ or the $SO^{+}$ part of the orthogonal group of an even lattice $L$ in $V$, or the discriminant kernel of such a group. Given $m\in\mathbb{Z}$, an \emph{automorphic form of weight $m$ with respect to $\Gamma$} is defined to be a (complex valued) function $\Phi$ on $K_{\mathbb{R}}+iC$ for which the equation \[\Phi(MZ)=J(M,Z)^{m}\Phi(Z),\qquad\mathrm{or\ equivalently}\qquad\Phi[M]_{m}(Z)=\Phi(Z),\] holds for all $M\in\Gamma$ and $Z \in K_{\mathbb{R}}+iC$. Using the standard argument, such a function is equivalent to a function on $P^{+}$ which is $-m$-homogenous (with respect to the action of $\mathbb{C}^{*}$) and $\Gamma$-invariant, as considered, for example, in \cite{[B]}.
We now consider some differential operators on functions on $K_{\mathbb{R}}+iC$. Given a basis for $K_{\mathbb{R}}$, we write $\partial_{x_{k}}$ for $\frac{\partial}{\partial x_{k}}$ (for $1 \leq k \leq b_{-}$). Similarly, $\partial_{y_{k}}$ stands for the coordinates of the imaginary part from $C$. The notation for the derivatives $\partial_{z_{k}}=\frac{1}{2}(\partial_{x_{k}}-i\partial_{y_{k}})$ and $\partial_{\overline{z_{k}}}=\frac{1}{2}(\partial_{x_{k}}+i\partial_{y_{k}})$ will be further shortened to $\partial_{k}$ and $\partial_{\overline{k}}$ respectively.
The operator $I=\sum_{k}x_{k}\partial_{x_{k}}$ multiplies a homogenous function on $K_{\mathbb{R}}$ by its homogeneity degree, and is thus independent of the choice of basis (indeed, it has an intrinsic Lie-theoretic description). The operators \[D^{*}=\sum_{k}y_{k}\partial_{k}\quad\mathrm{and}\quad\overline{D^{*}}=\sum_{k} y_{k}\partial_{\overline{k}}\] from \cite{[Na]} are intrinsic as well, and they are also invariant under translations in the real part of $K_{\mathbb{R}}+iC$. If the basis for $K_{\mathbb{R}}$ is \emph{orthonormal}, i.e., orthogonal with the first vector having norm 1 and the rest having norm $-1$, then the \emph{Laplacian of $K_{\mathbb{R}}$}, denoted $\Delta_{K_{\mathbb{R}}}$, is defined to be $\partial_{x_{1}}^{2}-\sum_{k=2}^{b_{-}}\partial_{x_{k}}^{2}$. It is independent of the choice of the orthonormal basis (though using a basis which is not orthonormal it takes different forms), and it is invariant under the action of $O(K_{\mathbb{R}})$ as well as under translations in $K_{\mathbb{R}}$. With complex coordinates it has three counterparts, \[\Delta_{K_{\mathbb{C}}}^{h}=\partial_{1}^{2}-\sum_{k=2}^{b_{-}}\partial_{k}^{2 },\quad\Delta_{K_{\mathbb{C}}}^{\overline{h}}=\partial_{\overline{1}}^{2}-\sum_{ k=2}^{b_{-}}\partial_{\overline{k}}^{2},\quad\mathrm{and}\quad\Delta_{K_{\mathbb {C}}}^{\mathbb{R}}=\partial_{1}\partial_{\overline{1}}-\sum_{k=2}^{b_{-}} \partial_{k}\partial_{\overline{k}},\] which we call the \emph{holomorphic Laplacian of $K_{\mathbb{C}}$} (of Hodge weight $(2,0)$), the \emph{anti-holomorphic Laplacian of $K_{\mathbb{C}}$} (of Hodge weight $(0,2)$), and the \emph{real Laplacian of $K_{\mathbb{C}}$} (of Hodge weight $(1,1)$), respectively. These operators have the same invariance and independence properties as $\Delta_{K_{\mathbb{R}}}$. Note that the appropriate combinations appearing in \cite{[Bru]} and \cite{[Na]} can be identified as our operators $\frac{1}{2}\Delta_{K_{\mathbb{C}}}^{h}$, $\frac{1}{2}\Delta_{K_{\mathbb{C}}}^{\overline{h}}$, and $\Delta_{K_{\mathbb{C}}}^{\mathbb{R}}$ respectively, expressed in a basis which is not orthonormal. We shall indeed discuss and generalize the operators $\Delta_{1}$ and $\Delta_{2}$ of \cite{[Na]} in Proposition \ref{RLcomp} below.
The weight changing operators and their defining property are given in \begin{thm} For any integer $m$ define $R_{m}^{(b_{-})}$ to be the operator \[(Y^{2})^{\frac{b_{-}}{2}-m-1}\Delta_{K_{\mathbb{C}}}^{h}(Y^{2})^{m+1-\frac{b_{ -}}{2}}=\Delta_{K_{\mathbb{C}}}^{h}-\frac{i(2m+2-b_{-})}{Y^{2}}D^{*}-\frac{ m(2m+2-b_{-})}{2Y^{2}}.\] In addition, define \[L^{(b_{-})}=(Y^{2})^{2}\overline{R_{0}}=(Y^{2})^{\frac{b_{-}}{2}+1}\Delta_{K_{ \mathbb{C}}}^{\overline{h}}(Y^{2})^{1-\frac{b_{-}}{2}}=(Y^{2})^{2}\Delta_{K_{ \mathbb{C}}}^{\overline{h}}+iY^{2}(2-b_{-})\overline{D^{*}}.\] Then the equalities \[(R_{m}^{(b_{-})}F)[M]_{m+2}=R_{m}^{(b_{-})}\big(F[M]_{m}),\qquad(L^{(b_{-})} F)[M]_{m-2}=L^{(b_{-})}\big(F[M]_{m})\] hold for every $\mathcal{C}^{2}$ function $F$ on $K_{\mathbb{R}}+iC$ and any $M \in O^{+}(V)$. \label{wcop} \end{thm} The different descriptions of $R_{m}^{(b_{-})}$ and $L^{(b_{-})}$ coincide by Lemma \ref{LappowMov} below. Theorem \ref{wcop} has the following standard \begin{cor} If $\Phi$ is an automorphic form of weight $m$ on $G(V) \cong K_{\mathbb{R}}+iC$ then $R_{m}^{(b_{-})}\Phi$ and $L^{(b_{-})}\Phi$ are automorphic forms on $K_{\mathbb{R}}+iC$ which have weights $m+2$ and $m-2$ respectively. \label{wt+-2} \end{cor} In correspondence with Theorem \ref{wcop} and Corollary \ref{wt+-2} we call $R_{m}^{(b_{-})}$ and $L^{(b_{-})}$ the \emph{weight raising operator of weight $m$} and the \emph{weight lowering operator} for automorphic forms on Grassmannians of signature $(2,b_{-})$ respectively. As already mentioned in the Introduction, these operators may also be given a Lie-theoretic description (see Section \ref{Proofswcop} for more details). However, the explicit operators appearing in Theorem \ref{wcop} are more useful for our applications.
We shall make use of the operator \[D^{*}\overline{D^{*}}-\frac{\overline{D^{*}}}{2i}=\overline{D^{*}}D^{*}+\frac{
D^{*}}{2i}=\sum_{k,l}y_{k}y_{l}\partial_{k}\partial_{\overline{l}},\] which we denote $|D^{*}|^{2}$. Lemma 2.2 of \cite{[Ze2]} shows that
\[\Delta_{m,n}^{(b_{-})}=8|D^{*}|^{2}-4Y^{2}\Delta_{K_{\mathbb{C}}}^{\mathbb{R}} -4im\overline{D^{*}}+4inD^{*}+2n(2m-b_{-})\] is the \emph{weight $(m,n)$ Laplacian} on $K_{\mathbb{R}}+iC$, and the \emph{weight $m$ Laplacian} $\Delta_{m}^{(b_{-})}$ is just $\Delta_{m,0}^{(b_{-})}$ (this extends the corresponding assertion of \cite{[Na]}, since his operator $\Delta_{1}$ is our $\Delta_{0}^{(b_{-})}$ divided by 8). The constants are normalized such that \begin{equation} \Delta^{(b_{-})}_{m,n}(Y^{2})^{t}=(Y^{2})^{t}\Delta^{(b_{-})}_{m+t,n+t} \label{Y2Delta} \end{equation} holds for every $m$, $n$, and $t$ (see the remark after Lemma \ref{LappowMov} below). The relations between $R_{m}^{(b_{-})}$, $L^{(b_{-})}$, and the corresponding Laplacians are given by \begin{prop} The equalities \[\Delta_{m+2}^{(b_{-})}R_{m}^{(b_{-})}-R_{m}^{(b_{-})}\Delta_{m}^{(b_{-})}=(2b_ {-}-4m-4)R_{m}^{(b_{-})}\] and \[\Delta_{m-2}^{(b_{-})}L^{(b_{-})}-L^{(b_{-})}\Delta_{m}^{(b_{-})}=(4m-2b_{-} -4)L^{(b_{-})}\] hold for every $m\in\mathbb{Z}$. \label{LapRmL} \end{prop} We recall that an automorphic form of weight $m$ on $K_{\mathbb{R}}+iC$ is said to have eigenvalue $\lambda$ if it is annihilated by $\Delta_{m}^{(b_{-})}+\lambda$ (i.e., eigenvalues are of $-\Delta_{m}^{(b_{-})}$). Hence Proposition \ref{LapRmL} has the following \begin{cor} If $F$ is an automorphic form of weight $m$ on $K_{\mathbb{R}}+iC$ which has eigenvalue $\lambda$ then the automorphic forms $R_{m}^{(b_{-})}F$ and $L^{(b_{-})}F$ have eigenvalues $\lambda+4m-2b_{-}+4$ and $\lambda-4m+2b_{-}+4$ respectively. \label{evmov} \end{cor}
By evaluating compositions of the weight changing operators one shows \begin{prop} The combination \[\Xi_{m}^{(b_{-})}=(Y^{2})^{2}\Delta_{K_{\mathbb{C}}}^{h}\Delta_{K_{\mathbb{C}} }^{\overline{h}}-iY^{2}(2m+2-b_{-})D^{*}\Delta_{K_{\mathbb{C}}}^{\overline{h}} +iY^{2}(2-b_{-})\overline{D^{*}}\Delta_{K_{\mathbb{C}}}^{h}+\] \[+\frac{(2-b_{-})(2m+2-b_{-})}{2}Y^{2}\Delta_{K_{\mathbb{C}}}^{\mathbb{R}} -\frac{m(2m+2-b_{-})}{2}Y^{2}\Delta_{K_{\mathbb{C}}}^{\overline{h}}\] commutes with all the weight $m$ slash operators as well as with the Laplacian $\Delta_{m}^{(b_{-})}$. The commutator of the global weight raising operator and the weight lowering operator is \[\big[R^{(b_{-})},L^{(b_{-})}\big]_{m}=R_{m-2}^{(b_{-})}L^{(b_{-})}-L^{(b_{-})} R_{m}^{(b_{-})}=\frac{m\Delta_{m}^{(b_{-})}}{2}-\frac{mb_{-}(2m-2-b_{-})}{4}.\] \label{RLcomp} \end{prop} Proposition \ref{RLcomp} provides another proof to Lemma 2.2 of \cite{[Ze2]} about $\Delta_{m}^{(b_{-})}$. It also implies that $\Xi_{m}^{(b_{-})}$ preserves the spaces of automorphic forms of weight $m$ for all $m\in\mathbb{Z}$ and for every discrete subgroup $\Gamma$ of cofinite volume in $O^{+}(V)$. It also commutes with $\Delta_{m}^{(b_{-})}$, hence preserves eigenvalues of such automorphic forms. By rank considerations, one can probably show that the ring of differential operators which commute with all the slash operators of weight $m$ is generated by $\Delta_{m}^{(b_{-})}$ and $\Xi_{m}^{(b_{-})}$, hence is a polynomial ring in two variables (if $b_{-}>1$). This assertion should also follow from part (3) of Theorem 3.3 of \cite{[Sh5]} (since the rank of the symmetric space $G(V)$ is 2 if $b_{-}>1$), though I have not verified this in detail. As $\Delta_{0}^{(b_{-})}$ is $8\Delta_{1}$ and $\Xi_{0}^{(b_{-})}$ is $16\Delta_{2}$ in the notation of \cite{[Na]}, Proposition \ref{RLcomp} generalizes the main result of that reference to other weights. A similar argument yields results of the same sort for $(m,n)$, where a possible normalization for $\Xi_{m,n}^{(b_{-})}$ is $(Y^{2})^{-n}\Xi_{m-n}^{(b_{-})}(Y^{2})^{n}$ for which an equality similar to Equation \eqref{Y2Delta} holds. We shall not need these results in what follows.
We now consider compositions of the weight raising operators. The natural $l$th power of $R_{m}^{(b_{-})}$ is the composition \[(R_{m}^{(b_{-})})^{l}=R_{m+2l-2}^{(b_{-})}\circ\ldots \circ R_{m}^{(b_{-})}.\] The general formulae for the resulting operator seems too complicated to write as a combination of $\Delta_{K_{\mathbb{C}}}^{h}$, $D^{*}$, and $\frac{1}{Y^{2}}$ with explicit coefficients. However, we can establish the properties given in the following \begin{prop} $(i)$ The operator $(R_{m}^{(b_{-})})^{l}$ takes automorphic forms of weight $m$ on $G(L_{\mathbb{R}})$ to automorphic forms of weight $m+2l$. $(ii)$ In case the former automorphic form is an eigenfunction with eigenvalue $\lambda$, the latter is also an eigenfunction, and the corresponding eigenvalue is $\lambda+l(4m+4l-2b_{-})$. $(iii)$ The operator $(R_{m}^{(b_{-})})^{l}$ can be written as \[(R_{m}^{(b_{-})})^{l}=\sum_{c=0}^{l}\sum_{a=0}^{c}A_{a,c}^{(l)}\frac{(iD^{*})^ {c-a}(\Delta_{K_{\mathbb{C}}}^{h})^{l-c}}{(-Y^{2})^{c}},\] where $A_{0,0}^{(0)}=1$ and given the coefficients $A_{a,c}^{(l)}$ for given $l$, the coefficient $A_{a,c}^{(l+1)}$ of the next power $l+1$ is defined recursively as \[\sum_{s=0}^{a}\binom{c-s}{a-s}A_{s,c}^{(l)}+(2m+4l-2c+4-b_{-})\bigg(A_{a,c-1}^ {(l)}+\frac{m+2l-c+1}{2}A_{a-1,c-1}^{(l)}\bigg).\] $(iv)$ For $a=0$ the coefficients $A_{0,c}^{(l)}$ are given by the explicit formula \[A_{0,c}^{(l)}=\frac{l!\cdot2^{c}}{(l-c)!}\binom{m+l-\frac{b_{-}}{2}}{c}.\] \label{Rmpowl} \end{prop} The binomial symbol appearing in part $(iv)$ of Proposition \ref{Rmpowl} is the \emph{extended binomial coefficient}: Indeed, for two non-negative integers $x$ and $n$ we have \[\binom{x}{n}=\frac{1}{n!}\prod_{j=0}^{n-1}(x-j),\] a formula which makes sense for $x\in\mathbb{R}$ (as well as $x$ in any $\mathbb{Q}$-algebra).
Part $(i)$ of Proposition \ref{Rmpowl} follows immediately from Corollary \ref{wt+-2}. For part $(ii)$ Corollary \ref{evmov} shows that the application of $R_{m+2r}$ (for $0 \leq r \leq l-1$) to an eigenfunction adds $4m+8r+4-2b_{-}$ to the eigenvalue, so the assertion follows from evaluating \[\sum_{r=0}^{l-1}(4m+8r+4-2b_{-})=l(4m+4l-2b_{-}).\] The proofs of parts $(iii)$ and $(iv)$ are given in Section \ref{Proofswcop}.
We recall that $M=\binom{a\ \ b}{c\ \ d} \in SL_{2}(\mathbb{R})$ defines the holomorphic map
\[M:\tau\in\Big[\mathcal{H}=\big\{\tau=x+iy\in\mathbb{C}\big|y>0\big\}\Big] \mapsto\frac{a\tau+b}{c\tau+d},\quad\mathrm{with}\quad j(M,\tau)=c\tau+d,\] the latter being the factor of automorphy of this action. Modular forms of weight $(k,l)$ (or just weight $k$ if $l=0$) with respect to a discrete subgroup $\Gamma$ of $SL_{2}(\mathbb{R})$ with cofinite volume (with respect to the invariant measure $\frac{dxdy}{y^{2}}$) are functions $f:\mathcal{H}\to\mathbb{C}$ which are invariant under the corresponding weight $(k,l)$ slash operators for elements of $\Gamma$. The weight $(k,l)$ Laplacian is \[\Delta_{k,l}=4y^{2}\partial_{\tau}\partial_{\overline{\tau}}-2iky\partial_{ \overline{\tau}}+2ily\partial_{\tau}+l(k-1),\] normalized such that $\Delta_{k}=\Delta_{k,0}$ annihilates holomorphic functions and the Laplacians commute with powers of $y$ as in Equation \eqref{Y2Delta}. The \emph{Shimura--Maa\ss\ operators} \[\delta_{k}=y^{-k}\partial_{\tau}y^{k}=\partial_{\tau}+\frac{k}{2iy} \quad\mathrm{and}\quad y^{2}\partial_{\overline{\tau}}\] (note the different normalization from \cite{[Bru]} and \cite{[Ze2]}!) take modular forms of weight $k$ to modular forms of weight $k+2$ and $k-2$ respectively, or more precisely, satisfy an appropriate commutation relation with the slash operators for all the elements of $SL_{2}(\mathbb{R})$. They also change Laplacian eigenvalues (again, with respect to $-\Delta_{k}$ rather than $\Delta_{k}$): $\delta_{k}$ adds $k$ to the eigenvalue, while $y^{2}\partial_{\overline{\tau}}$ subtracts $k-2$ from it. Moreover, the powers of the Shimura--Maa\ss\ operators are given by, e.g., Equation (56) in \cite{[Za]}, stating that \[\delta_{k}^{l}=\delta_{k+2l-2}\circ\ldots\circ\delta_{k}=\sum_{r=0}^{l}\frac{ l!}{(l-r)!}\binom{k+l-1}{r}\frac{\partial_{\tau}^{l-r}}{(2iy)^{r}}\] (for arbitrary $k$, not necessarily integral and non-negative). Theorem \ref{wcop} and Proposition \ref{LapRmL} show that our weight changing operators $R_{m}^{(b_{-})}$ and $L^{(b_{-})}$ have similar properties. However, our operators are differential operators of order 2 while the Shimura--Maa\ss\ operators are of order 1. This is why the results of Propositions \ref{RLcomp} and \ref{Rmpowl} are more complicated than the fact that $\delta_{k-2}y^{2}\partial_{\overline{\tau}}$ is just $\frac{\Delta_{k}}{4}$, the commutator $[\delta,y^{2}\partial_{\overline{\tau}}]_{k}$ being simply $\frac{k}{4}$, and Equation (56) of \cite{[Za]}.
Nonetheless, the operators $R_{m}^{(b_{-})}$ and $L^{(b_{-})}$ for small values of $b_{-}$ are closely related to the Shimura--Maa\ss\ operators. Indeed, for $b_{-}=1$ the group $SO_{2,1}^{+}$ is $PSL_{2}(\mathbb{R})$ and the tube domain $K_{\mathbb{R}}+iC$ is just $\mathcal{H}$. We have \[J(M,\tau)=j^{2}(M,\tau),\quad\mathrm{hence}\quad [M]_{m}=[M]_{2m}^{\mathcal{H}}\quad\mathrm{and}\quad\Delta_{m}^{(1)}=\Delta_{ 2m}\] (the same assertions hold for the operators involving anti-holomorphic weights). Our operators $R_{m}^{(1)}$ and $L^{(1)}$ are \emph{squares} of the Shimura--Maa\ss\ operators, namely \[R_{m}^{(1)}=\delta_{2m}^{2}=\delta_{2m+2}\delta_{2m}\quad\mathrm{and}\quad L^{ (1)}=(y^{2}\partial_{\overline{\tau}})^{2}.\] Note that in this case \[\Xi_{m}^{(1)}=\frac{(\Delta_{2m})^{2}}{16}-\frac{m\Delta_{2m}}{8}\in\mathbb{C} [\Delta_{m}^{(1)}=\Delta_{2m}],\] in accordance with the rank of the group being 1 rather than 2 (in particular, in the notation of \cite{[Na]} we have $\Delta_{2}=\frac{\Delta_{1}}{4}$ in this case).
For $b_{-}>1$ many authors (including \cite{[Bru]} and \cite{[Na]}) take the basis for $K_{\mathbb{R}}$ as two elements spanning a hyperbolic plane together with an orthogonal basis of elements of norm $-2$. In elements of the positive cone $C$, the first two coordinates are positive. In particular, for $b_{-}=2$ we have $K_{\mathbb{R}}+iC\cong\mathcal{H}\times\mathcal{H}$, with $\tau=x+iy$ and $\sigma=s+it$ being the two coordinates. The group $SO_{2,2}^{+}$ is an order 2 quotient of $SL_{2}(\mathbb{R}) \times SL_{2}(\mathbb{R})$, acting on $G(V)\cong\mathcal{H}\times\mathcal{H}$ through \[(M,N):(\tau,\sigma)\mapsto(M\tau,N\sigma)\quad\mathrm{with}\quad J\big((M,N),(\tau,\sigma)\big)=j(M,\tau)j(N,\sigma).\] It follows that \[[M,N]_{m}=[M]_{m,\tau}^{\mathcal{H}}[N]_{m,\sigma}^{\mathcal{H}}\quad\mathrm{ and}\quad\Delta_{m}^{(2)}=2\Delta_{m,\tau}+2\Delta_{m,\sigma}\] (which extend to the operators with anti-holomorphic weights as well). Our operators are \[R_{m}^{(2)}=2\delta_{m,\tau}\delta_{m,\sigma},\quad L^{(2)}=8y^{2}t^{2}\partial_{\overline{\tau}}\partial_{\overline{\sigma}} \quad\mathrm{and}\quad\Xi_{m}^{(2)}=\Delta_{m,\tau}\Delta_{m,\sigma}.\] In both cases $b_{-}=1$ and $b_{-}=2$ the assertions of this Section follow from properties of the Shimura--Maa\ss\ operators (note that $Y^{2}$ is $2y^{2}$ for $b_{-}=1$). When $b_{-}=2$ the special orthogonal group of a negative definite subspace is also $SO(2)$, which makes the theory of automorphic forms more symmetric.
Working with $b_{-}=3$ in this model yields another coordinate $z=u+iv$. The positivity of $y$, $t$, and $yt-v^{2}$ is equivalent to \[\Pi=\binom{\tau\ \
z}{z\ \ \sigma}\quad\mathrm{being\ in}\quad\mathcal{H}_{2}=\big\{\Pi=X+iY \in M_{2}(\mathbb{C})\big|\Pi=\Pi^{t},\ Y\gg0\big\}.\] Hence $K_{\mathbb{R}}+iC$ is identified with the Siegel upper half-plane of degree 2. The group $SO_{2,3}^{+}$ is $PSp_{4}(\mathbb{R})$, with the symplectic action and the factor of automorphy (hence the slash operators) from the theory of Siegel modular forms. In this case \[R_{m}^{(3)}=-\frac{M_{m}}{Y^{2}},\quad L^{(3)}=-Y^{2}N_{0},\quad\mathrm{and}\quad\Delta_{m}^{(3)}=2Tr(\Omega_{m,0})\] in the notation of \cite{[Ma1]} and \cite{[Ma2]} for degree 2 (for weight $(m,n)$ the latter assertion extends to the modified Laplacian $\widetilde{\Delta}_{m,n}^{(3)}$ presented in Section \ref{Proofswcop}). The operator $\Delta_{K_{\mathbb{C}}}^{h}$ is also a constant multiple of the operator $\mathbb{D}$ considered, for example, in \cite{[CE]} and \cite{[Ch]}.
\section{Images of Theta Lifts under $R_{m}^{(b_{-})}$ and $L^{(b_{-})}$ \label{Lifts}}
For natural $r$, $s$, $t$, and $l$ we define the polynomials \[P_{r,s,t}(\mu,Z)=\frac{(\mu,Z_{V})^{r}(\mu,\overline{Z_{V}})^{t}}{(Y^{2})^{s}}
\quad\mathrm{and}\quad P_{r,s,t}^{(l)}(\mu,Z)=P_{r,s,t}(\mu,Z)(\mu_{-}^{2})^{l}.\] As a functions of $\mu \in V$, the polynomial $P_{r,s,t}(\mu,Z)$, considered, e.g., in \cite{[Ze2]}, is homogenous of degree $(r+t,0)$ with respect to the element of $G(V)$ represented by $Z$, while $P_{r,s,t}^{(l)}(\mu,Z)$ has homogeneity degree $(r+t,2l)$. Equation (5) of \cite{[Ze2]} extends from $P_{r,s,t}=P_{r,s,t}^{(0)}$ to the more general polynomials $P_{r,s,t}^{(l)}$: The equality \begin{equation} P_{r,s,t}^{(l)}(MZ,\mu)=j(M,Z)^{s-r}\overline{j(M,Z)}^{s-t}P_{r,s,t}^{(l)}(Z,M^{ -1}\mu) \label{Prstlmod} \end{equation} holds for every $\mu \in V$, $Z \in K_{\mathbb{R}}+iC$, $M \in O^{+}(V)$, and $r$, $s$, $t$, and $l$ from $\mathbb{N}$.
We shall assume that $V=L_{\mathbb{R}}$ for some fixed even lattice $L$ (of signature $(2,b_{-})$), and consider the theta function of $L$ which is based on the polynomial $P_{r,s,t}^{(l)}$. These are (vector-valued) functions of $\tau=x+iy\in\mathcal{H}$ and $Z \in K_{\mathbb{R}}+iC$, which are sums of expressions of the form \begin{equation} F_{r,s,t}^{(l)}(\tau,Z,\mu)=e^{-\Delta_{v}/8\pi y}(P_{r,s,t}^{(l)})(\mu,Z)\mathbf{e}\bigg(\tau\frac{\mu_{+}^{2}}{2}+\overline{ \tau}\frac{\mu_{-}^{2}}{2}\bigg). \label{Frstldef} \end{equation} Here $\mu_{\pm}$ are the parts of $\mu \in V$ which lie in the spaces $v_{\pm}$ according to the element of $G(V)$ corresponding to $Z$, $\Delta_{v}$ is the Laplacian on $V$ which corresponds to the \emph{majorant} of that element (i.e., to the bilinear form in which the sign on the pairing on $v_{-}$ is inverted to be positive as well), and $\mathbf{e}(w)=e^{2\pi iw}$ for every complex $w$. A simple and direct calculation proves \begin{lem} $(i)$ We have the equality $\mu_{+}^{2}=P_{1,1,1}(\mu,Z)$. In addition, the following equalities hold: \[(ii)\quad\Delta_{v_{+}}P_{r,s,t}=4rtP_{r-1,s-1,t-1}.\quad(iii)\quad\Delta_{v_{ -}}(\mu_{-}^{2})^{l}=2l(2l+b_{-}-2)(\mu_{-}^{2})^{l-1}.\] \label{DeltavpmPrstl} \end{lem} Part $(i)$ of Lemma \ref{DeltavpmPrstl} shows that we can write the exponent in Equation \eqref{Frstldef} as the constant $\mathbf{e}\big(\overline{\tau}\frac{\mu^{2}}{2}\big)$ (independent of $Z$) times $e^{-2\pi yP_{1,1,1}}$. Since the differences in the indices in part $(ii)$ of Lemma \ref{DeltavpmPrstl} remain the same, $l$ does not affect the weight of modularity of $P_{r,s,t}^{(l)}$, and $P_{1,1,1}$ is invariant (by Equation \eqref{Prstlmod}), we find that replacing $P$ by $F$ in Equation \eqref{Prstlmod} still yields a valid equation. Let $L^{*}=Hom(L,\mathbb{Z})$ be the dual lattice of $L$ and $L^{*}/L$ the (finite) \emph{discriminant group} of $L$. Then the theta function $\Theta_{L,r,s,t}^{(l)}$ is the $\mathbb{C}[L^{*}/L]$-valued function defined by \[\Theta_{L,r,s,t}^{(l)}(\tau,Z)=\sum_{\gamma \in L^{*}/L}\theta_{\gamma+L,r,s,t}^{(l)}(\tau,Z)e_{\gamma},\ \ \theta_{\gamma+L,r, s,t}^{(l)}(\tau,Z)=\sum_{\mu\in\gamma+L}F_{r,s,t}^{(l)}(\tau,Z,\mu)\] (this function is $\Theta_{L}(\tau,0,0;v,P_{r,s,t}^{(l)})$ in the notation of \cite{[B]}, where $v \in G(L_{\mathbb{R}})$ corresponds to $Z \in K_{\mathbb{R}}+iC$). The extension of Equation \eqref{Prstlmod} to $\Theta_{L,r,s,t}^{(l)}$ shows that $\Theta_{L,r,s,t}^{(l)}$ is automorphic of weight $(s-r,s-t)$ as a function of $Z \in K_{\mathbb{R}}+iC$, and Theorem 4.1 of \cite{[B]} shows that as a function of $\tau\in\mathcal{H}$ it is a vector-valued modular form of weight $\big(1+r+t,2l+\frac{b_{-}}{2}\big)$ and the \emph{Weil representation} $\rho_{L}$. The latter is a representation of the metaplectic double cover $Mp_{2}(\mathbb{Z})$ of $SL_{2}(\mathbb{Z})$, which is defined by sending the generators $T$ and $S$ of $Mp_{2}(\mathbb{Z})$ lying over the elements $\binom{1\ \ 1}{0\ \ 1}$ and $\binom{0\ \ -1}{1\ \ \ \ 0}$ of $SL_{2}(\mathbb{Z})$ respectively to \[\rho_{L}(T)(e_{\gamma})=\mathbf{e}(\gamma^{2}/2)e_{\gamma},\] \[\rho_{L}(S)(e_{\gamma})=\frac{\zeta_{8}^{b_{-}-b_{+}}}{\sqrt{\Delta_{L}}}\sum_ {\delta \in L^{*}/L}\mathbf{e}(-(\gamma,\delta))e_{\delta}\] respectively. For the properties of $\rho_{L}$ see \cite{[Ze1]}, as well as the reference cited there. The space $\mathbb{C}[L^{*}/L]$ comes with a Hermitian pairing $\langle\cdot,\cdot\rangle_{\rho_{L}}$ in which the $e_{\gamma}$ are orthonormal, and $\rho_{L}$ is a unitary representation with respect to this pairing. The operation of complex conjugation on $\Theta_{L,r,s,t}^{(l)}$ interchanges $r$ and $t$ and sends $\tau$ to $-\overline{\tau}$ (this is equivalent to multiplying the bilinear form on $V$ by $-1$, but as we rather stay in the signature $(2,b_{-})$ setting, we prefer this anti-holomorphic operation on $\tau$). It also replaces $\rho_{L}$ by its dual representation, but we shall consider the effect of complex conjugation only for the automorphy in the $Z$ variable.
We are interested in the action of the operators $R_{m}^{(b_{-})}$ and $L^{(b_{-})}$ on theta kernels, and the resulting differential properties of the associated theta lifts. Several proofs will involve comparisons of these actions on theta kernels with the actions of the operators $\delta_{k}$ and $y^{2}\partial_{\overline{\tau}}$ on these theta kernels (multiplied by the appropriate powers of $y$). The latter are given (in a more general context) in Equations (6a) and (6b) of \cite{[Ze2]}. As $P_{r,s,t}^{(l)}=P_{r,s,t}(\mu_{-}^{2})^{l}$, Lemma \ref{DeltavpmPrstl} shows that in our case these equations take the form \begin{equation} \delta_{k}y^{\frac{b_{-}}{2}+2l}\Theta_{L,r,s,t}^{(l)}=\pi iy^{\frac{b_{-}}{2}+2l}\Theta_{L,r+1,s+1,t+1}^{(l)}+\frac{il(2l+b_{-}-2)} {8\pi}y^{\frac{b_{-}}{2}+2l-2}\Theta_{L,r,s,t}^{(l-1)} \label{deltakTheta} \end{equation} (where $k=1-\frac{b_{-}}{2}+r+t-2l$) and \begin{equation} y^{2}\partial_{\overline{\tau}}y^{\frac{b_{-}}{2}+2l}\Theta_{L,r,s,t}^{(l)}=\pi iy^{\frac{b_{-}}{2}+2l+2}\Theta_{L,r,s,t}^{(l+1)}+\frac{irt}{4\pi}y^{\frac{b_{-} }{2}+2l}\Theta_{L,r-1,s-1,t-1}^{(l)} \label{lowerTheta} \end{equation} (note again the different normalization of these operators).
Recall that given a modular form $F$ of weight $1+r+t-\frac{b_{-}}{2}-2l$ and representation $\rho_{L}$, possibly with exponential growth at the cusps, its \emph{theta lift with respect to the polynomial $P_{r,s,t}^{(l)}$} is defined in \cite{[B]}, \cite{[Ze2]}, and others as follows. For $w>1$ let
\[D_{w}=\big\{\tau\in\mathcal{H}\big||\Re\tau|\leq1/2,|\tau|\geq1,\Im\tau\leq w\big\},\] and assume that \[\lim_{w\to\infty}\int_{D_{w}}y^{1+r+t-\sigma}\langle F(\tau),\Theta_{L}(\tau,v,p_{v})\rangle_{\rho_{L}}\frac{dxdy}{y^{2}}\] exists for $\Re\sigma\gg0$ and defines a holomorphic function of $\sigma$ on some right half-plane, which may be extended to a meromorphic function of $\sigma$ for all $\sigma\in\mathbb{C}$. Then the theta lift $\Phi_{L,r,s,t}^{(l)}(F,Z)$ is the constant term of the expansion of this meromorphic function at $\sigma=0$. Now, the modular form $F$ has a Fourier expansion of the sort \begin{equation} F(\tau)=\sum_{\gamma \in L^{*}/L}\sum_{n\in\mathbb{Q}}c_{n,\gamma}(y)q^{n}e_{\gamma}, \label{Fourier} \end{equation} where $q^{n}$ denotes $\mathbf{e}(n\tau)$ and the $c_{n,\gamma}$ are smooth functions of $y=\Im\tau$, which vanish unless $n\in\frac{\gamma^{2}}{2}+\mathbb{Z}$. The modular forms which are usually considered also satisfies the condition that $c_{n,\gamma}=0$ unless $n\gg-\infty$ (for $F$ which is holomorphic on $\mathcal{H}$ this means at most a pole at the cusp, and no essential singularity).
The relations between the action of the (classical) Shimura--Maa\ss\ operators on the lifted modular form and the action of these operators on the theta kernel used for the theta lift are given in the following \begin{lem} Let $F_{\pm}$ be a modular form of weight $k\pm2=1+r+t-\frac{b_{-}}{2}-2l\pm2$ and representation $\rho_{L}$, with Fourier expansion as in Equation \eqref{Fourier}, and assume that the the regularized theta lifts $\Phi_{L,r,s,t}^{(l)}(Z,\delta_{k}F_{-})$ and $\Phi_{L,r,s,t}^{(l)}(Z,y^{2}\partial_{\overline{\tau}}F_{+})$ are well-defined. Assume that the growth condition $c_{\gamma,n}(y)=o(e^{\varepsilon y})$ as $y\to\infty$ holds for every $\gamma$, $n$, and $\varepsilon>0$, and that $c_{0,0}(y)$ is $o(y^{T})$ as $y\to\infty$ for some $T$. Then the theta lift $\Phi_{L,r,s,t}^{(l)}(Z,\delta_{k}F_{-})$ coincides, up to an additive constant which may appear only if $r=t$, with the value at $Z$ of the theta lift of $F_{-}$ with respect to $-y^{2}\partial_{\overline{\tau}}\Theta_{L,r,s,t}^{(l)}$. The same assertion holds for $\Phi_{L,r,s,t}^{(l)}(Z,y^{2}\partial_{\overline{\tau}}F_{+})$ and the theta lift of $F_{+}$ with respect to $-\delta_{k}\Theta_{L,r,s,t}^{(l)}$. \label{trans} \end{lem}
\begin{proof} See Lemmas 3.4 and 3.6 of \cite{[Ze2]} as well as the argument proving Lemma 2.7 of that reference. Note the factors of $2i$ distinguishing our operators here from those of \cite{[Ze2]}, and observe that the theta function is conjugated in the integral defining the theta lift. \end{proof}
The complex conjugation of $\Theta_{L,r,s,t}^{(l)}$ in the definition of the theta lift implies that $\Phi_{L,r,s,t}^{(l)}(Z,F)$ is automorphic of weight $(s-t,s-r)$. We shall thus consider only the case $r=s$, where the automorphy in (the corresponding) Equation \eqref{Prstlmod} involves only $j(M,Z)$ and not its complex conjugate. As with $P_{r,s,t}$, we may omit the superscript $(l)$ in case $l=0$. In the same manner as in Section \ref{Operators}, we shall postpone most of the (calculational) proofs to Section \ref{Proofsact}. Only the assertions about theta lifts will be proved here.
The first assertion we are interested in is \begin{prop} The action of $R_{m}^{(b_{-})}$ takes $y^{\frac{b_{-}}{2}}\overline{\Theta_{L,m,m,0}}$ to $4\pi i$ times the complex conjugate of $y^{2}\partial_{\overline{\tau}}\big(y^{\frac{b_{-}}{2}}\Theta_{L,m+2,m+2,0} \big)$. \label{LSO} \end{prop}
We remark that Proposition \ref{LSO} may be formulated in terms of comparing the actions of elements from the universal enveloping algebras of $\mathfrak{sl}_{2}(\mathbb{R})$ and $\mathfrak{so}(V)\cong\mathfrak{so}_{2,b_{-}}$ on the theta kernel. However, unlike Proposition 2.3 of \cite{[Ze2]} (and Proposition 4.5 of \cite{[Bru]}), which compare the action of order 2 elements of both universal enveloping algebras, here the one from the algebra of $\mathfrak{so}_{2,b_{-}}$ has order 2 while the element from $\mathfrak{sl}_{2}$ has order 1.
We can now prove establish the first property of the theta lift from \cite{[Ze2]}.
\begin{thm} Assume that $b_{-}$ is even, and let $f$ be a weakly holomorphic modular form of weight $1-\frac{b_{-}}{2}-m$ and representation $\rho_{L}$. Consider the modular form $F=\frac{1}{(2\pi i)^{m}}\delta_{1-\frac{b_{-}}{2}-m}^{m}f$, of weight $k=1-\frac{b_{-}}{2}+m$, and its theta lift $\Phi_{L,m,m,0}(Z,F)$ considered in Theorem 3.9 of \cite{[Ze2]}. The image of the latter automorphic form under $\frac{1}{(8\pi^{2})^{b_{-}/2}}(R_{m}^{(b_{-})})^{b_{-}/2}$ is a meromorphic automorphic form of weight $m+b_{-}$ on $K_{\mathbb{R}}+iC$, whose singularities are poles of order $m+b_{-}$ along special divisors. \label{merb-even} \end{thm}
\begin{proof} Proposition \ref{LSO} yields the equality \[\frac{1}{8\pi^{2}}R_{m}^{(b_{-})}y^{\frac{b_{-}}{2}}\overline{\Theta_{L,m,m,0} }(\tau,Z)=\frac{i}{2\pi}\overline{y^{2}\partial_{\overline{\tau}}y^{\frac{b_{-}} { 2}}\Theta_{L,m+2,m+2,0}(\tau,Z)}\] for every $\tau\in\mathcal{H}$ and $Z \in K_{\mathbb{R}}+iC$. As $F$ (as well as its images under any power of $\delta_{k}$) satisfies the conditions of Lemma \ref{trans}, we establish the equality \[\frac{1}{8\pi^{2}}R_{m}^{(b_{-})}\Phi_{L,m,m,0}(Z,F)=\Phi_{L,m+2,m+2,0}\bigg(Z ,\frac{1}{2\pi i}\delta_{k}F\bigg).\] Repeating this argument, we get \[\frac{1}{(8\pi^{2})^{l}}(R_{m}^{(b_{-})})^{l}\Phi_{L,m,m,0}(Z,F)=\Phi_{L,m+2l, m+2l,0}\bigg(Z,\frac{1}{(2\pi i)^{l}}\delta_{k}^{l}F\bigg)\] for any $l\in\mathbb{N}$. Consider now the case $l=\frac{b_{-}}{2}$. Then $\widetilde{F}=\frac{1}{(2\pi i)^{b_{-}/2}}\delta_{k}^{b_{-}/2}F$ is $\frac{1}{(2\pi i)^{m+b_{-}/2}}\delta_{k-2m}^{m+b_{-}/2}f$ with $f$ weakly holomorphic of weight $1-\frac{b_{-}}{2}-m$ (which is integral since $b_{-}$ is even). But then $\frac{1}{(2\pi i)^{m+b_{-}/2}}\delta_{k-2m}^{m+b_{-}/2}$ is just the operator $\big(\frac{\partial_{\tau}}{2\pi i}\big)^{m+b_{-}/2}$ (which takes $q^{n}$ from a Fourier expansion to $n^{m+b_{-}/2}q^{n}$---this is the reason for our normalization), so that the weight $1+\frac{b_{-}}{2}+m$ modular form $\widetilde{F}$ is again weakly holomorphic. Theorem 14.3 of \cite{[B]} now shows that our automorphic form of weight $m+b_{-}$, which we write as $\Phi_{L,m+b_{-},m+b_{-},0}(Z,\widetilde{F})$, is meromorphic on $K_{\mathbb{R}}+iC$, with poles of order $m+b_{-}$ along rational quadratic divisors associated with negative norm vectors in $L^{*}$ whose corresponding coefficients in Equation \eqref{Fourier} do not vanish. This completes the proof of the theorem. \end{proof}
We remark that in case the modular form $f$ is a harmonic weak Maa\ss\ form then the modular form $\widetilde{F}$ from the proof of Theorem \ref{merb-even} is again weakly holomorphic. Moreover, in case the image of $f$ under the operator $\xi_{k-2m}$ of \cite{[BF]} does not have a pole at the cusp, the theta lift has no additional singularities, and the result of Theorem \ref{merb-even} extends to this case. However, in the theta lift $\Phi_{L,m,m,0}(Z,F)$ itself one can still distinguish the case where $f$ is weakly holomorphic from the one where $F$ is such a harmonic weak Maa\ss\ form.
For the weight lowering operator $L^{(b_{-})}$, we do not have a nice equivalent to Proposition \ref{LSO}. However, we do have an interesting result concerning its $m$th power. We begin with \begin{lem} The image of $\Theta_{L,k,n,n}^{(l)}(-\overline{\tau},Z)$ under $L^{(b_{-})}$ is \[4\pi^{2}y^{2}\Theta_{L,k+2,n,n}^{(l+1)}(-\overline{\tau},Z)+n\bigg(2l+\frac{b_ {-}}{2}\bigg)\Theta_{L,k+1,n-1,n-1}^{(l)}(-\overline{\tau},Z)+\] \[+\frac{n(n-1)l\big(l-1+\frac{b_{-}}{2}\big)}{4\pi^{2}y^{2}}\Theta_{L,k,n-2,n-2 }^{(l-1)}(-\overline{\tau},Z).\] \label{Ltheta} \end{lem}
Lemma \ref{Ltheta} allows us to establish the following \begin{prop} For any $s\in\mathbb{N}$, the image of $\overline{\Theta_{L,m,m,0}}$ under $(L^{(b_{-})})^{s}$ attains, on $\tau$ and $Z$, the value \[\sum_{h}\binom{s}{h}\frac{\Gamma\big(s+\frac{b_{-}}{2}\big)}{ \Gamma\big(h+\frac{b_{-}}{2}\big)}\frac{m!(4\pi^{2}y^{2})^{h}}{(m-s+h)!} \Theta_{L,s+h,m-s+h,m-s+h}^{(h)}(-\overline{\tau},Z).\] \label{LsTheta} \end{prop}
The case $s=m$ in Proposition \ref{LsTheta} is of particular importance, as is shown in the following \begin{prop} The expression $(L^{(b_{-})})^{m}y^{\frac{b_{-}}{2}}\overline{\Theta_{L,m,m,0}}$ equals the complex conjugate of $(-4\pi i)^{m}\delta_{1-\frac{b_{-}}{2}-m,\tau}^{m}y^{\frac{b_{-}}{2}+2m}\Theta_{L,0,0,m }^{(m)}(\tau,Z)$. \label{Lb-Rtaum} \end{prop}
Automorphic forms of non-zero weight can never be real-valued, because complex conjugation yields an automorphic form with a different weight. However, multiplying the complex conjugate automorphic form by a power of $Y^{2}$ leads to an object which is comparable with the image of our automorphic form under the appropriate power of a weight changing operator, as these two functions do have the same weight. We shall thus say that an automorphic form $\Phi$, of positive weight $m$, is \emph{$m$-real} if its image under the $m$th power of the weight lowering operators $L^{(b_{-})}$ coincides with its complex conjugate multiplied by a positive multiple of $(Y^{2})^{m}$. We now show that the theta lifts from Theorem 3.9 of \cite{[Ze2]} are $m$-real, or more generally:
\begin{thm} Let $F$ be as in Theorem \ref{merb-even} (but without the restriction on the parity of $b_{-}$) , and assume that $F$ is an eigenfunction with respect to (minus) the Laplacian of weight $1-\frac{b_{-}}{2}+m$, with eigenvalue $\lambda=-\frac{mb_{-}}{2}$. Assume further that the Fourier coefficients $c_{\gamma,n}$ of $F$ appearing in Equation \eqref{Fourier} are real. Then applying the operator $(L^{(b_{-})})^{m}$ to $\frac{i^{m}}{2}\Phi_{L,m,m,0}(Z,F)$ yields the complex conjugate of $\frac{i^{m}}{2}\Phi_{L,m,m,0}(Z,F)$ multiplied by $m!\Gamma\big(m+\frac{b_{-}}{2}\big)(Y^{2})^{m}/\Gamma\big(\frac{b_{-}}{2} \big)$. \label{LmPhiconj} \end{thm}
\begin{proof} By Proposition \ref{Lb-Rtaum}, the image of $\frac{i^{m}}{2}\Phi_{L,m,m,0}$ under $(L^{(b_{-})})^{m}$ coincides with $\frac{i^{m}}{2}$ times the regularized integral of $F$ paired with the function \[(-4\pi i)^{m}\delta_{1-\frac{b_{-}}{2}-m,\tau}^{m}y^{\frac{b_{-}}{2}+2m}\Theta_{L,0,0,m }^{(m)}(\tau,Z).\] On the other hand, the fact that the first index in $P_{0,0,m}$ vanishes allows us to use Equation \eqref{lowerTheta} successively $m$ times and write \[(-\pi i)^{m}y^{\frac{b_{-}}{2}+2m}\Theta_{L,0,0,m}^{(m)}(\tau,Z)\quad\mathrm{as\ just}\quad(-y^{2}\partial_{\overline{\tau}})^{m}y^{\frac{b_{-}}{2}}\Theta_{L,0,0 ,m}(\tau,Z).\] As in the proof of Theorem \ref{merb-even}, we can write $(L^{(b_{-})})^{m}\Phi_{L,m,m,0}(Z,F)$, using Lemma \ref{trans}, as the theta lift $\frac{i^{m}}{2}\Phi_{L,0,0,m}\big(Z,4^{m}\delta_{1-\frac{b_{-}}{2}-m,\tau}^{m} (-y^{2}\partial_{\overline{\tau}})^{m}F\big)$. Now, as $F$ is an eigenfunction and $y^{2}\partial_{\overline{\tau}}$ takes eigenfunctions to eigenfunctions, we can replace each combination $-4\delta_{l}y^{2}\partial_{\overline{\tau}}$, starting from the inner pair, by the appropriate eigenvalue. As after applying $(-y^{2}\partial_{\overline{\tau}})^{r}$ the eigenvalue becomes $\lambda-r\big(m-r-\frac{b_{-}}{2}\big)$, the modular form we plug inside the latter lift is just $F$ multiplied by the scalar $\prod_{r=0}^{m-1}\big[\lambda-r\big(m-r-\frac{b_{-}}{2}\big)\big]$. Substituting the value of $\lambda$, the $r$th multiplier becomes just $(r-m)\big(r+\frac{b_{-}}{2}\big)$, and the product is $(-1)^{m}m!\Gamma\big(m+\frac{b_{-}}{2}\big)/\Gamma\big(\frac{b_{-}}{2}\big)$. Division by $m!\Gamma\big(m+\frac{b_{-}}{2}\big)(Y^{2})^{m}/\Gamma\big(\frac{b_{-}}{2}\big)$ thus gives $\frac{(-i)^{m}}{2}\Phi_{L,0,m,m}(Z,F)$, so that we need to show why $\Phi_{L,0,m,m}(Z,F)$ is the complex conjugate of $\Phi_{L,m,m,0}(Z,F)$. As the Fourier coefficients of $F$ are real, we obtain $\overline{F(\tau)}=F(-\overline{\tau})$. On the other hand, we have seen that complex conjugation on our theta function interchanges the indices $r$ and $t$ and replaces the variable $\tau$ by $-\overline{\tau}$. The required assertion now follows from the fact that powers of $y$ and the measure $\frac{dxdy}{y^{2}}$ are both preserved by the change of variable $\tau\mapsto-\overline{\tau}$. This completes the proof of the theorem. \end{proof}
We remark that the choice of $\lambda=-\frac{mb_{-}}{2}$ in Theorem \ref{LmPhiconj} is not crucial. Any choice of $\lambda$ for which the number $\prod_{r=0}^{m-1}\big[r\big(m-r-\frac{b_{-}}{2}\big)-\lambda\big]$ is positive will be sufficient for Theorem \ref{LmPhiconj} to hold (with the same proof). However, we chose this eigenvalue as it is the eigenvalue of the theta lifts from \cite{[Ze2]}.
\section{Proofs of the Properties of $R_{m}^{(b_{-})}$ and $L^{(b_{-})}$ \label{Proofswcop}}
In this Section we include the proofs of the properties of the weight raising and weight lowering operators appearing in Section \ref{Operators}.
We first introduce (following \cite{[Na]}) a convenient set of generators for $O^{+}(V)$. For $\xi \in K_{\mathbb{R}}$ we define the element $p_{\xi} \in SO^{+}(V)$ whose action is \[\big[\mu \in K_{\mathbb{R}}=\{z,\zeta\}^{\perp}\big]\mapsto\mu-(\mu,\xi)z, \quad\zeta\mapsto\zeta+\xi-\frac{\xi^{2}}{2}z,\quad z \mapsto z.\] Furthermore, given an element $A \in O(K_{\mathbb{R}})$ and a scalar $a\in\mathbb{R}^{*}$ such that $a>0$ if $A \in O^{+}(K_{\mathbb{R}})$ and $a<0$ otherwise, we let $k_{a,A} \in O^{+}(V)$ be the element acting as \[\big[\mu \in K_{\mathbb{R}}=\{z,\zeta\}^{\perp}\big] \mapsto A\mu,\quad\zeta-\frac{\zeta^{2}}{2}z\mapsto\frac{1}{a}\bigg(\zeta-\frac{\zeta^{2 }}{2}z\bigg),\quad z \mapsto az.\] For any $Z \in K_{\mathbb{R}}+iC$ we have \[p_{\xi}Z=Z+\xi,\quad J(p_{\xi},Z)=1,\quad k_{a,A}Z=aAZ,\quad\mathrm{and}\quad J(k_{a,A},Z)=\frac{1}{a}.\] Note that the relation between $A$ and the sign of $a$ is equivalent to preserving $C$ rather than mapping $Z$ into $K_{\mathbb{R}}-iC$---it appears that \cite{[Na]} ignored this point. Choose now an element of $G(K_{\mathbb{R}})$ in which the positive definite space is generated by the norm 1 vector $u_{1}$, and consider the involution $w \in SO^{+}(K_{\mathbb{R}})$ defined by \[\big[\mu \in K_{\mathbb{R}}=\{z,\zeta\}^{\perp}\big]\mapsto\mu-2(\mu,u_{1})u_{1}, \quad\zeta-\frac{\zeta^{2}}{2}z\mapsto-z,\quad z\mapsto-\bigg(\zeta-\frac{\zeta^{2}}{2}z\bigg)\] ($w$ inverts the positive definite space $\mathbb{R}u_{1}$). Its action on $K_{\mathbb{R}}+iC$ is through \[wZ=\frac{2}{Z^{2}}\big[Z-2(Z,u_{1})u_{1}\big]\quad\mathrm{with}\quad J(w,Z)=\frac{Z^{2}}{2}.\] The elements $k_{a,A}$ with $(a,A)$ in the index 2 subgroup of $R^{*} \times O(K_{\mathbb{R}})$ thus defined and $p_{\xi}$ for $\xi \in K_{\mathbb{R}}$ generate the stabilizer $St_{O^{+}(V)}(\mathbb{R}z)$ of the isotropic space $\mathbb{R}z$ in $O^{+}(V)$ as the semi-direct product of these groups. The fact that adding $w$ to $St_{O^{+}(V)}(\mathbb{R}z)$ generates $O^{+}(V)$ is now easily verified by considering the action on isotropic 1-dimensional subspaces of $V$.
Some useful relations are derived in the following \begin{lem} Let $K_{\mathbb{R}}$ be a non-degenerate vector space of dimension $b_{-}$, fix $\alpha\in\mathbb{C}$, and let $F$ be a $\mathcal{C}^{2}$ function which is defined on a neighborhood of a point $Z=X+iY \in K_{\mathbb{C}}$ with $Y^{2}>0$. Then the following relations hold: \[(Y^{2})^{-\alpha}\Delta_{K_{\mathbb{C}}}^{h}\big((Y^{2})^{\alpha} F\big)(Z)=\Delta_{K_{\mathbb{C}}}^{h}F(Z)-\frac{2i\alpha}{Y^{2}}D^{*}F(Z)-\frac{ \alpha(\alpha-1+\frac{b_{-}}{2})}{Y^{2}}F(Z)\] and \[(Y^{2})^{-\alpha}\Delta_{K_{\mathbb{C}}}^{\overline{h}}\big((Y^{2})^{\alpha} F\big)(Z)=\Delta_{K_{\mathbb{C}}}^{\overline{h}}F(Z)+\frac{2i\alpha}{Y^{2}} \overline{D^{*}}F(Z)-\frac{\alpha(\alpha-1+\frac{b_{-}}{2})}{Y^{2}}F(Z).\] \label{LappowMov} \end{lem} We remark that Lemma \ref{LappowMov} holds for $K_{\mathbb{R}}$ of arbitrary signature (not necessarily Lorentzian), but not negative definite (for $Y^{2}>0$ to be possible).
\begin{proof} The proof is obtained by a straightforward calculation, using an orthonormal basis for $K_{\mathbb{R}}$ and the action of $\partial_{k}$ and $\partial_{\overline{k}}$ on functions of $Y$ alone. \end{proof}
We remark that the third operator $\Delta_{K_{\mathbb{C}}}^{\mathbb{R}}$ bears a property similar to Lemma \ref{LappowMov}, which is used implicitly in Section 3 of \cite{[Ze2]} in order to prove Equation \eqref{Y2Delta}.
We can now present the \begin{proof}[Proof of Theorem \ref{wcop}] Multiply both sides of the desired assertion for $R_{m}^{(b_{-})}$, as well as the function $F$ there, by $(Y^{2})^{m}$. Lemma \ref{LappowMov}, the first definition of $R_{m}^{(b_{-})}$, and Equation \eqref{Y2mod} show that this yields the equivalent equality \[(R_{0}^{(b_{-})}F)[M]_{2,-m}=R_{0}^{(b_{-})}\big(F[M]_{0,-m}).\] Observe that conjugating the latter equation and multiplying by $(Y^{2})^{2}$ yields the required equality for $L^{(b_{-})}$. Hence we are reduced to proving only this equality. Moreover, $R_{0}^{(b_{-})}$ involves only holomorphic differentiations, which means that it commutes with the power of $\overline{J(M,Z)}$ coming from the anti-holomorphic weights. Hence we can take $m=0$, which implies that proving the equation \[(R_{0}^{(b_{-})}F)[M]_{2}=R_{0}^{(b_{-})}\big(F[M]_{0})\] (which the assertion for $R_{0}^{(b_{-})}$ in the formulation of the theorem) suffices for proving the theorem. Writing the arguments as $M^{-1}(Z)$ in both sides and using the cocycle condition brings the latter equation to the form \begin{equation} (R_{0}^{(b_{-})}F)(Z)J(M^{-1},Z)^{2}=(R_{0}^{(b_{-})})^{M^{-1}}F(Z). \label{R0red} \end{equation} By a standard argument it suffices to verify Equation \eqref{R0red} for $M^{-1}$ being one of the generators of $O^{+}(V)$ considered above. Equation \eqref{R0red} with $M^{-1}=p_{\xi}$ follows from the invariance of both $\Delta_{K_{\mathbb{C}}}^{h}$ and $D^{*}$ under translations of $X=\Re Z$ and the fact that $J(p_{\xi},Z)=1$. The action of $M^{-1}=k_{a,A}$ divides $\Delta_{K_{\mathbb{C}}}^{h}$ by $a^{2}$, leaves $D^{*}$ invariant, and divides $Y^{2}$ by $a^{2}$ (since $A \in O(K_{\mathbb{R}})$), which proves Equation \eqref{R0red} since $J(k_{a,A},Z)=\frac{1}{a}$. Finally, for $M^{-1}=w$ we have the equalities \[(\Delta_{K_{\mathbb{C}}}^{h})^{w}=\bigg(\frac{Z^{2}}{2}\bigg)^{2}\Delta_{K_{ \mathbb{C}}}^{h}-(b_{-}-2)\frac{Z^{2}}{2}D,\qquad(D^{*})^{w}=\frac{Z^{2}}{ \overline{Z}^{2}}D^{*}-\frac{2iY^{2}}{\overline{Z}^{2}}D\] with $D=\sum_{k}z_{k}\partial_{k}$ from \cite{[Na]} (the corresponding operator from \cite{[Na]} is $\frac{1}{2}\Delta_{K_{\mathbb{C}}}^{h}$ rather than $\Delta_{K_{\mathbb{C}}}^{h}$, while $\delta=\frac{Z^{2}}{2}$, $\overline{\delta}=\frac{\overline{Z}^{2}}{2}$, and $d=\frac{Y^{2}}{2}$ there). Using Equation \eqref{Y2mod} we thus find that applying $M^{-1}=w$ to the sum of $\Delta_{K_{\mathbb{C}}}^{h}$ and $\frac{i(b_{-}-2)}{Y^{2}}D^{*}$ (which is $R_{0}^{(b_{-})}$) multiplies it by $\big(\frac{Z^{2}}{2}\big)^{2}$ (as the coefficients in front of $D$ cancel), which establishes Equation \eqref{R0red} also for this case using the value of $J(w,Z)$. This completes the proof of the theorem. \end{proof}
In order to indicate what is the Lie-theoretic interpretation of the operators $R_{m}^{(b_{-})}$ and $L^{(b_{-})}$, we recall the vector $u_{1}$ we used for defining $w$ above, and take a vector $\tilde{u} \in K_{\mathbb{R}}$ of norm $-1$ which is orthogonal to $u_{1}$ (we assume here $b_{-}>1$, but for $b_{-}=1$ our operators are squares of the order 1 operators $\delta_{2m}$ and $y^{2}\partial_{\overline{\tau}}$, whose Lie-theoretic interpretation is given, e.g., in \cite{[Ve]}). These choices determine the parabolic subgroup of $SO^{+}(V)$ appearing in the following \begin{prop} Let $H_{K_{\mathbb{R}}}$ be the subgroup of $SO^{+}(K_{\mathbb{R}})$ consisting of those matrices which preserve the isotropic subspace $\mathbb{R}(u_{1}+\tilde{u})$ and whose action on the quotient $(u_{1}+\tilde{u})^{\perp}/\mathbb{R}(u_{1}+\tilde{u})$ is trivial. Define $H$ to be the group generated by all the elements $p_{\xi}$ with $\xi \in K_{\mathbb{R}}$ and by the elements $k_{a,A}$ with $a>0$ and $A \in H_{K_{\mathbb{R}}}$. Then the group $H$ operates freely and transitively on $K_{\mathbb{R}}+iC$. \label{parab} \end{prop} Let $K \cong SO(2) \times SO(b_{-})$ be the stabilizer, in $SO^{+}(V)$, of the element of $G(V)$ represented by $Z=iu_{1}$, and let $\mathfrak{k}$ be its Lie algebra. The action of a normalized generator of $\mathfrak{so}(2)\subseteq\mathfrak{k}$ on $\mathfrak{so}(V)_{\mathbb{C}}$ decomposes the latter space into the eigenspaces with eigenvalue 0 (this is precisely $\mathfrak{k}$) and $\pm i$ (complex conjugate spaces of dimension $b_{-}$ each). Hence the action on the space of products of two elements of $\mathfrak{so}(V)$ (inside its universal enveloping algebra, say) decomposes into eigenspaces with eigenvalues 0 and $\pm2i$. One verifies that in each of the $\pm2i$-eigenspaces, precisely one combination commutes with the part $\mathfrak{so}(b_{-})$ of $\mathfrak{k}$. As our automorphic forms correspond to functions on $SO(V)$ on which $SO(2) \subseteq K$ operates according to a specific character and $SO(b_{-})$ operate trivially (normalized suitably), these elements (of order 2) of the universal enveloping algebra of $\mathfrak{so}(V)$ lead to weight raising and weight lowering operators. One may then evaluate, using the interplay between the operations of $\mathfrak{k}$ and the Lie algebra of the group $H$ from Proposition \ref{parab}, the action of these operators, and find that they lead to our $R_{m}^{(b_{-})}$ and $L^{(b_{-})}$. However, the change of coordinates between $H_{K_{\mathbb{R}}}$ and $K_{\mathbb{R}}+iC$ in this evaluation is more tedious than one might have believe.
We also indicate briefly the connection between our operators and those of \cite{[Sh1]}. That reference defines, for every representation $\rho$ of $\mathbb{C}^{\times} \times GL_{b_{-}}(\mathbb{C})$ (a subgroup of which we identify the complexification of the compact subgroup $K$, which is isomorphic to the product $\mathbb{C}^{\times} \times SO(b_{-},\mathbb{C})$), a differential operator that roughly sends (vector-valued) automorphic forms with weight (i.e., representation) $\rho$ to automorphic forms having representation $\rho\otimes\omega$, where $\omega$ is the standard representation of that product on $\mathbb{C}^{b_{-}}$. This representation space is considered as the holomorphic cotangent space of $G(V)$, and the operator is, in fact, just the holomorphic differential map $d$, twisted by the image of a scalar $\eta$ and a matrix $\xi$ (both defined explicitly in \cite{[Sh1]}) via the representation $\rho$. Starting with the 1-dimensional representation which is the $m$th power of $\mathbb{C}^{\times}$ (this is the representation associated with our automorphic forms of weight $m$) and repeating this operation twice, we obtain an automorphic form with representation involving $\omega^{\otimes2}$. The idea is expressing the resulting automorphic form when $\omega$ is identified with $K_{\mathbb{C}}$, and using the bilinear form on the latter space in order to replace the $\omega^{\otimes2}$-valued automorphic forms by scalar-valued ones.
Now, we replace the coordinate denoted $z$ in \cite{[Sh1]} by $u=\sqrt{2}z$, considering it as lying in the complexified space $(v_{-})_{\mathbb{C}}$ associated to some base point for $G(V)$, and decompose it as some multiple $u_{z}$ of $z_{v_{-}}$ plus a vector $u_{\perp}$ which is perpendicular to $z_{v_{-}}$. Here $z$ is again the isotropic vector we used for defining $K_{\mathbb{R}}$. Choosing the positive part of $z$ appropriately (recall that the vector denoted $p(z)$ in \cite{[Sh1]} is not presented in the canonical form), we obtain that our norm 0 vector has pairing $1+u^{t}u-2u^{t}z_{v_{-}}$ with $z$ and its positive and negative $K_{\mathbb{C}}$ coordinates are $i(1-u^{t}u)$ and $2u_{\perp}$ respectively. It follows that the associated element $Z$ of $K_{\mathbb{C}}$ (which can be shown to be in $K_{\mathbb{R}}+iC$) satisfies $(Z+ie_{+})^{2}=\frac{-4}{1+u^{t}u-2u^{t}z_{-}}$ (where $e_{+}$ is the generator of the positive part of $K_{\mathbb{R}}$), so that the inverse map sends $Z$ to the vector obtained by multiplying the positive part of $-2\frac{Z+ie_{+}}{(Z+ie_{+})^{2}}$ by $i$, and adding $z_{v_{-}}$ to the result. Given an automorphic form $F$ of weight $m$ on $G(V)$, a very lengthy, tedious, and involved calculation gives us the expression for the $\omega^{\otimes2}$-valued automorphic form obtained from $F$ under the operator mentioned in the previous paragraph, and after applying the pairing we obtain an expression closely related to $(Z+ie_{+})^{2m}R_{m}^{b_{-}}[(Z+ie_{+})^{-2m}F]$. Indeed, the expression denoted by $\eta$ in \cite{[Sh1]} becomes $\frac{16Y^{2}}{|(Z+ie_{+})^{2}|^{2}}$ using our variable, so that multiplying by $\eta^{m}$ before applying the operator and by $\eta^{-m}$ afterwards corresponds to the operation involving $Y^{2m}$ appearing in the definition of $R_{m}^{b_{-}}$, as well as the additional operation with $(Z+ie_{+})^{2m}$. However, the details of this calculation are very long as well, and therefore we have chosen to state and prove Theorem \ref{wcop} more directly.
For calculational purposes it turns out convenient to introduce the operator \[\widetilde{\Delta}_{m,n}^{(b_{-})}=\Delta_{m,n}^{(b_{-})}-2n(2m-b_{-}),\] on which complex conjugation interchanges the indices $m$ and $n$. The operator \[(D^{*})^{2}-\frac{D^{*}}{2i}=\sum_{k,l}y_{k}y_{l}\partial_{k}\partial_{l}\] will also show up, so we denote it $\widetilde{(D^{*})^{2}}$. We now turn to the \begin{proof}[Proof of Proposition \ref{LapRmL}] Conjugating the desired equality for $R_{m}^{(b_{-})}$ by $(Y^{2})^{m}$, applying Equation \eqref{Y2Delta}, and taking the differences between the operators $\widetilde{\Delta}_{m,n}^{(b_{-})}$ and $\Delta_{m,n}^{(b_{-})}$ into consideration, we see that the asserted equality for $R_{m}^{(b_{-})}$ is equivalent to \[\widetilde{\Delta}_{2,-m}^{(b_{-})}R_{0}^{(b_{-})}-R_{0}^{(b_{-})}\widetilde{ \Delta}_{0,m}^{(b_{-})}=(2b_{-}+4m-4)R_{0}^{(b_{-})}.\] Moreover, multiplying the complex conjugate of the latter equation by $(Y^{2})^{2}$ and comparing $\widetilde{\Delta}_{2,-m}^{(b_{-})}$ with $\Delta_{2,-m}^{(b_{-})}$ yields the required property for $L^{(b_{-})}$ (with the index $m$ replaced by $-m$). Hence, as in the proof of Theorem \ref{wcop}, we are reduced to proving this single equation. In addition, the dependence on $m$ of the left hand side enters only through the difference $-4imD^{*}$ between the operators $\widetilde{\Delta}_{l,-m}^{(b_{-})}$ and $\Delta_{l}^{(b_{-})}$ with $l\in\{0,2\}$. As a simple calculation yields \[\quad\big[D^{*},\Delta_{K_{\mathbb{C}}}^{h}\big]=i\Delta_{K_{\mathbb{C}}}^{h} \quad\mathrm{and}\quad\bigg[D^{*},\frac{D^{*}}{Y^{2}}\bigg]=\frac{iD^{*}}{Y^{2} },\] it suffices to prove the equality for $m=0$ (i.e., the original assertion for $R_{0}^{(b_{-})}$): \[\Delta_{2}^{(b_{-})}R_{0}^{(b_{-})}-R_{0}^{(b_{-})}\Delta_{0}^{(b_{-})}=(2b_{- }-4)R_{0}^{(b_{-})}.\] The commutator of $\Delta_{0}^{(b_{-})}$ and $R_{0}^{(b_{-})}$ is evaluated using the equalities
\[\big[|D^{*}|^{2},\Delta_{K_{\mathbb{C}}}^{h}\big]=i\overline{D^{*}}\Delta_{K_{ \mathbb{C}}}^{h}+iD^{*}\Delta_{K_{\mathbb{C}}}^{\mathbb{R}}+\frac{\Delta_{K_{ \mathbb{C}}}^{\mathbb{R}}}{2},\]
\[\bigg[|D^{*}|^{2},\frac{D^{*}}{Y^{2}}\bigg]=\frac{3i|D^{*}|^{2}-i\widetilde{ (D^{*})^{2}}+D^{*}}{2Y^{2}},\quad\big[Y^{2}\Delta_{K_{\mathbb{C}}}^{\mathbb{R}}, \Delta_{K_{\mathbb{C}}}^{h}\big]=2iD^{*}\Delta_{K_{\mathbb{C}}}^{\mathbb{R}} +\frac{b_{-}}{2}\Delta_{K_{\mathbb{C} }}^{\mathbb{R}},\] \[\mathrm{and}\quad\bigg[Y^{2}\Delta_{K_{\mathbb{C}}}^{\mathbb{R}},\frac{D^{*}}{
Y^{2}}\bigg]=\frac{2i|D^{*}|^{2}-2i\widetilde{(D^{*})^{2}}+i\Delta_{K_{\mathbb{C }}}^{h}+i\Delta_{K_{\mathbb{C}}}^{\mathbb{R}}+(2-b_{-})D^{*}}{2Y^{2}}\] (which all follow from straightforward calculations). Applying the equalities \[\Delta_{2}^{(b_{-})}=\Delta_{0}^{(b_{-})}-8i\overline{D^{*}}\quad\mathrm{and}
\quad\overline{D^{*}}\circ\big(\frac{D^{*}}{Y^{2}}\big)=\frac{2|D^{*}|^{2}-iD^{* }}{2Y^{2}}\] and putting in the appropriate scalars now establishes the proposition. \end{proof}
Our next task is the \begin{proof}[Proof of Proposition \ref{RLcomp}] We begin by evaluating $R_{m-2}^{(b_{-})}L^{(b_{-})}$ written as \[R_{m-2}^{(b_{-})}(Y^{2})^{2}\Delta_{K_{\mathbb{C}}}^{\overline{h}}+R_{m-2}^{ (b_{-})}i(2-b_{-})Y^{2}\overline{D^{*}}=(Y^{2})^{2}R_{m}^{(b_{-})}\Delta_{K_{ \mathbb{C}}}^{\overline{h}}+i(2-b_{-})Y^{2}R_{m-1}^{(b_{-})}\overline{D^{*}}.\] Using the equalities \[\big[\Delta_{K_{\mathbb{C}}}^{h},\overline{D^{*}}\big]=-i\Delta_{K_{\mathbb{C}
}}^{\mathbb{R}}\quad\mathrm{and}\quad D^{*}\overline{D^{*}}=|D^{*}|^{2}+\frac{\overline{D^{*}}}{2i}\] we establish the equation \[R_{m-2}^{(b_{-})}L^{(b_{-})}=\Xi_{m}^{(b_{-})}+\frac{(2-b_{-})(2m-b_{-})}{8} \Delta_{m}^{(b_{-})},\] where $\Xi_{m}^{(b_{-})}$ is defined in the formulation of the proposition. We now decompose $R_{m}^{(b_{-})}$ in $L^{(b_{-})}R_{m}^{(b_{-})}$ (which is $(Y^{2})^{2}\overline{R_{0}^{(b_{-})}}R_{m}^{(b_{-})}$), yielding \[(Y^{2})^{2}\overline{R_{0}^{(b_{-})}}\Delta_{K_{\mathbb{C}}}^{h}-i(2m+2-b_{-} )Y^{2}\overline{R_{-1}^{(b_{-})}}D^{*}-\frac{m(2m+2-b_{-})}{2}Y^{2}\overline{R_{ -1}^{(b_{-})}}.\] The formulae \[\big[\Delta_{K_{\mathbb{C}}}^{\overline{h}},D^{*}\big]=i\Delta_{K_{\mathbb{C}}
}^{\mathbb{R}}\quad\mathrm{and}\quad\overline{D^{*}}D^{*}=|D^{*}|^{2}-\frac{D^{* }}{2i}\] now show that \[L^{(b_{-})}R_{m}^{(b_{-})}=\Xi_{m}^{(b_{-})}-\frac{b_{-}(2m+2-b_{-})}{8} \Delta_{m}^{(b_{-})}+\frac{mb_{-}(2m+2-b_{-})}{4}.\] The required commutation relation follows. As Theorem \ref{wcop} shows that the compositions $R_{m-2}^{(b_{-})}L^{(b_{-})}$ and $L^{(b_{-})}R_{m}^{(b_{-})}$ commute with all the slash operators of weight $m$, and Proposition \ref{LapRmL} implies that these operators commute with $\Delta_{m}$, the assertion about $\Xi_{m}^{(b_{-})}$ is also established. This proves the proposition. \end{proof}
Finally, we come to the \begin{proof}[Proof of parts $(iii)$ and $(iv)$ of Proposition \ref{Rmpowl}] We prove part $(iii)$ by induction (the case $l=0$ being trivial). If $(R_{m}^{(b_{-})})^{l}$ is presented by the asserted formula then $(R_{m}^{(b_{-})})^{l+1}$, which is $R_{m+2l}^{(b_{-})}(R_{m}^{(b_{-})})^{l}$, equals \[R_{m+2l}^{(b_{-})}\sum_{c=0}^{l}\sum_{s=0}^{c}A_{s,c}^{(l)}\frac{(iD^{*})^{c-s }(\Delta_{K_{\mathbb{C}}}^{h})^{l-c}}{(-Y^{2})^{c}}=\sum_{s,c}A_{s,c}^{(l)}\frac {R_{m+2l-c}^{(b_{-})}(iD^{*})^{c-s}(\Delta_{K_{\mathbb{C}}}^{h})^{l-c}}{(-Y^{2} )^{c}}.\] For each $c$, the term involving $\frac{D^{*}}{Y^{2}}$ (resp. $\frac{1}{Y^{2}}$) in $R_{m+2l-c}^{(b_{-})}$ takes the term with indices $c$ and $s$ (for $l$) to a multiple of the term with corresponding to $c+1$ and $s$ (resp. $c+1$ and $s+1$) for $l+1$. For $\Delta_{K_{\mathbb{C}}}^{h}$ we have \[\big[\Delta_{K_{\mathbb{C}}}^{h},iD^{*}\big]=\Delta_{K_{\mathbb{C}}}^{h} \quad\mathrm{hence}\quad\Delta_{K_{\mathbb{C}}}^{h}(iD^{*})^{c-s}=\sum_{a=s}^{c} \binom{c-s}{a-s}(iD^{*})^{c-a}\Delta_{K_{\mathbb{C}}}^{h},\] and we multiply the latter sum by $\frac{(\Delta_{K_{\mathbb{C}}}^{h})^{l-c}}{(-Y^{2})^{c}}$. This shows that $(R_{m}^{(b_{-})})^{l+1}$ can be expressed by the asserted formula. Putting in the multipliers $A_{s,c}^{(l)}$ from $(R_{m}^{(b_{-})})^{l}$ and the coefficients of $\frac{D^{*}}{Y^{2}}$ and $\frac{1}{Y^{2}}$ in $R_{m+2l-c}^{(b_{-})}$, summing over $c$ and $s$, and taking the coefficient in front of the term with indices $c$ and $a$ (and $l+1$) in the result, we obtain the recursive relation asserted in part $(iii)$. We now observe that for $a=0$ the recursive formula reduces to \[A_{0,c}^{(l+1)}=A_{0,c}^{(l)}+(2m+4l-2c+4-b_{-})A_{0,c-1}^{(l)}.\] Denote the asserted value of $A_{0,c}^{(l)}$ by $B_{0,c}^{(l)}$. As $A_{0,0}^{(0)}=1=B_{0,0}^{(0)}$, it suffices to show that the numbers $B_{0,c}^{(l)}$ satisfy the latter recursive formula. But the equality \[2(l-c+1)\bigg(m+l-c-\frac{b_{-}}{2}+1\bigg)+c(2m+4l-2c+4-b_{-} )=2(l+1)\bigg(m+l-\frac{b_{-}}{2}+1\bigg)\] holds for every $l$ and $c$ (and $m$ and $b_{-}$), and multiplication by $\frac{l!\cdot2^{c-1}}{c(l+1-c)!}$ and by the binomial coefficient $\binom{m+l-\frac{b_{-}}{2}}{c-1}$ yields the required recursive relation for the numbers $B_{0,c}^{(l)}$. This completes the proof of the proposition. \end{proof}
\section{Actions on Theta Kernels---Proofs \label{Proofsact}}
The main technical lemma, which will be required for the evaluations in most of the following proofs, is based on \begin{lem} Given $\mu \in L_{\mathbb{R}}$, the operators $R_{0}^{(b_{-})}$ and $L^{(b_{-})}$ take the function $P_{1,1,1}$ of $Z \in K_{\mathbb{R}}+iC$ to $-\frac{b_{-}}{2}P_{0,2,2}$ and $-\frac{b_{-}}{2}P_{2,0,0}$. \label{RmLP111} \end{lem}
\begin{proof} The commutation relation between powers of $Y^{2}$ and the operators $R_{m}^{(b_{-})}$ obtained from the first definition of the latter operators in Theorem \ref{wcop} and the fact that the latter operators involve only holomorphic differentiation allows us to write $R_{0}^{(b_{-})}P_{1,1,1}$ as $P_{0,1,1}R_{-1}^{(b_{-})}(\mu,Z_{V,Z})$. Hence we must evaluate the operation of $\Delta_{K_{\mathbb{C}}}^{h}$ and $D^{*}$ on $(\mu,Z_{V,Z})$. For the latter operator a simple calculation yields \[2iD^{*}(\mu,Z_{V,Z})=2i(\mu,Y_{V,Z})+2Y^{2}(\mu,z)=(\mu,Z_{V,Z})-(\mu, \overline{Z_{V,Z}})+2Y^{2}(\mu,z).\] The former operator is pure of weight 2, hence its action gives a non-zero result only on the part $-\frac{Z^{2}}{2}(\mu,z)$, and using an orthonormal basis one finds that this result is just $-b_{-}(\mu,z)$. Combining these results, we find that \[\bigg[R_{-1}^{(m)}=\Delta_{K_{\mathbb{C}}}^{h}+\frac{ib_{-}}{Y^{2}}D^{*}-\frac {b_{-}}{2Y^{2}}\bigg](\mu,Z_{V,Z})=-\frac{b_{-}}{2Y^{2}}(\mu,\overline{Z_{V,Z}}) ,\] from which the value of $R_{0}^{(b_{-})}P_{1,1,1}$ follows. The assertion about $L^{(b_{-})}P_{1,1,1}$ is a consequence of the value of $R_{0}^{(b_{-})}P_{1,1,1}$, since $P_{1,1,1}$ is a real function and $L^{(b_{-})}$ is the operator which is complex conjugate to $R_{0}^{(b_{-})}$, multiplied by $(Y^{2})^{2}$. This proves the lemma. \end{proof}
Another useful evaluation appears in the following \begin{lem} The holomorphic and anti-holomorphic $Z$-gradients of $P_{1,1,1}$ have, as vectors in $K_{\mathbb{C}}$, the norms $P_{0,2,2}\mu_{-}^{2}$ and $P_{2,2,0}\mu_{-}^{2}$ respectively. \label{gradnorm} \end{lem}
\begin{proof} $(\mu,\overline{Z_{V,Z}})$ is anti-holomorphic, and the holomorphic gradients of $(\mu,Z_{V,Z})$ and $Y^{2}$ are $\mu_{K_{\mathbb{R}}}-(\mu,z)Z$ and $-iY$ respectively, where $\mu_{K_{\mathbb{R}}}$ is the orthogonal projection of $\mu \in L_{\mathbb{R}}$ onto $K_{\mathbb{R}}=\{z,\zeta\}^{\perp}$. It follows that $P_{1,1,1}$ has holomorphic gradient \[P_{0,2,1}\big[Y^{2}(\mu_{K_{\mathbb{R}}}-(\mu,z)Z)+i(\mu,Z_{V,Z})Y\big].\] Now, the (easily evaluated) equalities \[\big(\mu_{K_{\mathbb{R}}}-(\mu,z)Z,Y\big)=(\mu,Y_{V,Z})-iY^{2}(\mu,z)\] and \[(\mu,z)^{2}Z^{2}-2(\mu,z)(\mu_{K_{\mathbb{R}}},Z)+2(\mu,z)(\mu,Z_{V,Z})=2(\mu, z)(\mu,\zeta)-\zeta^{2}(\mu,z)^{2}\] reduce to the norm of the latter gradient \[P_{0,2,2}\big[\mu_{K_{\mathbb{R}}}^{2}+2(\mu,\zeta)(\mu,z)-\zeta^{2}(\mu,z)^{2 }-P_{1,1,1}\big].\] But $\mu$ is $\big(\mu_{K_{\mathbb{R}}},\mu_{z},(\mu,\zeta)-\zeta^{2}\mu_{z}\big)$ in the $K_{\mathbb{R}}\times\mathbb{R}\times\mathbb{R}$ coordinates, so that the sum of the first three terms in the brackets is just $\mu^{2}$. Subtracting $P_{1,1,1}=\mu_{+}^{2}$ completes the proof of the first assertion, and the second assertion follows from complex conjugation since the function $P_{1,1,1}$ is real-valued. This proves the lemma. \end{proof}
For $\mu \in L_{\mathbb{R}}$ and $\tau=x+iy\in\mathcal{H}$ we denote the vector $\sqrt{2\pi y}\mu$ by $\tilde{\mu}$. Its norm is $2\pi y\mu^{2}$, and after choosing an element of $G(L_{\mathbb{R}})$, it decomposes into $\tilde{\mu}_{+}$ (of norm $2\pi y\mu_{+}^{2}$) and $\tilde{\mu}_{-}$ (whose norm is $2\pi y\mu_{-}^{2}$). We now prove
\begin{prop} Let $f:\mathbb{R}\to\mathbb{R}$ be a smooth function. Then the images of the function $f(\tilde{\mu}_{+})$ under $R_{0}^{(b_{-})}$ and $L^{(b_{-})}$ are $2\pi yP_{0,2,2}\big[\tilde{\mu}_{-}^{2}f''(\tilde{\mu}_{+})-\frac{b_{-}}{2}f'(\tilde{ \mu}_{+})\big]$ and $2\pi yP_{2,0,0}\big[\tilde{\mu}_{-}^{2}f''(\tilde{\mu}_{+})-\frac{b_{-}}{2}f'(\tilde{ \mu}_{+})\big]$ respectively. \label{1st} \end{prop}
\begin{proof} Both operators consist of a first order operator $D$ (a multiple of $D^{*}$ or of $\overline{D^{*}}$) and a second order operator $\Delta$ (which equals $\Delta_{K_{\mathbb{C}}}^{h}$ or $\Delta_{K_{\mathbb{C}}}^{\overline{h}}$). Then $D\big(f(T)\big)=DT \cdot f'(T)$, and $\Delta\big(f(T)\big)$ is the sum of $\Delta T \cdot f'(T)$ and an expression involving $f''(T)$. In our case $T=\tilde{\mu}_{+}=2\pi y\mu_{+}^{2}=2\pi yP_{1,1,1}$, so that the coefficient of $f'(T)$ is just $2\pi y$ times $R_{0}^{(b_{-})}P_{1,1,1}$ and $L^{(b_{-})}P_{1,1,1}$, and the latter expressions are evaluated using Lemma \ref{RmLP111}. The coefficients of $f''(T)$ coming from $\Delta$ being $\Delta_{K_{\mathbb{C}}}^{h}$ or $(Y^{2})^{2}\Delta_{K_{\mathbb{C}}}^{\overline{h}}$ are the norms (in $K_{\mathbb{C}}$) of the holomorphic and anti-holomorphic gradients of $T$, the latter being multiplied by $(Y^{2})^{2}$. For $T=2\pi yP_{1,1,1}$ these norms take the values given in Lemma \ref{gradnorm}, multiplied by $(2\pi y)^{2}$. Gathering these results together and substituting the value of $\tilde{\mu}_{-}^{2}$ completes the proof of the proposition. \end{proof}
We now turn to proving assertions concerning the images of theta lifts (or complex conjugates of theta functions), having only holomorphic weights of automorphy, under the operators $R_{m}^{(b_{-})}$ and $L^{(b_{-})}$. This was seen to boil down to the operation on the function $F_{r,s,t}^{(l)}$ from Equation \eqref{Frstldef}, with $\tau$ replaced by $-\overline{\tau}$, under the additional assumption $s=t$. The exponent was seen, using part (i) of Lemma \ref{DeltavpmPrstl} to be $\mathbf{e}\big(-\tau\frac{\mu^{2}}{2}\big)e^{-\tilde{\mu}_{+}^{2}}$, where the first multiplier is a constant (i.e., independent of $Z$). The polynomial part is evaluated in \begin{lem}
$(i)$ For any natural numbers $k$ and $n$ we have \[(2\pi y)^{n}P_{n-k,0,0}e^{-\Delta_{v_{+}}/8\pi y}(P_{k,n,n})e^{-2\pi yP_{1,1,1}}=(-1)^{k}\frac{d^{k}}{dT^{k}}(T^{n}e^{-T})\bigg|_{T=\tilde{\mu}_{+}^{ 2}}.\] $(ii)$ Applying $e^{\Delta_{v_{-}}/8\pi y}$ to $(2\pi y)^{l}(\mu_{-}^{2})^{l}$ yields $\sum_{p}\binom{l}{p}\big[\Gamma\big(l+\frac{b_{-}}{2}\big)/ \Gamma\big(p+\frac{b_{-}}{2}\big)\big]\big(\tilde{\mu}_{-}^{2}\big)^{p}$. \label{expT} \end{lem}
We allow the index $n-k$ appearing in Part (i) here to be negative, with the natural extension of the definition of $P_{r,s,t}$ to negative $r$. We remark that the expressions obtained in this part are just the generalized Laguerre polynomials $L_{k}^{(n-k)}$, multiplied by the exponents, and normalized appropriately.
\begin{proof} Multiple applications of part (ii) of Lemma \ref{DeltavpmPrstl} show that \[\frac{\Delta_{v_{+}}^{h}}{h!(-8\pi y)^{h}}P_{k,n,n}=\frac{k!n!P_{k-h,n-h,n-h}}{(k-h)!(n-h)!h!(-2\pi y)^{h}}.\] Multiplying by $(2\pi y)^{n}P_{n-k,0,0}$ and summing over $h$, the left hand side of the equation in part $(i)$ becomes just \[\sum_{h}\binom{k}{h}\frac{n!}{(n-h)!}(-1)^{h}(2\pi yP_{1,1,1})^{n-h}e^{-2\pi yP_{1,1,1}}.\] On the other hand, differentiating the product $T^{n}e^{-T}$ $k$ times with respect to $T$ yields \[\sum_{h=0}^{k}\binom{k}{h}\bigg(\frac{d}{dT}\bigg)^{k}T^{n}\cdot\bigg(\frac{d} {dT}\bigg)^{k-h}e^{-T}=\sum_{h=0}^{k}\binom{k}{h}\frac{n!T^{n-h}}{(n-h)!}(-1)^{ k-h}e^{-T},\] and substituting $T=\tilde{\mu}_{+}^{2}=2\pi yP_{1,1,1}$ yields the same expression multiplied by $(-1)^{k}$. This establishes part $(i)$. For part $(ii)$, applying part (iii) of Lemma \ref{DeltavpmPrstl} successively evaluates \[\Delta_{v_{-}}^{l-p}(\mu_{-}^{2})^{l}=\frac{4^{l-p}l!}{p!}\cdot\frac{ \Gamma\big(l+\frac{b_{-}}{2}\big)}{\Gamma\big(p+\frac{b_{-}}{2}\big)}(\mu_{-}^{2 })^{p}.\] Dividing this term by $(8\pi y)^{l-p}(l-p)!$, multiplying everything by $(2\pi y)^{l}$, and substituting $\tilde{\mu}_{-}^{2}=2\pi y\mu_{-}^{2}$ gives the asserted expression. This completes the proof of the lemma. \end{proof}
As $\tilde{\mu}_{-}^{2}=\tilde{\mu}^{2}-\tilde{\mu}_{+}^{2}$, Lemma \ref{expT} implies that the dependence of the expression $(2\pi y)^{n+l}P_{n-k,0,0}F_{k,n,n}^{(l)}(-\overline{\tau},Z,\mu)$ (or the corresponding theta function) on the variable $Z$ is only through the quantity $\tilde{\mu}_{+}^{2}$. For convenience, we gather these results in the following \begin{cor} Define the functions \[f_{k,n,p}^{(w)}(T)=(-1)^{k}\frac{d^{k}}{dT^{k}}(T^{n}e^{-T})\cdot(w-T)^{p},\] where $k$, $p$, and $n$ are natural numbers and $w\in\mathbb{R}$. Then the theta function $\Theta_{L,k,n,n}^{(l)}(-\overline{\tau},Z)$ equals \[\sum_{\mu \in L^{*}}\sum_{p}\binom{l}{p}\frac{\Gamma\big(l+\frac{b_{-}}{2}\big)}{ \Gamma\big(p+\frac{b_{-}}{2}\big)}\frac{f_{k,n,p}^{(\tilde{\mu}^{2})}(\tilde{\mu }_{+}^{2})}{(2\pi y)^{n+l}P_{n-k,0,0}}\mathbf{e}\bigg(-\tau\frac{\mu^{2}}{2}\bigg)e_{\mu+L}.\] \label{eDeltaPmu} \end{cor}
\begin{proof} Just substitute the value of $e^{-\Delta_{v}/8\pi y}(P_{k,n,n}^{(l)})$, which equals the product of $e^{-\Delta_{v_{+}}/8\pi y}(P_{k,n,n})$ and $e^{-\Delta_{v_{+}}/8\pi y}\big((\mu_{-}^{2})^{l}\big)$, from Lemma \ref{expT} into the expression defining the theta function. \end{proof}
We can now present the \begin{proof}[Proof of Proposition \ref{LSO}] As seen above, it suffices to consider the action of $R_{m}^{(b_{-})}$ only on the expression $P_{0,m,m}(\mu,Z)e^{-2\pi yP_{1,1,1}}$ with fixed $\mu$ (recall that $P_{0,m,m}$ is harmonic). The holomorphicity of the differentiation in $R_{m}^{(b_{-})}$ shows that the result is the same as $P_{0,m,m}R_{0}^{(b_{-})}e^{-\tilde{\mu}_{+}^{2}}$. By putting $f(T)=e^{-T}$, Proposition \ref{1st} evaluates $R_{0}^{(b_{-})}e^{-\tilde{\mu}_{+}^{2}}$ as $2\pi yP_{0,2,2}\big(\tilde{\mu}_{-}+\frac{b_{-}}{2}\big)e^{-\tilde{\mu}_{+}}$, and multiplying by $P_{0,m,m}$ yields \[R_{m}^{(b_{-})}P_{0,m,m}e^{-\tilde{\mu}_{+}^{2}}=4\pi^{2}y^{2}P_{0,m+2,m+2} \bigg[\mu_{-}^{2}+\frac{b_{-}}{4\pi y}\bigg]e^{-2\pi yP_{1,1,1}}.\] But the expression in parentheses is $e^{\Delta_{v_{-}}/8\pi y}(\mu_{-}^{2})$ by part $(ii)$ of Lemma \ref{expT}, and the harmonicity of $P_{0,m+2,m+2}$ allows us to put it also into the action of $e^{-\Delta_{v}/8\pi y}$ without affecting the resulting expression. Putting in the missing constant $y^{\frac{b_{-}}{2}}\mathbf{e}\big(-\tau\frac{\mu^{2}}{2}\big)e_{\mu+L }$ and summing over $\mu \in L^{*}$ we establish the equality \[R_{m}^{(b_{-})}y^{\frac{b_{-}}{2}}\Theta_{L,0,m,m}(-\overline{\tau},Z)=4\pi^{2 }y^{ 2+\frac{b_{-}}{2}}\Theta_{L,0,m+2,m+2}^{(1)}(-\overline{\tau},Z).\] But as $P_{m+2,m+2,0}$ is harmonic, Equation \eqref{lowerTheta} shows that applying the operator $-4\pi iy^{2}\partial_{\overline{\tau}}$ to $y^{\frac{b_{-}}{2}}\Theta_{L,m+2,m+2,0}(\tau,Z)$ yields the complex conjugate of the latter expression, and complex conjugation inverts the sign of $4\pi i$. This proves the proposition. \end{proof}
We now turn to the \begin{proof}[Proof of Lemma \ref{Ltheta}] Write the theta function $\Theta_{L,k,n,n}^{(l)}(-\overline{\tau},Z)$ as in Corollary \ref{eDeltaPmu}. It suffices to fix $\mu \in L^{*}$ and compare the coefficients of $\mathbf{e}\big(-\tau\frac{\mu^{2}}{2}\big)e_{\mu+L}$ in both sides. Take some $0 \leq p \leq l$, and apply Proposition \ref{1st} with the function $f=f_{k,n,p}^{(\tilde{\mu}^{2})}$. The powers of $2\pi y$ and $P_{1,0,0}$ from Corollary \ref{eDeltaPmu} and Proposition \ref{1st} merge to $(2\pi y)^{n+l-1}P_{n-2-k,0,0}$ in the denominator, and the remaining part of $L^{(b_{-})}f_{k,n,p}^{(\tilde{\mu}^{2})}$ is \[\binom{l}{p}\frac{\Gamma\big(l+\frac{b_{-}}{2}\big)}{\Gamma\big(p+\frac{b_{-}} {2}\big)}\bigg[\tilde{\mu}_{-}^{2}\big(f_{k,n,p}^{(\tilde{\mu}^{2})} \big)''(\tilde{\mu}_{+}^{2})-\frac{b_{-}}{2}\big(f_{k,n,p}^{(\tilde{\mu}^{2})} \big)'(\tilde{\mu}_{+}^{2})\bigg].\] As $\tilde{\mu}_{-}^{2}=\tilde{\mu}^{2}-\tilde{\mu}_{+}^{2}$, and as one easily evaluates \[(f_{k,n,p}^{w})'(T)=-pf_{k,n,p-1}^{w}(T)-f_{k+1,n,p}^{w}(T),\] the part in brackets in latter expression equals \begin{equation} f_{k+2,n,p+1}^{(\tilde{\mu}^{2})}(\tilde{\mu}_{+}^{2})+\bigg(2p+\frac{b_{-}}{2 }\bigg)f_{k+1,n,p}^{(\tilde{\mu}^{2})}(\tilde{\mu}_{+}^{2})+p\bigg(p+\frac{b_{-} }{2}-1\bigg)f_{k,n,p-1}^{(\tilde{\mu}^{2})}(\tilde{\mu}_{+}^{2}). \label{f'dec} \end{equation} We now write the denominator in the preceding constant as \[\frac{\big(p+\frac{b_{-}}{2}\big)}{\Gamma\big(p+1+\frac{b_{-}}{2}\big)}, \quad\frac{1}{\Gamma\big(p+\frac{b_{-}}{2}\big)},\quad\mathrm{and}\quad\frac{1}{ \big(p-1+\frac{b_{-}}{2}\big)\Gamma\big(p-1+\frac{b_{-}}{2}\big)}\] in front of the three terms in Equation \eqref{f'dec} respectively, and after taking the sum over $p$ and gathering the functions with the same index $p$ together, we see that the quotient $\Gamma\big(l+\frac{b_{-}}{2}\big)/\Gamma\big(p+\frac{b_{-}}{2}\big)$ multiplies \[\bigg(p-1+\frac{b_{-}}{2}\bigg)\binom{l}{p-1}f_{k+2,n,p}^{(\tilde{\mu}^{2})} +\bigg(2p+\frac{b_{-}}{2}\bigg)\binom{l}{p}f_{k+1,n,p}^{(\tilde{\mu}^{2})} +(p+1)\binom{l}{p+1}f_{k,n,p}^{(\tilde{\mu}^{2})}\] (where we have omitted the variable $\tilde{\mu}_{+}^{2}$). Using the identity $b\binom{a}{b}=a\binom{a-1}{b-1}$ we can write the latter expression as \[l\bigg[\binom{l-1}{p-2}f_{k+2,n,p}^{(\tilde{\mu}^{2})}(\tilde{\mu}_{+}^{2} )+2\binom{l-1}{p-1}f_{k+1,n,p}^{(\tilde{\mu}^{2})}(\tilde{\mu}_{+}^{2})+\binom{ l-1}{p}f_{k,n,p}^{(\tilde{\mu}^{2})}(\tilde{\mu}_{+}^{2})\bigg]+\] \begin{equation} +\frac{b_{-}}{2}\bigg[\binom{l}{p-1}f_{k+2,n,p}^{(\tilde{\mu}^{2})}(\tilde{\mu}_ {+}^{2})+\binom{l}{p}f_{k+1,n,p}^{(\tilde{\mu}^{2})}(\tilde{\mu}_{+}^{2})\bigg]. \label{fknpwcomb} \end{equation} Now, differentiating $k$ times and multiplying by $(w-T)^{p}$ takes the equality \[(T^{n}e^{-T})'=(nT^{n-1}-T^{n})e^{-T}\quad\mathrm{to}\quad f_{k,n,p}^{(w)} (T)-f_{k+1,n,p}^{(w)}(T)=nf_{k,n-1,p}^{(w)}(T).\] One application of this relation replaces $f_{k+1,n,p}^{(\tilde{\mu}^{2})}$ by $f_{k+2,n,p}^{(\tilde{\mu}^{2})}+nf_{k+1,n-1,p}^{(\tilde{\mu}^{2})}$, and we also obtain \[f_{k,n,p}^{(\tilde{\mu}^{2})}=f_{k+2,n,p}^{(\tilde{\mu}^{2})}+2nf_{k+1,n-1,p}^ {(\tilde{\mu}^{2})}+n(n-1)f_{k,n-2,p}^{(\tilde{\mu}^{2})}.\] Each of the terms in Equation \eqref{fknpwcomb} thus contributes to the total coefficient in front of $f_{k+2,n,p}^{(\tilde{\mu}^{2})}$, which using the classical properties of the binomial coefficients reduces to $\big(l+\frac{b_{-}}{2}\big)\binom{l+1}{p}$. Using the recursive property of the gamma function again, we obtain the coefficient $\binom{l+1}{p}\Gamma\big(l+1+\frac{b_{-}}{2}\big)/\Gamma\big(p+\frac{b_{-}}{2} \big)$, which together with \[\frac{1}{P_{n-2-k,0,0}(2\pi y)^{n+l-1}}=\frac{4\pi^{2}y^{2}}{P_{n-2-k,0,0}(2\pi y)^{n+l+1}}\] yields the coefficient appearing in front of $f_{k+2,n,p}^{(\tilde{\mu}^{2})}(\tilde{\mu}_{+}^{2})$ in the expansion of $4\pi^{2}y^{2}\Theta_{L,k+2,n,n}^{(l+1)}(-\overline{\tau},Z)$ in Corollary \ref{eDeltaPmu}. The total coefficient in front of the function $f_{k+1,n-1,p}^{(\tilde{\mu}^{2})}$ in Equation \eqref{fknpwcomb} becomes (again, using binomial identities) just $\big(2l+\frac{b_{-}}{2}\big)\binom{l}{p}$, and the gamma quotient and the powers of $2pi y$ and $P_{1,0,0}$ complete the formula for the second asserted term. For the remaining term $n(n-1)l\binom{l-1}{p}f_{k,n-2,p}^{(\tilde{\mu}^{2})}$ from Equation \eqref{fknpwcomb} we use the functional equation of the gamma function again to write $\Gamma\big(l+\frac{b_{-}}{2}\big)$ as $\big(l-1+\frac{b_{-}}{2}\big)\Gamma\big(l-1+\frac{b_{-}}{2}\big)$, and we also decompose \[P_{n-2-k,0,0}(2\pi y)^{n+l-1}=4\pi^{2}y^{2}P_{n-2-k,0,0}(2\pi y)^{n-2+l-1}.\] Corollary \ref{eDeltaPmu} then establishes the remaining asserted term in a similar manner. This completes the proof of the lemma. \end{proof}
We go on to the \begin{proof}[Proof of Proposition \ref{LsTheta}] We prove the assertion by induction on $s$. The case $s=0$ is trivial. Denote the asserted coefficient corresponding to the $h$th term in the expression for the image under $(L^{(b_{-})})^{s}$ by $a_{s,h}(y)$. We need to evaluate \[\sum_{h}a_{s,h}(y)L^{(b_{-})}\Theta_{L,s+h,m+s-h,m+s-h}^{(h)},\] and compare it with the asserted expression for $s+1$. Lemma \ref{Ltheta} shows that for each $h$ the $L^{(b_{-})}$-image of the corresponding theta function is a linear
combination of three theta functions, which correspond to the index $s+1$ and the indices $h-1$, $h$, and $h+1$. After applying the appropriate summation index changes, the coefficient which we get in front of $\Theta_{L,s+1+h,m-s-1+h,m-s-1+h}^{(h)}$ in $(L^{(b_{-})})^{s+1}\overline{\Theta_{L,m,m,0}}$ is \[4\pi^{2}y^{2}a_{s,h-1}(y)+(m-s+h)\bigg(2h+\frac{b_{-}}{2}\bigg)a_{s,h}(y)+\] \[+\frac{(m-s+h)(m-s+h+1)(h+1)\big(h+\frac{b_{-}}{2}\big)}{4\pi^{2}y^{2}}a_{s, h+1}(y).\] Substituting the values of $a_{s,t}$ for $t$ being $h-1$, $h$, and $h+1$, one easily sees that all three terms yield the same multiplier $\frac{m!(4\pi^{2}y^{2})^{h}}{(m-s-1+h)!}$. Applying the functional equation for the gamma function in the first and third term, we obtain that the remaining expression equals \[\frac{\Gamma\big(s+\frac{b_{-}}{2}\big)}{\Gamma\big(h+\frac{b_{-}}{2}\big)} \bigg[\bigg(h-1+\frac{b_{-}}{2}\bigg)\binom{s}{h-1}+\bigg(2h+\frac{b_{-}}{2} \bigg)\binom{s}{h}+(h+1)\binom{s}{h+1}\bigg].\] The same considerations we applied for evaluating the coefficient of $f_{k+2,n,p}^{(\tilde{\mu}^{2})}$ in Lemma \ref{Ltheta} show that the expression in brackets equals $\big(s+\frac{b_{-}}{2}\big)\binom{s+1}{h}$. Applying the functional equation of the gamma function once more, this yields the asserted value of $a_{s+1,h}$. This completes the proof of the proposition. \end{proof}
Finally, we come to the \begin{proof}[Proof of Proposition \ref{Lb-Rtaum}] We begin by proving that for any $q\in\mathbb{N}$, the action of the operator $(-4\pi i)^{q}\delta_{1-\frac{b_{-}}{2}+r+t-2l,\tau}^{q}$ sends $y^{\frac{b_{-}}{2}+2l}\Theta_{L,r,s,t}^{(l)}(\tau,Z)$ to \[\sum_{h=0}^{q}\binom{q}{h}(4\pi^{2})^{h}y^{\frac{b_{-}}{2}+2l-2q+2h}\frac{ l!\Gamma\big(l+\frac{b_{-}}{2}\big)}{(l-q+h)!\Gamma\big(l-q+h+\frac{b_{-}}{2} \big)}\Theta_{L,r+h,s+h,t+h}^{(l-q+h)}(\tau,Z).\] For $q=0$ the assertion is trivially true. We write the asserted function of $y$ preceding the theta function in the term corresponding to $h$ in the sum arising from the index $q$ as $\binom{q}{h}(4\pi^{2})^{h}b_{l-q+h}(y)$. Given that this assertion holds for $q$, we apply Equation \eqref{deltakTheta} for the operator $-4\pi i\delta_{1-\frac{b_{-}}{2}+r+t-2l+2q}$ acting on each term, and observe that the resulting theta functions correspond to the index $q+1$ and to the summation indices $h+1$ and $h$. Moreover, after the usual index change manipulations one sees that the total coefficient in front of the theta function with indices $q+1$ and $h$ is \[(4\pi^{2})^{h}\bigg[\binom{q}{h-1}b_{l-q+h-1}(y)+\binom{q}{h} (l-q+h)\bigg(l-q+h+\frac{b_{-}}{2}-1\bigg)\frac{b_{l-q+h-1}(y)}{y^{2}}\bigg].\] As the second term here is easily seen to be just $\binom{q}{h-1}b_{l-q+h-1}(y)$, the inductive assertion follows from the classical property of the binomial coefficients. With $r=s=0$ and $t=l=q=m$ the general formula from above becomes \[\sum_{h}\binom{m}{h}\frac{\Gamma\big(m+\frac{b_{-}}{2}\big)}{ \Gamma\big(h+\frac{b_{-}}{2}\big)}\frac{m!(4\pi^{2}y^{2})^{h}}{h!}y^{\frac{b_{-} }{2}}\Theta_{L,h,h,m+h}^{(h)}(\tau,Z).\] On the other hand, Putting $m=s$ in Proposition \ref{LsTheta}, multiplying by $y^{\frac{b_{-}}{2}}$ (which commutes with differential operators in the variable $Z$), and taking the complex conjugate of the result, yields precisely the same expression. This proves the proposition. \end{proof}
\noindent\textsc{Einstein Institute of Mathematics, the Hebrew University of Jerusalem, Edmund Safra Campus, Jerusalem 91904, Israel}
\noindent E-mail address: [email protected]
\end{document} | arXiv |
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Multi-spectral imaging of vegetation for detecting CO2 leaking from underground
Joshua H. Rouse1 nAff2,
Joseph A. Shaw1,
Rick L. Lawrence3,
Jennifer L. Lewicki4,
Laura M. Dobeck5,
Kevin S. Repasky1 &
Lee H. Spangler5
Environmental Earth Sciences volume 60, pages 313–323 (2010)Cite this article
Practical geologic CO2 sequestration will require long-term monitoring for detection of possible leakage back into the atmosphere. One potential monitoring method is multi-spectral imaging of vegetation reflectance to detect leakage through CO2-induced plant stress. A multi-spectral imaging system was used to simultaneously record green, red, and near-infrared (NIR) images with a real-time reflectance calibration from a 3-m tall platform, viewing vegetation near shallow subsurface CO2 releases during summers 2007 and 2008 at the Zero Emissions Research and Technology field site in Bozeman, Montana. Regression analysis of the band reflectances and the Normalized Difference Vegetation Index with time shows significant correlation with distance from the CO2 well, indicating the viability of this method to monitor for CO2 leakage. The 2007 data show rapid plant vigor degradation at high CO2 levels next to the well and slight nourishment at lower, but above-background CO2 concentrations. Results from the second year also show that the stress response of vegetation is strongly linked to the CO2 sink–source relationship and vegetation density. The data also show short-term effects of rain and hail. The real-time calibrated imaging system successfully obtained data in an autonomous mode during all sky and daytime illumination conditions.
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The increased consideration of geological sequestration for reducing greenhouse gas emissions is accompanied by a need for methods to detect CO2 if it leaks from the ground into the atmosphere. Leakage of CO2 from the ground into the atmosphere (referred to as 'seepage' by Oldenburg and Unger 2003) not only represents compromised security of the sequestration facility, but also can result in health, safety, and environmental risks, including asphyxiation of humans or animals (Oldenburg et al. 2003). Whereas relatively low leak rates do not create immediate health or safety risks, they still require detection because of the significant amounts of gas that could potentially escape over a long time period (Oldenburg et al. 2003).
Within a broad class of optical remote sensing techniques, measurements can be made to directly detect the CO2 gas through optical absorption (Sakaizawa et al. 2009; Humphries et al. 2008; Repasky et al. 2006; Koch et al. 2004; Cremers et al. 2001). Alternatively, it might be possible to detect the gas indirectly through changes in the reflectance spectrum of vegetation. An extreme natural example of this occurred at Mammoth Mountain, California, where large amounts of escaping CO2 killed wide swaths of forest vegetation (Lewicki et al. 2007a; Farrar et al. 1995, 1999). Extensive studies of the tree kill at Mammoth Mountain have demonstrated the utility of satellite, airborne, and ground-based multispectral and hyperspectral remote sensing for detecting high rates of CO2 leakage indirectly through vegetation reflectance measurements (Martini and Silver 2002; Martini et al. 2000; Farrar et al. 1999).
The use of passive hyperspectral vegetation sensing for early detection and spatial mapping of CO2 leaks from underground storage facilities was formally proposed by Pickles and Cover (2004). More recently, a hyperspectral scanning imager was used with decision-tree data analysis to detect CO2 leakage (Keith et al. 2009) in a 2007 subsurface CO2 release (Lewicki et al. 2007b, 2009) at the Zero Emissions Research and Technology (ZERT) field site at Montana State University in Bozeman, Montana (Spangler et al., this issue; Lewicki et al. 2007b). A portable hyperspectral sensor was also used to detect changes in vegetation reflectance spectra correlated with CO2 leakage from the ZERT subsurface releases in 2007 and 2008 (Male et al. 2010).
Leakage detection with passive spectral imaging is based on detection of plant stress caused by elevated soil CO2 concentration. Whereas slightly elevated CO2 concentrations in the atmosphere can increase plant health and growth (Rogers et al. 1994; Kimball et al. 1993; Bazzaz and Fajer 1992), high concentrations of CO2 in the root zone can kill plants through asphyxiation (Qi et al. 1994) and soil acidification (McGee and Gerlach 1998). The CO2 concentration for air inside soil under typical conditions is only several percent (concentration at 0.5 m depth), and when this concentration becomes greater than about 10–20%, root development in plants is suppressed (Farrar et al. 1999). This produces decreased uptake of water and other nutrients, leading to stress or ultimately death of the plants. The root-zone concentration of CO2 required to kill vegetation varies with exposure time, soil moisture, soil nutrient content, and plant species. Vegetation at Mammoth Mountain was observed to exhibit stress when the soil CO2 concentration rose to 20% or greater (Farrar et al. 1999).
This paper presents results of a study using multispectral imaging to detect changes in the spectral reflectance of vegetation that are correlated with proximity to the subsurface CO2 release region. The method described here provides detection with a system that is potentially simpler and lower cost than hyperspectral imaging, and that provides continuous daytime operation in clear and cloudy weather through the use of real-time calibration using a reflectance panel deployed in the field with the imager. The use of a platform-mounted imager has demonstrated the potential of a remotely deployed instrument that avoids the need of airborne or satellite imagery and provides much higher-spatial resolution, although at the expense of reduced overall spatial coverage.
The balance of this paper is organized as follows. The methodology is described, including a description of the imaging system, its calibration, its continuous outdoor deployment at the ZERT field site in Bozeman, Montana during shallow subsurface CO2 releases in the summers of 2007 and 2008, and the data processing. Next the results of a statistical study are shown, focusing on a regression analysis of the green, red, and near-infrared (NIR) reflectances and the Normalized Difference Vegetation Index (NDVI) over time. Finally, the significance and implications of the results are discussed and the paper is concluded with suggestions for future work.
Experimental layout
During the summers of 2007 and 2008, shallow subsurface CO2 release experiments were conducted at the ZERT site located just west of Montana State University in Bozeman, Montana, at 45.66°N, 111.08°W (Spangler et al., this issue). Two vegetation test strips, one mown and one unmown, were oriented perpendicular to a 100-m long, perforated pipe buried approximately 1.3–2.5 m below land surface to simulate homogeneous leakage from a horizontal well (Spangler 2009; Lewicki et al. 2007b), as indicated in Fig. 1. The CO2 release rates were 0.1 tons/day in the 2007 experiment (9 July–23 July) and 0.3 tons/day in the 2008 experiment (7 July–7 August). Both releases produced five to six "hot spots" of elevated soil CO2 flux aligned along the surface trace of the horizontal well (e.g., Figure 1 in Lewicki et al. 2007b; this issue). We assumed that the measured CO2 flux had positive correlation with CO2 concentration.
CO2 flux map interpolated based on measurements made at the black dots for the 2008 release experiment at the ZERT facility in Bozeman, Montana (Lewicki et al., this issue). The inset box indicates the approximate area observed by the multispectral imager within the vegetation test area, divided into three test regions (near, middle, and far relative to the well location). The white line indicates the surface trace of the buried CO2 release pipe
Multispectral images of the vegetation test strips were obtained both before and after the releases in both years. As shown in Figs. 2 and 3, the imager was placed on a 3-m high platform, pointed downward at a 45° angle below the horizon to view both vegetation strips from 10 m northwest of the pipe to 0.5 m northwest of the pipe in 2007, and from 10 m northwest of the pipe to 1 m southeast of the pipe in 2008. Multi-spectral imager data were analyzed in three regions, with region 1 nearest the horizontal well, region 2 in the middle, and region 3 located farthest from the well (these regions are indicated by the numbered white boxes in Fig. 1 and also marked on the photograph in Fig. 3).
Layout of the mown and unmown vegetation test segments and the multispectral imager during the 2007 (left) and 2008 (right) CO2 release experiments. The 55° full-angle imager field of view (FOV) is indicated by the red lines
Layout of the three test regions within the mown and unmown segments of the vegetation test strip. The horizontal red line indicates the sub-surface pipe from which gas was released and the "hot spot" label indicates a location of particularly high CO2 flux, as indicated also in Fig. 1 (J. Shaw photo)
A portion of the vegetation test strip was mowed several weeks before the release started and therefore remained free of tall grass that senesced more rapidly than the underlying grasses and plants. Conversely, by the end of the experiment, the underlying vegetation in the unmown segment could only be seen by imaging through a veil of tall, mostly dead grass. This distinct difference can be seen in Fig. 3 and also in the photographs shown in Fig. 4, which were taken near the beginning and end of the 2008 release experiment. The colored circles on the right-hand side of Fig. 4 indicate the approximate locations of three individual plants we report on later.
Photographs of the vegetation test area taken from atop the multispectral imager platform on a 3 July 2008 and b 9 August 2008 to visually illustrate the changes in the condition of the vegetation in the mown (center) and unmown (right) segments of the test area (J. Shaw photos). The colored circles in b indicate the approximate locations of three individual plants discussed later
For data analysis, the mown and unmown segments were both divided into three regions at progressively greater distance from the pipe, as shown in Figs. 1 and 3. Region 1 was near the pipe in 2007 (0.5 m from the pipe at closest approach) and directly over the pipe in 2008. Regions 2 and 3 were each 4.5 m long. Regions 1 and 2 were used as test areas and region 3 was the control, since its location farthest from the pipe was expected to provide a background reading unaffected by the leaking CO2.
Multispectral imaging system and calibration
The experiments used a three-chip multispectral imager (model MS3100 from Geospatial Systems, Inc.) with customized wide-angle optics. Images were calibrated to obtain spectral band-averaged reflectances by periodically viewing a photographic grey card in 2007 and by continuously viewing a Spectralon reflectance panel in 2008. The camera splits incoming light via prisms and dichroic surfaces into green (500–580 nm), red (630–710 nm), and NIR (735–865 nm) bands that approximately match the corresponding Landsat satellite bands. Each of the three detector arrays is 7.6 mm × 6.2 mm with a pixel size of 4.65 μm × 4.65 μm, which results in 1,392 × 1,040 pixels in each image. Owing to the small detector array size, a fisheye lens adapter was added to a 20-mm f/2.8 Nikkor lens to increase the full-angle horizontal field of view from 24° to 55°. The imaging system was operated in 8-bit mode, giving 256 digital numbers (DN) for each pixel. In 2008, a computer program was employed to adjust the camera's integration time to achieve real-time exposure control based on the brightness of the pixels that were continually viewing a Spectralon reflectance calibration panel in one portion of the image.
Rigorous characterization and calibration of the entire imaging system were performed prior to deployment in the field (Rouse 2008). Parameters that were characterized included DN versus integration time, DN versus gain, DN versus f/number, DN versus radiance, DN versus camera temperature, DN versus polarization angle, pixel bleeding during readout, pixel recovery when viewing a dim scene after a bright one, and spatial non-uniformity of the camera response. For the characterizations, the imager was operated looking into an integrating sphere to ensure spatially uniform radiance across the camera's field of view. Under test conditions that simulated the expected field conditions the imager stayed within a highly linear operating range; the only characteristic that required correction was the spatial non-uniformity, partly a result of the ultra-wide-angle optics used to increase the field of view. When the imager viewed a uniformly illuminated reflectance panel, there was a nearly 50% fall off of brightness from the center to the edge of the image. To correct this, a unique linear signal-dependent calibration function was applied to each pixel of each detector array. Gain and offset terms were found for each of these calibration functions from a linear fit to five images: one dark image, images of 50 and 99% reflectance panels illuminated by direct sunlight, and images of the same two panels illuminated by diffuse shaded skylight.
Each non-uniformity-adjusted vegetation image was calibrated radiometrically with a reflectance-panel image according to Eq. 1. In this equation, subscript λ indicates the wavelength (color band), DN is the digital number for the pixel viewing the indicated object, and ρ is the reflectance (known for the calibration target, and calculated for the scene). For the 2007 experiment, a nominally 18% reflective photographic grey card was imaged every time the imager's integration time was changed because of changes in the ambient light. In the 2008 experiment, the images were calibrated by imaging a 99% reflective Spectralon panel that was part of every image. This made it possible to calibrate each image during the experiment with higher accuracy and, coupled with the previously mentioned auto-exposure algorithm, enabled operation even in highly variable weather conditions
$$ \rho _{\lambda } = \left( {{\frac{{{\text{DN}}_{{\lambda ,\,{\text{scene}}}} - {\text{DN}}_{{\lambda ,\,{\text{dark}}}} }}{{{\text{DN}}_{{\lambda ,{\text{calibration}}\,{\text{target}}}} - {\text{DN}}_{{\lambda ,\,{\text{dark}}}} }}}} \right)\rho _{{\lambda ,\,{\text{calibration}}\,{\text{target}}}}. $$
Spectral image analysis
Calibrated reflectance data from the green, red, and NIR bands were averaged over the three regions in the mown and unmown vegetation test strips and analyzed with a linear regression technique, along with the NDVI, calculated from the red and NIR bands according to Eq. 2 (Rouse et al. 1974).
$$ {\text{NDVI}} = {\frac{{\rho_{\text{NIR}} - \rho_{\text{red}} }}{{\rho_{\text{NIR}} + \rho_{\text{red}} }}}. $$
The NDVI exploits the sharp relative increase of reflectance that occurs near a wavelength of 700 nm because of strong chlorophyll absorption at visible wavelengths and high reflectance at the NIR wavelengths related to leaf structure. As vegetation becomes stressed, its NIR reflectance falls, its red reflectance rises, and the red edge near 700 nm shifts to shorter wavelengths and becomes less steep (Horlerd et al. 1983; Carter 1993; Jensen 2000; Carter and Knapp 2001). Reflectance measurements of healthy and stressed plants showed that the sensitivity of reflectance to plant stress is maximized in the 685–700 nm spectral bands (Carter 1993; Carter and Knapp 2001). These studies also showed that specific stress agents do not have spectral 'signatures,' so one should be able to detect changes in chlorophyll content, leaf anatomy, or water content by analyzing both the red and NIR portions of the spectrum, or by analyzing an index that combines these bands (Jordan 1969; Carter and Knapp 2001).
The NDVI is one particular vegetation index that combines the red and NIR band reflectances in a manner that captures the key information related to vegetation stress (Rouse et al. 1974). Experiments comparing eddy covariance measurements of ecosystem CO2 fluxes with the NDVI measured with multispectral imaging has shown high correlation (correlation coefficient = −0.981), and that the NDVI captured the effects of changing environmental conditions such as drought, recovery, and fire on the carbon flux (Fuentes et al. 2006).
This study relies on a linear regression technique applied to time series of the imager's band reflectances and NDVI data. In addition to analyzing single spectral band reflectances and NDVI, spectral band and NDVI combinations were statistically analyzed to find the best possible combination to model vegetation stress. Time was set as the response variable, spectral bands and/or NDVI were set as predictor variables, and region number was set as the categorical variable. The full linear regression model for a general fit can be seen in Eq. 3.
$$ y = \beta_{0} + \beta_{\text{G}} x_{\text{G}} + \beta_{\text{R}} x_{\text{R}} + \beta_{\text{NIR}} x_{\text{NIR}} + \beta_{\text{NDVI}} x_{\text{NDVI}} + \beta_{\tau } \tau + \beta_{{{\text{G}},\tau }} (x_{\text{G}} \times \tau ) \, + \beta_{{{\text{R}},\tau }} (x_{\text{R}} \times \tau ) + \beta_{{{\text{NIR}},\tau }} (x_{\text{NIR}} \times \tau ) + \beta_{{{\text{NDVI}},\tau }} (x_{\text{NDVI}} \times \tau ). $$
In Eq. 3, y is the decimal date (response variable), β 0 is the y-intercept (linear regression coefficient), β G is the slope for the green band, x G is the green reflectance (predictor variable), β R is the slope for the red band, x R is the red reflectance, β NIR is the slope for the NIR band, x NIR is the NIR reflectance, β NDVI is the slope for the NDVI, x NDVI is the computed NDVI value, β τ is the slope for the region, τ is the region number (categorical variable), and β G,τ, β R,τ, β NIR,τ, and β NDVI,τ are the slope terms for the interaction terms. All the β values were calculated in linear regression analysis software.
To find the best band combination, coefficients of determination (R 2) and p values were analyzed to determine how well the regressions fit the data, if the regression was significant, and if the spectral difference in vegetation regions was statistically separable. The possible differences in the spectral combinations were explored via both the intercepts and slopes from the linear regressions; a difference in slope indicates a different vegetation response to stress and a difference in intercept most likely means that the vegetation started at different health values. The variables were selected using a stepwise regression selection procedure based on an extra-sum-of-squares F test.
The regression analysis showed that the NDVI was the most consistent choice for explaining the variability in vegetation health and was strongest for statistically separating regions. A single best regression equation was used in this study (the form of which is shown in Eq. 4) for analysis of each year's data and individual plants to standardize the analysis.
$$ {\text{Date}} = \beta_{0} + \left( {\beta_{\text{NDVI}} \times {\text{NDVI}}} \right) + \tau_{{{\text{i}},1 - 2}} + \left( {\tau_{{{\text{s}},1 - 2}} \times {\text{NDVI}}} \right) + \tau_{{{\text{i}},1 - 3}} + \left( {\tau_{{{\text{s}},1 - 3}} \times {\text{NDVI}}} \right) + \tau_{{{\text{i}},2 - 3}} + \left( {\tau_{{{\text{s}},2 - 3}} \times {\text{NDVI}}} \right). $$
Here, date is the response variable, β 0 is the offset, β NDVI is the slope, NDVI is the predictor variable, the three τ i variables account for differing intercepts (and therefore different starting conditions among regions), while the three τ s variables account for the different slopes (and therefore rates of response) among the regions.
The NDVI was most consistent in that it had the highest R 2 values for both the 2007 and 2008 unmown segments, but had slightly lower values (by less than 0.065) than combinations of red-NIR and green-NIR-NDVI for the mown segments and the individual plants, respectively. More importantly, the NDVI alone was best able to statistically separate vegetation regions in every case. The NDVI allows more effective modeling of interactions between spectral responses and regions, although spectral band combinations are best for explaining variability in the reflectance spectrum itself (Maynard et al. 2006; Lawrence and Ripple 1998).
After exploring various combinations of band reflectances and NDVI, linear regressions only involving NDVI were used, even though according to Robinson et al. (2004) a regression involving a spectral band interaction term (such as NDVI, which involves both NIR and red reflectances) without the individual bands is not statistically ideal. This was done because when the band reflectances were included, the ability of the regression to statistically separate regions and the significance of each of the bands and NDVI toward the regression were diminished. It was also found from the 2007 and 2008 field experiments that the use of a robust calibration technique increases the accuracy of a plant stress detection system, enough that the effects of a small increase in CO2 concentration, rain, and hail are all detectable, even in cloudy conditions.
In the rest of this section, results are shown in the form of tables and graphs of regression parameters and lines plotted with the measured data. This is done separately for the mown and unmown segments of the vegetation test area for both the 2007 and 2008 experiments. Although the regressions were calculated as time-versus-NDVI (or reflectance), we plot the results with time on the abscissa to simplify physical understanding of the temporal evolution. These results are interpreted and discussed further in the next section of the paper.
2007 experiment results
The 2007 experiment resulted in a limited amount of useful data because of calibration difficulties caused by an unexpectedly non-Lambertian grey card reflectance (some grey cards were found to be quite Lambertian, while others were found to be highly specular). Even still, images selected with the grey card held at the proper angle gave a sufficiently diffuse, approximately 18% reflection that provided a usable calibration for the imager. NDVI data obtained from image regions that should be separable did turn out to be statistically separable (p value less than 0.05) with high coefficients of determination (R 2), from 0.4472 to 0.7256.
2007 results for the mown segment
Results from the application of the previously described statistical analysis to 2007 mown-segment data are shown in tables and graphs in this subsection. Table 1 lists the R 2 and p values that indicate moderately high correlation of the NDVI and time (used as the response variable). Table 2 lists the p values that distinguish between regions at different distance from the buried pipe. The data and resulting regression lines for mown-segment 2007 data are shown graphically in Fig. 5. This graph shows the trend of weaker plant stress in region 2 relative to region 1 (nearest the pipe), measured by the change in NDVI over time during the CO2 release, although these differences were marginally significant and not significant at α = 0.05. The blue regression line for region 3 appears skewed because of the lack of variability of NDVI over time in this control region located farthest from the pipe and both regions 1 and 2 were statistically significantly different than region 3 (α = 0.05).
Table 1 2007 mown segment date-versus-NDVI regression R 2 and p values
Table 2 2007 mown segment date-versus-NDVI regression p values that distinguish between vegetation regions
Plot of 2007 mown-segment date-versus-NDVI regression lines and measured data for regions 1 (green), 2 (red), and 3 (blue)
2007 results for the unmown segment
Results of the statistical analysis applied to 2007 data in the unmown segment are shown in tables and graphs in this subsection. Table 3 lists the R 2 and p values that indicate high correlation of the NDVI and time (the response variable). Table 4 lists the p values that distinguish between regions at different distance from the buried pipe. The data and resulting regression lines for 2007 data in the unmown segment are shown graphically in Fig. 6. This graph again shows the trend of weaker plant stress in region 2 relative to region 1 (nearest the pipe), measured by the change in NDVI over time during the CO2 release. Differences between regions 1 and 2 again were marginally significant, while differences between regions 1 and 3 were significant, although regions 2 and 3 were not significantly different.
Table 3 2007 unmown segment date-versus-NDVI regression R 2 and p values
Table 4 2007 unmown segment date-versus-NDVI regression p values that distinguish between vegetation regions
Plot of 2007 unmown segment date-versus-NDVI regression lines and measured data for regions 1 (green), 2 (red), and 3 (blue)
In the 2008 experiment, better calibration techniques led to good data being collected every day the system was operated correctly. NDVI data obtained from image regions that should be separable did, in fact, turn out to be statistically separable (p value less than 0.05) with high coefficients of determination (R 2). For the 2008 data, the date-versus-NDVI regression plots include horizontal green and red lines to indicate the beginning and end of the gas release, respectively. Similarly, rain events are indicated with dashed black horizontal lines and hail events are shown as solid black horizontal lines.
Results of the statistical analysis applied to 2008 data in the mown segment are shown in tables and graphs in this subsection. Table 5 lists the R 2 and p values that indicate high correlation of the NDVI and time (the response variable). Table 6 lists the p values that distinguish between regions at different distance from the buried pipe. The data and resulting regression lines for 2008 data in the mown segment are shown graphically in Fig. 7. This graph shows a trend of increasing plant vigor throughout the release experiment. However, the rate of increase is lower in regions 1 relative to region 2, while regions 1 and 3 are not statistically different in this case, compatible with the hypothesis that higher CO2 flux in region 1 (near the release pipe; Fig. 1) was inhibiting plant vigor relative to the gains made in region 2. Plant vigor is also correlated with frequent and heavy rain events (Fig. 7).
2008 Mown segment date-versus-NDVI regression lines and measured data for regions 1 (green), 2 (red), and 3 (blue). The start and end of the CO2 release are marked by the vertical green and red line lines. Rain events are marked by vertical black dashed lines and hail events are marked by vertical black solid lines
Results of the statistical analysis applied to 2008 data in the unmown segment are shown in tables and graphs in this subsection. Table 7 lists the R 2 and p values that indicate very high correlation of the NDVI and time (the response variable). Table 8 lists the p values that distinguish between regions at different distance from the buried pipe. The data and resulting regression lines for 2008 data in the unmown segment are shown graphically in Fig. 8. In contrast to the increasing plant vigor observed over time in the mown segment, this figure indicates a steady decrease of vigor (increase of plant stress) over the course of the 2008 release experiment. This difference can be understood by observing the veil of tall, senesced prairie grass that partially obscures the imager's view to the relatively healthy underlying vegetation, as is seen in the photographs of Fig. 3. Despite this veil of senesced tall grass, the NDVI still indicates a higher rate of increased plant stress in region 1 (near the pipe) relative to regions 2 and 3 (but essentially indistinguishable rates between regions 1 and 2).
2008 Unmown segment date-versus-NDVI regression lines and measured data for regions 1 (green), 2 (red), and 3 (blue). Vertical lines indicate start (green), stop (red), rain (black dashed), and hail (black solid), as in Fig. 7
2008 results for individual plants within unmown segment
The same statistical analysis also was applied to 2008 data at the locations of three individual plants that had been observed closely with a hand-held hyperspectral sensor (Male et al. 2010). This was done to assess the ability of the platform-mounted multispectral imager to monitor detail at the single-plant level. The results for these three individual plants are shown in this subsection. Table 9 lists the R 2 and p values that indicate high correlation of the NDVI and time (the response variable) for these individual plants. Table 10 lists the p values that distinguish between individual plants and the unmown region 3. The data and resulting regression lines for the individual plants are shown graphically in Fig. 9. The stress for all three plants increased at similar rates during the period of the release, but stopped increasing almost immediately after the CO2 stopped flowing.
Table 9 Individual plants' date-versus-NDVI regression R 2 and p values
Table 10 Individual plants' date-versus-NDVI regression p values that distinguish between individual plants and the unmown region 3
2008 Individual plant date-versus-NDVI regression lines and measured data for the unmown segment. Plant locations are adjacent to hot spot, as indicated approximately with colored circles in Fig. 4b. Vertical lines are the same as in Figs. 7 and 8
The data obtained during the 2007 and 2008 release experiments indicate the ability of a multispectral imaging system to detect plant stress or nourishment that is correlated with increased CO2 concentrations resulting from a simulated leak. Results from both 2007 and 2008 are consistent with the hypothesis that elevated CO2 flux in region 1 (near the pipe) leads to increased plant stress that manifests itself as a higher rate of decay or lower rate of increase in the NDVI. With increased CO2 levels, and depending on sink–source balance, there is a noticeable increased stress or decrease in plant vigor. The connection between NDVI and CO2 levels is supported further by the logarithmic map of CO2 flux measured at the ZERT field site during the 2008 release experiment (Lewicki et al., this issue), which is shown in Fig. 1. The east side of the vegetation test area is on the edge of a hot spot where the vegetation was entirely dead within several weeks of the beginning of the release. Overall, the CO2 flux is above-background levels in the eastern areas of regions 1 and 2 (within about 5 m of the release pipe) and at background levels in region 3.
Owing to a veil of tall, senesced prairie grass obscuring a clear view of the underlying vegetation (see Fig. 3), the stress signatures were suppressed in the data for the unmown segment. Nevertheless, there was statistically significant separation of the regression lines for regions 1 and 2 relative to region 3 (Fig. 8). Ongoing studies are exploring this further through higher-spatial resolution processing of multispectral imager data.
For the 2008 experiment, soil moisture and precipitation data were also available to help determine how the change in plant health was driven by rain and hail events. Although the regression results show that the overall NDVI trends are highly correlated with increased CO2 fluxes, short-term effects of rain and hail can also be seen as short-term correlated fluctuations in rain, soil moisture, and measured NDVI. As indicated in the plot of soil moisture data versus time shown in Fig. 10, there was generally a drying trend, punctuated by large short-term increases in soil moisture after rain and hail events that actually resulted in the net change being slightly positive (i.e., the soil moisture at the end of the release was slightly higher than at the beginning of the release). The net increase of soil moisture throughout the release suggests that the stress seen in the NDVI plots is not caused primarily by a lack of soil moisture, although there are other factors that also need to be explored in future studies. In agreement with arguments about sink–source importance (Arp 1991), the dense vegetation in the unmown segment had to compete more than the vegetation in the mown segment, shown by smaller relative changes in soil moisture, thereby allowing the mown segment to become healthier.
Time-series plot of volumetric soil moisture near the vegetation test area during the 2008 release experiment. The general drying trend was offset by significant periodic rainfall, which led to the soil moisture being comparable at the beginning (7 July 2008 = day 189) and end (7 Aug 2008 = day 220) of the release
It is useful to note that the improved calibration and operation methods in the 2008 experiment made it possible to see effects of rain and hail in the multispectral data. The NDVI plots show rain events as dashed black lines and hail events as solid black lines. These plots, along with the precipitation and soil moisture data (shown in Fig. 10), show that directly after a hail storm the vegetation NDVI decreases, but as the water saturates the ground and the moisture is taken in by the vegetation, within a day or two the detectable plant vigor increases. This increase in vigor can also be seen immediately after significant rainfalls without hail (dashed horizontal lines).
A platform-mounted multispectral imager detected statistically significant changes in the reflectance spectra of vegetation that is strongly correlated with leaking CO2. This was shown by linking increased CO2 concentration with plant stress or nourishment, ruling out the effects of water, considering sink–source balance, and using NDVI to show that regions of stressed vegetation and regions of non-stressed vegetation are statistically separable.
The multispectral imager was run in a mostly autonomous mode, in both clear and cloudy conditions, whereas some other sensors require sustained periods of clear skies to operate reliably. This was made possible by placing the imager in a weatherproof box and by placing a high-quality reflectance calibration target in the field of view of the imager so that it is imaged along with the vegetation in every image. The robust calibration and continuous operating mode also allowed finer-scale, short-term effects caused by rain and hail to be observed in the data.
It was found that this system was best able to detect stress effects from the CO2 leak when the vegetation region in question was left unmown. This is because of the competing sink–source relationship between densely and less densely packed vegetation. When the vegetation is left in its natural state, there will be a higher density of vegetation, and drying during warm summer months will create more competition for water, thereby throwing off the sink–source relationship and enhancing the effects from CO2. The CO2 effects can still be observed in the mown segment, but the correlation was stronger for the unmown segment data.
This study suggests that future work is warranted to produce an even lower-cost system that can operate autonomously for robust multispectral vegetation stress detection. Such a system with sufficiently low cost could be reproduced to, for example, sample multiple portions of a large area overlying a sequestration facility. Work is ongoing at Montana State University to develop and test exactly such a system, using a low-cost camera with a filter wheel, custom-wide angle optics, and integrated microcontroller for data acquisition.
NDVI:
ZERT:
Zero Emissions Research and Technology
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This paper was prepared with the support of the U.S. Department of Energy, under Award No. DE-FC26-04NT42262. However, any opinions, findings, conclusions, or recommendations expressed herein are those of the authors and do not necessarily reflect the views of the DOE. The authors express gratitude to the many colleagues who made working at the ZERT site productive and enjoyable.
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Joshua H. Rouse
Present address: ITT Space Systems Division, Rochester, NY, 14606, USA
Electrical and Computer Engineering Department, Montana State University, Bozeman, MT, 59717, USA
Joshua H. Rouse, Joseph A. Shaw & Kevin S. Repasky
Land Resources and Environmental Sciences Department, Montana State University, Bozeman, MT, 59717, USA
Rick L. Lawrence
Earth Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA
Jennifer L. Lewicki
Chemistry and Biochemistry Department, Montana State University, Bozeman, MT, 59717, USA
Laura M. Dobeck & Lee H. Spangler
Joseph A. Shaw
Laura M. Dobeck
Kevin S. Repasky
Lee H. Spangler
Correspondence to Joseph A. Shaw.
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Rouse, J.H., Shaw, J.A., Lawrence, R.L. et al. Multi-spectral imaging of vegetation for detecting CO2 leaking from underground. Environ Earth Sci 60, 313–323 (2010). https://doi.org/10.1007/s12665-010-0483-9
Issue Date: March 2010
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On the security of compressed encryption with partial unitary sensing matrices embedding a secret keystream
Nam Yul Yu ORCID: orcid.org/0000-0001-9265-712X1
EURASIP Journal on Advances in Signal Processing volume 2017, Article number: 73 (2017) Cite this article
The principle of compressed sensing (CS) can be applied in a cryptosystem by providing the notion of security. In this paper, we study the computational security of a CS-based cryptosystem that encrypts a plaintext with a partial unitary sensing matrix embedding a secret keystream. The keystream is obtained by a keystream generator of stream ciphers, where the initial seed becomes the secret key of the CS-based cryptosystem. For security analysis, the total variation distance, bounded by the relative entropy and the Hellinger distance, is examined as a security measure for the indistinguishability. By developing upper bounds on the distance measures, we show that the CS-based cryptosystem can be computationally secure in terms of the indistinguishability, as long as the keystream length for each encryption is sufficiently large with low compression and sparsity ratios. In addition, we consider a potential chosen plaintext attack (CPA) from an adversary, which attempts to recover the key of the CS-based cryptosystem. Associated with the key recovery attack, we show that the computational security of our CS-based cryptosystem is brought by the mathematical intractability of a constrained integer least-squares (ILS) problem. For a sub-optimal, but feasible key recovery attack, we consider a successive approximate maximum-likelihood detection (SAMD) and investigate the performance by developing an upper bound on the success probability. Through theoretical and numerical analyses, we demonstrate that our CS-based cryptosystem can be secure against the key recovery attack through the SAMD.
Compressed sensing (CS) [1–4] is a novel data acquisition scheme that samples a signal at a sub-Nyquist rate, which allows simultaneous data acquisition and compression. The original signal can be faithfully recovered from the measurement samples, if it is sparse with respect to a particular basis and sampled via a random projection. With efficient measurement and stable reconstruction, the CS technique has been of interest in a variety of research fields, e.g., communications [5–7], sensor networks [8–10], image processing [11–13], and radar [14].
Recently, a great deal of attention has been paid to the CS technique for data confidentiality in information security field. A CS-based cryptosystem encrypts a plaintext through a CS measurement process by keeping the sensing matrix secret. Then, the ciphertext can be decrypted by a CS reconstruction process. Thus, the CS-based cryptosystem performs simultaneous data acquisition and encryption at physical layer. Such a lightweight cryptosystem is particularly attractive for secure communications in wireless sensor networks, where the resources are not sufficient for providing data confidentiality by conventional encryption.
The security potential of compressed sensing was hinted by Candes and Tao [3], where the measurement samples were referred to as a weakly encrypted ciphertext. In [15], Rachlin and Baron proved that the CS-based cryptosystem cannot be perfectly secure but might be computationally secure. Orsdemir et al. [16] showed that it is computationally secure against a key search technique via an algebraic approach. Subsequently, many researchers have studied the security of CS-based cryptosystems for practical applications, which will be discussed with more details in Section 2.3. For a comprehensive review of CS techniques in information security, readers are referred to [17].
In this paper, we study the computational security of a CS-based cryptosystem that encrypts a plaintext with a partial unitary sensing matrix embedding a secret keystream. The keystream to be embedded is obtained by a keystream generator of stream ciphers, which ensures fast and efficient generation of the keystream. Assuming that the keystream is part of the original one with an extremely long period, we renew it at each encryption, which leads to a one-time sensing (OTS) cryptosystem. Then, the initial seed (or state) of the original keystream generator is essentially the secret key of the CS-based cryptosystem. With the sensing matrix, we demonstrate that the CS-based cryptosystem theoretically guarantees a stable and robust CS decryption for a legitimate recipient.
For security analysis, we first use probability metrics to investigate the security in a statistical manner. The total variation (TV) distance [18] between probability distributions of ciphertexts conditioned on a pair of plaintexts is examined as a security measure for the indistinguishability [19] of our CS-based cryptosystem. We investigate the TV distance by developing upper bounds on the relative entropy [20] and the Hellinger distance [21], which demonstrates that our CS-based cryptosystem can be computationally secure in terms of the indistinguishability, as long as the keystream length for each encryption is sufficiently large with low compression \(\left (\frac {M}{N} \right)\) and sparsity \(\left (\frac {K}{N} \right)\) ratios.
Next, we analyze the security of our CS-based cryptosystem by examining the resistance against a cryptanalytic attack. We consider a potential chosen plaintext attack (CPA) from an adversary to recover the key of our CS-based cryptosystem. In the CPA, the adversary needs to restore a keystream embedded in CS encryption, which is nontrivial unlike in stream ciphers, since the keystream is not outstanding from a known plaintext-ciphertext pair. Associated with the key recovery attack, we show that the security of our CS-based cryptosystem is based on the mathematical intractability of a constrained integer least-squares (ILS) problem. For a sub-optimal, but feasible key recovery attack, we consider a successive approximate maximum-likelihood (ML) detection (SAMD) for the adversary's CPA and investigate the performance by developing an upper bound on the success probability. Finally, theoretical analysis and numerical results reveal that our CS-based cryptosystem can be secure against the key recovery attack through the SAMD.
This paper is organized as follows. Section 2 reviews the CS principle, discusses some known CS-based cryptosystems, and summarizes the contributions of this paper. In Section 3, we describe a mathematical model of the CS-based cryptosystem proposed by this paper. We discuss a theoretical guarantee of CS decryption for a legitimate recipient by the cryptosystem. In Section 4, we analyze the indistinguishability of our CS-based cryptosystem, to demonstrate the computational security. Section 5 introduces an adversary's potential CPA strategy for key recovery, where we describe the details and examine the performance of SAMD. Section 6 presents numerical results to demonstrate the reliability and the security of our CS-based cryptosystem. Finally, concluding remarks will be given in Section 7.
A matrix (or a vector) is represented by a boldface upper (or lower) case letter. U T and |U| denote the transpose and the determinant of a matrix U, respectively. tr(U) denotes the trace of a matrix U or the sum of all diagonal entries of U. U(k,t) is an entry of an M×N matrix U in the kth row and the tth column, where 0≤k≤M−1 and 0≤t≤N−1. μ(U) denotes the maximum magnitude of the entries of U, i.e., \(\mu (\mathbf {U}) = \underset {k, t}{\max } |\mathbf {U}(k,t)|\). diag(s) is a diagonal matrix whose diagonal entries are from a vector s. An identity matrix is denoted by I, where the dimension is determined in the context. W is a conventional N×N Walsh-Hadamard matrix, where W W T=W T W=N I. Also, D denotes a discrete-cosine transform (DCT) matrix, where D D T=D T D=N I. For a vector \(\mathbf {x} = (x_{0},\cdots, x_{N-1})^{T} \in \mathbb {R}^{N}\), the l p -norm of x is denoted by \( || \mathbf {x} ||_{p} = \left (\sum _{k=0}^{N-1} |x_{k}|^{p} \right)^{\frac {1}{p}} \), where 1≤p<∞. If the context is clear, ||x|| denotes the l 2-norm of x. A vector \(\mathbf {n} \sim {\mathcal {N}} \left (\mathbf {0}, \sigma ^{2} \mathbf {I}\right)\) is a Gaussian random vector with mean 0=(0,⋯,0)T and covariance σ 2 I. Finally, \(\mathbb {E}[\!\cdot ]\) denotes the average of a random vector or a random matrix.
Table 1 summarizes the abbreviations of this paper.
Table 1 Abbreviations
Compressed sensing
Compressed sensing (CS) [1–3] is to recover a sparse signal from the measurements that are believed to be incomplete. A signal \(\mathbf {x} \in \mathbb {R}^{N}\) is called K-sparse with respect to a sparsifying (orthonormal) basis Ψ if θ=Ψ x has at most K nonzero entries, where K≪N. The sparse signal x is linearly measured by \(\mathbf {r} = \boldsymbol {\Phi } \mathbf {x} + \mathbf {n} = \boldsymbol {\Phi } \boldsymbol {\Psi }^{T} \boldsymbol {\theta } + \mathbf {n} \in \mathbb {R}^{M} \), where Φ is an M×N measurement matrix with M≪N and \(\mathbf {n} \in \mathbb {R}^{M}\) is a measurement noise. The CS theory states that if the sensing matrix A=Φ Ψ T obeys the restricted isometry property (RIP) [2], a stable and robust reconstruction of θ can be guaranteed from the incomplete measurement r. The CS reconstruction is accomplished by solving the l 1-minimization problem of
$$ \boldsymbol{\widehat{\theta}} = \underset{\boldsymbol{\theta}}{\text{argmin}} || {\boldsymbol{\theta}} ||_{1} \text{subject to } ||\mathbf{A} {\boldsymbol{\theta}} - \mathbf{r}||_{2} \leq \epsilon $$
with convex optimization or greedy algorithms [4]. For simplicity, this paper assumes Ψ=I, or that x is sparse in canonical basis, which yields the sensing matrix of A=Φ.
Prior works on CS-based cryptosystems
Since the foundational works of [15] and [16], there have been many research efforts on CS-based cryptosystems. Bianchi, Bioglio, and Magli [22, 23] analyzed the security of a noiseless CS-based cryptosystem utilizing random Gaussian sensing matrices in an OTS manner. In [24], a similar analysis has been made for a noiseless CS-based cryptosystem having a circulant sensing matrix for efficient CS processes. Cambareri et al. [25] proposed a CS-based cryptosystem that supports multiclass encryption using a random Bernoulli matrix and its class-dependent variations. In spite of exploiting different security measures, i.e., indistinguishability [23] and asymptotic spherical security [25], the security analyses of [23] and [25] showed that the statistical properties of ciphertexts reveal only the information about the energy of the plaintexts. The security of the multiclass encryption scheme has been further investigated in [26] against a known plaintext attack (KPA), by examining the average number of candidate solutions matching a plaintext-ciphertext pair.
In addition to the secret sensing matrix, a CS-based cryptosystem may employ an extra cryptographic primitive, which can be considered as a product cipher. For instance, scrambling or random permutation has been additionally accomplished, before [27] or after [28] CS encryption. In [29], nonlinear diffusion has been added to quantized ciphertexts. Zhang et al. [30] proposed a bi-level protected CS (BLP-CS), where the sparsifying basis and the sensing matrix are generated with different secret keys. In the BLP-CS, the knowledge of both the sparsifying basis and the sensing matrix is required for CS decryption.
To gain a resistance against KPA and CPA, a CS-based cryptosystem normally operates in an OTS manner, by renewing the sensing matrix at each encryption. As the renewal requires the additional complexity and can quickly waste up the cryptographic resource for generating each sensing matrix, a CS-based cryptosystem reusing the sensing matrix during multiple encryptions has also been of interest. However, it is insecure against KPA and CPA, since an adversary can easily recover the sensing matrix with N linearly independent plaintexts by solving the system of linear equations [15]. While reusing the same sensing matrix, the BLP-CS [30] attempted to overcome the weakness and to achieve a CPA-resistance by ensuring a RIPless reconstruction for an adversary.
CS-based cryptosystems can work in a framework of physical layer security [31]. The emerging technology of physical layer security is a promising paradigm for enhancing wireless security [32], by exploiting the randomness of wireless channel characteristics. In [33], Agrawal and Vishwanath derived sufficient conditions for secret communications via CS in a wiretap channel. Reeves at al. [34] investigated the secrecy capacity of a wiretap channel employing CS. Dautov and Tsouri [35] used the received signal strength indicator (RSSI) from wireless channels for secure key establishment in a CS-based cryptosystem, where the shared key can be used to form a common sensing matrix in a sender and a recipient. In practice, a variety of CS-based cryptosystems concerning the security and privacy of multimedia, imaging, and smart grid data have been suggested and studied in [36–39].
Summary of contributions
The main results of this paper are summarized in comparison with prior works. Our CS-based cryptosystem encrypts a plaintext with a partial unitary sensing matrix embedding a secret keystream, which is used only once for each encryption. Thus, it operates in an OTS manner, similar to those of [22–25], but different from the BLP-CS [30]. It can further reduce the consumption of the cryptographic resource by renewing only the keystream of length N, not replacing the entire M×N sensing matrix, at each encryption. Unlike the BLP-CS, our CS-based cryptosystem uses only a single cryptographic primitive, or the secret keystream, while keeping the sparsifying basis public. Furthermore, the secret keystream can be efficiently generated by a keystream generator of stream ciphers. Based on the RIP analysis, the knowledge of the sensing matrix, or equivalently the keystream, theoretically guarantees a reliable CS decryption.
In security analysis, we obtain the result by two different approaches. On the one hand, we demonstrate the indistinguishability of our CS-based cryptosystem, by investigating the TV distance between probability distributions of a pair of ciphertexts. This statistical approach seems like the analysis of [23], but we use a new probability metric of the Hellinger distance [21] to characterize the TV distance. On the other hand, we consider a potential CPA from an adversary for key recovery of our CS-based cryptosystem. By formulating the CPA as an NP-hard problem, we show that the success of the CPA is computationally infeasible for a sufficiently large keystream length. In addition, we introduce a sub-optimal but feasible CPA strategy and investigate the performance with the highest possible success probability. Finally, the CPA performance turns out to be quite poor even under an optimistic scenario, which guarantees the security against the CPA for our CS-based cryptosystem. The second type of security analysis is new in this paper.
Mathematical model
CS encryption with a partial unitary sensing matrix
A CS-based cryptosystem encrypts a sparse plaintext \(\mathbf {x} \in \mathbb {R}^{N}\) through the CS measurement process with a sensing matrix \(\boldsymbol {\Phi } \in \mathbb {R}^{M \times N}\), which produces the ciphertext \(\mathbf {r} = \boldsymbol {\Phi } \mathbf {x} + \mathbf {n} \in \mathbb {R}^{M}\), where \(\mathbf {n} \sim \mathcal {N}\left (\mathbf {0}, \sigma ^{2} \mathbf {I}\right)\) is a measurement noise. This paper proposes a CS-based cryptosystem that employs a partial unitary sensing matrix Φ embedding a secret keystream, as defined in Definition 1.
The sensing matrix1 of our CS-based cryptosystem is defined by
$$ \boldsymbol{\Phi} = \frac{1}{\sqrt{M}} \mathbf{R}_{\Omega} \mathbf{U} = \frac{1}{\sqrt{MN}} \mathbf{R}_{\Omega} \mathbf{U}_{1} {diag}(\mathbf{s}) \mathbf{U}_{2}. $$
In (1), R Ω is a public random subsampling operator that selects M rows out of N ones uniformly at random, where the selected indices are specified by Ω={ω 0,⋯,ω M−1}. Also, \(\mathbf {U}_{i} \in {\mathbb {R}}^{N \times N} \) is a unitary matrix, i.e., \(\mathbf {U}_{i}^{T} \mathbf {U}_{i} = \mathbf {U}_{i} \mathbf {U}_{i}^{T} = N \mathbf {I}\) for i=1 and 2, respectively. In particular, each entry of U 1 has unit magnitude, i.e., |U 1(k,t)|=1 for all 0≤k,t≤N−1. Finally, \(\mathbf {U} = \frac {1}{\sqrt {N}} \mathbf {U}_{1} { diag}(\mathbf {s}) \mathbf {U}_{2} \) is also unitary for s∈{−1,+1}N, where s is a secret keystream to be embedded in Φ for each CS encryption.
In this paper, we use U 1=H, or an N×N Hadamard matrix that employs a binary m-sequence [40] of period N−1=2n−1 for a positive integer n, i.e., \(\mathbf {d} = \left (d_{0}, \cdots, d_{2^{n}-2} \right)\), where d k ∈{0,1}. For 0≤k,t≤N−1, each entry of H is given by
$$\mathbf{H}(k, t) = \left\{ \begin{array}{ll} 1, & \quad \text{if}\ k=0\ \text{or}\ t=0, \\ (-1)^{d_{k+t-2}}, & \quad \text{otherwise,} \end{array} \right. $$
where the index k+t−2 is computed modulo 2n−1. From the structure, H is symmetric, or H T=H. As d has the ideal two-level autocorrelation [40], i.e.,
$$\sum\limits_{k = 0}^{2^{n}-2} (-1)^{d_{k} + d_{k + \tau}} = \left\{ \begin{array}{ll} 2^{n}-1, & \quad \text{if } \tau = 0, \\ -1, & \quad \text{if } 1 \leq \tau \leq 2^{n}-2, \end{array} \right. $$
where k+τ is computed modulo 2n−1, it is obvious that H H T=H T H=N I. Since H is public, the structure and the initial state of an n-stage linear feedback shift register (LFSR) generating the binary m-sequence d are publicly known.
Keystream generation for CS encryption
In the sensing matrix Φ of (1), we assume that s is a segment of length N from the original keystream of an extremely long period, which enables to renew the keystream s at each CS encryption. For fast and efficient keystream generation, one may employ an LFSR-based nonlinear keystream generator of stream ciphers. For example, we may consider the combinatorial sequence generator [41], the filtering sequence generator [42], the clock-controlled generator [43, 44], the shrinking generator [45], and the self-shrinking generator (SSG) [46], each of which presents a simple structure but a remarkable resistance against various attacks. For more details on keystream generators and stream ciphers, see [47] and [48]. Regarding the keystream of our CS-based cryptosystem, we make the following assumption.
Assumption 1
An original keystream from a stream cipher is designed to have nice pseudorandomness properties [40] such as balance, large period, low autocorrelation, and large linear complexity. With the properties, we assume that each element of the keystream s takes +1 or −1 independently and uniformly at random, which facilitates the security analysis of our CS-based cryptosystem.
When we employ a keystream generator to produce the keystream s, the initial seed (or state) of the generator is essentially the key of our CS-based cryptosystem. The key should be kept secret between a sender and a legitimate recipient, whereas the structure of the keystream generator can be publicly known. For secure key exchange, we may establish a separate secure channel, or use the key establishment via the RSSI from wireless channels as in [35].
CS decryption
For CS decryption, a noisy ciphertext \(\mathbf {r} = \boldsymbol {\Phi } \mathbf {x} + \mathbf {n} \in {\mathbb {R}}^{M} \) is available for an adversary as well as a legitimate recipient, where \(\mathbf {n} \sim {\mathcal {N}}\left (\mathbf {0}, \sigma ^{2} \mathbf {I}\right)\) is a measurement noise. A legitimate recipient of the ciphertext r, who knows Φ, attempts to recover the plaintext x by conducting a CS reconstruction. Meanwhile, an adversary will make various attempts to recover the plaintext x or the keystream s, with no knowledge of Φ.
Proposition 1 presents the reliability and the stability of our CS-based cryptosystem for a legitimate recipient, which is from the RIP result [49, 50] of a partial unitary sensing matrix.
[49, 50] For a legitimate recipient, our CS-based cryptosystem theoretically guarantees a stable decryption of a K-sparse plaintext with bounded errors, as long as \(M = \mathcal {O}\left (\mu ^{2}(\mathbf {U}) \cdot K\log ^{4}N \right)\).
When U 1=H, numerical experiments revealed that \(\mu (\mathbf {U}) = {\mathcal {O}}\left (\sqrt {\log N}\right) \) for i) U 2=W or ii) U 2=D, if each entry of the keystream s takes +1 or −1 uniformly at random. In this case, if \(M = \mathcal {O}\left (K\log ^{5}N\right)\), Proposition 1 guarantees a stable decryption.
Table 2 summarizes a symmetric-key CS-based cryptosystem proposed in this paper.
Table 2 Symmetric-key CS-based cryptosystem
A CS-based cryptosystem cannot be perfectly secure [15] but is believed to be computationally secure [15, 16]. In this section, we analyze the computational security of our CS-based cryptosystem by studying the notion of indistinguishability [19].
Assume that a cryptosystem produces a ciphertext by encrypting one of two possible plaintexts. The cryptosystem is said to have the indistinguishability, if no adversary can determine in polynomial time which of the two plaintexts corresponds to the ciphertext, with probability significantly better than that of a random guess [51]. In short, if a cryptosystem has the indistinguishability, an adversary is unable to learn any partial information of the plaintext in polynomial time from a given ciphertext.
In specific, let us consider an indistinguishability experiment [51] with a constraint of K-sparse plaintexts. First of all, an adversary creates a pair of plaintexts x 1 and x 2 with at most K nonzero entries per each. Then, our CS-based cryptosystem produces a ciphertext r=Φ x h +n by randomly selecting h, where h=1 or 2. Given r, the adversary attempts to figure out which plaintext, x 1 or x 2, was encrypted for the ciphertext, by carrying out a polynomial time test \({\mathcal {D}}: \mathbf {r} \rightarrow h \in \{1, 2\}\).
In this paper, we make use of the total variation (TV) distance [18] to evaluate the performance of the indistinguishability experiment. Let d TV(p 1,p 2) be the TV distance between the probability distributions p 1=Pr(r|x 1) and p 2=Pr(r|x 2). Then, it is readily checked from [52] that the probability that an adversary can successfully distinguish the plaintexts by some kind of the binary hypothesis test \({\mathcal {D}}\) is bounded by
$$ p_{d} \leq \frac{1}{2} + \frac{d_{\text{TV}} (p_{1}, p_{2}) }{2}. $$
Therefore, if d TV(p 1,p 2) approaches to zero, the probability of success will be at most that of a random guess, which leads to the indistinguishability of a cryptosystem. Consequently, one can argue that a cryptosystem with d TV(p 1,p 2) closer to zero would be more secure in terms of the indistinguishability. Since computing d TV(p 1,p 2) directly is difficult [53], we compute two probability metrics instead to bound the TV distance, which ultimately examines the indistinguishability of our CS-based cryptosystem.
In [23] and [24], the relative entropy (or the Kullback-Leibler divergence [20]) has been used to quantify the indistinguishability. Precisely, the relative entropy of two probability distributions gives an upper bound on the TV distance by Pinsker's inequality [54] or the refinements [55], which ultimately bounds the success probability of the indistinguishability experiment by (2).
In (1), one may assume that the entries of Φ are asymptotically Gaussian for a sufficiently large N, since each one can be seen as the sum of independent random variables weighted by each entry of s. Along with the Gaussian noise n, we assume that r, conditioned on x 1 (or x 2), is a jointly Gaussian random vector. Also, \( {\mathbb {E}}[\boldsymbol {\Phi }] = \frac {1}{\sqrt {MN}} \mathbf {R}_{\Omega } \mathbf {U}_{1} \cdot {\mathbb {E}} [\!\text {diag}(\mathbf {s})] \cdot \mathbf {U}_{2} = \mathbf {0}\) for a given R Ω , as each entry of s takes ±1 with probability 1/2 under Assumption 1. Thus, \( {\mathbb {E}}\left [\!\mathbf {r} | \mathbf {x}_{h}\right ] = {\mathbb {E}}[\boldsymbol {\Phi }] \cdot \mathbf {x}_{h} + {\mathbb {E}}[\!\mathbf {n}] = \mathbf {0}\). With the Gaussian random vector r, the relative entropy between p 1=Pr(r|x 1) and p 2=Pr(r|x 2) has the following closed-form expression [56]
$$ D\left(p_{1} || p_{2}\right) = \frac{1}{2} \left[ \log \frac{|\mathbf{C}_{2}|}{|\mathbf{C}_{1}|} + \text{tr} \left(\mathbf{C}_{2}^{-1} \mathbf{C}_{1}\right) - M \right], $$
where C 1 and C 2 are the covariance matrices of r conditioned on x 1 and x 2, respectively. By measuring the relative entropy by (3), we obtain an upper bound on the TV distance, i.e.,
$$ d_{\text{TV}} (p_{1}, p_{2}) \leq \min \left(\sqrt{\frac{D(p_{1} || p_{2})}{2}}, \ 1 \right) $$
by Pinsker's inequality. In (4), the upper bound is set to be at most 1, since d TV(p 1,p 2)∈[ 0,1].
In what follows, we present an upper bound on the relative entropy with some constraints on plaintexts, which subsequently yields an analytic upper bound on the maximum TV distance by (4).
In our CS-based cryptosystem, assume that each plaintext x has at most K nonzero entries with the constant energy \(\mathcal {E}_{x} = || \mathbf {x} ||^{2}\). Then, the relative entropy of (3) is bounded by
$$ {}D(p_{1} || p_{2})\! \leq\! \frac{M}{2} \left(\!K \mu^{2} (\mathbf{U}_{2})\! \cdot \text{PNR}\,-\,\log\! \left(K \mu^{2} (\mathbf{U}_{2})\! \cdot\! \text{PNR}\! +\! 1 \right) \right)\!, $$
where \({PNR} = \frac {\mathcal {E}_{x}}{M \sigma ^{2}}\) is the plaintext-to-noise power ratio (PNR).
See the Appendices. □
In Theorem 1, μ(U 2)=1 if U 2=W, while \(\mu (\mathbf {U}_{2}) = \sqrt {2}\) if U 2=D. However, if \(\mathbf {U}_{2} = \sqrt {N} \mathbf {I}\), the upper bound increases as N for \(\mu (\mathbf {U}_{2}) = \sqrt {N}\). Thus, Theorem 1 implies that one must not use \(\mathbf {U}_{2} = \sqrt {N} \mathbf {I}\), to achieve the indistinguishability of our CS-based cryptosystem.
To ensure a reliable CS decryption for a legitimate recipient, our CS-based cryptosystem can set \(K = {\mathcal {O}} \left (\frac {M}{ \mu ^{2} (\mathbf {U}) \log N} \right)\) for nonuniform CS recovery [57], which yields the following corollary.
Corollary 1
In our CS-based cryptosystem with U 1=H and N=2n, assume U 2=W or D, where \(\mu (\mathbf {U})={\mathcal {O}}(\sqrt {\log N})\). In Theorem 1, if \(K \leq \frac {c M}{n^{2}} \) with a constant c, then
$$ \begin{aligned} D(p_{1} || p_{2}) & \leq \frac{M}{2} \left(\frac{c M \mu^{2} (\mathbf{U}_{2})}{n^{2}} \cdot \text{PNR}\right. \\ &\quad - \log \left.\left(\frac{c M \mu^{2} (\mathbf{U}_{2})}{n^{2}} \cdot \text{PNR} + 1 \right) \right). \end{aligned} $$
Thus, if the keystream length N is sufficiently large with given M and PNR, our CS-based cryptosystem will have low relative entropy, which contributes to the indistinguishability against an adversary, while guaranteeing the reliability for a legitimate recipient.
Hellinger distance
To bound the TV distance, we may use another probability metric, the Hellinger distance [21]. In our CS-based cryptosystem, recall that the ciphertext r, conditioned on x h , is assumed to be a jointly Gaussian random vector with zero mean and the covariance matrix C h , where h=1 or 2. Then, the Hellinger distance for the multivariate Gaussian distributions p 1 and p 2 is given by [58, 59]
$$ d_{\mathrm{H}} (p_{1}, p_{2}) = \sqrt{1- \frac{|\mathbf{C}_{1}|^{\frac{1}{4}} |\mathbf{C}_{2}|^{\frac{1}{4}}}{|\mathbf{C}_{3}|^{\frac{1}{2}}} }, $$
where \(\mathbf {C}_{3} = \frac {\mathbf {C}_{1} + \mathbf {C}_{2}}{2}\). The Hellinger distance is particularly useful by giving both upper and lower bounds on the TV distance [60], i.e.,
$$ {}d_{\mathrm{H}}^{2} (p_{1}, p_{2}) \leq d_{\text{TV}} (p_{1}, p_{2}) \leq d_{\mathrm{H}}(p_{1}, p_{2}) \sqrt{2 - d_{\mathrm{H}}^{2} (p_{1}, p_{2})}. $$
In what follows, we present an upper bound on the Hellinger distance of (6), which leads to an analytic upper bound on the maximum TV distance by (7).
Recall the assumptions and definitions of Theorem 1. In our CS-based cryptosystem, the Hellinger distance of (6) is bounded by
$$ d_{\mathrm{H}} (p_{1}, p_{2}) \leq \sqrt{1 - \left(\frac{2\sqrt{K \mu^{2} (\mathbf{U}_{2}) \cdot \text{PNR} + 1}}{K \mu^{2} (\mathbf{U}_{2})\cdot \text{PNR} + 2} \right)^{\frac{M}{4} }}, $$
where \(\text {PNR} = \frac {\mathcal {E}_{x}}{M \sigma ^{2}}\).
In our CS-based cryptosystem with U 1=H and N=2n, assume U 2=W or D, where \(\mu (\mathbf {U}) = {\mathcal {O}}\left (\sqrt {\log N}\right)\). In Theorem 2, if \(K \leq \frac {c M}{n^{2}}\) with a constant c, then
$$ d_{\mathrm{H}} (p_{1}, p_{2}) \leq \sqrt{1 - \left(\frac{2n \sqrt{c M \mu^{2} (\mathbf{U}_{2}) \cdot \text{PNR} + n^{2}}} {c M \mu^{2} (\mathbf{U}_{2}) \cdot \text{PNR} + 2n^{2}} \right)^{\frac{M}{4} }}. $$
Thus, if the keystream length N is sufficiently large with given M and PNR, our CS-based cryptosystem will have low Hellinger distance, which contributes to the indistinguishability against an adversary, while guaranteeing the reliability for a legitimate recipient.
Remark 1
Theorems 1 and 2 suggest that the relative entropy and the Hellinger distance will approach to zero as PNR decreases. Accordingly, our CS-based cryptosystem will have low TV distance by (4) and (7) at low PNR. Similarly, the TV distance will be low when M and K are small, respectively. Consequently, our CS-based cryptosystem can be indistinguishable at low PNR for small M and K.
When N=2n increases, Corollaries 1 and 2 suggest that if M is fixed, the relative entropy and the Hellinger distance will decrease at a given PNR by reducing \(K = {\mathcal {O}} \left (\frac {M}{n^{2}} \right)\), which will be confirmed by numerical results of Section 5. On the other hand, if M increases with \(M = {\mathcal {O}} \left (K n^{2}\right) \) for a given K, numerical results reveal that they also decrease over N at a given PNR, which contradicts Theorems 1 and 2. This observation implies that there is a room to improve the bounds of the theorems. Combined with Remark 1, the TV distance will be low if the keystream length N is sufficiently large with low compression \(\left (\frac {M}{N} \right)\) and sparsity \(\left (\frac {K}{N} \right)\) ratios, which leads to the asymptotic indistinguishability of our CS-based cryptosystem.
Potential key recovery attack
In this section, we consider a potential key recovery attack in which an adversary attempts to recover the key of our CS-based cryptosystem. In the CPA, the adversary tries to restore a keystream from a ciphertext (stage 1) and then to recover the original key from the restored keystream via algebraic cryptanalysis (stage 2). With a sufficiently long key, we assume that the number of keystream bits required for the algebraic cryptanalysis, denoted by D, is much larger than the ciphertext length M. For a convenience of analysis, we assume D=N, which means that the adversary needs to restore a keystream of full length N from stage 1. Figure 1 illustrates the potential CPA from an adversary for key recovery. This section discusses the adversary's strategy for keystream recovery in stage 1. Once a keystream is successfully restored through stage 1, a known cryptanalysis [47, 48] can be carried out in stage 2 for key recovery, which will not be discussed in this paper.
An adversary's chosen plaintext attack for key recovery against our CS-based cryptosystem
Mathematical intractability of keystream recovery
In stage 1 of the CPA, an adversary needs to observe a correct N-bit keystream from a ciphertext that has been encrypted by a chosen plaintext. We assume that the adversary will choose a plaintext x such that each entry of \(\widehat {\mathbf {x}} = \mathbf {U}_{2} \mathbf {x} \) is nonzero for a unitary matrix U 2. Then, the corresponding ciphertext is given by
$$ \begin{aligned} \mathbf{r} = \boldsymbol{\Phi} \mathbf{x} + \mathbf{n} & = \frac{1}{\sqrt{MN}} \mathbf{R}_{\Omega} \mathbf{U}_{1} \text{diag}(\mathbf{s}) \mathbf{U}_{2} \mathbf{x} + \mathbf{n}\\ & = \frac{1}{\sqrt{MN}} \mathbf{R}_{\Omega} \mathbf{U}_{1} \text{diag}(\widehat{\mathbf{x}}) \mathbf{s} + \mathbf{n}\\ & = \mathbf{A} \mathbf{s} +\mathbf{n}, \end{aligned} $$
where \(\mathbf {A} = \frac {1}{\sqrt {MN}} \mathbf {R}_{\Omega } \mathbf {U}_{1} \text {diag}(\widehat {\mathbf {x}})\). Unlike in stream ciphers, restoring the keystream s from the known plaintext-ciphertext pair is not a trivial task, since s is hidden under compression in r.
From the ciphertext r of (9), an adversary needs to find a most likely keystream, which is equivalent to a maximum-likelihood (ML) estimate of
$$ \widehat{\mathbf{s}} = \underset{\mathbf{s} \in \{-1, +1\}^{N}}{\text{argmin}} || \mathbf{r} - \mathbf{A} \mathbf{s} ||^{2}. $$
Finding the ML solution of (10) is known as a constrained integer least-squares (ILS) problem, which is also called a closest vector problem (CVP) [61] in lattices. For a general A, the constrained ILS problem is proven to be NP hard [62].
To find a most likely keystream of (10), an exhaustive ML search requires the complexity of \({\mathcal {O}}\left (2^{N}\right)\), which would be computationally infeasible if the keystream length N is sufficiently large. Alternatively, the generalized sphere decoding (GSD) algorithms [63–65] can find an ML solution to the ILS problem of the underdetermined system with M<N. However, as it has the complexity exponential in N−M [63–65], the GSD cannot be applicable to the ILS problem with M≪N. To the best of our knowledge, there is no polynomial-time algorithm to find an ML solution of (10) with M≪N for a sufficiently large N.
In summary, the computational security of our CS-based cryptosystem against the key recovery attack is brought by the mathematical hardness that no polynomial-time algorithm is known to find an ML solution to the underdetermined ILS problem. In fact, the mathematical intractability of the ILS problem has been exploited by public-key cryptosystems [66–68]. In our symmetric-key CS-based cryptosystem, it also ensures that if the keystream length N is sufficiently large with M≪N, no adversary will be able to find a most likely keystream of length N in polynomial time, which demonstrates the computational security of our CS-based cryptosystem against the key recovery attack.
Successive approximate maximum-likelihood detection (SAMD)
In Section 5.1, we demonstrated that the ML detection would be infeasible for keystream recovery, as long as the keystream length is sufficiently large. As an alternative, we consider a sub-optimal, but feasible keystream recovery process for the CPA. Instead of restoring an N-bit keystream at once, we assume that an adversary attempts to restore a disjoint J-bit segment2 of the keystream from each detection, where J≪N, and repeats the detection \(\lceil \frac {N}{J} \rceil \) times successively to restore the keystream of full length N. In this subsection, we describe the details of the successive detection process for keystream recovery.
For a convenience of analysis, we assume a chosen plaintext such that \(\widehat {\mathbf {x}} = \left (\sqrt {MN}, \cdots, \sqrt {MN}\right)^{T}\) in (9), which yields A=R Ω U 1 for our analysis3. In the keystream recovery, an adversary has a freedom to choose the value of J and the J-bit positions of a keystream to be restored at the ith detection. Let Θ i ⊂{0,⋯,N−1} be a set of indices, where |Θ i |=J if 1≤i≤n s −1 and |Θ i |=N−(n s −1)J if i=n s , respectively, for \(n_{s} = \lceil \frac {N}{J} \rceil \). Also, Θ a ∩Θ b =ϕ for a≠b, where ϕ is an empty set, and \(\phantom {\dot {i}\!}\Theta _{1} + \cdots + \Theta _{n_{s}} = \{0, \cdots, N-1 \}\).
Let \(\mathbf {s}_{\Theta _{i}} \in \{ -1, +1 \}^{|\Theta _{i} |}\) be a |Θ i |-bit vector, where the entries are taken from the indices of Θ i in the keystream s. At the ith detection, an adversary attempts to find \(\mathbf {s}_{\Theta _{i}}\) from the ciphertext r of (9). With \(\mathbf {s}_{\Theta _{1}}, \cdots, \mathbf {s}_{\Theta _{i-1}}\) that have been detected from the previous detections, the ith detection should use a new ciphertext r i by subtracting their contribution from r, i.e.,
$$ \mathbf{r}_{i} = \mathbf{r} - \sum\limits_{h=1}^{i-1} \mathbf{R}_{\Omega} \mathbf{U}_{1} \mathbf{R}_{\Theta_{h}}^{T} \widehat{\mathbf{s}}_{\Theta_{h}}, $$
where \(\widehat {\mathbf {s}}_{\Theta _{h}}\) is an estimate from the hth detection. In (11), \(\mathbf {R}_{\Theta _{h}}^{T}\) is an N×J column selection operator that selects J columns of U 1 whose indices are specified by Θ h . Let Δ i ={0,⋯,N−1}∖(Θ 1+⋯+Θ i ), where \(\Delta _{n_{s}} = \phi \), and \(\mathbf {R}_{\Delta _{i}}^{T}\) be an N×(N−i J) column selection operator whose indices are specified by Δ i . By assuming \(\widehat {\mathbf {s}}_{\Theta _{h}} = \mathbf {s}_{\Theta _{h}}\) for 1≤h≤i−1, we have from (11)
$$ \begin{aligned} \mathbf{r}_{i} & = \mathbf{R}_{\Omega} \mathbf{U}_{1} \mathbf{R}_{\Theta_{i}}^{T} \mathbf{s}_{\Theta_{i}} + \mathbf{R}_{\Omega} \mathbf{U}_{1} \mathbf{R}_{\Delta_{i}}^{T} \mathbf{s}_{\Delta_{i}} + \mathbf{n} \\ & = \mathbf{m}_{i} + \mathbf{w}_{i} + \mathbf{n}, \end{aligned} $$
where \(\mathbf {m}_{i} = \mathbf {R}_{\Omega } \mathbf {U}_{1} \mathbf {R}_{\Theta _{i}}^{T} \mathbf {s}_{\Theta _{i}}\) corresponds to a desired component to be detected at the ith detection, \(\mathbf {w}_{i} = \mathbf {R}_{\Omega } \mathbf {U}_{1} \mathbf {R}_{\Delta _{i}}^{T} \mathbf {s}_{\Delta _{i}} \) is an interfering component from the keystream segments that have not been detected yet, and \(\mathbf {n} \sim {\mathcal N} (\mathbf {0}, \sigma ^{2} \mathbf {I})\) is a Gaussian random noise.
In (12), \(\mathbf {w}_{n_{s}} = \mathbf {0}\) since \(\Delta _{n_{s}} = \phi \). On the other hand, if 1≤i≤n s −1, each entry of w i is taken from the sum of N−i J column vectors of R Ω U 1, each of which is weighted by the entry of \(\mathbf {s}_{\Delta _{i}}\). Since each entry of \(\mathbf {s}_{\Delta _{i}}\) takes +1 or −1 randomly and independently under Assumption 1, w i will follow the jointly Gaussian distribution by the central limit theorem [69]. By noting that w i +n can be modeled as a Gaussian random vector for 1≤i≤n s , r i is also Gaussian for a given \(\mathbf {s}_{\Theta _{i}}\). Then,
$$ \begin{aligned} {\mathbb{E}}\left[\mathbf{r}_{i} | \mathbf{s}_{\Theta_{i}} \right] &= {\mathbb{E}}\left[\mathbf{m}_{i} | \mathbf{s}_{\Theta_{i}} \right] + {\mathbb{E}}\left[\mathbf{w}_{i} | \mathbf{s}_{\Theta_{i}} \right] + {\mathbb{E}}\left[\mathbf{n} | \mathbf{s}_{\Theta_{i}} \right]\\ &= \mathbf{R}_{\Omega} \mathbf{U}_{1} \mathbf{R}_{\Theta_{i}}^{T} \mathbf{s}_{\Theta_{i}} = \mathbf{m}_{i}, \end{aligned} $$
where \({\mathbb {E}}\left [\mathbf {w}_{i} | \mathbf {s}_{\Theta _{i}} \right ] = {\mathbb {E}}\left [\mathbf {n} | \mathbf {s}_{\Theta _{i}} \right ] = \mathbf {0}\), since \(\mathbf {s}_{\Theta _{i}}\) is independent of w i and n, respectively. Also, the covariance of r i is given by
$$ {}{\mathbb{E}}\!\left[\!\left(\mathbf{r}_{i} \,-\, \mathbf{m}_{i}\right)\!\left(\mathbf{r}_{i} \,-\, \mathbf{m}_{i}\right)^{T}\! | \mathbf{s}_{\Theta_{i}}\!\right] \!\!= \!{\mathbb{E}}\!\left[\!\left(\mathbf{w}_{i} \,+\, \mathbf{n}\right) \!\left(\mathbf{w}_{i} \,+\, \mathbf{n}\right)^{T}\!\right]\! =\! \mathbf{K}_{i} + \sigma^{2} \mathbf{I}, $$
where w i and n are independent. In (14),
$$ \begin{aligned} \mathbf{K}_{i} = {\mathbb{E}}\left[\mathbf{w}_{i} \mathbf{w}_{i}^{T}\right] & = \mathbf{R}_{\Omega} \mathbf{U}_{1} \mathbf{R}_{\Delta_{i}}^{T} \cdot {\mathbb{E}}\left[\mathbf{s}_{\Delta_{i}}\mathbf{s}_{\Delta_{i}}^{T}\right] \cdot \mathbf{R}_{\Delta_{i}} \mathbf{U}_{1}^{T} \mathbf{R}_{\Omega}^{T} \\ & = \mathbf{R}_{\Omega} \mathbf{U}_{1} \mathbf{R}_{\Delta_{i}}^{T} \cdot \mathbf{R}_{\Delta_{i}} \mathbf{U}_{1}^{T} \mathbf{R}_{\Omega}^{T}, \end{aligned} $$
where \({\mathbb {E}}\left [\mathbf {s}_{\Delta _{i}} \mathbf {s}_{\Delta _{i}}^{T}\right ] = \mathbf {I}\). Since K i does not depend on \(\mathbf {s}_{\Theta _{i}}\), the covariance of r i in (14) is equal for all possible \(\mathbf {s}_{\Theta _{i}} \in \{-1, +1\}^{|\Theta _{i}|}\) at each ith detection. Under the Gaussian model of r i with equal covariance, we can apply the ML decision rule [70] at the ith detection, which yields
$$ \widehat{\mathbf{s}}_{\Theta_{i}} = \underset{\mathbf{s}_{\Theta_{i}} \in \{-1, +1\}^{|\Theta_{i}|}}{\text{argmin}} \left(\mathbf{r}_{i} - \mathbf{m}_{i} \right)^{T} \left(\mathbf{K}_{i} + \sigma^{2} \mathbf{I} \right)^{-1} \left(\mathbf{r}_{i} - \mathbf{m}_{i} \right). $$
In (11) and (12), we assumed that all the estimates \(\widehat {\mathbf {s}}_{\Theta _{h}}\), 1≤h≤i−1, from the previous detections are correct, and then ignored the estimation errors \(\mathbf {s}_{\Theta _{h}} - \widehat {\mathbf {s}}_{\Theta _{h}}\) while subtracting the contribution from r. Therefore, (16) cannot be a true ML detection, but an optimistic approximation to the adversary.
Finally, the adversary carries out the approximate ML detection of (16) n s times successively for 1≤i≤n s and restores the full N-bit keystream by combining the disjoint |Θ i |-bit estimates of \(\widehat {\mathbf {s}}_{\Theta _{i}} \). Throughout this paper, the detection process is called a successive approximate ML detection (SAMD). In what follows, we present an upper bound on the success probability of the SAMD.
In the SAMD, recall the approximate ML decision rule of (16) applied at each ith detection for 1≤i≤n s , where \(n_{s} = \lceil \frac {N}{J} \rceil \). Let λ min (K i ) be the minimum eigenvalue of the covariance matrix K i in (15). Let P succ be the probability that an N-bit keystream can be successfully restored by the SAMD. Then,
$$ P_{succ} \leq \prod_{i=1}^{n_{s}} \left(1 - Q \left(\sqrt{\frac{M \mu^{2} (\mathbf{U}_{1})}{ \lambda_{min} (\mathbf{K}_{i}) + \sigma^{2}}} \right) \right) \triangleq P_{succ, UB}, $$
where \(Q(x) = \frac {1}{\sqrt {2 \pi }} \int _{x}^{\infty } e^{-\frac {t^{2}}{2}} dt\).
Theorem 3 shows the result for a general unitary matrix U 1, which suggests that our CS-based cryptosystem should choose an N×N unitary matrix U 1 such that μ(U 1) is as small as possible, regardless of N, in order to degrade the performance of the SAMD. In this paper, μ(U 1)=1 from U 1=H.
The upper bound on the success probability of Theorem 3 represents the highest possible performance that the SAMD can achieve with no estimation errors at each detection, which is an optimistic scenario for an adversary. In reality, the actual probability of success will be much lower than the upper bound, due to estimation errors and error propagation through detections. If an adversary finds a solution of (16) via an exhaustive search, the complexity of each detection of the SAMD will be \({\mathcal O} \left (2^{J}\right)\) with J≪N.
Minimum eigenvalues of K i
Theorem 3 implies that minimizing λ min(K i ) can improve the performance of the SAMD. At the ith detection of the SAMD, it is an adversary that determines the selection operator \(\mathbf {R}_{\Theta _{i}}\). Therefore, if the adversary appropriately chooses Θ i (or equivalently Δ i ) to minimize λ min(K i ), the success probability of the SAMD can be improved. In this paper, we consider three possible selections for Θ i that the adversary may choose reasonably.
Uniform selection: \(\Theta _{i} = \left \{i-1, \ \lfloor \frac {N}{J} \rfloor +i-1, \ \cdots, \ (J-1)\lfloor \frac {N}{J} \rfloor +i-1 \right \}\).
Consecutive selection: Θ i ={(i−1)J, (i−1)J+1, ⋯, i J−1}.
Random selection: Θ i selects the J indices from {0,⋯,N−1}∖(Θ 1+⋯+Θ i−1) uniformly at random.
Each selection is valid for 1≤i≤n s −1, and \(\Theta _{n_{s}} = \{0, \cdots, N-1\} \setminus (\Theta _{1} + \cdots + \Theta _{n_{s}-1})\), where \(n_{s} = \lceil \frac {N}{J} \rceil \). To further minimize λ min(K i ), the adversary might be able to develop a more sophisticated selection of Θ i by exploiting the structure of R Ω and U 1. However, we leave this issue open for future research. Regarding the selection operator, we have the following assumption.
Once an adversary chooses a value of J and a type of selection, we assume that they will be fixed through the entire detections of the SAMD.
Intuitively, the larger J will ensure better detection performance for the SAMD, since a longer keystream segment that can be subtracted from each detection may contribute less interference. The intuition will be justified by the numerical results of Section 6. In this regard, Assumption 2 is valid, since the adversary's reasonable option is to fix the value of J to the largest possible one allowed by the computing power. In addition, the numerical results of Section 6 show that λ min(K i ) is not so affected by the type of selections, which also supports Assumption 2.
In what follows, we present a theoretical lower bound on λ min(K i ) for 1≤i≤n s , if Θ i is a random selection.
In our CS-based cryptosystem with U 1=H, assume that an adversary chooses a random selection for Θ i in the ith detection of the SAMD, where \(1 \leq i \leq n_{s} = \lceil \frac {N}{J} \rceil \). Let \(I_{T} = \lceil \frac {N - c_{2} M\log M}{J} \rceil \) for a constant c 2>0. Then,
$$ \lambda_{\min}(\mathbf{K}_{i}) \geq \left\{ \begin{array}{ll} \left(\sqrt{N - iJ} - \sqrt{c_{1} M \log M} \right)^{2}, & \quad \text{if } i < I_{T}, \\ 0, & \quad \text{if } i \geq I_{T} \end{array} \right. $$
with high probability, where c 1 is a constant with 0<c 1<c 2.
The numerical results of Section 6 show that the lower bound also holds for uniform and consecutive selections. Using the bound, Corollary 3 presents a further upper bound on the success probability of the SAMD, which is straightforward from Theorems 3 and 4 with μ(H)=1.
In our CS-based cryptosystem with U 1=H, if an adversary chooses a random selection for Θ i , 1≤i≤n s during the SAMD, P succ, UB in Theorem 3 is bounded by
$$\begin{aligned} {}P_{\mathrm{succ, UB}} \!\leq\! & \left(\!1\! - Q \left(\sqrt{\frac{M }{\sigma^{2}}} \right) \right)^{n_{s} - I_{T} + 1} \\ & \cdot \prod_{i=1}^{I_{T} -1}\!\! \left(\! 1\! -\! Q\! \left(\! \sqrt{\frac{M }{(\sqrt{N - iJ} \,-\,\! \sqrt{c_{1} M \log M})^{2}\,+\,\sigma^{2}}}\! \right) \right) \\ & \triangleq P_{\text{succ}, \mathrm{U}^{2}\mathrm{B}}, \end{aligned} $$
where \(I_{T} = \lceil \frac {N - c_{2} M\log M}{J} \rceil \) for constants c 1 and c 2 with 0<c 1<c 2.
Numerical results
This section presents numerical results to demonstrate the reliability and the security of our CS-based cryptosystem. In numerical experiments, each plaintext x has at most K nonzero entries, where the positions are chosen uniformly at random and the coefficients are taken from the Gaussian distribution. In CS encryption, \(\boldsymbol {\Phi }= \frac {1}{\sqrt {MN}} \mathbf {R}_{\Omega } \mathbf {U}_{1} \text {diag}(\mathbf {s}) \mathbf {U}_{2}\), where U 1=H, and U 2=W or D. Also, the secret keystream s is generated by the self-shrinking generator [46] of a 128-stage LFSR. For CS decryption, the CoSaMP recovery algorithm [71] has been employed for a legitimate recipient to decrypt each ciphertext with the knowledge of Φ.
CS decryption of a legitimate recipient
Figure 2 demonstrates the performance of CS decryption of a legitimate recipient, where the plaintext length is N=1024 and the ciphertext length is M=48. The figure sketches the normalized mean squared error (NMSE), defined by \(\text {NMSE} = {\mathbb {E}} \left [ \frac {||\mathbf {x} - \widehat {\mathbf {x}} ||^{2}}{||\mathbf {x}||^{2}} \right ]\), where x and \(\widehat {\mathbf {x}}\) are original and decrypted plaintexts, respectively. We examine the performance with total 10000 plaintexts at a given PNR, where each one has at most K=4 nonzero entries. For comparison, we sketch the performance of CS reconstruction with a random Gaussian sensing matrix for Φ. The figure shows that the performance of our CS decryption is as good as that of CS recovery with a random Gaussian sensing matrix. As a consequence, it demonstrates that our CS-based cryptosystem guarantees a reliable CS decryption for a legitimate recipient.
The normalized mean squared error (NMSE) of CS decryption for a legitimate recipient
Indistinguishability
Figure 3 displays the upper and lower bounds of TV distance over PNR with U 2=W, where N=1024, M=48, and K=4. In the figure, the relative entropy of (3) and the Hellinger distance of (6) were computed using the covariance matrix of (19). Averaged over 10,000 pairs of randomly generated plaintexts (x 1,x 2) with at most K nonzero entries per each, the relative entropy and the Hellinger distance yield the bounds of (4) and (7) on the TV distance, respectively. For comparison, we also sketch the theoretical upper bounds on the TV distance, which are obtained by the maximum relative entropy of (5) and the maximum Hellinger distance of (8), respectively. The figure shows that the TV distance approaches to zero as noise level grows, which implies that our CS-based cryptosystem can be indistinguishable at low PNR. As PNR increases, however, we observe that the upper and lower bounds increase and finally converge to certain levels, respectively. More extensive simulations agreed with the implication of Remark 1 that the CS-based cryptosystem will have lower TV distances with less PNR, M, and K. We made similar observations of the TV distance when U 2=D and/or each plaintext has bipolar nonzero entries.
The upper and lower bounds of total variation distance over PNR
Figure 4 depicts the upper bounds on the success probability of an adversary in the indistinguishability experiment, where the best- and worst-case upper bounds of (2) are from the minimum and maximum achievable TV distances of (7), respectively, obtained by the Hellinger distance (6). In the figure, U 2=W and PNR=25 dB. With a given ciphertext length M=48, the maximum sparsity is set as \(K = \left \lfloor c M / \log _{2}^{2} N \right \rfloor \) for each N=2n, to ensure a reliable nonuniform CS decryption for a legitimate recipient, where c=8.5. For comparison, we sketch the empirical success probability of CS decryption by a legitimate recipient, where a decrypted plaintext has been declared as a success if \(\frac {||\mathbf {x} - \widehat {\mathbf {x}} ||^{2}}{||\mathbf {x}||^{2}} < 10^{-2}\). The figure reveals that the adversary's success probability approaches to that of a random guess as the keystream length N increases, while a legitimate recipient maintains its reliability.
The success probability of legitimate recipient and adversary for a given M
Figure 5 also displays the upper bounds on the success probability of an adversary in the indistinguishability experiment. At this time, the ciphertext length is kept as \(M = \left \lceil c K \log _{2}^{2} N \right \rceil \) for each N=2n with a given K=4, where c=0.12. As in Fig. 4, it also reveals that the adversary's success probability approaches to 0.5 as the keystream length N increases, while a legitimate recipient maintains its reliability. In conclusion, the empirical results of Figs. 4 and 5 show that if the keystream length N is sufficiently large with low compression \(\left (\frac {M}{N} \right)\) and sparsity \(\left (\frac {K}{N} \right)\) ratios, our CS-based cryptosystem can be computationally secure in terms of the indistinguishability, while guaranteeing a reliable CS decryption for a legitimate recipient.
The success probability of legitimate recipient and adversary for a given K
Performance of SAMD
Figure 6 sketches the minimum eigenvalues of the covariance matrix K i of (15) at the ith detection for various J∈{32,48,64,80}, where N=1024 and M=48. For comparison, it also sketches the lower bound of Theorem 4, where c 1=0.5 and c 2=1. For each i, we tested with 100,000 pairs of (Ω,Θ i ) for random subsampling and selection operators R Ω and \(\mathbf {R}_{\Theta _{i}}\), where Θ i had been fixed through the tested pairs in case of uniform and consecutive selections. In each subfigure, λ min(K i ) is sketched over 1≤i≤n s −1, where \(n_{s} = \lceil \frac {N}{J} \rceil \). Figure 6 shows that if J increases, λ min(K i ) decreases faster over i, which suggests that the detection performance will be improved as J increases. It is plausible because if more keystream bits are detected from the ith detection with no estimation errors, more interfering components can be subtracted from the (i+1)th detection. In addition, it appears that the minimum eigenvalues are irrelevant to the types of Θ i , which means that an adversary may expect no benefits from a particular selection of Θ i . Finally, Fig. 6 demonstrates that the lower bound of Theorem 4 is valid, not only for random selection but also for uniform and consecutive selections.
The minimum eigenvalues of K i at the ith detection of the SAMD
Figure 7 displays the upper bounds on the success probability of the SAMD for keystream recovery. For comparison, it also sketches the theoretical upper bound of Corollary 3 for random selection Θ i . In view of the adversary's bounded computing power, we set J≤128, where the complexity of each detection in the SAMD will be \({\mathcal {O}}\left (2^{J}\right)\) by an exhaustive search. Since λ min(K i ) has similar values for different types of Θ i 's in Fig. 6, the upper bounds of Fig. 7 are also similar for every selection types. Moreover, the upper bounds increase over J, which is obvious from the sharp decline of λ min(K i ) over J, observed from Fig. 6. However, even if an adversary chooses a large value of J, the upper bounds on the success probability are still significantly low, which implies that the potential of the SAMD to restore a correct N-bit keystream is pessimistic. Note that this is the result of an optimistic scenario, and in reality, the actual probability of success of the SAMD will be much lower than the upper bounds, due to estimation errors and their propagation through the SAMD.
The upper bounds on the success probability of the SAMD
This paper has proposed a CS-based cryptosystem that encrypts a plaintext with a partial unitary sensing matrix embedding a secret keystream. We demonstrated that our CS-based cryptosystem can offer a theoretically and empirically reliable decryption performance for a legitimate recipient, which is the first contribution of this paper. Then, we examined the indistinguishability of our CS-based cryptosystem by studying the TV distance as a security measure. To investigate the TV distance, we developed upper bounds on the relative entropy and the Hellinger distance, respectively. From the second contribution, we showed that our CS-based cryptosystem can be computationally secure in terms of the indistinguishability, as long as the keystream length for each encryption is sufficiently large with low compression and sparsity ratios.
In addition, we considered a potential CPA from an adversary to recover the key of our CS-based cryptosystem. The computational security of our CS-based cryptosystem against the CPA is based on the mathematical hardness that no polynomial-time algorithm is known to find an ML solution to the underdetermined ILS problem for keystream recovery. As a sub-optimal approach, we introduced the SAMD for an adversary to restore a secret keystream in polynomial time. In the third contribution, we developed an upper bound on the success probability of the SAMD and demonstrated that the performance of the keystream recovery through the SAMD is very pessimistic. In conclusion, our CS-based cryptosystem with a partial unitary sensing matrix embedding a secret keystream can be secure against the CPA, while guaranteeing a stable and robust decryption for a legitimate recipient.
1 This paper assumes that a plaintext x is sparse in canonical basis, or Ψ=I. In general, if a plaintext x is sparse with respect to an arbitrary orthonormal basis Ψ, i.e., x=Ψ T θ, the sensing matrix A=Φ Ψ T maintains the form of (1) by considering U 2 Ψ T as a new unitary matrix U 2.
2 In the last detection, \((N - (\lceil \frac {N}{J} \rceil - 1) J)\)-bit segment will be restored, where \(\lceil \frac {N}{J} \rceil \) denotes the nearest integer greater than or equal to \(\frac {N}{J}\).
3 Under this assumption, numerical results showed that the upper bound on the success probability of the successive detection is more favorable for an adversary than that of \(\widehat {\mathbf {x}}\) with arbitrary nonzero entries.
Proof of Theorem 1
We give a brief sketch for the proof of Theorem 1, as the underlying technique is similar to that of Theorem 1 in [72]. Similar to Lemma 1 of [72], the covariance matrix of r is given by
$$ \mathbf{C}_{h} = {\mathbb{E}}\left[\mathbf{r} \mathbf{r}^{T} | \mathbf{x}_{h}\right] = \mathbf{R}_{\Omega} \widetilde{\mathbf{C}}_{h} \mathbf{R}_{\Omega}^{T} + \sigma^{2} \mathbf{I}, $$
where \( \widetilde {\mathbf {C}}_{h} = \frac {1}{N} \mathbf {U}_{1}^{T} \text {diag}\left (\frac {|\widehat {\mathbf {x}}_{h}|^{2} }{M} \right) \mathbf {U}_{1} \) for \(\widehat {\mathbf {x}}_{h} = \mathbf {U}_{2} \mathbf {x}_{h}\). Let λ 1(C h )≥⋯≥λ M (C h ) be the eigenvalues of C h , while \(\lambda _{1}(\widetilde {\mathbf {C}}_{h}) \geq \cdots \ge \lambda _{N} \left (\widetilde {\mathbf {C}}_{h}\right)\) be the eigenvalues of \(\widetilde {\mathbf {C}}_{h}\). With \(\widehat {\mathbf {x}}_{h} = \mathbf {U}_{2} \mathbf {x}_{h} = \left (\widehat {x}_{h,0}, \cdots, \widehat {x}_{h,N-1}\right)^{T}\), let v h =(v h,0,⋯,v h,N−1)T, where \(v_{h,k} = |\widehat {x}_{h, \pi (k)}|^{2}\) for k=0,⋯,N−1, and π(k) is a permutation for v h,0≥⋯≥v h,N−1. From the definition of \(\widetilde {\mathbf {C}}_{h}\), it is clear that \(\lambda _{t}\left (\widetilde {\mathbf {C}}_{h}\right) = \frac {v_{h, t-1}}{M} \ge 0\) for t=1,⋯,N.
In (19), \(\widehat {\mathbf {C}}_{h} = \mathbf {R}_{\Omega } \widetilde {\mathbf {C}}_{h} \mathbf {R}_{\Omega }^{T} \) is an M×M principal submatrix of \(\widetilde {\mathbf {C}}_{h} \), where successive application of the interlacing inequality [73] leads to \( \lambda _{t+N-M} \left (\widetilde {\mathbf {C}}_{h}\right) \leq \lambda _{t} \left (\widehat {\mathbf {C}}_{h}\right) \leq \lambda _{t} \left (\widetilde {\mathbf {C}}_{h}\right)\) for 1≤t≤M. Thus, \( \underset {h}{\min } \ \underset {\mathbf {x}_{h}}{\min } \ \lambda _{M} \left (\widehat {\mathbf {C}}_{h}\right) = \underset {h}{\min } \ \underset {\mathbf {x}_{h}}{\min } \ \lambda _{N} \left (\widetilde {\mathbf {C}}_{h}\right) = 0\) from v h,N−1≥0. On the other hand, \( \underset {h}{\max } \ \underset {\mathbf {x}_{h}}{ \max } \ \lambda _{1} \left (\widehat {\mathbf {C}}_{h}\right) = \underset {h}{\max } \ \underset {\mathbf {x}_{h}}{ \max } \ \lambda _{1} \left (\widetilde {\mathbf {C}}_{h}\right) = \underset {h}{\max } \ \underset {\mathbf {x}_{h}}{ \max } \ \frac {v_{h, 0}}{M}\). By the Cauchy-Schwarz inequality, we obtain \( \frac {v_{h, 0}}{M} = \frac {|\widehat {x}_{h, \pi (0)}|^{2}}{M} = \frac {1}{M} \left | \sum _{k \in \mathcal {S}} x_{h, k} \mathbf {U}_{2}(\pi (0), k) \right |^{2} \leq \frac {K \mu ^{2} (\mathbf {U}_{2}) \cdot \mathcal {E}_{x}}{M}\), where \(\mathcal {S}\) is the set of nonzero entries of x h with \(|\mathcal {S}| \leq K\). As \(\lambda _{t} (\mathbf {C}_{h}) = \lambda _{t} \left (\widehat {\mathbf {C}}_{h}\right) + \sigma ^{2}\) from \(\mathbf {C}_{h} = \widehat {\mathbf {C}}_{h} + \sigma ^{2} \mathbf {I}\), we have
$$ {\displaystyle \begin{array}{cc}{\lambda}_{\mathrm{min}}& =\underset{h}{\min \limits}\kern1em \underset{{\mathbf{x}}_h}{\min \limits}\kern1em {\lambda}_M\left({\mathbf{C}}_h\right)={\sigma}^2,\\ {}{\lambda}_{\mathrm{max}}& =\underset{h}{\max \limits}\kern1em \underset{{\mathbf{x}}_h}{\max \limits}\kern1em {\lambda}_1\left({\mathbf{C}}_h\right)=\frac{K{\mu}^2\left({\mathbf{U}}_2\right)\cdotp {\mathcal{E}}_x}{M}+{\sigma}^2,\end{array}} $$
where h=1 or 2.
Meanwhile, the upper bound on \(\text {tr} \left (\mathbf {C}_{2}^{-1} \mathbf {C}_{1} \right)\) in Lemma 3 of [72] yields
$$ {}\begin{aligned} D(p_{1} || p_{2}) & \leq \frac{1}{2} \sum\limits_{t=1}^{M} \left(\log \frac{\lambda_{M+1-t}(\mathbf{C}_{2})}{\lambda_{t}(\mathbf{C}_{1})} + \frac{\lambda_{t}(\mathbf{C}_{1})}{\lambda_{M+1-t}(\mathbf{C}_{2})} - 1 \right) \\ & = \frac{1}{2} \sum\limits_{t=1}^{M} \, f(z_{t}), \end{aligned} $$
where f(z)=z− logz−1 and \(z_{t} = \frac {\lambda _{t} (\mathbf {C}_{1})}{ \lambda _{M+1-t} (\mathbf {C}_{2})} > 0\). With λ min and λ max in (20), define \( \tau = \frac {\lambda _{\max }}{\lambda _{\min }} = \frac {K \mu ^{2} (\mathbf {U}_{2}) \mathcal {E}_{x}}{M \sigma ^{2}} + 1 > 1\). Similar to the proof of Theorem 1 in [72], \(D(p_{1} || p_{2}) \leq \frac {M}{2} f(\tau)\), which yields (5).
We use definitions and notations in the proof of Theorem 1. Let λ 1(C 3)≥⋯≥λ M (C 3) be the eigenvalues of \(\mathbf {C}_{3} = \frac {\mathbf {C}_{1} + \mathbf {C}_{2}}{2}\). Clearly, the eigenvalues of C 1, C 2, and C 3 are positive by (20) and the Weyl inequality [73]. In (6), let \( \Gamma = \frac {|\mathbf {C}_{1}|^{\frac {1}{2}} |\mathbf {C}_{2}|^{\frac {1}{2}}}{|\mathbf {C}_{3}|} \triangleq \frac {\Gamma _{n}}{\Gamma _{d}}\). Then,
$$ \begin{aligned} \Gamma_{d} = \prod_{t=1}^{M} \lambda_{t} (\mathbf{C}_{3}) &\leq \left(\frac{\sum_{t=1}^{M} \lambda_{t} (\mathbf{C}_{3})}{M} \right)^{M} = \left(\frac{\text{tr} (\mathbf{C}_{3})}{M} \right)^{M}\\& = \left(\frac{\text{tr} (\mathbf{C}_{1}) + \text{tr} (\mathbf{C}_{2})}{2M} \right)^{M}, \end{aligned} $$
where the inequality is from the arithmetic mean-geometric mean inequality. For h=1 or 2, the tth diagonal entry of \(\widetilde {\mathbf {C}}_{h} = \frac {1}{N} \mathbf {U}_{1}^{T} \text {diag}\left (\frac {|\widehat {\mathbf {x}}_{h}|^{2} }{M} \right) \mathbf {U}_{1} \) is given by \(\frac {1}{MN} \sum _{k=0}^{N-1} |\widehat {x}_{h,k}|^{2} \mathbf {U}_{1}^{2} (k, t) = \frac {1}{MN} || \widehat {\mathbf {x}}_{h} ||^{2} = \frac {1}{M} ||\mathbf {x}_{h}||^{2} = \frac {\mathcal {E}_{x}}{M},\) where \(\mathbf {U}_{1}^{2} (k, t) = 1\) for 0≤t≤N−1. Note that \(\widehat {\mathbf {C}}_{h} = \mathbf {R}_{\Omega } \widetilde {\mathbf {C}}_{h} \mathbf {R}_{\Omega }^{T}\) has the same diagonal entry of \(\widetilde {\mathbf {C}}_{h}\). Thus, from \(\mathbf {C}_{h} = \widehat {\mathbf {C}}_{h} + \sigma ^{2} \mathbf {I}\), we have
$$ \text{tr}(\mathbf{C}_{h}) = \text{tr}(\widehat{\mathbf{C}}_{h}) + M \sigma^{2} = \mathcal{E}_{x} + M \sigma^{2}, $$
where (21) becomes
$$ \Gamma_{d} \leq \left(\frac{\mathcal{E}_{x}}{M} + \sigma^{2} \right)^{M}. $$
In Γ n , the geometric mean-harmonic mean inequality yields
$$ |\mathbf{C}_{h}|^{\frac{1}{2}} = \left(\prod_{t=1}^{M} \lambda_{t} (\mathbf{C}_{h}) \right)^{\frac{1}{2}} \ge \left(\frac{1}{\frac{1}{M} \sum_{t=1}^{M} \lambda_{t}^{-1} (\mathbf{C}_{h})} \right)^{\frac{M}{2}}, $$
where h=1 or 2. By the Kantorovich inequality [74],
$$ \begin{aligned} \frac{1}{M} \sum\limits_{t=1}^{M} \lambda_{t}^{-1} (\mathbf{C}_{h}) & \leq \frac{M}{4 \ \text{tr}(\mathbf{C}_{h})} \left(\frac{\lambda_{1} (\mathbf{C}_{h})}{\lambda_{M} (\mathbf{C}_{h})} + \frac{\lambda_{M} (\mathbf{C}_{h})}{\lambda_{1} (\mathbf{C}_{h})} \!+ 2 \right) \\ & = \frac{M}{4 \ \text{tr}(\mathbf{C}_{h})} \left(\frac{\lambda_{\max}}{\lambda_{\min}} + \frac{\lambda_{\min}}{\lambda_{\max}} + 2 \right) \\ & = \frac{M}{4 \ \text{tr}(\mathbf{C}_{h})} \left(\tau + \frac{1}{\tau} + 2 \right), \end{aligned} $$
where λ 1(C h ) and λ M (C h ) have been replaced by λ max and λ min of (20), respectively. In (25), \(\tau = \frac {\lambda _{\max }}{\lambda _{\min }} = \frac {K \mu ^{2}(\mathbf {U}_{2}) \cdot \mathcal {E}_{x}}{M \sigma ^{2}} + 1 = K \mu ^{2}(\mathbf {U}_{2}) \cdot \text {PNR} + 1\). By (22), (24), and (25),
$$ \Gamma_{n} \geq \left(\frac{4 \sqrt{\text{tr}(\mathbf{C}_{1}) \cdot \text{tr}(\mathbf{C}_{2})} }{M(\tau + \frac{1}{\tau} + 2)} \right)^{M} = \left(\frac{4 \left(\frac{\mathcal{E}_{x}}{M} + \sigma^{2} \right) }{\tau + \frac{1}{\tau} + 2} \right)^{M}. $$
By combining Γ d and Γ n , (23) and (26) yield
$$ \begin{aligned} \Gamma = \frac{\Gamma_{n}}{\Gamma_{d}} &\geq \frac{\left(\frac{4 \left(\frac{\mathcal{E}_{x}}{M} + \sigma^{2} \right) }{\tau + \frac{1}{\tau} + 2} \right)^{M}} {\left(\frac{\mathcal{E}_{x}}{M} + \sigma^{2} \right)^{M}} = \left(\frac{2\sqrt{\tau} }{\tau + 1} \right)^{\frac{M}{2}} \\&= \left(\frac{2\sqrt{K \mu^{2} (\mathbf{U}_{2})\cdot \text{PNR}+1} }{ K \mu^{2} (\mathbf{U}_{2}) \cdot \text{PNR}+ 2} \right)^{\frac{M}{2}}. \end{aligned} $$
Finally, the proof is completed by \(d_{\mathrm {H}} (p_{1}, p_{2}) = \sqrt {1 - \Gamma ^{\frac {1}{2}}}\).
In (15), K i is the Gram matrix, or \(\mathbf {K}_{i} = \mathbf {A}_{i}^{T} \mathbf {A}_{i}\) with \(\mathbf {A}_{i} = \mathbf {R}_{\Delta _{i}} \mathbf {U}_{1}^{T} \mathbf {R}_{\Omega }^{T}\) for 1≤i≤n s −1, where λ min(K i )≥0, since K i is positive semi-definite [73]. Let \(\mathbf {s}_{\Theta _{i}}\) and \(\mathbf {s}_{\Theta _{i}} '\) be a pair of correct and wrong J-bit segments from a keystream s at the index set Θ i , respectively. From (13), \({\mathbb {E}}\left [\mathbf {r}_{i} | \mathbf {s}_{\Theta _{i}} \right ] = \mathbf {m}_{i} = \mathbf {R}_{\Omega } \mathbf {U}_{1} \mathbf {R}_{\Theta _{i}}^{T} \mathbf {s}_{\Theta _{i}} \) and \({\mathbb {E}}\left [\mathbf {r}_{i} | \mathbf {s}_{\Theta _{i}} ' \right ] = \mathbf {m}_{i} ' = \mathbf {R}_{\Omega } \mathbf {U}_{1} \mathbf {R}_{\Theta _{i}}^{T} \mathbf {s}_{\Theta _{i}} '\), respectively. Also, (14) yields \({\mathbb {E}}\left [\left (\mathbf {r}_{i} - \mathbf {m}_{i} \right)\left (\mathbf {r}_{i} - \mathbf {m}_{i} \right)^{T} | \mathbf {s}_{\Theta _{i}} \right ] = {\mathbb {E}}\left [\left (\mathbf {r}_{i} - \mathbf {m}_{i} ' \right)\left (\mathbf {r}_{i} - \mathbf {m}_{i} '\right)^{T} | \mathbf {s}_{\Theta _{i}} '\right ] = \mathbf {K}_{i} + \sigma ^{2} \mathbf {I}\). Assuming that r i is a Gaussian random vector, the binary hypothesis detection of Section 3.2 in [70] reveals that the pairwise error probability that \(\mathbf {s}_{\Theta _{i}} '\) is incorrectly detected by the ith detection is
$$ \begin{aligned} {}\text{Pr}\left[\!\mathbf{s}_{\Theta_{i}} \!\rightarrow\! \left. \mathbf{s}_{\Theta_{i}} ' \right| \mathbf{s}_{\Theta_{i}}, \mathbf{s}_{\Theta_{i}} ' \right] & \geq Q \left(\frac{|| \mathbf{m}_{i} - \mathbf{m}_{i} '||}{2 \sqrt{\lambda_{\min}(\mathbf{K}_{i})+\sigma^{2}}} \right) \\ & = Q \left(\frac{|| \mathbf{R}_{\Omega} \mathbf{U}_{1} \mathbf{R}_{\Theta_{i}}^{T} \left(\mathbf{s}_{\Theta_{i}} - \mathbf{s}_{\Theta_{i}}'\right)||} {2 \sqrt{\lambda_{\min}(\mathbf{K}_{i})+\sigma^{2}}} \right)\!. \end{aligned} $$
We assume that the pairwise error event occurs only for a specific \(\mathbf {s}_{\Theta _{i}}'\), which is closest to \(\mathbf {s}_{\Theta _{i}}\), and ignore all the other \(\mathbf {s}_{\Theta _{i}}'\). In other words, we take into account only a single \(\mathbf {s}_{\Theta _{i}}'\), where \(\mathbf {s}_{\Theta _{i}} - \mathbf {s}_{\Theta _{i}}'\) has the nonzero entry (+2 or −2) at one position, or equivalently \(|| \mathbf {s}_{\Theta _{i}} - \mathbf {s}_{\Theta _{i}} ' || = 2\) for a given \(\mathbf {s}_{\Theta _{i}}\). This assumption, similar to the one in [75], is favorable for an adversary. From (27), the error probability under the assumption is given by
$$ {}\begin{aligned} P_{e}^{(i)} & \,=\, \sum_{\mathbf{s}_{\Theta_{i}}}\! \text{Pr} \left[\! \mathbf{s}_{\Theta_{i}}\right] \!\cdot\! \sum_{\mathbf{s}_{\Theta_{i}} ' } \text{Pr}\left[\! \mathbf{s}_{\Theta_{i}} \rightarrow\! \mathbf{s}_{\Theta_{i}} '\! \mid\! \mathbf{s}_{\Theta_{i}}, \mathbf{s}_{\Theta_{i}} ' \right] \!\cdot\! \text{Pr} \left[ \mathbf{s}_{\Theta_{i}}' \mid \mathbf{s}_{\Theta_{i}} \right] \\ & =\! \sum_{\mathbf{s}_{\Theta_{i}}}\! \text{Pr} \left[ \mathbf{s}_{\Theta_{i}}\right] \!\cdot\! \text{Pr}\left[ \mathbf{s}_{\Theta_{i}} \!\rightarrow\! \mathbf{s}_{\Theta_{i}} ' \mid \mathbf{s}_{\Theta_{i}}, \mathbf{s}_{\Theta_{i}} ', ||\mathbf{s}_{\Theta_{i}} - \mathbf{s}_{\Theta_{i}} ' || \,=\, 2\! \right] \\ & =\! \text{Pr}\left[ \mathbf{s}_{\Theta_{i}} \rightarrow \mathbf{s}_{\Theta_{i}} ' \mid \mathbf{s}_{\Theta_{i}}, \mathbf{s}_{\Theta_{i}} ', ||\mathbf{s}_{\Theta_{i}} - \mathbf{s}_{\Theta_{i}} ' || = 2 \right] \\ & \geq Q \left(\frac{\sqrt{\sum_{k=0}^{M-1} 4 \left| \mathbf{U}_{1} \left(\omega_{k}, \theta_{i, \tau}\right) \right|^{2} }}{2 \sqrt{\lambda_{\min}(\mathbf{K}_{i})+\sigma^{2}}} \right) \\ & = Q \left(\sqrt{\frac{M \mu^{2}(\mathbf{U}_{1})}{\lambda_{\min}(\mathbf{K}_{i})+\sigma^{2}}} \right), \end{aligned} $$
where ω k ∈Ω and θ i,τ ∈Θ i . In (28), we assumed that \(\mathbf {s}_{\Theta _{i}}\) and \(\mathbf {s}_{\Theta _{i}} '\) differ only at a position corresponding to the column index θ i,τ of U 1. Note that \(P_{e}^{(i)}\) is under the assumption that all the estimates from previous i−1 detections have been subtracted with no errors to yield r i of (12). Then, the success probability of the ith detection is
$$ \begin{aligned} P_{s}^{(i)} & = \text{Pr} \left[ \widehat{\mathbf{s}}_{\Theta_{i}} = \mathbf{s}_{\Theta_{i}} \mid \widehat{\mathbf{s}}_{\Theta_{1}} = \mathbf{s}_{\Theta_{1}}, \cdots, \widehat{\mathbf{s}}_{\Theta_{i-1}} = \mathbf{s}_{\Theta_{i-1}} \right] \\ & = 1 - P_{e}^{(i)} \leq 1 - Q \left(\sqrt{\frac{M \mu^{2}(\mathbf{U}_{1})}{\lambda_{\min}(\mathbf{K}_{i})+\sigma^{2}}} \right), \end{aligned} $$
where 1≤i≤n s . If a correct N-bit keystream is to be restored, all the component detections should be successful. Thus, the success probability of the SAMD is
$$ {}\begin{aligned} P_{\text{succ}} &= \text{Pr} \left[\widehat{\mathbf{s}}_{\Theta_{1}} = \mathbf{s}_{\Theta_{1}}, \cdots, \widehat{\mathbf{s}}_{\Theta_{n_{s}}} = \mathbf{s}_{\Theta_{n_{s}}} \right] \\ & = \prod_{i=1}^{n_{s}} \text{Pr} \left[ \widehat{\mathbf{s}}_{\Theta_{i}} = \mathbf{s}_{\Theta_{i}} \mid \widehat{\mathbf{s}}_{\Theta_{1}} = \mathbf{s}_{\Theta_{1}}, \cdots, \widehat{\mathbf{s}}_{\Theta_{i-1}} = \mathbf{s}_{\Theta_{i-1}} \right] \\ & = \prod_{i=1}^{n_{s}} P_{s}^{(i)}. \end{aligned} $$
Finally, we obtain the upper bound of (17) by combining (29) and (30), which completes the proof.
In (15), let \(\mathbf {A}_{i} = \mathbf {R}_{\Delta _{i}} \mathbf {H}^{T} \mathbf {R}_{\Omega }^{T}\) with U 1=H. Then, the singular values of A i are equal to the square roots of the eigenvalues of \(\mathbf {K}_{i} = \mathbf {A}_{i}^{T} \mathbf {A}_{i}\), where λ min(K i )≥0 for all i's. In other words, if σ min(A i ) denotes the minimum singular value of A i , then \(\lambda _{\min }(\mathbf {K}_{i}) = \sigma _{\min }^{2} (\mathbf {A}_{i})\).
To examine σ min(A i ) for 1≤i≤n s −1, we first define \(\mathbf {B}_{i} = \mathbf {H}^{T} \mathbf {R}_{\Omega }^{T}\). Then, B i is an N×M matrix satisfying \( \mathbf {B}_{i}^{T} \mathbf {B}_{i} = \mathbf {R}_{\Omega } \mathbf {H} \cdot \mathbf {H}^{T} \mathbf {R}_{\Omega }^{T} = N \mathbf {I}\), which means that each column of B i is mutually orthogonal. Also, it is clear that the l 2-norm of each row of B i is \(\sqrt {M}\), since each entry of B i is ±1. If Θ i is a random selection, so is Δ i , where \(\phantom {\dot {i}\!}\mathbf {A}_{i} = \mathbf {R}_{\Delta _{i}} \mathbf {B}_{i}\) is an (N−i J)×M matrix obtained by randomly subsampling (N−i J) rows from B i , where the selected row indices are specified by Δ i . For such a matrix A i , Corollary 5.55 of [4] shows that for every t≥0,
$$ \sigma_{\min} (\mathbf{A}_{i}) \geq \sqrt{N-iJ} - t \sqrt{M} $$
with probability at least \(1-2M e^{-ct^{2}}\) for a constant c>0. The corollary assumed that \(t \geq \sqrt {c_{1} \log {M}}\) and N−i J>c 2 M logM for the bound to be nontrivial and nonnegative, where 0<c 1<c 2. Thus, the bound of (31) is valid only for \(i <\lceil \frac {N - c_{2} M\log M}{J} \rceil = I_{T}\), and we set σ min(A i )≥0 if i≥I T , which gives the bound of (18) from \(\lambda _{\min }(\mathbf {K}_{i}) = \sigma _{\min }^{2} (\mathbf {A}_{i})\).
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This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (no. NRF-2017R1A2B4004405).
School of Electrical Engineering and Computer Science, Gwangju Institute of Science and Technology (GIST), Gwangju, Korea
Nam Yul Yu
Correspondence to Nam Yul Yu.
The author declares that he has no competing interests.
Yu, N. On the security of compressed encryption with partial unitary sensing matrices embedding a secret keystream. EURASIP J. Adv. Signal Process. 2017, 73 (2017). https://doi.org/10.1186/s13634-017-0508-6
Compressed encryption
Integer least-squares (ILS) problem
Total variation distance
Stream ciphers | CommonCrawl |
\begin{document}
\title[nowhere vanishing Hessian determinant]{Homogeneous functions with nowhere vanishing Hessian determinant} \author{Connor Mooney} \address{Department of Mathematics, UC Irvine} \email{\tt [email protected]}
\begin{abstract} We prove that functions that are homogeneous of degree $\alpha \in (0,\,1)$ on $\mathbb{R}^n$ and have nowhere vanishing Hessian determinant cannot change sign. \end{abstract} \maketitle
\section{Introduction}
Let $n \geq 2$, and let $\Omega \subset \mathbb{R}^n$ be the cone over a domain $\Sigma \subset \mathbb{S}^{n-1}$ that has nonempty boundary. Let $$p_n := \max\{1,\, (n-1)(n-2)\}.$$ In this paper we show:
\begin{thm}\label{Main} If there exists a function $u : \Omega \rightarrow \mathbb{R}$ that satisfies: \begin{enumerate}[label=(\roman*)] \item $u$ is homogeneous of degree $\alpha \in (0,\,1)$, \item $u \in W^{2,\,p}_{loc}(\Omega) \cap W^{1,\,\infty}((\Omega \cap B_2) \backslash B_1)$ for some $p > p_n$ (with $p = p_n$ allowed if $n \leq 3$), \item $u > 0$ in $\Omega$ and $u = 0$ on $\partial \Omega$, and \item either $\det D^2u$ or $-\det D^2u$ is locally strictly positive in $\Omega$, \end{enumerate} then $\mathbb{R}^n \backslash \Omega$ is a convex cone, and $\Sigma$ contains a closed hemisphere. \end{thm}
\noindent By locally strictly positive we mean bounded below by positive constants almost everywhere on compact sets, where the constants may depend on the sets.
An immediate consequence of Theorem \ref{Main} is:
\begin{thm}\label{Main2} If $u: \mathbb{R}^n \rightarrow \mathbb{R}$ satisfies: \begin{enumerate}[label=(\roman*)] \item $u$ is homogeneous of degree $\alpha \in (0,\,1)$, \item $u \in W^{2,\,p}_{loc}\left(\mathbb{R}^n \backslash \{0\}\right)$ for some $p > p_n$, and \item either $\det D^2u$ or $-\det D^2u$ is locally strictly positive in $\mathbb{R}^n \backslash \{0\}$, \end{enumerate} then $u$ does not change sign. \end{thm} \noindent Indeed, if $u$ changes sign then we may apply Theorem \ref{Main} to the sets $\{u > 0\}$ and $\{-u > 0\}$ to get a contradiction.
\begin{rem} Theorem \ref{Main2} is special to the cases $\alpha \in (0,\,1)$. Indeed, when $\alpha \notin [0,\,1]$ and $\alpha < k^2$ for some nonzero integer $k$, the $\alpha$-homogeneous functions $$u = r^{\alpha}\cos(k\theta)$$ are sign-changing and have nowhere vanishing Hessian determinant on $\mathbb{R}^2 \backslash \{0\}$. We also remark that $0$-homogeneous functions have vanishing Hessian determinant on the rays where they achieve their maxima, and $1$-homogeneous functions have identically vanishing Hessian determinant. \end{rem}
Apart from its own interest, Theorem \ref{Main2} is motivated by the question of when interior gradient estimates hold for solutions to the special Lagrangian equation \begin{equation}\label{sLag} F(D^2u) := \sum_{k = 1}^n \tan^{-1}(\lambda_k(D^2u)) = \Theta(x) \in \left(-n\frac{\pi}{2},\, n\frac{\pi}{2}\right). \end{equation} Here $u$ is a function on a domain in $\mathbb{R}^n$ and $\lambda_k(D^2u)$ denote the eigenvalues of $D^2u$. Equation (\ref{sLag}) prescribes the mean curvature of the gradient graph of $u$ in $\mathbb{R}^n \times \mathbb{R}^n$. In particular, this graph is volume-minimizing when $\Theta$ is constant.
The existence of continuous viscosity solutions to the Dirichlet problem for (\ref{sLag}) is known in certain situations (see e.g. \cite{HL}, \cite{CP}), and there are many fascinating open questions concerning the regularity of these solutions. For example, is not known whether they are locally Lipschitz if either $\Theta$ is a constant with $|\Theta| < (n-2)\frac{\pi}{2}$ (they are if $|\Theta| \geq (n-2)\frac{\pi}{2}$, see \cite{WY}) or if $\Theta$ is Lipschitz. Classical proofs of interior gradient estimates for elliptic PDEs involve differentiating the equation once, so is reasonable to ask if interior gradient estimates for (\ref{sLag}) hold under such conditions on $\Theta$. A first attempt to {\it disprove} the validity of such estimates could be to build a function $u$ that is homogeneous of degree $\alpha \in (0,\,1)$, smooth away from the origin, and has nowhere vanishing Hessian determinant. Then $F(D^2u)$ would behave near the origin like a constant plus smooth function on the sphere times $|x|^{2-\alpha}$, which is $C^1$, while $u$ has unbounded gradient. Taking $u = |x|^{\alpha}$ appears to do the trick, but this function is not a viscosity solution at the origin; one needs $u$ to change sign to prevent this issue. Theorem \ref{Main2} precludes the existence of such functions, and hence can be viewed as evidence in favor of a positive result.
\begin{rem} One might also try to build one-homogeneous functions $u$ on $\mathbb{R}^n$ such that $\sigma_{n-1}(D^2u)$ is nowhere vanishing, since in that case $F(D^2u)$ is Lipschitz at the origin. Here $\sigma_k(D^2u)$ denotes the $k^{th}$ symmetric polynomial of the eigenvalues. It is not hard to show that such functions are necessarily convex or concave (see Section \ref{Preliminaries}), and thus do not solve the equation at the origin. Interestingly, there exist nonlinear one-homogeneous functions $u$ on $\mathbb{R}^3$ whose Hessians are either indefinite or $0$ at every point (see \cite{M}), so that $F(D^2u)$ tends to zero at the origin along rays, but $F(D^2u)$ is not continuous at the origin for these examples. \end{rem}
The paper is organized as follows. In Section \ref{Preliminaries} we recall some preliminary results about one-homogeneous functions and about maps with integrable dilatation, which are natural analogues of quasi-conformal maps in higher dimensions. In Section \ref{Proof} we prove Theorem \ref{Main}. The idea of the proof is to study the geometry of the gradient image of the one-homogeneous function $u^{\frac{1}{\alpha}}$. Our analysis is partly inspired by the beautiful arguments in \cite{HNY} used to show the linearity of one-homogeneous functions on $\mathbb{R}^3$ that solve linear uniformly elliptic equations.
\section*{Acknowledgments}
This research was supported by NSF grant DMS-1854788.
\section{Preliminaries}\label{Preliminaries}
In this section we recall a few results about one-homogeneous functions and about maps with integrable dilatation.
Let $v$ be a one-homogeneous function on $\mathbb{R}^n$ that, away from the origin, is locally $W^{2,\,p}$ for some $p \geq 1$. Euler's formula for homogeneous functions says that \begin{equation}\label{Euler} v(x) = \nabla v(x) \cdot x. \end{equation}
Here and below we let $r := |x|$ and we denote points in $\mathbb{S}^{n-1}$ by $\omega$. Writing $$v = r\,g(\omega)$$ and choosing a coordinate system where $\omega$ is the last direction, we have \begin{equation}\label{OneHomogHessian} D^2v(\omega) =
\begin{bmatrix}
\nabla_{\mathbb{S}^{n-1}}^2g + g\,I_{n-1 \times n-1} & 0 \\
0 & 0
\end{bmatrix}. \end{equation} Here and below, $\nabla_{\mathbb{S}^{n-1}}$ and $\nabla_{\mathbb{S}^{n-1}}^2$ denote the usual gradient and Hessian operators on the sphere. It is sometimes useful to represent $v$ in $\{x_n > 0\}$ by a function $\bar{v}$ on $\mathbb{R}^{n-1}$ defined by $$\bar{v}(y) := v(y,\,1),$$ so that $$v(x',\,x_n) = x_n\bar{v}\left(\frac{x'}{x_n}\right).$$ Taking the Hessian yields $$D^2v(y,\,1) =
\begin{bmatrix}
D^2\bar{v} & -D^2\bar{v} \cdot y \\
-D^2\bar{v} \cdot y & y^T\cdot D^2\bar{v} \cdot y
\end{bmatrix}.
$$ Using this we calculate \begin{equation}\label{DetRelation}
\sigma_{n-1}(D^2v)(y,\,1) = \text{tr}(\text{cof}(D^2v))(y,\,1) = (1+|y|^2)\det D^2\bar{v}(y). \end{equation} Here the operator $\sigma_k$ denotes the $k^{th}$ symmetric polynomial of the eigenvalues. It is easiest to verify this formula after rotating in the $y$ variables so that $D^2\bar{v}$ is diagonal. If $v$ is $C^2$ in a neighborhood of $e_n$ and $D^2\bar{v}(0)$ is nonsingular, then we can represent the gradient image of $v$ near $e_n$ as the graph of a function $w$ using the relation $$w(\nabla \bar{v}(y)) = \partial_nv(y,\,1) = \bar{v} - y \cdot \nabla \bar{v},$$ i.e. $w$ is the (negative) Legendre transform of $\bar{v}$. One differentiation gives $$\nabla w(\nabla\bar{v}(y)) = -y,$$ and another gives \begin{equation}\label{II} D^2w(\nabla \bar{v}(y)) = -(D^2\bar{v})^{-1}(y). \end{equation} In particular, the second fundamental form at $\nabla v(e_n)$ of the image under $\nabla v$ of a small ball around $e_n$ is $(D^2\bar{v})^{-1}(0)$. From this it is easy to see that if $v$ is locally $C^2$ away from the origin and $\sigma_{n-1}(D^2v)$ is nowhere vanishing, then $v$ is either convex or concave. Indeed, it suffices to show that either $D^2v \geq 0$ or $-D^2v \geq 0$ at some point. By (\ref{II}) this is true at the inverse image under $\nabla v$ of any point on $\nabla v(\mathbb{S}^{n-1})$ that is touched from one side by a hyperplane.
We now recall a few facts about maps of integrable dilatation. Let $$\varphi: U \subset \mathbb{R}^n \rightarrow \mathbb{R}^n$$ be a map in $W^{1,\,n}_{\text{loc}}(U)$ such that $\det D\varphi > 0$ almost everywhere. The dilatation $K$ of $\varphi$ is defined by the ratio
$$K(x) := \frac{|D\varphi|^n}{\det D\varphi}.$$ If $K$ is bounded and $n = 2$ then $\varphi$ is quasi-conformal, hence continuous and either open or constant by classical results. Reshetnyak extended this result to mappings with bounded dilatation in all dimensions \cite{R}. The boundedness required in Reshetnyak's theorem has since been relaxed to integrability in certain $L^p$ spaces:
\begin{thm}[{\bf Iwaniec-\v{S}ver\'{a}k}, \cite{IS}]\label{OM1} If $n = 2$ and $K \in L^1_{loc}(U)$, then $\varphi$ is continuous and either open or constant. \end{thm}
\begin{thm}[{\bf Manfredi-Villamor}, \cite{MV}]\label{OM2} If $n \geq 3$ and $K \in L^p_{loc}(U)$ for some $p > n-1$, then $\varphi$ is continuous and either open or constant. \end{thm}
\noindent It is conjectured that the latter result holds in the equality case $p = n-1$ (see \cite{IS}), and there are counterexamples when $p < n-1$ due to Ball (see \cite{B}).
\section{Proof of Theorem \ref{Main}}\label{Proof}
In this final section we prove the main theorem.
\begin{proof}[{\bf Proof of Theorem \ref{Main}}] We write $$u = r^{\alpha}f(\omega).$$ In a coordinate system where the last direction is $\omega \in \Sigma$, the Hessian of $u$ at $\omega$ can be written \begin{equation}\label{alphaHessian} D^2u = \begin{bmatrix}
\nabla_{\mathbb{S}^{n-1}}^2f + \alpha\,f\,I_{n-1 \times n-1} & (\alpha - 1)\nabla_{\mathbb{S}^{n-1}}f \\
(\alpha - 1)\nabla_{\mathbb{S}^{n-1}}f & \alpha(\alpha - 1)f
\end{bmatrix}.
\end{equation} Subtracting the multiple $\frac{\left(\nabla_{\mathbb{S}^{n-1}}f\right)_k}{\alpha f}$ of the last row from the $k^{th}$ row in (\ref{alphaHessian}) for $k \leq n-1$ and taking the determinant we arrive at \begin{equation}\label{uHessDet} \begin{split} \det D^2u &= \alpha(\alpha -1)f^{2-n} \cdot \\ &\left[\det\left(f\nabla_{\mathbb{S}^{n-1}}^2f + \left(\frac{1}{\alpha}-1\right)\nabla_{\mathbb{S}^{n-1}}f \otimes \nabla_{\mathbb{S}^{n-1}}f + \alpha f^2I_{n-1 \times n-1}\right)\right]. \end{split} \end{equation} Now let $v$ be the one-homogeneous function defined by $$v = \begin{cases} u^{1/\alpha}, \quad \omega \in \Sigma \\ 0, \quad \text{otherwise.} \end{cases}$$ At $\omega \in \Sigma$ we compute the Hessian of $v$ in the same coordinates as above, using the formula (\ref{OneHomogHessian}):
\begin{equation}\label{vHessDet0}
D^2v = \frac{1}{\alpha}f^{\frac{1}{\alpha}-2}
\begin{bmatrix}
f\nabla_{\mathbb{S}^{n-1}}^2f + \left(\frac{1}{\alpha}-1\right)\nabla_{\mathbb{S}^{n-1}}f \otimes \nabla_{\mathbb{S}^{n-1}}f + \alpha f^2I_{n-1 \times n-1} & 0 \\
0 & 0
\end{bmatrix}.
\end{equation} We conclude from (\ref{uHessDet}) and (\ref{vHessDet0}) that \begin{equation}\label{vHessDet}
\sigma_{n-1}(D^2v) = \frac{f^{\frac{n-1}{\alpha} - n}}{\alpha^n(\alpha-1)}\det D^2u \end{equation} on $\Sigma$. In particular, either $\sigma_{n-1}(D^2v)$ or $-\sigma_{n-1}(D^2v)$ is locally strictly positive in $\Omega$. We also have by standard embeddings and the fact that $$\frac{1}{\alpha} > 1$$ that $v \in C^1(\mathbb{S}^{n-1})$. Indeed, by homogeneity we may view $v$ as a function of $n-1$ variables away from $0$. When $n \geq 4$ the Sobolev exponent $p_n$ is thus supercritical. In the case $n = 3$ it is critical and, denoting by $\bar{v}$ the restriction of $v$ to a hyperplane tangent to $\mathbb{S}^{n-1}$, we may apply the continuity assertion in Theorem \ref{OM1} to either $\nabla \bar{v}$ or its reflection over a line. Here we used the fact that $\sigma_{n-1}(D^2v)$ is nowhere vanishing in $\{v > 0\}$ and the relation (\ref{DetRelation}). In the case $n = 2$ we use that $W^{1,\,1}$ embeds to continuous on the line.
Now let $K := \nabla v\left(\mathbb{S}^{n-1}\right)$. For $\nu \in \mathbb{S}^{n-1}$, slide the hyperplane $\{x \cdot \nu = t\}$ (starting with $t$ large, and decreasing $t$) until it touches $K$ at some point $p_{\nu}$. Since $0 \in K$ we have $p_{\nu} \cdot \nu \geq 0.$ We claim that \begin{equation}\label{PositiveSet} p_{\nu} \cdot \nu > 0 \Rightarrow (\nabla v)^{-1}(p_{\nu}) \cap \mathbb{S}^{n-1} = \{\nu\}. \end{equation} To show the implication (\ref{PositiveSet}) it suffices to show that $(\nabla v)^{-1}(p_{\nu}) \cap \mathbb{S}^{n-1} \subset \{\nu,\,-\nu\}$, since by (\ref{Euler}) we have $$\nabla v(\omega) \cdot \omega = v(\omega) \geq 0$$ for all $\omega \in \mathbb{S}^{n-1}$. Assume by way of contradiction that $p_{\nu} \cdot \nu > 0$ but $(\nabla v)^{-1}(p_{\nu}) \cap \mathbb{S}^{n-1}$ is not contained in $\{\nu,\,-\nu\}$. After a rotation we may assume that $\nu = e_1$, and after another rotation in the $x_2,\,...,\,x_n$ variables we may assume that $\nabla v(\tilde{\nu}) = p_{\nu}$ for some $\tilde{\nu} \in \mathbb{S}^{n-1}$ such that $\tilde{\nu}_n > 0$. For $y \in \mathbb{R}^{n-1}$ we let $$\bar{v}(y) = v(y,\,1).$$ By construction, $\partial_1\bar{v}$ has a local maximum at $\tilde{\nu}/\tilde{\nu}_n$ (here we identify points on the hyperplane $\{x_n = 1\}$ with $\mathbb{R}^{n-1}$), and $\bar{v} > 0$ at this point, since $\nabla v(\tilde{\nu}) = p_{\nu}$ is nonzero. However, using (\ref{DetRelation}) we see that either $\det D^2\bar{v}$ or $-\det D^2\bar{v}$ is locally strictly positive in $\{\bar{v} > 0\}$. Theorems (\ref{OM1}) and (\ref{OM2}), applied to either $\nabla \bar{v}$ or its reflection over a hyperplane, imply that $\nabla \bar{v}$ is an open mapping in $\{\bar{v} > 0\}$, which contradicts that $\partial_1\bar{v}$ has a local maximum in this set.
Finally, let $\text{co}(K)$ denote the convex hull of $K$, and let $w$ be the support function of $\text{co}(K)$, that is, $$w(x) := \sup_{y \in \text{co}(K)} (y \cdot x).$$ The implication (\ref{PositiveSet}) implies that $v = w$. Indeed, it is clear that $0 \leq v \leq w$, and for $\nu \in \mathbb{S}^{n-1} \cap \{w > 0\}$ we have $$w(\nu) = p_{\nu} \cdot \nu = \nabla v(\nu) \cdot \nu = v(\nu).$$ Because either $\sigma_{n-1}(D^2v)$ or $-\sigma_{n-1}(D^2v)$ is locally strictly positive in $\Omega$, the set $\text{co}(K)$ has non-empty interior. Indeed, if not, then $v = w$ is translation-invariant in some direction orthogonal to $\text{co}(K)$, which along with the one-homogeneity of $v$ implies that $\sigma_{n-1}(D^2v) \equiv 0$. We conclude that $\{v > 0\} \cap \mathbb{S}^{n-1}$ contains some closed hemisphere, completing the proof. \end{proof}
\begin{figure}\label{fig1}
\end{figure}
\begin{rem} The proof in fact shows that $u$ is the $\alpha^{th}$ power of the support function of a convex set that has nonempty interior and $0$ in its boundary (note that if $0$ were not in the boundary of $\text{co}(K)$ then $w = v > 0$ on all of $\mathbb{S}^{n-1}$), and that $\mathbb{R}^n \backslash \Omega$ is the reflection through the origin of the convex dual to the tangent cone of this set at the origin (see Figure \ref{fig1}). \end{rem}
\end{document} | arXiv |
The spatial and temporal reconstruction of a medieval moat ecosystem
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Olga Antczak-Orlewska ORCID: orcid.org/0000-0001-9392-623X1,2,
Daniel Okupny ORCID: orcid.org/0000-0002-8836-60443,
Andrzej Kruk ORCID: orcid.org/0000-0003-1452-79064,
Richard Ian Bailey ORCID: orcid.org/0000-0001-9870-410X4,
Mateusz Płóciennik ORCID: orcid.org/0000-0003-1487-66982,
Jerzy Sikora ORCID: orcid.org/0000-0003-4494-29835,
Marek Krąpiec ORCID: orcid.org/0000-0003-4270-16686 &
Piotr Kittel ORCID: orcid.org/0000-0001-6987-79687
Moats and other historical water features had great importance for past societies. The functioning of these ecosystems can now only be retrieved through palaeoecological studies. Here we aimed to reconstruct the history of a stronghold's moat during its period of operation. Our spatio-temporal approach allowed mapping of the habitat changes within a medieval moat for the first time. Using data from four cores of organic deposits taken within the moat system, we describe ecological states of the moat based on subfossil Chironomidae and Ceratopogonidae assemblages. We found that over half (57%) of the identified dipteran taxa were indicative of one of the following ecological states: limnetic conditions with or without periodic water inflow, or marshy conditions. Samples representing conditions unfavourable for aquatic insects were grouped in a separate cluster. Analyses revealed that the spatio-temporal distribution of midge assemblages depended mostly on depth differences and freshwater supply from an artificial channel. Paludification and terrestrialization did not happen simultaneously across the moat system, being greatly influenced by human activity. The results presented here demonstrate the importance of a multi-aspect approach in environmental archaeology, focusing not only on the human environment, but also on the complex ecology of the past ecosystems.
The diversity of habitats and of the mutual connections between organisms and their environment can now be studied in a wide variety of ecosystems, including historical features no longer present. Moats were artificial bodies of water providing defence for inhabited strongholds in medieval times. Ecological studies of moats can therefore provide a window into historical influences of human activity on natural ecosystems, broadening our knowledge of the functioning of such features, of past human societies, and of our coexistence with (seemingly distant) animal communities. Ecological studies can be carried out on moats that have either been preserved from historical times (e.g.1,2) or have been recently reconstructed. However, these modern moats do not perform the same function today as in the Middle Ages or Early Modern Period3, and their aquatic ecosystems may therefore differ. The medieval stronghold's inhabitants influenced water conditions by changing its trophic state, creating new habitats by placing construction elements, modifying its hydromorphology and sedimentation of defined types of water deposits, and their enrichment with selected metals (see4,5). What were the nature, scale, and consequences of such activities? The only way to get a clear window into those past ecosystems is through palaeoecological studies.
Geochemical studies of historical layers (such as core samples) conducted so far have in many cases confirmed their indicator role in environmental archeology6. The clearly higher Pb and Cu concentrations in the sediments of medieval moats have been used to determine the course of historical watercourses. However, the grain and mineral composition of sediments influences their susceptibility to metal sorption7. Therefore, geochemical mapping of archaeological sites, aiming to assess the distribution and characteristics of metallic pollutants in the environment, should take into account the location of the studied cores.
The chosen location of cores also influences palaeoecological studies. Typically, palaeoecological studies are based on one core of sediment, taken from the deepest (mostly central) part of the (palaeo)lake, on the assumption that the subfossil remains of the organisms once living in different parts of the water body are passively transported over time and deposited into the deepest area of the lake bottom8. This way, representative biotic records can be obtained with only one drilling, reducing cost and time of analyses. However, in the case of macrofossils, diatoms, or cladocerans, littoral taxa can be underrepresented in sequences derived from one core9,10,11. Analogously, chironomid head capsules accumulate mostly near the habitat of the larvae, and offshore transport is primarily observed in the less well preserved early instars12, which may produce some biases. Generally, if the subfossil record is expected to adequately represent the environment of sedimentation, and track particular events, multiple coring within the (palaeo)lake or mire is needed8,13.
In environmental archaeology, it is recommended to take at least two cores for palaeoecological analyses—one from the studied site (on-site profile) and one from an area separated from direct human activity (off-site) (e.g.14,15). This allows examination of both local and regional vegetation history, and also of environmental information, such as past water level or temperature changes. Due to differences in deposition between aquatic and terrestrial systems, "wet sites" or "wet features" (such as a moat) allow examination with use of a wider range of ecological proxy analyses, based on organic deposits within cores sampled directly from archaeological trenches.
Our on-site palaeoecological investigations were an integral part of an archaeological study of a stronghold's moat system in Rozprza (central Poland; see 'Study system' section, below)5,16. The main sediment core from the studied moat has already been investigated in detail (see5). The results indicated that when the stronghold was inhabited (14th-15th century AD), the moat was infilled with shallow water with possible episodes of rinsing through an artificial channel from the nearby river. Changes in water trophic state were tracked using chironomid-based models, and the Chironomidae-inferred summer temperatures of the Late Vistulian came from the nearby palaeomeander17. The Holocene chironomid sequence in the palaeomeander core was too low in abundance for quantitative reconstructions. However, even profiles with a depauperate biotic record can be used to obtain some ecological information, especially if treated simultaneously with other analyses. Therefore, we took additional cores from potentially different habitats (cf.18) within the moat, in order to obtain a more complete picture of environmental changes and human impact on the ecosystem in both time and space.
Though chironomids have proven to be useful in environmental archaeology (e.g.19,20,21,22,23), they have not previously been used for spatial reconstructions of artificial features. Moats, barays and other anthropogenic ponds have more limited potential to accumulate most of the subfossil remains in one point than lakes, mostly because of the specific basin morphology. Therefore, a carefully constructed spatial approach is required, taking into consideration habitat mosaics within the moat ecosystem. Midge larvae (Diptera: Chironomidae and Ceratopogonidae), especially Chironomidae, represent an ideal proxy for spatial reconstruction of artificial features. They are ecologically diverse, sensitive to environmental changes, and indicative of particular ecosystem conditions and processes24,25. Therefore, their distribution in shallow reservoirs with complex morphometry may be diversified and earlier studies based on one sediment profile5 need to be complemented in order to cover the spatial aspect.
Our study addresses the general problem of lack of replicability in the environmental sciences. The issue is particularly visible when deriving data from remote places or from the past (e.g. in geology, oceanography, palaeontology, archaeology). In palaeoecology, replicability is applied when studying and comparing the same phenomenon/event (e.g. Younger Dryas) in different sites. However, few studies replicate from the same locality, collecting cores several metres apart. On the few occasions that this small-scale replication has been carried out, important variation has been identified (e.g.26).
Considering these issues, our main goal was to reconstruct changes in habitat distribution across the stronghold's moat system over time. We aimed to examine: (1) the extent to which the Chironomidae and Ceratopogonidae subfossils from the different cores reflected differences in past community structure and environmental conditions in different parts of the moat (spatial aspect), and (2) the extent to which midge communities changed simultaneously and/or evenly throughout the moat basin (temporal/time aspect). We hypothesised that even in such a small and shallow water body as a moat, the differences in midge community structure between cores and time periods would be significant and, thus, the habitat mosaic and other aspects of stronghold functioning could be more fully reconstructed using multiple cores.
Rozprza is located in central Poland, about 60 km south of Łódź in the Piotrków Plain. The study site is situated in the middle reach of the Luciąża River valley, a tributary of the Pilica River in the Vistula River basin (Fig. 1).
(A) Study site location in the territory of Poland. (B) The satellite picture of the contemporary surroundings of the study area. The red arrow indicates the stronghold's remnants (source: Google Earth, modified). (C) Aerial photo of the study area. The locations of the studied profiles (cores) are marked by yellow dots. Arrows indicate the traces of artificial elements visible in the terrain relief—moats (dashed lines) and presumed artificial canal (dotted lines) (photo: P. Wroniecki, 2015).
Nowadays, the late medieval stronghold remnants with their moat system are situated in an area covered by fields and meadows between the Rozprza and Łochyńsko villages (51° 18′ 07″ N; 19° 40′ 04″ E). The poorly preserved traces of moats, ramparts and baileys are however still visible in the field (Fig. 1C) and on Digital Elevation Models (DEMs). The study site is located on the valley floor with regulated channels of the Luciąża, Rajska and Bogdanówka rivers, as well as a dense network of drainage canals (Fig. 1B).
The studied fortress was established on the Plenivistulian fluvial terrace remnant, in the area of the widely spread valley floor. Such a location of the motte-and-bailey was reasonable for defensive reasons, as the sandy terrace remnant was protected by the surrounding swampy areas within the valley floor16,27. However, the hillock of the terrace remnant occupied by the motte in the late Middle Ages was very low (up to 1 m).
The late medieval motte-and-bailey timber castle at Rozprza was built about 1330 AD and replaced an earlier timber and earth ringfort of unclear chronology (between 11th and 13th century AD). Motte-and-bailey castles were common in western Europe already in the 11th century28 but introduced to Poland much later, in the 13th century29. In the 14th and 15th centuries AD it was one of the most popular types of rural noble residences.
The main moat of the Rozprza motte-and-bailey was established ca. 1330 AD and was later filled with organic (gyttja and peat) and partially inorganic deposits containing rich remains of wood (Fig. 2, Supplementary Fig. S1B–D). The fill of the main moat was the subject of a detailed palaeoenvironmental study by Kittel et al.5. The accumulation of overbank silty sandy organic mud took place within the moat ditch system as late as in the 18th or 19th c. AD5,16. The main moat had a width of 17–21 m and a trapezoidal cross-section with a depth of 0.5 m, up to ca. 1.3 m. Wooden constructions were recorded near the inner slope of the moat ditch—in the form of a palisade created by two rows of vertical, sharpened wooden poles, and horizontal beams lying behind them. Those constructions were covered with thick layers of slope deposits (sand with organic mud). Many large chunks of wood (branches and boughs) were recorded in peat and organic mud of the upper unit of the moat fill close to the inner slope of the main moat16.
Core correlation with reference to: modelled chronology, midge-inferred SOM subcluster zones (symbols are used as in the Fig. 3), lithology, grain-size composition, lithogeochemical results and statistical relations of selected elements. Midge concentration refers to Chironomidae and Ceratopogonidae head capsules (hc). The data points of the mean age ± 1σ (AD) are given for each core. Question mark means uncertain date of the secondary moat establishment. Geochemical periods were designated on the variable results with respect to average values Na/K (mean = 0.12), Fe/Mn (mean = 60) and Cu/Zn (mean = 0.27) ratios.
The second moat had a width of about 11 m and a depth of ca. 0.5 m. In the trapezoid cross-section it has a flat bottom. The fill of the moat consisted of sand with organic admixtures at the very bottom, limnic deposits, peat with silts, sands and organic mud, and also slope wash deposits on the moat slopes. Generally, these sediments are characterised by a massive structure, dark or grey-brown uniform colour, resulting from a strong concentration of dispersed organic matter and iron compounds (Supplementary Fig. S1A). The chronology of the feature formation has been estimated to 1485–1634 AD based on 14C data (Table 1). It demonstrates that the secondary moat was built probably in the early 16th century AD. However, the analysis of most archaeological small finds obtained during field works, mostly pottery fragments, estimated their age to the period between 14th and a mid-15th century AD. Evidence of human activity on the stronghold in the 16th century AD is also very limited. Therefore, an establishment of the second moat in the 15th century AD cannot be excluded (see16).
Table 1 The results of radiocarbon dating of the organic deposits of Rozprza moats.
Results and interpretation
Chronology of moat deposit accumulation
In total six radiocarbon dates were used for the construction of the age-depth model for the RP W3(2) core from the deepest studied part of the main moat (Table 1, Supplementary Table S1)5,16. Based on dendrochronological data from a wooden fragment from the moat's bottom, establishment of the main moat was defined to ca. 1330 AD. In the early phase, the moat was filled with gyttja. In the first half of the 16th century AD, a sedentation of peat began. In the early 18th c. AD the peat was covered with overbank alluvia (Fig. 2).
A comparable pattern of evolution of the main moat was reconstructed based on the age-depth model for the RP W3(4) core (Table 1, Supplementary Table S2). The lacustrine deposition was replaced by peat sedentation in the mid-15th c. AD. The accumulation of overbank alluvia may have been initiated in the first decades of the 18th c. AD. The fill of the western part of the moat was covered in 1944 AD by an embankment from the destroyed stronghold mound5.
The chronology of the main moat fills in the RP W1 core confirms an establishment of this moat in the 1st half of the 14th c. AD. Moreover, the eastern part of the wet defensive system was filled with peat from its beginning up to 16th c. AD (Table 1, Supplementary Table S3). The upper part of the moat fill in the RP W1 area was probably removed during melioration works in 20th c. AD.
The absolute chronology of the fill of the secondary moat, studied in the RP F2 core, demonstrates that this additional defensive ditch has been established most probably in the first half of the 16th c. or possibly in the late 15th c. AD (Table 1, Supplementary Table S4). From 18th c. AD, an effect of flooding is visible, recorded by sandy admixtures in organic deposits of the moat fill (cf.16).
Dipterans and environment: identification of the moat phases
The samples from depths of 45-39 cm in RP W3(4) core and the samples from depths of 67 and 49-21 cm in RP F2 core were empty, i.e. without any taxon present. The self-organising map (SOM) allowed clusters of non-empty core samples with similar community composition to be produced. The taxa significantly associated with them were then identified with Indicator Species Analysis.
Two main clusters were distinguished in the output layer of the SOM: X and Y, comprising the respective pairs of subclusters: X1 and X2, and Y1 and Y2 (Fig. 3). The subclusters were ordered according to the gradient observed in the number of indicator species (Fig. 4). Subcluster X1 represents unfavourable conditions for Chironomidae development, mostly overbank deposits (see Fig. 2). It contains surface samples from RP W3 cores (from 51 cm depth in W3(2) and from 55 cm in the W3(4) profile), samples from 91 to 87 cm of W3(2) core depth, 85–83 cm and 73 cm of W3(4) core depth, most samples from RP W1 core, and the whole sequence of the non-empty samples from the second moat (RP F2 core). Subcluster X2 included samples from 83 to 55 cm of RP W3(2) core, 75 cm and 63–57 cm of RP W3(4) core, as well as one sample (29 cm depth) from RP W1 core. They were associated with high organic matter (OM) content (mean 59.9%) and slightly acidic (mean pH = 6.6), probably telmatic (marshy) conditions (Fig. 2). Cluster Y represents limnetic conditions dominated by fully aquatic midges and low values (i.e. < 20) of TOC/N ratio for deposits. The latter suggests that organic matter mainly came from algal phytoplankton49,81. Subcluster Y2 reflects habitat with higher detrital (K, Mg, Ca) and sulphide (Cu, Fe) element concentrations, grouping bottom samples from the RP W3(2) core up to 95 cm and samples of 39–37 cm depth from the RP W1 core (Fig. 3). Samples grouped in the subcluster Y1 (81–77 cm and 71–65 cm core RP W3(4) depth, sample from 21 cm core RP W1 depth) are associated with lower sulphide element content (Fig. 3).
Seventy nine non-empty core samples assigned to 42 self-organising map (SOM) output neurons (A1–G6). The neurons are arranged into a two-dimensional lattice (7 × 6). Clusters (X and Y) and subclusters (X1, X2, Y1 and Y2; shown in different degrees of greyness) of neurons have been identified with the use of hierarchical cluster analysis. Sample codes are arranged as follows: first two signs stand for core symbol (W1—RP W1, W2—RP W3(2), W4—RP W3(4), F2—RP F2), followed by numbers referring to depth (in cm b.g.l.).
Fifty five dipteran taxa significantly (p ≤ 0.05) associated with SOM subclusters X2, Y1 and Y2 (respectively, 8, 19 and 28 taxa). No palaeoindicator was significantly associated with subcluster X1. The shading is scaled independently for each taxon; it is darker for a stronger association in virtual core samples. Maximum observed indicator value (IndVal) is shown above each taxon plane; IndVals and their significance levels were calculated on the basis of real core samples. The plane for Procladius (56***), which is indicative of subcluster Y1, is not presented for graphical reasons; it resembles the plane for Ablabesmyia.
A total of 55 (57%) dipteran taxa were significantly associated with a certain subcluster, i.e. they were indicators of its respective environmental conditions (Fig. 4). Among these, 24 exhibited IndVals significant at p ≤ 0.001, 20 at p ≤ 0.01, and 11 at p ≤ 0.05. An upward trend was observed in the number of such taxa for subclusters in the order X1, X2, Y1, Y2. No palaeoindicator was significantly associated with subcluster X1, eight taxa were significantly associated with X2, 19 with Y1 and 28 with Y2. Therefore, this order of subclusters corresponds to increasingly favourable conditions for development of a rich biota.
The most indicative (at p < 0.001) morphotypes for subcluster X2 were Limnophyes-Paralimnophyes and Parametriocnemus-Paraphaenocladius, which are associated with the semi terrestrial habitats with slightly acidic water30. Ceratopogonid species grouped as Dasyhelea-type seem to have similar preferences31, while Chironomini taxa linked to X2 prefer shallow, muddy water bodies and can occur in seasonal surface water. Chironomids associated with Y1 were mostly typical of warm, productive, littoral habitats, and many of them are phytophilous (e.g. Glyptotendipes pallens-type, Lauterborniella). However, also associated with this subcluster were Tanytarsus lugens-type and Paratanytarsus austriacus-type, often recorded in cold, oligotrophic conditions. Morphotypes significant to subcluster Y2 included both taxa associated with warm, eutrophic stagnant water (e.g. Micropsectra pallidula-type, Cladotanytarsus mancus-type, Cryptochironomus) and those preferring meso- and oligotrophic conditions (e.g. Psectrocladius barbatipes-type and Bezzia-type). Many of them, such as Zavreliella and Polypedilum sordens-type are associated with macrophytes. Moreover, several chironomids associated with running water (such as Nanocladius rectinervis-type, Corynoneura coronata-type and Psectrotanypus varius) were recorded with a significant IndVal in this subcluster. This differentiation is confirmed by the results of the chemical composition filling from the upper part of the RP W3(2) core (Fig. 2), because the rich organic sediments (OM even above 90%) are covered by acidic deposits with organic matter content below 7% and very variable concentration of lithophilic elements (for example K range 0.17–2 mg/g). The changes in time of sorption capacity were probably caused by changes in the porosity of the sediments that accumulated in the moat. This feature is the result of the difference between natural and dry bulk density and it is particularly modified by the content of very fine fraction in the sediments32. In turn, the increased abundance in nutrients results from a high proportion of the clay fraction, which in the sediments from RP W3(2) often exceeds 3%, with a maximum of 6.45% (Fig. 2). These features of the biogenic accumulation environment influenced conditions for the development of vegetation and chironomids.
Canonical correspondence analysis (CCA) was carried out to detect midge (Chironomidae and Ceratopogonidae) geochemical signal correlations. Axis 1 (Ax1) explained 11.3% and Axis 2 (Ax2) 4.2% of species data variance among individual core samples. For the species-environment relationship variance, 42.1% was explained by Ax1 and 16.0% by Ax2. The analysis (Fig. 5) demonstrated that pH, Ca, Pb, Fe and organic matter (p < 0.01), as well as Cu and K (p < 0.05), were significant in shaping midge assemblages in the moat, with 6.4% of the variance explained by pH, 4.6% by Ca, 4.2% by Pb, and 3.0% by Fe. Organic matter (OM) and Cu both explained 2.3%, while K explained 1.5% of the total variance. Pb was positively correlated and pH negatively correlated with Ax1. The rest of the variables were positively correlated with Ax2.
CCA biplot showing changes in the moat states expressed by SOM subclusters represented by indicative dipteran taxa (triangles) and sediment samples (circles), under a gradient of environmental variables (A). Variables correlated with Ax1 are shown as red arrows, while those correlated with Ax2 are shown as blue arrows. Mn and Zn were not significant for the analysis. Taxa and samples associated with each subcluster are coloured differently. Sample codes given on zoom (B) are arranged as follows: first two signs stand for core symbol (W1—RP W1, W2—RP W3(2), W4—RP W3(4), F2—RP F2), followed by numbers referring to depth (in cm b.g.l.). For full names of taxa see Supplementary Table S5.
The samples grouped in the X1 subcluster generally represented conditions unfavourable for aquatic biota. Many of them were characterised by relatively high Pb, probably reflecting increased denudation processes after stronghold abandonment and increased flooding activity in the 18th–19th centuries AD (cf.5). This series describes monofraction of the mineral admixture (share of the sand mainly ranges between 70–90% and Mz for 70% number of samples is 1.6–2.2 phi; Fig. 2). According to Kittel et al.21, in the absence of a clear boundary between individual layers, identification of flood activity should include changes of colour sediments, caused by admixture of decomposed and diffused organic matter (Supplementary Fig. S1). In our case, organic matter values varied little among X1 subcluster samples (mean for this section 29.3%) and corresponded with light-grey horizon (overbank organic mud and overbank sandy organic mud vide:5; Fig. 2) a dozen cm thick. The overbank deposits accumulated in rapidly changing oxygenation of water. Such conditions stimulated microbial decomposition of organic matter and allowed accumulation of sulphide elements, such as Cu, Mn and Fe (Fig. 2). The taxa typical of the telmatic phase of the moat (X2) were associated with low pH. Moreover, Limnophyes-Paralimnophyes, Parametriocnemus-Paraphaenocladius and Polypedilum sordens-type prefer habitats with high organic matter and iron compounds content. However, among samples classified to the X2 cluster, those from RP W3(2) core were more related to acidic conditions than those from RP W3(4). Generally more alkaline conditions are preferred by the chironomids indicative of the limnetic stage of the moat, in particular to subcluster Y2 (Fig. 5). Those taxa (e.g. Zavreliella and Cricotopus bicinctus-type) prefer habitats with high Ca and K values. Several phytophilous taxa (such as Paratanytarsus penicillatus-type, Corynoneura coronata-type and C. arctica-type) seemed to be more associated with sulphide elements (Cu, Zn), than with pH level. Alkaline conditions are also important to several taxa indicative of the Y1 subcluster (like Tanytarsus pallidicornis-type 2), but unlike Y2, these taxa are associated with low element levels (Fig. 5). The occurrence of allochthonic mineral matter with variable grain-size parameters (Supplementary Fig. S2) in the Y1 samples may be responsible for the increased lithophilic elements (mainly Mg and K) content in deposits (Fig. 2).
The high concentrations of Ca (often above 60 mg/g) (Fig. 2) are documented in the deposition environment of the hypergenic zone in Central Europe (cf.33). The intensive chemical denudation and leaching of mineral substrate of variable origins were confirmed in the catchment of Luciąża River valley, the surface geology and mineralogy of which was documented in detail by Wachecka-Kotkowska34. Moreover, the documented sediment structures and textures (Fig. 2, Supplementary Figs. S1 and S2) prove the supply of mineral matter in two different accumulation environments. The first one is related to upper flow regime (mainly samples from RP F2 and RP W3(4)), the second one in the lower flow regime conditions, which is related to the sorting by absorbing detrital material by selective deposition of fine grains from the suspension35.
In most of the studied profiles the concentration of sulphide elements was low (Cu: 3.49–57.2 μg/g; Zn: 3.12–210 μg/g; Fe: 1.2–99.3 mg/g) (Fig. 2), but irrespective of lithology, these results are typical for a river valley environment in Central Europe36. The stratigraphy differentiation in deposit chemistry indicated that enrichment of Cu and Fe took place during the changes of sedimentation type from organic rich to mineral input or increased humification. Precipitation of colloidal forms of these elements was dependent on changes in the local groundwater level37. In the Luciąża River valley the water budget was represented by water flowing underground into the alluvia coming from the post-glacial areas surrounding the Rozprza stronghold (e.g. Radomsko and Dobryszyce Hills), the water of the Luciąża river system, and precipitation water that did not participate in the evapotranspiration processes.
Testing differences among cores in relative taxon abundance
To examine whether an individual core can be considered representative of the whole site, we tested for differences in relative dipteran taxon abundances among cores, controlling for sample volume and sample age effects. Diagnostics revealed no significant deviations from model assumptions for the fitted poisson family generalised linear mixed model (GLMM). Fixed effect model selection based on AICc revealed the full model to be the best-fitting model, including an interaction between core and species (LL = − 3727.7, d.f. = 294, AICc = 8079.1, weight = 1). This indicated a significant difference among cores RP W1, RP W3(2) and RP W3(4) in chironomid relative species abundance distributions. Therefore, there may be some error associated with extrapolating results from a single core across a whole site. The species with the biggest differences among cores in relative abundance were Chironomus plumosus-type and Dicrotendipes notatus-type, which both are common, often with high share in the samples. However, they had much higher relative abundance in core RP W3(2) than RP W1 (Supplementary Table S5).
The results based on chironomid and ceratopogonid assemblages generally confirm three main stages of the moat history: limnetic, telmatic and terrestrial. In addition to the previous study5, however, we reveal variation in moat habitat overall and in temporal habitat changes across the moat system.
The afore-mentioned three stages of the moat are visible only in its deepest, south-western part (both RP W3 cores) (Fig. 2). Here the ecological processes, such as paludification, lasted longer and the habitat changes were less dynamic, resulting in more stable conditions for biota. However, the limnetic phase was of a different nature in the RP W3(2) and RP W3(4) cores, which were located close (12.5 m) to each other. While fresh water from the artificial canal (see Figs. 1C, 6,5) firstly supplied the southern part of the moat, the inflow may have been higher in RP W3(2) than in RP W3(4), probably because of its greater depth (Fig. 6). The significant presence of rheophilic taxa in the former core (indicative for cluster Y2) supports this. Moreover, the sediment chemistry record (Fig. 2), in particular values of Fe/Mn ratio, suggest higher oxygenation in this part of the moat, which may indicate the course of the water current. A crucial factor in this case could also be the structure of the moat bottom closely related to the groundwater level, which determines the habitat diversification of the plant cover. These processes could lead to the aggregation of soils grains/sediment into concretions and lumps, which when combined with Fe and decomposed organic matter, can lead to development of dense hardly permeable zones38. Such a situation in the studied area would have a direct impact on the disturbance of vertical water movement and the possibility of plant rooting, determining the specific geochemical cycle between moats-plants-sediments. While habitats in both RP W3 cores during the limnetic phase, with high pH and dense vegetation, could support well-functioning biotic communities, conditions in the shallower part of RP W3(4) were slightly less favourable for midges. There, the limnetic stage is reflected mainly in the Y1 subcluster, interrupted by single samples with lower concentration of midge larvae. The telmatic phase in this core could have started earlier than in RP W3(2), as indicated by peat deposits and the X2 cluster. This effect may be caused by the location of RP W3(4) close to the moat edge, resulting in faster sedentation and shallowing of the moat bottom.
Phases and episodes of different moat states over time: (A) limnetic stage only in the southern (deeper) part of the main moat (1329—ca. 1450/1500 AD); (B) – limnetic stage in the whole main moat, episodes of higher water level in the NE part of the moat, indicated by Y2, X2 and Y1 subclusters in the core RP W1 (ca. 1370–1380 AD, ca. 1400–1410 AD, ca. 1430–1440 AD); (C) construction of the southern secondary moat, telmatic stage (ca. 1500—ca. 1710/1750 AD); (D) terrestrial stage in the both moats up to their covering with the material from the stronghold's mound in 20th century AD (ca. 1750–1944 AD).
The north-eastern part of the main moat had worse conditions for chironomid development, as is shown by cluster X1 containing most samples from the RP W1 core (Fig. 2). Generally, the RP W1 core is characterised by relatively uniform lithology, consisting mostly of peat (with periodical supply of the mineral fraction, including sands). It is the result of much shallower conditions in this part of the main moat resulting in dominating sedentation of peat. During the first phase of the moat history (up to ca. 1440 AD), episodes of more complex midge larvae assemblages were recorded, as reflected by clusters Y2, X2 and Y1 (Fig. 2), and higher chironomid richness and abundance (Supplementary Fig. S5). They may indicate some limnetic episodes also in the NE part of the moat (Fig. 6), though not as clear and stable as in its deeper southern part. Despite slight differences in the concentration of most elements, they are confirmed by selected geochemical indicators (e.g. increase of Fe/Mn ratio to 82). These episodes are also accompanied by a clear decrease in the values of denudation indicators (i.e. Na/K from 0.12 to 0.04 and Ca/Mg from 0.03 to 0.01) (Fig. 2). The record of the RP W1 core ends approx. 1560 AD, probably because of the anthropogenic removal of the top parts of the moat filling during drainage works in the 20th c. AD.
Generally, the history of the main moat lasted for six centuries, since its establishment in ca. 1329 AD till the earthwork from 1944 AD, when the moat remnants were levelled (Fig. 6). In fact, the moat was only functioning as a defensive water body for the first 120–170 years of its existence. Even then it was fully aquatic only in its deepest, south-western part, with only several recorded episodes of higher water level in the whole feature (Fig. 6B). Its status changed in the 15th century, when it dried up (or was drained), becoming a kind of wetland, and ca. 300 years later it was fully filled with organic deposits and covered by overbank mineral matter.
The second moat (RP F2 core) was functioning briefly, as it was created not earlier than in the 2nd half of 15th century, and most probably ca. 1500 AD (see Fig. 2, Supplementary Table S4). In fact, it is not certain whether it was built as a functional moat, or possibly as a dry ditch (e.g. for melioration or defensive purposes). The chironomid scarcity and high admixture of sands (Figs. 2, 5) rather support the latter possibility. Moreover, the secondary moat was much narrower (ca. 11 m) than the main one, so active slope processes provided a constant supply of mineral matter (mostly sands).
The results of habitat reconstruction confirm the modelling (GLMM) outcome that one core does not show the entire history of the moat. This is because a moat is not a typical water body—not only very small and shallow, but also with a specific shape, which hinders transport of fossils to the deepest part of the moat. The steep, almost vertical banks, artificial channels, depth differences and many other features had a crucial impact on the sediment spatial composition and, hence, also on moat functioning. Anthropogenic wood and other artefacts and ecofacts in the bottom can serve as an additional habitat, e.g. for periphyton development.
Besides Rozprza, only a few moats in Europe had hitherto been studied using palaeoecological analyses (e.g.39,40,41), and they mostly focused on human economy and functioning, rarely touching the issue of the moat ecology itself. Moreover, some of these studied defensive objects were dry (e.g. in Prague42 and Gdańsk43). The external moat of the Czermno stronghold44 seems to be comparable with the Rozprza site, though they cover different time spans. Both features evidenced relatively fast peat sedentation and paludification in comparison with natural water bodies (cf.17). The moat system in the Tum (Łęczyca) stronghold has been well studied, including palynological and plant macrofossil analyses from its different parts45,46. However, no further spatial reconstruction of the environmental conditions within the feature was provided.
Moats and similar human-made features are hardly comparable with natural ecosystems. While the multiple coring approach is sometimes undertaken i.e. to track past water-level changes47,48, in such archaeological sites as Rozprza motte, various factors need to be considered, in particular human impact. In waterlogged sites, such as the Serteya Neolithic pile-dwelling, the human–environment relation can be tracked alongside the quantitatively reconstructed climatic background22,49. Palaeoecological methods are of great importance while tracking the history of the cities, like Gdańsk50 and London51 with the use of profiles of wet sediments.
In the majority of archaeological sites, if palaeoecological studies are undertaken, they focus on the surroundings of the excavations, mainly because of the lack of wet organic sediments to take core from (e.g.52,53). Another issue is the cost and time, which need to be taken into consideration with any additional core. In our case, the cores of sediments were taken as monoliths directly from the walls of archaeological trenches, and the profiles were relatively short, which was a great convenience. The additional cores, however, were examined only with respect to lithological and geochemical composition, accompanied by chironomid (and ceratopogonid) analysis. With the use of macrofossil analysis, habitat diversity could be even more accurately mapped, which is worth considering in future research.
To sum up, multiple cores are required to get a complete picture of the spatio-temporal changes within the ecosystem. The environmental reconstructions from the deepest part of the moat (RP W3(2)) presented in Kittel et al.5 are substantiated here, and the results are largely consistent with the core taken from the same trench (RP W3(4)). However, the sequences from the second moat (RP F2) and the NE part of the main moat (RP W1) significantly complement the reconstruction and help give a better understanding of the functioning of moat ecosystems and motte-and-bailey strongholds.
This study represents the first reconstruction of moat habitats during its functioning that considers spatial variation. It is likely that many similar water bodies could be investigated this way, broadening our knowledge about past societies and ecology of such human-made ecosystems.
The research in Rozprza began with a non-destructive survey carried out in 2013–2015. Methods included analytical field walking, aerial photography, geophysical and geochemical prospecting, as well as thorough geological mapping. A dense network of cores taken with a hand auger resulted in elaboration of detailed cross-sections of the ringfort vicinity. Thanks to this investigation it was possible to localise some archaeological and palaeogeographical features27,54. This led to the next extensive, interdisciplinary investigation. This fieldwork was conducted in 2015–2016, with the use of archaeological trenches, geological outcrops and a wide range of palaeoecological studies. They aimed to reconstruct the environmental conditions and settlement history of the mediaeval stronghold at Rozprza5,16,55.
The procedure of exposing trenches included removing successive 10-cm layers of sediments, distinguishing stratigraphic units within them, and wet-sifting with a 4 × 4 mm sieve in order to collect archaeological artefacts and ecofacts. All trench walls and collected features were thoroughly documented as orthophotos. The sediments for palaeoecological analyses were collected from the trench walls as monoliths using metal boxes with dimensions of 50 × 10 × 10 cm (Supplementary Fig. S1). Thanks to this method, the undisturbed structure of the sediments was preserved.
The RP W1 profile was collected from the deepest section of the trench 1/2015. This trench, with dimensions 2.5 × 12 m, was exposed in the eastern part of the main moat (Fig. 1C). Wooden vertical posts, associated with numerous fragments of wood, were revealed in the bottom of the moat. The moat was shallow here, reaching up to 50 cm depth. Two cores of sediments were collected from the trench 3/2015 (1.5 × 25 m), situated in the south-western part of the main moat. The RP W3(2) profile was taken from the deepest part of the main moat and RP W3(4) from its shallower part. Trench 3/2015 exposed the very well preserved moat fill, adjoining the outer rampart and the motte mound, allowing for their full cross-section. The RP F2 profile was taken from the thoroughly deepened and purified wall of the drainage ditch, which currently crosses the secondary moat. The deposits were collected from the deepest part of the smaller southern moat (Fig. 1C).
Digital reconstruction of the main moat relief
The 3D reconstruction model of the bottom of the main moat was prepared within the GIS environment (Qgis, SAGA GIS and PlanlaufTerrain softwares) using point cloud of Airborne Laser Scanning (ALS) already accessible via the Geoportal.gov.pl web service. This was supplemented with results of detailed coring (80 drillings 1 to 2 m apart) of the moat as well as results of excavation of archaeological trenches 1/2015 and 3/2015. Contemporary bare earth points covering the moat in the ALS derived point cloud were replaced by points with the height values of the surface of mineral bedrock indicating the original bottom of the moat. Subsequently, all the points were interpolated to obtain a Digital Elevation Model of the stronghold area with the main moat virtually reconstructed and emptied. This allowed for modelling water circulation and subsequent changes of moat states.
Geochemical and sedimentological analysis
Detailed geochemical tests covered material from the four cores presented here (133 samples from 4 cores) (Fig. 2). The basic physical and chemical parameters were the following: organic matter content (LOI—loss on ignition), calcium carbonate (CaCO3) content (volumetric measurement of CO2 from conversion of CaCO3 by 10% HCl), biophilic elements such as: total organic carbon (TOC), total nitrogen (TN) and total sulphur (TS) content (combustion method at the Rapid CS cube and VarioMax analyser—Elementar), and reaction (pH in distilled water). All parameters were measured in 2-cm resolution according to the procedure by Tolksdorf et al.56. Ash material without organic matter (remaining after LOI) was dissolved with concentrated 65% HNO3, 10% HCl and H2O2 in a Berghof Speedwave microwave mineralizer. Elements with palaeoenvironmental significance (Na, K, Ca, Mg, Fe, Mn, Cu, Zn and Pb) identified in the resulting solution were marked by the atomic absorption spectroscopy (AAS) method with use of Solar Unicam and following procedure after Borówka37.
When interpreting the individual elements of the chemical composition and grain size distribution for the mineral fraction for all four cores, the variable character of the sediment accumulation conditions was taken into account57,58. Palaeoenvironmental conditions responsible for the sedimentation of the studied deposits were interpreted by determining the quantitative ratios of the elements (such as: Na/K, Fe/Mn and Cu/Zn) with the assumption that the individual lithogeochemical components came from different sources (cf.49).
The grain size composition of mineral ash (treated as terrigenous silica) remaining after solution was prepared as in Clift et al.59, using a Mastersizer 3000 laser particle-size analyser (Malvern). The grain-size data were stored and processed using GRADISTAT software v. 8.060.
Chironomidae and Ceratopogonidae analysis
The samples for midge analysis were taken as contiguous 2-cm slices of the sediment from each profile, apart from the RP W3(2) core, from which they were collected with 4 cm resolution. The number of samples analysed in each profile was similar (ranging between 23 in RP W1 and 27 in RP W3(2)), while sample volume varied between 5 and 70 cm3.
Chironomidae preparation methods followed Brooks et al.25. The sediments were passed through a 63 μm mesh sieve. If head capsule (hc) concentration in the sediments was low, kerosene flotation was used following the procedure of Rolland and Larocque61. Processed sediment was scanned under a stereo-binocular microscope. Where applicable, a minimum of 50 (preferably 100) chironomid head capsules from each sample were picked and mounted in Euparal®. Identification of chironomids followed Schmid62, Brooks et al.25, and Andersen et al.63, while ceratopogonids were divided into two morphotypes distinguished by Walker64. Ecological preferences of identified taxa are based mainly on Brooks et al.25, Vallenduuk and Moller Pillot65, Moller Pillot30,66, and Luoto31. The midge sequences are presented on stratigraphic diagrams (Supplementary Figures S3–S6) created with C2 software67.
Radiocarbon and dendrochronological dating
The chronology of the Rozprza moat system was estimated using radiocarbon (14C) and dendrochronological methods. Both dendrochronological and conventional radiocarbon dating of organic material using the LSC technique were performed in the Laboratory of Absolute Dating in Kraków (Poland). A few wood fragments sampled during moat system exploration16 were dendrochronologically dated using standard procedures68.
A total of 15 samples from moats of the Rozprza motte-and-bailey were collected for radiocarbon (14C) dating (Table 1). Thirteen of these were sampled from three cores of the main moat and two were from the southern secondary moat (cf.16). For the full cross-section of the deepest part of the main moat, seven dates were obtained for the RP W3(2) core and four for RP W3(4)5. Two more 14C datings were made for the RP W1 core from the eastern shallow part of the main moat, and a further two for the RP F2 core in the southern additional moat.
Twelve samples of bulk organic deposits (organic mud, peat or gyttja) were dated using the liquid scintillation technique (LSC) and three samples of selected terrestrial plant macrofossils dated using the accelerator mass spectrometry technique (AMS). All samples were chemically pre-treated using the AAA (acid–alkali–acid) method. The procedure for conventional radiocarbon dating of organic material using the liquid scintillation counting method (LSC) included the standard synthesis of benzene from organic samples69. 14C measurements were carried out with a 3-photomultiplier spectrometer, the HIDEX 300SL and Quantulus 1220. Organic samples dated using the AMS technique were combusted, purified, and graphitised with Fe catalyst70. The mixture of graphite and Fe powder was pressed into a target holder and measured with the AMS system at the Centre for Applied Isotope Studies at the University of Georgia, USA or in the Accelerator Mass Spectrometry Laboratory (D-AMS laboratory code) in Seattle, USA (see71 for details).
Calibrated radiocarbon ages (BC/AD) were made using the IntCal20 radiocarbon calibration dataset72 and the OxCal 4.4.2 calibration software73,74. The age-depth curves for studied cores were elaborated based on the OxCal P_Sequence model75. The age-depth models were obtained separately for four studied cores. More detailed chronology was obtained for the longest RP W3(2) core, and the new model slightly differs from that published by Kittel et al.5. A dendrochronological date from a fragment of wooden ecofact found in the very bottom of the main moat was included into RP W3(2) and RP W3(4) age-depth models (cf.5). For an estimation of absolute chronology of selected palaeoenvironmental events, the probability distributions of the modelled calendar ages for 1-cm intervals of deposits were calculated (Supplementary Tables S1–S4). These were used for estimation of absolute chronology of selected palaeoenvironmental events.
Statistical data analyses
Self-organising map and indicator species analysis
Patterns in the dipteran assemblages were recognized with Kohonen's (unsupervised) artificial neural network (ANN), also referred to as a self-organising map (SOM)76,77. Artificial neural networks (ANNs) are simple structural and functional models of the brain. ANNs have many advantages, which allow a researcher to apply them to "difficult" data. They do not require any a priori specification of the model underlying a studied phenomenon because they learn it based on the processed data. They are also robust to noise in data78,79. This is important for the purposes of the present study, because taxa abundances in field samples do not reflect exactly the original abundances of populations80. Additionally, in palaeoecological research the long time separating the living populations and their sampling, and resulting decomposition and fragmentation additionally enhance the problem81. ANNs are also robust to non-linear relationships between variables and to non-normal distributions in data82,83. This is also crucial in this study because the counts of rare species cannot be effectively normalised by any transformation due to their absence in most samples and therefore strongly skewed variable distributions82,84. Furthermore, dipteran assemblages are shaped by many abiotic and biotic factors that are related in complex ways. The SOM application in palaeoecology and its advantage in comparison to classical zonation methods (CONISS and Optimal Partitioning) are well described by Płóciennik et al.85.
Kohonen's ANNs are constructed from data processing units (neurons) arranged in two layers: an input layer used for data input, and an output layer responsible for data structuring and output. The data used for the SOM analysis comprised log-transformed abundances of 97 taxa recorded in 79 non-empty core samples.
They were displayed on the input layer comprising 97 neurons (one input neuron per taxon). The output neurons were arranged as a two-dimensional rectangular lattice. The number of output neurons should be close to 5√n, where n is the number of samples; in this case the result was 44 (see86). Therefore, the final size of the lattice was 7 × 6 (= 42) neurons.
Each input neuron repeatedly transmitted signals to each output neuron. These signals were strengthened or weakened by modifying the weight of the connections between neurons. On this basis, a virtual dipteran core sample (DCS) was created in each output neuron.
The distance between virtual DCSs on the two-dimensional lattice exhibited their mutual dissimilarity, i.e. virtual DCSs in distant output neurons differed considerably while those in neighbouring output neurons were similar. The latter might not be true when the neighbouring output neurons were in different (sub)clusters as the virtual DCSs, and respective output neurons, were additionally clustered with hierarchical cluster analysis (with Ward algorithm and Euclidean distance)78,86,87.
Finally, each real DCS was assigned to the best matching virtual DCS and the respective output neuron. Therefore, the mutual distance of the real DCSs on the two-dimensional lattice was a derivative of the mutual dissimilarity and position of virtual DCSs: significantly dissimilar real DCSs were located in distant neurons, while similar real DCSs were located in the same neuron or in adjoining neurons83.
The batch training algorithm was chosen for the purpose of network training, because it does not require any training rate factor to be specified78. The network training and the clustering of virtual DCSs were performed with the use of the SOM Toolbox88 developed by the Laboratory of Information and Computer Science at the Helsinki University of Technology (http://www.cis.hut.fi/projects/somtoolbox/).
The SOM Toolbox allows the associations between dipteran taxa and SOM regions to be visualised in the form of greyness gradients over the two-dimensional lattice83. This visualisation may facilitate the formulation of ecological conclusions as taxa with the same patterns of greyness usually co-occurred and exhibited similar habitat preferences.
However, the SOM Toolbox does not provide a statistical verification of those associations. For this reason, the untransformed dipteran abundance data were subjected to Indicator Species Analysis (ISA): the associations between each dipteran taxon and each subcluster of output neurons, and its respective environmental conditions, were expressed in a numeric form with the indicator values (IndVals)89. IndVals complement the visualisation in the form of greyness gradients. An IndVal (range 0–100%) of taxon i in all real DCSs of subcluster j is a product of three values: (1) Aij—a measure of specificity, i.e. the mean abundance of taxon i in real DCSs assigned to subcluster j divided by the sum of its average abundances in all subclusters (%), (2) Fij—a measure of fidelity, i.e. the frequency of occurrence of taxon i (%) in real DCSs assigned to subcluster j, and (3) the constant 100 in order to produce the percentages:
$${\text{IndVal}}_{ij} = {\text{A}}_{ij} \times {\text{F}}_{ij} \times {1}00$$
$${\text{A}}_{{{\text{ij}}}} = {\text{ taxon}}\;{\text{abundance}}_{ij} /{\text{ taxon}}\;{\text{abundance}}_{i.}$$
$${\text{F}}_{ij} = {\text{ N}}\;{\text{real}}\;{\text{core}}\;{\text{samples}}_{ij} /{\text{N}}\;{\text{real}}\;{\text{core}}\;{\text{samples}}_{.j}$$
The maximum IndVal (100%) was observed when all real DCSs with taxon i were assigned to subcluster j and when taxon i was present in all real DCSs assigned to subcluster j89. Significant maximum IndVals, and therefore significant associations of individual taxa with a given SOM subcluster (and its respective environmental conditions), were identified with Monte Carlo randomisation statistics. The significance level was calculated as the proportion of randomised trials with IndVal exceeding or equal to the observed IndVal. The above calculations were performed in PC-ORD90.
Generalised linear mixed model
We asked whether chironomid taxon composition was consistent across sediment cores, so that results from a single core could be extrapolated across the whole site. Following Hadfield et al.91, we used a poisson family generalised linear mixed model (GLMM) with log link to test whether relative species abundances differed between cores RP W1, RP W3(2) and RP W3(4). Core RP F2 was excluded due to low temporal overlap of RP F2 sample ages with other cores, particularly core RP W1 (Fig. 2). Three samples with no age estimate from radiocarbon dating were excluded. Chironomid morphotypes that were absent from a core, age category, or sample were included as zero counts. After reducing samples to only those with strongly overlapping ages among the three cores, sample age was converted to a factor with 6 levels, to allow for nonlinearities in changes in species abundance over time. All resulting age categories were represented by all studied cores. This resulted in a dataset with 5141 individual species counts.
The response variable in the generalised linear mixed model (GLMM) was the untransformed count of individuals of each chironomid morphotype in each sample. To control for variation in sediment volume among samples, we included ln(sample volume cm3) as an offset (logged because we fit a poisson model with log link), so that the fixed effect parameter estimates represented the effects of predictors on chironomid counts per unit sediment volume. Fixed effect predictors included core, chironomid taxon (morphospecies), and their interaction. Sample age category and its pairwise interactions with core and morphotype were included as random effects to control for temporal variation in abundances. The core: morphospecies interaction fixed effect therefore tested for differences among cores in relative species abundance, controlling for differences in overall abundance among cores and morphospecies (fixed main effects), sample volume (offset), and any influence of sample age (random effects). All analyses were run in R version 4.1.292. GLMM was performed using the R package glmmTMB93. Model diagnostics were performed in DHARMa94, and fixed effect model selection based on AICc carried out in MuMIn95. All random effects were included in every model.
Post-hoc comparison of relative species abundance differences among cores for individual chironomid morphotypes, based on the estimated core:morphospecies interaction, were carried out in the R package phia96. This package is not compatible with mixed effects models and so these analyses are based on a GLM model including fixed effects only, run using base R's glm function.
Canonical correspondence analysis
Because the performed DCA for all four combined cores dataset revealed long biological data gradients (4.499 on Ax 1 and 4.787 on Ax 2 [SD units]), Canonical Correspondence Analysis (CCA) was selected to compare geochemical and biotic variable patterns. Due to autocorrelation, Na and Mg content were excluded from the further analysis. The CCA was performed on square-root transformed data with downweighting rare taxa, biplot scaling and inter-sample distance. The significance of environmental variables relating to the biota was tested with the Monte Carlo permutation with automatic selection and permutation under full model.
The datasets analysed during the current study are available from the corresponding author on reasonable request.
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We would like to thank dr hab. R. Stachowicz-Rybka for macrofossil selection for AMS dating and P. Wroniecki for providing an aerial photograph of the study site. The research project has been financed by a grant from the National Science Centre, Poland based on the decision No. DEC-2013/11/B/HS3/03785 and by an IDUB UGrants First (decision No. 533-D000-GF09-22).
Laboratory of Palaeoecology and Archaeobotany, Department of Plant Ecology, Faculty of Biology, University of Gdansk, 59 Wita Stwosza St., 80-308, Gdańsk, Poland
Olga Antczak-Orlewska
Department of Invertebrate Zoology and Hydrobiology, Faculty of Biology and Environmental Protection, University of Lodz, 12/16 Banacha St., 90-237, Lodz, Poland
Olga Antczak-Orlewska & Mateusz Płóciennik
Institute of Marine and Environmental Sciences, University of Szczecin, 18 Mickiewicza St., 70-383, Szczecin, Poland
Daniel Okupny
Department of Ecology and Vertebrate Zoology, Faculty of Biology and Environmental Protection, University of Lodz, 12/16 Banacha St., 90-237, Lodz, Poland
Andrzej Kruk & Richard Ian Bailey
Department of Historical Archaeology and Weapon Studies, Institute of Archaeology, University of Lodz, 65 Narutowicza St., 90-131, Lodz, Poland
Jerzy Sikora
Faculty of Geology, Geophysics and Environmental Protection, AGH University of Science and Technology, 30 Mickiewicza St., 30-059, Kraków, Poland
Marek Krąpiec
Department of Geology and Geomorphology, Faculty of Geographical Sciences, University of Lodz, 88 Narutowicza St., 90-139, Lodz, Poland
Piotr Kittel
Andrzej Kruk
Richard Ian Bailey
Mateusz Płóciennik
O.A.-O. developed the concept of the paper, conceived the study, and contributed chironomid data. D.O. contributed geochemical data. A.K. and R.I.B. provided statistical analyses. P.K. supervised the project, analysed lithology of the cores and the geomorphology of the area. P.K. and J.S. conducted the fieldwork at the study site. J.S. analysed the archaeological context. M.K. provided radiocarbon and dendrological dates. O.A.-O., P.K., D.O. and M.P. interpreted the data and wrote the manuscript with contributions from all authors. O.A.-O., J.S., A.K., D.O. and P.K. created the figures and tables. All authors reviewed the manuscript.
Correspondence to Olga Antczak-Orlewska.
Antczak-Orlewska, O., Okupny, D., Kruk, A. et al. The spatial and temporal reconstruction of a medieval moat ecosystem. Sci Rep 12, 20679 (2022). https://doi.org/10.1038/s41598-022-24762-w | CommonCrawl |
Fuel value indices of selected woodfuel species used in Masindi and Nebbi districts of Uganda
Samuel Ojelel1,
Tom Otiti2 &
Samuel Mugisha1
Biomass currently meets more than 97% of the total energy requirements in Uganda. However, contrary to this heavy reliance on biomass, there is paucity of information regarding the fuel value indices (FVIs) of woodfuel species used in different locations of the country such as Masindi and Nebbi districts. This study therefore sought to identify ten woodfuel species commonly used by the communities in these two districts and examine their FVIs from basic properties, namely; moisture content, density and gross calorific value.
A semi-structured interview using a checklist of guiding questionnaire was conducted to generate a woodfuel species list. The familiarity index (FI) was calculated for each species and then used to rank ten commonly used species for further analysis. The moisture content, density and gross calorific value of the selected species were determined in triplicate. The FVI of each species was then determined from these basic properties. One-way ANOVA, Pearson product moment correlation, and Spearman rank correlation coefficient analyses were performed in SPSS ver.16.0 to examine the variation and relationship of variables.
Ten woodfuel species belonging to seven families and eight genera were identified as commonly used species. Combretum collinum was mentioned by every respondent as a suitable woodfuel species. A significant variation in moisture content and density was recorded among the species (F ( df = 9) = 92.927, p = 0.0001) unlike in gross calorific value (F ( df = 9) = 1.400, p = 0.253). There was a positive correlation between density and gross calorific value (r = 0.895, n = 30, p = 0.0001) and a negative correlation between moisture content and gross calorific value (r = −0.518, n = 30, p = 0.003). The FVIs obtained ranged from 1.10 in Ficus natalensis to 13.09 in Albizia grandibracteata. There was also a positive relationship (rho = 0.62) between FVIs and FIs using Pearson rank correlation coefficient.
Moisture content and density are important properties in the selection of woodfuel species than gross calorific value. On the proposition of the FVIs, A. grandibracteata is a suitable woodfuel species than F. natalensis. These findings fit well into the ongoing efforts by Government and Civil Society Organizations to encourage woodlot management to ensure the sustainability of woodfuel in the country.
Woodfuel is the largest energy source for the world's population in developing countries and the demand is likely to continue [1,2]. The use of woodfuel is both an ancient and modern tradition that is not likely to change in the next decades [3]. Woodfuel is particularly significant to the poor and rural households [3,4]. The greatest portion of woodfuel is derived from natural forests, and the dwindling forest cover has made its availability a matter of critical concern in most developing countries [5,6]. The limited availability and high cost of alternative energy resources in Africa continue to make firewood and charcoal the major energy resource [7]. The heavy reliance on woodfuel has escalated the woodfuel demand leading to forest cover decline and inevitable woodfuel shortages in many African countries [8-10].
In Uganda, biomass is the most significant source of energy with a wide range of applications from domestic use such as heating, cooking, and lighting to small-scale industrial use in bakeries, tea processing, tobacco curing, lime and brick making, fish smoking, jaggeries, and distilleries [8,9,11]. It currently accounts for more than 97% the country's energy supplies [8,12]. The daily per capita consumption rate is estimated to be 4 kg [11]. In the urban areas, people use charcoal more than firewood because the former has lower transport costs per unit energy and higher energy content per ton [13]. However, it is produced inefficiently thus contributing to the scarcity of bioenergy resources.
The relentless demand for biomass energy in Uganda has been attributed to the increasing population, growing industrial sector, as well as increased rate of urbanization [14]. This shows that socioeconomic factors are largely responsible for the dependence on biomass energy resources which are available at low (or even zero) cost while commercial fuels such as liquefied petroleum gas, kerosene, and diesel are often beyond the acquisitive power of most poor people [15]. The socioeconomic characteristics of local communities in Uganda such as age, gender, wealth, education, heterogeneity of the population, household size, land holding, and distance from the forest are believed to greatly influence the level of dependence on firewood [16].
The contribution of natural forests and woodlands to biomass energy supplies in Uganda has been recognized [9,17]. However, assessment of woodfuel FVIs from basic properties remains inadequately addressed. This information is important in aiding the identification of high fuel potential woodfuel species to enhance existing woodfuel stocks so that the low-cost energy source is maintained [10]. There is evidence that the improvement of existing stocks is cheaper than establishment of new plantations or woodlots [18]. Indeed, the management of landscapes with the aim of producing woodfuel or maintaining existing stocks requires that woodfuel species with qualities acceptable to rural communities are identified [19]. Besides providing woodfuel, the management of indigenous woodlands is believed to be more ecologically sound and offers better chances for the conservation of biodiversity [9]. It is with this background that this study sought to answer the questions: (i) Which woodfuel species are commonly used by the communities in Masindi and Nebbi districts? (ii) What are their FVIs computed from basic properties (gross calorific value (GCV), density (D), and moisture content (MC))?. This information is vital in prioritizing high fuel-potential tree species for utilization, management, and improvement to ensure a steady supply of woodfuel. This study therefore fits well into the ongoing Government and Civil Society campaigns to maintain woodlots so as to ensure a balance between supply and demand of woodfuel in the country.
This study was conducted in Masindi and Nebbi districts of Uganda. Nebbi district is located in the north western region of Uganda between 2° 20′ and 2° 40′ N and 31° 0′0 and 31°.20′ E. It lies at an altitude of 945 to 1,219 m above sea level. It is bordered by the Democratic Republic of Congo in the East. Meanwhile, Masindi district is located in western Uganda and lies between 1° 22′ and 2° 20′N and 31° 22′ and 32° 23′E and altitude in the range of 621 to 1,158 m above sea level. These two districts were selected based on their current extensive savannah biomass stocks that supply the major neighboring urban areas.
Selection of informants
A multistage sampling approach was used to select the informants. In this technique, two sub-counties from Masindi and Nebbi districts (Pakanyi and Nebbi, respectively) were randomly selected; one parish was selected randomly from each sub-county, and four villages were chosen from each parish. A systematic random sampling technique was used to select 10 respondents in each village for the interview totaling to 80. A semi-structured questionnaire was administered to women and girls as the key informants in order to elicit responses on trees used for woodfuel purposes. The free listing technique was applied whereby the respondents were asked to mention any tree that comes to their mind until they could not mention any more species. From the responses, the FI was calculated for each species according to [20], and the ten species with the highest FI (Table 1) were chosen for basic property analysis.
Table 1 Ten commonly used woodfuel species
Wood sample collection and preparation
Wood samples measuring 30 cm in length and 2.5-cm diameter were collected for each of the ten woodfuel species shown in Table 1. These samples were prepared according to [15] whereby each was sub-divided into triplicates of length 10 cm, marked and weighed immediately in the field, and brought for analysis in the laboratory. The MC, D, and GCV were determined as described in the subsequent sections.
Moisture content (MC)
The triplicate samples were dried in an electric oven at 100°C until they attained a constant weight. The MC was then determined on dry weight basis according to Equation 1 [21]:
$$ \mathrm{M}\mathrm{C}=\frac{Ww-Wo}{Wo}\times 100\% $$
where Ww = wet weight and Wo = oven-dry weight.
Density (D)
This was determined as a ratio between oven-dry weight (g) and volume (cm3) using Equation 3. The procedure followed the technique described in [22] whereby wood volume is first determined by immersing the samples in water for 5 days (120 h) to achieve total saturation, allowed to drain for 5 min, and their diameters measured using a vernier caliper. The volume was calculated using Equation 2:
$$ \mathrm{Volume}=\frac{\pi \times {D}^2\times L}{4} $$
where D = average diameter and L = length of the sample.
The samples were again dried at 100°C until attaining a constant weight. The D was then calculated according to Equation 3:
$$ \mathrm{Density}=\frac{\mathrm{DW}}{\mathrm{SV}} $$
where DW = dry weight (g) and SV = sample volume (cm3)
Gross calorific value
The GCV of the wood samples was determined using a Gallenkamp autobomb calorimeter (SG96/02/536, Gallenkamp and Company Ltd, London, UK). The method is based on combustion in the 'bomb' chamber. When the sample is burned, the resulting heat is measured by the increase in the temperature of water surrounding the bomb. In this study, 1g of wood sample was pelleted using a briquette press and weighed in a crucible. The pellet was then connected to the firing wire fitted between the electrodes with the aid of a cotton thread. The circuit was tested and the bomb was filled with oxygen to a pressure of 3,000 Pa (30 bar). The calorimeter vessel was filled with water (total weight 3 kg) at 21°C to 23°C; the prepared bomb was placed inside the calorimeter vessel, and then the calorimeter vessel was placed into the water jacket. The machine was switched on and left for a while (10 to 15 min) to warm up. The initial temperature of the water was recorded before firing and after 10 to 15 minutes, the final temperature reached was recorded. Benzoic acid was used as a standard. The GCV was then calculated using Equation 4:
$$ \mathrm{G}\mathrm{C}\mathrm{V}\kern0.5em =\left[\left(FT- IT\right)\kern0.5em \times \kern0.5em 10\cdot 82\right]\kern0.5em -\kern0.5em 0.086/\left(\mathrm{weight}\kern0.5em \mathrm{of}\kern0.5em \mathrm{sample}\right) $$
where 10.82 = heat capacity of the calorimeter in kJ/K, 0.086 = combined energy value of nickel wire and cotton in kJ/g, FT = final temperature, and IT = initial temperature
Fuel value index (FVI)
This was calculated by considering D and GCVs as positive characters and MC as the negative character using Equation 5 [19]:
$$ \mathrm{F}\mathrm{V}\mathrm{I}=\frac{\mathrm{Gross}\kern0.5em \mathrm{calorific}\kern0.5em \mathrm{value}\kern0.5em \left(\mathrm{K}\mathrm{J}/\mathrm{g}\right)\kern0.5em \mathrm{Density}\kern0.5em \left(\mathrm{g}/\mathrm{cm}\right)}{\mathrm{Moisture}\kern0.5em \mathrm{content}\left(\%\right)} $$
SPSS ver. 16.0 was used to analyze the results. The following tests were carried out: (i) one-way ANOVA on the variability of MC, D, and GCV, (ii) Pearson product correlation coefficient on the relationship between D and GCV and MC and GCV, and (iii) Pearson rank correlation coefficient on the relationship between FVIs and FIs.
Preferred woodfuel species
Table 1 shows the list of ten commonly used woodfuel species with their FIs. The species are distributed among eight genera and seven families. The family Mimosaceae has three species; Combretaceae has two while the rest have one species each. The genus Albizia and Combretum have two species each and the others have one. Combretum collinum (FI = 100) was mentioned by all respondents as a desirable woodfuel species. This is attributed to its availability in the surrounding environment and apparent possession of desirable woodfuel properties which has been reported elsewhere [10,13,23,24]. Tamarindus indica was least mentioned as woodfuel because the benefits from its fruits are seen to outweigh those resulting from woodfuel use.
Woodfuel basic properties and FVIs
There was a significant variation in the MC among the species at p < 0.05 level (F ( df = 9) = 92.927, p = 0.0001), and the effect size calculated using eta squared was 0.976. The MC ranged from 36.18% in A. grandibracteata to 69.41% in Ficus natalensis (Table 2). Assessment of the Pearson product moment correlation coefficient between MC and GCV showed a negative correlation (r = −0.518, n = 30, p = 0.003) with 26.83% coefficient of determination. This confers with the widely held acknowledgement that MC adversely affects the calorific value of the wood [10,25]. However, a one-to-one examination showed that F. natalensis deviates from this widely held inference conferring with the trend that [6] also reported. Water in green tissues exists primarily in the form of free water filling the capillaries and as water of constitution (e.g., water absorbed in lignocellulose) [6]. The variation in the number and size of wood capillaries and water of constitution is a key determinant of MC variation across species [26]. This therefore implies that species with high MC have a large number and size of wood capillaries and consequently water of constitution than those with low MC.
Table 2 Basic properties and FVIs of woodfuel species used in Masindi and Nebbi districts
There was a significant variation at p < 0.05 level (F (df = 9) = 11.528, p = 0.0001) in the D of woodfuel species and the effect size was 0.84 determined using eta squared. The D values ranged from 0.23 g/cm3 in F. natalensis to 0.64 g/cm3 in A. grandibracteata (Table 2). A strong positive correlation between D and GVC was obtained using Pearson product moment coefficient (r = 0.895, n = 30, p = 0.0001) with a coefficient of determination of 80%. This trend was also reported by [27,28]. The D values obtained in this study are, however, lower than those reported in [9] due to the use of small-dimension samples in this study which are less dense than the whole branches used in the previous study (Table 3). Wood density describes the proportion of a stem that is tissue and cell walls (xylem conduit walls) and space within cell walls (xylem conduit apertures) [29]. Woodfuel species with high D have high fiber tissue per unit volume and thick fiber walls than those with low D [29], more carbon and energy content per unit volume [30], and produce long-lasting embers [19,31]. However, like the findings of [32], this investigation also found that F. natalensis with a low D (Table 2) has a high GCV. Although it was not investigated during study, high GCVs have been attributed to high concentrations of extractives and lignins in wood [31] and is therefore a plausible explanation for this exception.
Table 3 Comparison between current and [9] results
The variation in GCVs of woodfuel species was insignificant at p < 0.05 level (F (df = 9) = 1.400, p = 0.253) which is also reported by [25]. The values obtained ranged from 20.05 KJ/g in A. grandibracteata to 29.46 KJ/g in F. natalensis (Table 2). Woodfuel species with high GCVs have high concentrations of extractives and lignins in wood while a predominance of sugar units such as cellulose and hemicellulose leads to lower GCVs [31].
Fuel value index
The FVIs ranged from 1.10 in F. natalensis to 13.09 in A. grandibracteata (Table 2). There was a positive but insignificant relationship between FVIs and FIs (r = 0.62, n = 10, p = 0.056). This positive relationship between FVIs and FIs indicates that local preference is partly informed by physical (basic) properties of the wood. The values generated here are lower than those obtained by the authors in [9] who considered D and MC only (Table 3). The variation in the geographical location of the studies is also a plausible source of this difference. The FVI is an insightful parameter for screening woodfuel species [10], and it can be concluded that A. grandibracteata with the highest FVI possesses better woodfuel properties. However, authors in [19] assert that rural people are very discerning about woodfuel requirements and therefore socially acceptable features also need to be considered to generate woodfuel species checklist in a given locality.
The woodfuel species used in Masindi and Nebbi districts show significant variations in MC and D. These two variables are therefore more important in assessing woodfuel species. The results further emphasize that a single basic property is not sufficient to justify selection of the most suitable woodfuel species thereby conferring with authors in [10]. On the proposition of the FVIs, A. grandibracteata is a more suitable woodfuel species than F. natalensis. This study therefore lays an important premise for selection of high fuel potential woodfuel species for possible inclusion into reforestation, afforestation, and agroforestry initiatives currently encouraged by the Government and Civil Society Organizations to address woodfuel shortage in Uganda generally. However, before such species prioritization is done, it is recommended that socially defined features that shape local peoples' perception of acceptable woodfuel species need to be considered [6,19] which was not undertaken in this study. This will lead to the generation of woodfuel species checklist that is socially acceptable and with high fuel potential in a given area.
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The authors acknowledge the support by Makerere University Staff Development Program and DAAD In-Country Scholarship program for Uganda for SO. We thank Robert Nuwamanya, the laboratory technician at College of Agricultural and Environmental Sciences, Makerere University for his technical guidance in GCV determination. We also thank the communities of Masindi and Nebbi for willingly participating in our interviews.
Department of Biological Science, Makerere University, P.O. Box 7062, Kampala, Uganda
Samuel Ojelel & Samuel Mugisha
Department of Physics, Makerere University, P.O. Box 7062, Kampala, Uganda
Tom Otiti
Samuel Ojelel
Samuel Mugisha
Correspondence to Samuel Ojelel.
SO was responsible for data collection, laboratory experiments, analysis, and drafting of the manuscript. TO and SM were in charge of the study design, reviewing the manuscript draft, and overseeing the research. All the authors have therefore contributed substantially to this manuscript. All authors have read and approved the final manuscript.
TO is a professor in the Department of Physics, Makerere University. He is also the chairman, Uganda National Energy Development Organization (UNEDO); member of African Energy Policy Research Network (AFREPREN); group leader, Solar Energy Research Group, Physics Department, Makerere University; Member of the International Energy Foundation (IEA) and a member of the Scientific and Technical Advisory Panel (STAP) of the Global Environment Facility (GEF). SM (PhD) is a lecturer in the Department of Biological Sciences of Makerere University. SO is a teaching assistant in the Department of Biological Sciences, Makerere University and also a post graduate student in the same Department.
Ojelel, S., Otiti, T. & Mugisha, S. Fuel value indices of selected woodfuel species used in Masindi and Nebbi districts of Uganda. Energ Sustain Soc 5, 14 (2015). https://doi.org/10.1186/s13705-015-0043-y
DOI: https://doi.org/10.1186/s13705-015-0043-y
Woodfuel | CommonCrawl |
Algorithms for Molecular Biology
Enumeration of minimal stoichiometric precursor sets in metabolic networks
Ricardo Andrade1,2,
Martin Wannagat1,2,
Cecilia C. Klein1,2,
Vicente Acuña3,
Alberto Marchetti-Spaccamela1,4,
Paulo V. Milreu5,
Leen Stougie1,6,7 &
Marie-France Sagot1,2
Algorithms for Molecular Biologyvolume 11, Article number: 25 (2016) | Download Citation
What an organism needs at least from its environment to produce a set of metabolites, e.g. target(s) of interest and/or biomass, has been called a minimal precursor set. Early approaches to enumerate all minimal precursor sets took into account only the topology of the metabolic network (topological precursor sets). Due to cycles and the stoichiometric values of the reactions, it is often not possible to produce the target(s) from a topological precursor set in the sense that there is no feasible flux. Although considering the stoichiometry makes the problem harder, it enables to obtain biologically reasonable precursor sets that we call stoichiometric. Recently a method to enumerate all minimal stoichiometric precursor sets was proposed in the literature. The relationship between topological and stoichiometric precursor sets had however not yet been studied.
Such relationship between topological and stoichiometric precursor sets is highlighted. We also present two algorithms that enumerate all minimal stoichiometric precursor sets. The first one is of theoretical interest only and is based on the above mentioned relationship. The second approach solves a series of mixed integer linear programming problems. We compared the computed minimal precursor sets to experimentally obtained growth media of several Escherichia coli strains using genome-scale metabolic networks.
The results show that the second approach efficiently enumerates minimal precursor sets taking stoichiometry into account, and allows for broad in silico studies of strains or species interactions that may help to understand e.g. pathotype and niche-specific metabolic capabilities. sasita is written in Java, uses cplex as LP solver and can be downloaded together with all networks and input files used in this paper at http://sasita.gforge.inria.fr/.
The question of which metabolites an organism needs from its environment (henceforth called the sources) in order to grow or to produce a given set of metabolites (henceforth called the targets) is crucial for both fundamental and applied reasons. This indeed enables to define the growth conditions of organisms in the laboratory, as well as the minimal media necessary for the production of compounds of biotechnological interest (for instance, ethanol). More recently, great interest in establishing which nutrients are exchanged among different organisms in communities such as present in the human gut has also been raised by the interest to develop new strategies for fighting infection that rely on the use of probiotics instead of antibiotics [1]. However the latter requires that: (1) such exchanges are computed in a very efficient way in genome-scale metabolic networks; (2) all possible minimal sets of sources are identified for a given target set of interest in order to fully understand the interactions that may take place among the organisms in a community, as well as the alternative niches that may with time develop for some such organisms.
Early attempts at enumerating all minimal precursor sets (minimal sets of sources) were based only on topology (henceforth called topological precursor sets). Stoichiometry was thus not taken into account, leading to possibly many unfeasible solutions [2–5]. The algorithm of Romero and Karp was based on a backtrack traversing of the metabolic graph from the target compounds to the seeds while Handorf et al. tested the reachability of the target from a heuristically defined collection of sets of sources. Neither enumerated all minimal precursor sets. Cycles, although omnipresent in metabolic networks (e.g. Krebs cycle), were not included until the method of Cottret et al. [4]. However, the latter algorithm could be applied only to small networks due to a high memory requirement; subsequently, Acuña et al. [5] allowed the enumeration of all minimal precursor sets of networks of about 1000 reactions. The authors also pointed out that the enumeration of precursor sets and of precursor cut sets could be done simultaneously in quasi-polynomial total time. Precursor cut sets are a set of sources such that, if they are eliminated, then the target set of interest can no longer be produced by any combination of the remaining sources.
The approach of Zarecki et al. [6] takes stoichiometry into account and consists of two steps. First, the size of a set of sources of minimal cardinality that allows the production of a target is determined solving a mixed integer linear programming problem. In a second step, the authors identify a single set of sources of the determined size such that the sum of the molecular weight of the compounds is minimal.
To our knowledge, there are two algorithms that attempt to enumerate all minimal precursor sets with stoichiometry (henceforth called stoichiometric precursor sets) [7, 8].
Imieliński et al. [7] propose a method that first enumerates all extreme semipositive conservation relations (ESCR), that is the extreme rays of the cone defined by the transposed stoichiometric matrix. The precursor sets are then obtained by the enumeration of hitting sets of the ESCRs. As the authors state, this approach is impractical for genome-scale metabolic networks since it is impossible to enumerate all ESCRs with the current algorithms [7]. Consequently, a method is proposed that enumerates a subset of the ESCRs (those that do not contain water) to obtain (via hitting sets) minimal precursor sets that contain water. These solutions are physiologically minimal (all media contains water), but not necessarily the theoretically minimal.
The method of Eker et al. [8] is based on logical and linear constraint solving and on computational boolean algebra. The authors formulated two different constraint models, that were called steady-state and machinery-duplicating. Their steady-state model requires a non-negative net production of all compounds that are on the path from the precursors to the target. Observe that the term steady-state is usually used to denote a slightly different model where all compounds that are on the path from the precursors to the target cannot accumulate. Their machinery-duplicating model is more restrictive as it requires a strict positive net production of these compounds. Notice that a set of sources that allows the production of the target(s) in the machinery-duplicating model, allows also the production of the same target(s) in the steady-state model. A toy example illustrates the difference between the two models (see Fig. 1). In this network, we have a source (p), a target (t), internal compounds (a, b, c), and three reactions (\(r_{1:} \,p + a \rightarrow c, r_{2:}\, c \rightarrow b, r_{3:}\, b \rightarrow a + t\)). Following their steady-state model, the source p is a precursor of t. The compounds a, b, and c have a zero net production when we assign a positive flux value 1 to each reaction. In the machinery-duplicating model, there is no precursor set that enables to produce t: indeed, no flux would fulfil the condition of a strict positive net production of a, b, and c. However, this type of cycle resembles the Krebs cycle that plays an essential role in the production of energy in aerobic organisms. To reveal the similarity between the toy example and the Krebs cycle, let the compound a take the role of oxaloacetate (which is regenerated through the Krebs cycle), the source p feed the cycle as acetyl-CoA, the compound b be any compound on the Krebs cycle such as e.g. citrate or succinate, and the target t any by-product of the Krebs cycle such as NADH or carbon dioxide. We argue that the machinery-duplicating model is too restrictive as therein cycles of the type shown in Fig. 1 are not captured.
Network with one source p and one target t illustrating the difference between the two models used by Eker et al. [8], and the limitation of the machinery-duplicating model. The source p is a precursor set for the production of the target if the steady-state model is assumed. In this toy example, the target can not be produced following the machinery-duplicating model
An approach widely used in flux-balance analysis (FBA) [9] to model an organism's growth condition is to include a so-called biomass reaction, that consumes in the right amounts every compound needed for a cell to duplicate. Such a reaction has a single product, an artificial compound (representing the duplicated cell) that can be modelled as a target compound in our enumeration approach: a cell can grow and duplicate if it can produce this target compound.
The objective of this paper is twofold. On the theoretical level, we show the relationship between topological and stoichiometric precursor sets and we discuss some complexity results. On the methodological level, we provide two algorithms that enumerate all minimal stoichiometric precursor sets. The first one is based on the above mentioned relationship between topological and stoichiometric precursor sets. Although interesting in terms of theory, it however is not efficient in practice. The second approach, called sasita, uses a similar approach as in [10–12]. Therein the authors enumerate reaction subsets solving recursively mixed integer linear programming (MILP) problems. The reaction subsets correspond to alternate flux distributions to obtain an identical value of the objective function [10], the k-shortest elementary flux modes (EFM) [11] or the smallest minimal cut sets (MCS) [12]. All approaches enumerate minimal reaction sets. Here we consider minimality of the set of compounds.
sasita enables to enumerate all stoichiometrically feasible minimal precursor sets in both models, steady-state and machinery-duplicating, as in [8]. A natural question is to compare our results with those obtained in [8]. Unfortunately, the algorithm of Eker et al. [8] was not publicly available for testing. We nevertheless tried to reproduce the authors' results by incorporating their definitions in our method, but we were unable to obtain the same results, even using the same Escherichia coli network they made available. More surprisingly, we found a precursor set that is minimal with respect to the solutions found by Eker et al.
In the enumeration of minimal precursor sets for a given target set, Eker et al. [8] are able to work with genome-scale metabolic networks and are exhaustive, but their method is very time and memory consuming. The authors indeed indicate that it required 3 days of execution on a 24-core (with Hyper threading) 2.67 GHz Intel X5650 Xeon CPU-model processor, using the machinery-duplicating model on an Escherichia coli network composed of 2314 unidirectional reactions (no reversible reactions) of which 388 were transport reactions, to enumerate 787 solutions.
We show that we can apply sasita on big networks like the iJO1366 reconstruction of the Escherichia coli K-12 MG1655 with 3646 reactions and 2258 compounds.
In the "Background" section, we provide basic definitions; in particular, we extend ideas from topological precursor sets as defined in [5] in order to incorporate stoichiometry, and we discuss the relationship between topological and stoichiometric solutions. We then describe the sasita algorithm for enumerating all stoichiometrically feasible minimal precursor sets. In the "Results" section, we discuss in detail the comparison with respect to Eker's et al. proposal. Here we observe that sasita is the first publicly available software to enumerate minimal stoichiometric precursors sets both with the steady state and the machinery duplicating model. Experiments show that sasita can be applied to large genome-scale metabolic networks and we discuss the obtained results.
Definitions and properties
A metabolic network is composed of a set of compounds together with the reactions that transform them. The following example represents a metabolic network with four compounds and two reactions (the values before each compound in a reaction are the stoichiometric coefficients of the reaction):
$$\begin{aligned} r_{1:}& \,1.0 \, c_{1} + 2.0 \, c_{2} \rightarrow 1.0 \, c_{3}, \\ r_{2:}& \,3.0 \, c_{3} \rightarrow 1.0 \, c_{4}. \end{aligned}$$
In the following, we use directed hypergraphs [13, 14] to model a metabolic network. A metabolic network is characterised by a pair \(\mathcal {N}=(\mathcal {C},\mathcal {R})\), where \(\mathcal {C}\) is the set of vertices (representing metabolic compounds) and \(\mathcal {R}\) is a set of hyperarcs (representing metabolic reactions). All reactions are considered to be irreversible. Reversible reactions are thus split into a forward and a backward reaction. A stoichiometric matrix S associated to \(\mathcal {N}\) is a matrix containing the stoichiometric coefficients of each reaction with the reactions in \(\mathcal {R}\) as its columns and the compounds in \(\mathcal {C}\) as its rows. We define \(\mathcal {X}\subseteq \mathcal {C}\) as the set of source compounds and \(\mathcal {T}\subseteq \mathcal {C}\) as the set of target compounds. For simplicity, we assume that sources are not produced by any reaction; it is easy to verify such condition by adding for each metabolite x in \(\mathcal {X}\) a dummy metabolite \(x'\) and a dummy reaction producing one x from one \(x'\). Replacing x by their representative \(x'\) in the set \(\mathcal {X}\) produces an equivalent network (in terms of what a set of sources is able to produce) with the desired property (see [5]).
Topologically, a reaction \(r \in \mathcal {R}\) is defined by its substrates \({\textit{S}ubs(r)}\) and its products \({\textit{P}rod(r)}\), suggesting the interpretation of a reaction as an hyperarc with \({\textit{S}ubs(r)}\) as the set of tail nodes and \({\textit{P}rod(r)}\) as the set of head nodes of a reaction r. In the above example, \({\textit{S}ubs(r_{1})} = \{c_{1},c_{2}\}\) and \({\textit{P}rod(r_{1})} = \{c_{3}\}\). The network \(\mathcal {N}\) can then be seen as a directed hypergraph with stoichiometric coefficients associated with each hyperarc. Given a subset of reactions \(F\subseteq \mathcal {R}\), we denote by \({\textit{S}ubs(F)}\) and \({\textit{P}rod(F)}\) the union of the substrates and products, respectively, of the reactions in F.
Given a network \(\mathcal {N}\) and the sets of source \(\mathcal {X}\) and target \(\mathcal {T}\) compounds, we loosely define a precursor set as a set \(X \subseteq \mathcal {X}\) that can produce all the targets of \(\mathcal {T}\) using a subset of the reactions in \(\mathcal {R}\). The concept of a factory was introduced in [5]; a topological factory is defined as follows:
A set \(F \subseteq \mathcal {R}\) is a topological factory from \(X \subseteq \mathcal {X}\) to \(T \subseteq \mathcal {T}\) if \(T \cup {\textit{S}ubs(F)} \subseteq {\textit{P}rod(F)} \cup X\); i.e., if T and every substrate of every reaction in F is either a source or is produced by some reaction in F.
A set \(X \subseteq \mathcal {X}\) is a topological precursor set (TPS) for \(\mathcal {T}\) if there exists a topological factory from X to \(\mathcal {T}\).
Extending this definition to include stoichiometry requires that any substrate of a reaction in F, should be either produced at least in a same quantity by one or more reactions also in F, or the substrate should be a source compound. Observe that, if the flux vector \(v \in \mathbb {R}^{|\mathcal {R}|}\) denotes the flux of every reaction in the network per time unit, then \(Sv\in \mathbb {R}^{|\mathcal {C}|}\) is the vector of net production of all compounds in the network for the flux v. Furthermore, \((Sv)_{A}\) specifies the net production of the compounds in a set A.
A stoichiometric factory (S-factory) from \(X \subseteq \mathcal {X}\) to \(T \subseteq \mathcal {T}\) is a set \(F \subseteq \mathcal {R}\), such that there exists a flux vector \(v \ge 0\) satisfying:
\(v_{i} \left\{ \begin{array}{ll} > 0 & \quad i \in F\\ = 0 & \quad \text {otherwise,} \end{array} \right.\)
\((Sv)_{\mathcal {C}\setminus X} \ge 0\),
\((Sv)_{T} > 0\).
A S-factory from X to T is minimal if it does not contain any other S-factory from X to T.
It is possible to adapt Definition 2 to the steady-state assumption by replacing the greater-than-or-equal sign by the equal sign in the second constraint. Thus, the steady-state constraint is put on the compounds in \(\mathcal {C}\) that are neither in X nor in \(\mathcal {T}\).
A set \(X \subseteq \mathcal {X}\) is a stoichiometric precursor set (SPS) of T if there exists a S-factory from X to T. A SPS of T is minimal if it does not contain any other SPS of T.
The following facts summarise the main differences between TPSs and SPSs:
Every S-factory is a topological factory. Every SPS is also a TPS.
Not every topological factory is a S-factory. Not every TPS is a SPS.
Given these facts, it is clear that any S-factory always contains a topological factory. A natural question that arises is whether we can decompose an S-factory into a set of topological factories. We show that this is not true:
There exist minimal S-factories which are not the union of minimal topological factories.
There exist minimal SPSs which do not consist of a union of minimal TPSs.
The first fact is a direct consequence of the definitions of SPS and TPS. The remaining facts are illustrated using Fig. 2 that has two minimal TPSs (\(\{p1\}\) and \(\{p3\}\)), and two minimal SPSs (\(\{p1\}\) and \(\{p2, p3\}\)) of the target set \(\{t\}\). Observe that \(\{p3\}\) is a (minimal) TPS but is not a SPS (fact 2). The minimal stoichiometric factory from p1 to t consists in the set of reactions r1, r2, r3, and r4, while the minimal topological factory from p1 to t does not contain the reaction r2 from a to b (fact 3). The minimal SPSs \(\{p2, p3\}\) cannot be obtained as combinations of any minimal TPSs (fact 4). Figure 2 gives an intuition about the facts; similar characteristics can be found in real metabolic networks as well.
Illustration of facts 2–4. The stoichiometric values are all equal to one. There are two minimal TPSs: \(\{p1\}\) (obtained from the topological factory \(\{r_1,\ r_3,\ r_4\}\)), and \(\{p3\}\) (obtained from the topological factory \(\{r_7,\ r_6,\ r_5\}\)). The source p2 does not take part of a minimal topological factory because its consumption involves the consumption of the source p3, which forms already a minimal TPS. There are two minimal SPSs: \(\{p1\}\) (obtained from the stoichiometric factory \(\{r_1,\ r_2,\ r_3,\ r_4\}\)), and \(\{p2, p3\}\) (obtained from the stoichiometric factory \(\{r_8\}\))
Figure 2 shows an example where it is indeed not possible to obtain the minimal S-factory that contains r1, r2, r3, and r4 from minimal topological factories. However, it is true that every minimal S-factory is a union of minimal topological factories of the many-to-one transformed network defined as follows:
Given \(\mathcal {N}=(\mathcal {C},\mathcal {R})\), the many-to-one transformation of \(\mathcal {N}\) is the metabolic network \(\Psi (\mathcal {N})=(\mathcal {C},\Psi (\mathcal {R}))\) such that for each reaction \(r\in \mathcal {R}\) and for each metabolite \(a\in {\textit{P}rod(r)}\), there is a reaction \(r_a\) in \(\Psi (\mathcal {R})\) such that \({\textit{S}ubs(r_a)}={\textit{S}ubs(r)}\) and \({\textit{P}rod(r_a)}=\{a\}\).
Given a reaction \(r \in \mathcal {R}\) with \({\textit{P}rod(r)}=\{a_1,\ldots ,a_k\}\), we denote by \(\Psi (r)=\{r_1,\ldots ,r_k\}\) the set of k reactions in \(\Psi (\mathcal {R})\) that correspond to the many-to-one transformation of r, that is, \({\textit{S}ubs(r_i)}={\textit{S}ubs(r)}\) and \({\textit{P}rod(r_i)}=\{a_i\}\). Furthermore, we extend this definition to sets of reactions, that is, if \(R\subseteq \mathcal {R}\), we denote \(\Psi (R)=\cup _{r\in R} \Psi (r)\).
It is clear that all minimal topological factories in \(\mathcal {N}\) are also among the minimal topological factories in the many-to-one network \(\Psi (\mathcal {N})\) if we would retransform the many-to-one reactions into their original hyper-reactions. However, there are additional minimal topological factories in \(\Psi (\mathcal {N})\) that are not minimal in \(\mathcal {N}\) after the retransformation. From this, we claim that every minimal S-factory in \(\mathcal {N}\) is a union of minimal topological factories of \(\Psi (\mathcal {N})\). The following definition and lemmas will provide the basis for the proof of this statement.
In a hypergraph, we define a (simple) path \(p=(M,R)\) from s to t as a chain of different metabolites \(M=(m_0,\ldots ,m_n)\) and a chain of different reactions \(R=(r_1,\dots ,r_n)\) such that:
\(m_0=s\) and \(m_n=t\);
\(m_i \in Subs(r_{i+1}),\ i\in \{0,\ldots ,n-1\}\);
\(m_i \in Prod(r_i),\ i\in \{1,\ldots ,n\}\).
Lemma 1
Given a minimal S-factory H from \(X \in \mathcal {X}\) to \(T \in \mathcal {T}\) in the network \(\mathcal {N}\) , for every reaction \(r \in H\) , there is always at least one path in H from one of the products of r to some metabolite in \(\mathcal {T}\).
Let us suppose without loss of generality that \(|T| = 1\). We are going to prove the lemma by contradiction. Suppose that there is a reaction \(r \in H\) such that \({\textit{P}rod(r)}=\left\{ p_{1},\dots ,p_{k}\right\}\), and that for all \(p_{i} \in {\textit{P}rod(r)}\) there is no path from \(p_{i}\) to T in H. Since there is no path to T in H that includes r, if \(r' \in H\) is a reaction that consumes one product of r, then there cannot be a path to T that includes \(r'\) (in fact if such a path exists then there is also a path that includes r). By repeating the same reasoning, consider the set of reactions \(I_{r}\) corresponding to the reactions in H that consume the products of r and of the reactions that consume the products of those reactions and so on. Let \(\overline{H}= H\setminus I_{r}\); we will argue that \(\overline{H}\) remains a stoichiometric factory.
Consider S the stoichiometric matrix associated to N and v the positive vector associated with H. Removing r from H means \(v_{r}=0\). Consider \(\overline{v}\), a vector with the same values of v except for the components of v corresponding to the reactions of \(I_{r}\) (the fluxes corresponding to the reactions in \(I_{r}\)) which should be zero. Since any of the reactions of \(I_{r}\) lead to T, we have that:
$$\begin{aligned} S\overline{v}\ge 0, \quad (S\overline{v})_{T} \ge 1. \end{aligned}$$
Since \(\overline{v_{k}}> 0\) for all \(k \in \overline{H}\), \(\overline{H}\subset H\) is a stoichiometric factory, which is a contradiction because H is minimal. \(\square\)
Given a set of reactions \(H\subseteq \Psi (\mathcal {R})\) and the set of sources \(X={\textit{S}ubs(H)}\cap \mathcal {X}\) , then H is a minimal topological factory from X to \(\mathcal {T}\) if and only if the two following statements are true:
For every metabolite m in \({\textit{S}ubs(H)}\setminus X\) there is exactly one reaction in H that produces m;
For every metabolite m in \({\textit{P}rod(H)}\) there exists a path from m to \(t\in \mathcal {T}\) contained in H.
We first prove that if H is a minimal topological factory from X to \(\mathcal {T}\), then both statements (1 and 2) above hold. By definition, H is a topological factory from X to \(\mathcal {T}\) if and only if any metabolite in \(\mathcal {T}\) and in \({\textit{S}ubs(H)}\) is a source in X or is produced by some reaction in H. Let m be a metabolite in \({\textit{S}ubs(H)}\setminus X\). Then by definition there is a reaction r in H that produces m. Suppose however that there is another reaction \(r'\) in H also producing m. Then \({\textit{P}rod(H\setminus \{r'\})}= {\textit{P}rod(H)}\). Thus, \(H\setminus \{r'\}\) would still be a topological factory from X to \(\mathcal {T}\) which contradicts the minimality and thus proves Statement 1.
Now let m be a metabolite in \({\textit{P}rod(H)}\). We show that there is a path from m to some target in \(\mathcal {T}\). By contradiction, suppose that there is no such path. Consider then the following iterative process. Starting from \(M=\mathcal {T}\) and \(R=\emptyset\), consider all reactions \(H'\) of H that produce the metabolites in M. Then add to R all reactions in \(H'\) and to M all substrates of \(H'\), and repeat the process until no reaction is added. Clearly, for all \(m \in M\), either \(m \in \mathcal {X}\) or m is produced by some reaction in R, and therefore R is a topological factory from X to \(\mathcal {T}\). Since all reactions in \(\mathcal {R}\) are also in H and H is a minimal topological factory, \(R=H\) and m must have been included in M in some iteration. Clearly from that iteration, we can recover a path from m to some metabolite in \(\mathcal {T}\) by going backwards in the described process and therefore Statement 2 above holds.
In order to prove the opposite implication, we first observe that, if both statements are true, then by Definition 1, the set H corresponds to a topological factory from X to \(\mathcal {T}\). Therefore, we only need to show that it corresponds to a minimal topological factory. Let \(H'\subseteq H\) be a minimal topological factory from X to \(\mathcal {T}\). By contradiction, suppose that \(H'\ne H\), then there is a reaction r in \(H\setminus H'\). Let a be the product of r. By hypothesis, there is a path from a to \(\mathcal {T}\) in H. However each reaction in the path is the only one in H producing the metabolites composing its products. Clearly the last reaction in the path (which produces a target) must also belong to \(H'\). Thus, at some point in the path, there is a metabolite which is the product of a reaction in H which is not in \(H'\), and is the substrate of a reaction in \(H'\). There is thus a substrate of a reaction in \(H'\) that is not produced by any reaction in \(H'\), which is a contradiction with the fact that H is a topological factory from X to \(\mathcal {T}\). Therefore, \(H'=H\) and the minimality is proved. \(\square\)
The following theorem shows that any minimal S-factory is the union of minimal topological factories in the many-to-one network.
Theorem 1
For any minimal S-factory \(H \subseteq \mathcal {R}\) from X to T in \(\mathcal {N}\) , there exists a set of minimal topological factories \(F_{1}, \ldots , F_{k}\) from X to T in \(\Psi (\mathcal {N})\) such that:
\(F_{1}, \ldots , F_{k}\subseteq \Psi (H)\);
For each reaction r in H there is \(i\in \{1,\ldots ,n\}\) such that \(\Psi (r)\cap F_i\ne \emptyset\).
From a given minimal S-factory H, we select a reaction \(r \in H\). By Lemma 1, we know that there is a path from at least one product m of r to one target compound t. Clearly, there is a path \(p=(M_p,R_p)\) from m to t in \(\Psi (\mathcal {N})\). Since \(\Psi (\mathcal {N})\) is a many-to-one network, every metabolite in \(M_p\) is produced by only one reaction in \(R_p\). We show that p can be extended to a topological factory from X to t. Starting from the set \(R_0=R_p\), we consider the set of metabolites \(M_0={\textit{S}ubs(R_0)}\setminus {\textit{P}rod(R_0)}\), that is, the set of substrates that are not produced by any reaction in the set. Let c be any metabolite in \(M_0\). In the S-factory H in \(\mathcal {N}\), there exists a reaction h that produces c. Let \(h_c\) be the many-to-one reaction in \(\Psi (h_c)\) that produces c, that is, \({\textit{P}rod(h_c)}=\{c\}\). We define \(R_1=R_0\cup \{h_c\}\) as the new set of reactions and \(M_1={\textit{S}ubs(R_1)}\backslash {\textit{P}rod(R_1)}\). We repeat this process defining \(R_{i+1}\) and \(M_{i+1}\) by choosing any metabolite in \(M_i\) until \(M_{i+1}\) is empty. By construction, the set of reactions \(F=R_{i+1}\) satisfies the two properties of Lemma 2, and therefore F is a minimal topological factory from X to t contained in \(\Psi (H)\). Repeating this process for every reaction \(r\in H\), we obtain a set \(F_{1}, \ldots , F_{k}\) of topological factories from X to t satisfying the desired properties. \(\square\)
The theorem suggests that a straightforward idea to enumerate all \(SPSs\) is to enumerate minimal topological factories in \(\Psi (\mathcal {N})\) and then just build combinations thereof checking their stoichiometric feasibility in \(\mathcal {N}\). The combinations can be done in the following way. We check all combinations of k minimal topological factories for feasibility, starting with \(k=1\). Before incrementing k, we test if there is a minimal \(SPS\) (with respect to the already obtained \(SPSs\)) that can be build from at least \(k+1\) minimal topological factories. This however is in general not an efficient approach because (i) many topological factories in \(\Psi (\mathcal {N})\) are not part of a \(SPS\), (ii) the powerset of all topological factories in \(\Psi (\mathcal {N})\) has to be built to obtain \(SPSs\). Issue (i) is illustrated in the network of Fig. 3a. There are n minimal topological factories in \(\Psi (\mathcal {N})\). One contains only \(\psi (r_1)\). The other minimal topological factories contain each \(\{\psi (r_t), \psi (r_a), \psi (r_b)\}\) and one of the reactions in \(\{\psi (r_2),\dots , \psi (r_n)\}\), respectively. The only \(SPS\) consists of \(p_1\) and can be obtained directly from the minimal topological factories of \(\Psi (\mathcal {N})\). The enumeration of the minimal topological factories that contain one of the reactions in \(\{\psi (r_2),\dots , \psi (r_n)\}\) may be time consuming, for nothing since none of them yields a \(SPS\). Indeed, the number of minimal topological factories in \(\Psi (\mathcal {N})\) can be much higher than the number of SPSs in \(\mathcal {N}\). Issue (ii) is depicted in Fig. 3b. There are n minimal topological factories in \(\Psi (\mathcal {N})\). Only the combination of all n minimal topological factories yield a \(SPS\) in \(\mathcal {N}\). However, all other combinations (that can be huge for large values of n) have to be considered and tested for feasibility.
a A network with \(\mathcal {R}= \{r_t, r_a, r_b, r_1,\dots , r_n\}\). Reaction \(r_i\) with \(i = 2,\dots ,n\) consumes \(p_i\) and produces compound c. \(\mathcal {T}= \{t\}\), \(\mathcal {X}= \{p_1,\dots ,p_n\}\). All stoichiometric values are equal to one. There is one minimal \(SPS\) (\(\{p_1\}\)) and n minimal topological factories in \(\psi (\mathcal {N})\) . One contains only \(\psi (r_1)\). The other minimal topological factories contain each \(\{\psi (r_t), \psi (r_a), \psi (r_b)\}\) and one of the reactions in \(\{\psi (r_2),\dots , \psi (r_n)\}\), respectively. b In this network, the set of compounds is given by \(\mathcal {C}= \{a,b,t,c_1,\dots ,c_n, p_1,\dots ,p_n\}\). The compounds \(p_1,\dots ,p_n\) are the sources and t is the target. The stoichiometric values are equal to 1 if not stated otherwise. Beside the reactions \(r_{a_1}: a \rightarrow t\) and \(r_{a_2}: a \rightarrow b\), there is the reaction \(r'\) that consumes \(n-1\) b and produces \(\{c_1,\dots ,c_n\}\) (1 each). Furthermore, there are n reactions with \({\textit{S}ubs(r_i)} = \{c_i, p_i\}\) and \({\textit{P}rod(r_i)} = \{a\}\), with \(i=1,\dots ,n\). The dots in the Figure illustrate the products \(c_2,\dots ,c_{n-1}\) of \(r'\) that are not shown for simplicity. The reactions \(r_2,\dots ,r_{n-1}\) are not shown for the same reason. There are n minimal topological factories in \(\psi (\mathcal {N})\), each containing the reactions \(\{\psi (r_{a_1}), \psi (r_{a_2}), \psi (r')\}\) and one of the many-to-one reactions of \(\{\psi (r_1),\dots ,\psi (r_n)\}\), respectively. The only minimal \(SPS\) contains all sources
We now discuss the main complexity results for finding and enumerating \(SPSs\). The next theorem shows that deciding whether a set is a \(SPS\) can be done efficiently.
Given a network \(\mathcal {N}\) , a subset \(X\subseteq \mathcal {X}\) of sources and a target set \(\mathcal {T}\) , we can decide in polynomial time whether X is a SPS for \(\mathcal {T}\).
We are going to show that it suffices to solve a linear optimisation problem to decide whether X is a SPS for \(\mathcal {T}\).
Consider the network \(\overline{\mathcal {N}}= (\overline{\mathcal {C}}, \overline{\mathcal {R}})\) with \(\overline{\mathcal {C}}= \mathcal {C}\cup \{t\}\) and \(\overline{\mathcal {R}}= \mathcal {R}\cup \{\overline{r}\}\). We add a compound t and a reaction \(\overline{r}\) to the network \(\mathcal {N}\). The reaction \(\overline{r}\) is build as follows: \({\textit{S}ubs(\overline{r})} = \mathcal {T}\) and \({\textit{P}rod(\overline{r})} = \{t\}\). Also, the values of all stoichiometric coefficients of \(\overline{r}\) are one. Consider now the following optimisation problem:
$$\begin{aligned} \begin{array}{cll} &{}{\displaystyle \text {Maximize}\,\,f(v) = \left( Sv\right) _{t}}\\ &{}s.t.\\ &{}\qquad \qquad \left( Sv\right) _{\mathcal {C}\setminus X} \ge 0,\\ &{}\qquad \qquad {\displaystyle v_{i} \ge 0,} &{} i=1,\dots , |\overline{\mathcal {R}}|, \end{array}\qquad \qquad \qquad \qquad {(M1)} \end{aligned}$$
where S represents the stoichiometric matrix of \(\overline{\mathcal {N}}\).
If \(v^{*}\) is a solution to M1 and \(f(v^{*}) > 0\) then the support of \(v^{*}\) is a stoichiometric factory from X to \(\mathcal {T}\) and X is a stoichiometric precursor set for \(\mathcal {T}\). \(\square\)
As concerns the problem of enumerating all solutions, we first observe that the proof that enumerating all minimal TPSs cannot be done in polynomial total time (that is, in the size of the input and the number of solutions) unless P = NP given in [5] can be immediately applied to show that enumerating all minimal SPSs cannot be done in polynomial total time unless P=NP. The same observation holds for enumerating all minimal cut sets (SCSs), which we define as follows:
A set \(X \subseteq \mathcal {X}\) is a stoichiometric cut set (SCS) (topological cut set (TCS)), if \(\mathcal {X}\setminus X\) is not a stoichiometric precursor set (topological precursor set).
We now show that the simultaneous enumeration of minimal SPSs and SCSs can be done in quasi-polynomial time. Notice that in [5], a quasi-polynomial time algorithm to simultaneously enumerate all TPSs and TCSs was presented by formulating the problem with a monotone boolean formula and then using a result of [15]. Such approach is possible even in the case of SPSs and SCSs.
The set of minimal SPSs and the set of minimal SCSs can be enumerated in total quasi-polynomial time.
Define the Boolean function \(f{:}\,2^\mathcal {S}\rightarrow \{0,1\}\) as \(f(X)=1\) if X is a SPS and \(f(X)=0\) otherwise. Clearly, this function is monotone: if \(f(X)=1\) then \(f(Y)=1\) for any set \(Y\supseteq X\). The collection \(\mathcal {P}\) of minimal SPSs is the collection of all minimal sets in \(\mathcal {S}\) that evaluate to 1 and the collection \(\mathcal {C}\) of minimal SCSs is the collection of all minimal sets whose complement in \(\mathcal {S}\) evaluates to 0. In the context of monotone Boolean functions, minimal SPSs correspond to the prime implicants and minimal SCSs to the prime implicates of f. In [15], a general algorithm is proposed to jointly enumerate prime implicants and prime implicates of any Boolean function. The algorithm and time analysis are rather technical and we only give a brief description of the incremental algorithm applied to our case. Briefly, given two collections of solutions already found, that is, of collections \((\mathcal {P}',\mathcal {C'})\) of SPSs and SCSs, the algorithm finds a set \(X\subseteq \mathcal {S}\) such that X does not contain any minimal SPS in \(\mathcal {P}'\) and \(\mathcal {S}\setminus X\) does not contain any minimal SCS in \(\mathcal {C'}\) (or proves that such set does not exist). Since either X is a SPS or \(\mathcal {S}\setminus X\) is a SCS, we have found a new solution not in \((\mathcal {P}',\mathcal {C'})\). Such a new solution is found in time \(O(n(\tau +n))+m^{O(\log m)}\) where \(n=|\mathcal {S}|\), m is the number of partial solutions already found (i.e. \(m=|\mathcal {P}'| +|\mathcal {C}'|\)) and \(\tau\) is the time needed to evaluate f. Since \(\tau\) is polynomial, we conclude the proof. \(\square\)
Relation to previous work
The paper in the literature that comes closest to ours is [8]. In fact, one of their definitions coincides completely with our definition of SPS. However, their work concentrates on a more restrictive model, which they call machinery-duplicating. The underlying idea of the latter is that each compound involved in a path from the precursor set to the target set should be produced in strictly positive amount, allowing a cell to therefore duplicate itself. We translate their definition by using the concept of factory (cf. Definition 2).
A MD-stoichiometric factory from \(X \subseteq \mathcal {X}\) to \(T \subseteq \mathcal {T}\) is a set \(F \subseteq \mathcal {R}\), if there exists a flux vector \(v \ge 0\) with \(Y = {\textit{S}ubs(F)} \backslash X\) satisfying:
\(v_{i} \left\{ \begin{array}{ll} > 0 &{} \quad i \in F\\ = 0 &{} \quad\text {otherwise,} \end{array} \right.\)
\((Sv)_{\mathcal {C}\setminus X} \quad \ge 0\),
\((Sv)_{T \cup Y} \quad > 0\).
A set \(X \subseteq \mathcal {X}\) is a MD-stoichiometric precursor set (MD-SPS) if there exists a MD-stoichiometric factory from X to \(\mathcal {T}\).
Comparing this definition to Definition 2, clearly any MD-SPS is a SPS, but not the other way around. Moreover, not every minimal MD-SPS is a minimal SPS. In their work, the authors claim that for the growth of a colony of cells, one must consider that not only the biomass compounds should be produced in positive amount, but also all the reactants of every reaction with nonzero flux that are not sources. However, as we already mentioned, cycles like the one in Fig. 1 are considered unfeasible according to the machinery-duplicating model. Yet cycles with this structure are present in real networks and play an important role in metabolism, such as in the urea or the Krebs cycle.
Enumerating precursor sets via MILP
In the following section, we describe how to enumerate all minimal \(SPS\) and MD-SPS using a MILP approach similar to [10–12]. The authors of these papers describe methods that enumerate reaction subsets by recursively solving MILP problems. Therein, solutions obtained in a previous step are excluded from the solution space.
Enumeration of minimal \(SPS\)
We now present a practical method to enumerate all minimal stoichiometric precursor sets that allow to produce the set \(\mathcal {T}\) in a positive amount. We iteratively solve a series of optimisation problems: at each iteration a mixed integer linear programming (MILP) problem is solved to obtain a minimal precursor set X; then we define a new MILP by adding a constraint that removes the obtained solution X and all the sets that contain it from the feasible set. We keep repeating this process until all solutions are found.
We need some additional definitions. For each source compound \(x_j \in \mathcal {X}\), we add to \(\mathcal {R}\) a reaction, which we call source-pool reaction, that produces \(x_j\) from nothing (with stoichiometric coefficient 1). We denote this new set by \(\overline{\mathcal {R}}\) and the set containing all source-pool reactions by \(\overline{\mathcal {R}}_{\mathcal {X}}\). This set of reactions allows to model the availability of the source compounds since the upper bounds on their fluxes are linked to the amount of each source that is available. In the sequel \(\overline{S}\) denotes the stoichiometric matrix S obtained by adding the columns given by the set of reactions \(\overline{\mathcal {R}}_{\mathcal {X}}\) and v the flux vector, U is an upper bound constant for the values of each flux, \(\epsilon\) is a vector of size \(|\mathcal {T}|\) with an arbitrarily small positive real number in all coordinates, b is the vector of binary variables associated with each compound in \(\mathcal {X}\), and we assume that \(b_j=1\) (\(b_j=0\)) implies that compound \(x_j\) is used (not used) to produce the target.
Given a network \(\mathcal {N}=\left\{ \mathcal {C},\mathcal {R}\right\}\), a stoichiometric matrix S, a set \(\mathcal {X}\subseteq \mathcal {C}\) of sources and a set \(\mathcal {T}\subseteq \mathcal {C}\) of targets, we first define the following optimisation problem (1) to find the first minimal solution:
$$\begin{aligned} \begin{array}{cll} {\displaystyle \min f = } &{} {\displaystyle \sum _{j=1}^{|\overline{\mathcal {R}}_\mathcal {X}|} b_{j}}\\ s.t &{} \left( \overline{S}v\right) _{\mathcal {T}} \ge \epsilon , &{} \\ &{} \overline{S}v \ge 0 \\ &{} {\displaystyle b_{j} = 0 \leftrightarrow v_{j}= 0,} &{} \quad \forall j \in \overline{\mathcal {R}}_\mathcal {X}\\ &{} {\displaystyle b_{j} \in \left\{ 0,1\right\} ,} &{} \quad \forall j \in \overline{\mathcal {R}}_\mathcal {X}\\ &{} {\displaystyle 0 \le v_{i} \le U,} &{} \quad \forall i \in \overline{\mathcal {R}}\end{array} \end{aligned}$$
Model (1) is similar to the first MILP presented in [6]. The first set of constraints requires to produce the target at least in a quantity \(\epsilon\). Instead of putting a small value for \(\epsilon\) one could also put e.g. the maximum biomass yield. In this sense we enumerate all minimal precursor sets that allows for the maximal production of biomass. The third set of constraints (constraints \({\displaystyle b_{j} = 0 \leftrightarrow v_{j}= 0}\)) denotes the fact that \(v_j\)—the flux associated to the source compound \(x_j\)—is positive if and only if \(b_j=1\). These constraints can be formulated as a MILP as follows:
$$\begin{aligned} \left. \begin{array}{l} b_{j} \le v_{j} \\ v_{j} \le U b_{j} \end{array}\right\} \quad\text { for } \forall j \in \overline{\mathcal {R}}_\mathcal {X}, \end{aligned}$$
If \(b_{j} = 1\), we have \(v_{j} \ge 1\), which will force us to have at least one unity of the source compound j.
Since the objective function of (1) minimises \(\sum _j b_j\), then the optimal solution \(S^*\), is a precursor set of minimum cardinality. We now show how to modify the MILP to obtain all other minimal precursor sets. To this goal let the pair \((v^{*},b^{*})\) be an optimal solution to Problem (1). Let \(I_{b^{*}}\) be the support of \(b^{*}\); we consider the following constraint:
$$\begin{aligned} \sum _{j \in I_{b^{*}}} b_{j} \le |I_{b^{*}}| - 1, \end{aligned}$$
Constraint (3) excludes the solution \((v^{*},b^{*})\) and all the solutions that contain \(b^{*}\) from the set of solutions of (1). Hence, adding to (1) constraints in the form (3) gives a new instance of the MILP whose solution is a new precursor set that is not included in the previously obtained ones and is minimal (though not necessarily of minimum cardinality). By repeating this procedure, which is a standard technique in mixed integer linear programming, we iteratively enumerate all minimal solutions.
If the obtained problem has no feasible solution, then we claim that we have found all minimal precursor sets.
MILP constraints for MD-SPS
In the work of Eker et al. [8], the machinery-duplicating model is defined through the use of linear constraints and boolean operators. If a set of sources is a MD-SPS, this implies it is a feasible solution according to their model. The authors also present a method to enumerate all MD-SPSs. Suppose we are given a set \(\{X_1,\ldots , X_k\}\) of precursor sets that were already found. Their method consists in finding a minimal subset of sources Y that verifies two conditions: (1) Y has at least one source in common with each precursor set in \(\{X_1,\ldots ,X_k\}\); and (2) the complement of Y must be able to produce the target according to the machinery-duplicating model. If one can find such a subset Y, a minimal precursor set can be obtained by taking the complement \(\overline{Y}\) of Y, and finding one minimal subset of \(\overline{Y}\). If no Y verifying the above conditions can be found, all minimal precursor sets have been enumerated and the algorithm stops.
Our method could also be adapted to consider the machinery-duplicating model presented by Eker et al. [8]. The machinery-duplicating constraint is defined as:
$$\begin{aligned} \left( \overline{S}v\right) _{j} > 0 \vee \bigwedge _{i \in Q_{j}} v_i = 0, \end{aligned}$$
where \(Q_{j}\) is the set of indices of reactions that use the compound j as a substrate. This can be reformulated into MILP constraints as:
$$\begin{aligned} \left( \overline{S}v\right) _{j} &\ge \overline{\epsilon }-D_{j} E_{j}, \\ \sum\limits _{i \in Q_{j}} v_i & \le D_{j} (1 -E_{j}), \end{aligned}$$
where \(\overline{\epsilon }\) is an arbitrarily small positive real number, \(D_{j}\) is a constant that can take any value greater or equal to \(U|Q_{j}|\) and \(E_{j}\) is an artificial binary variable. Adding (5) to Problem (1) allows us to enumerate all minimal stoichiometric precursor sets that respect the machinery-duplicating model.
In this section, we present the experiments we realised and discuss the results we obtained. We start by comparing the method we developed with the one of Eker et al. [8]. We then show the performance of sasita versus the approach where minimal \(SPSs\) are obtained from combinations of minimal topological factories in the many-to-one network. Finally, we apply sasita to some genome-scale metabolic networks, obtained from Monk et al. [16]. The objective of this last part is both to illustrate how our method can be used and to validate it by reproducing the findings of the authors.
All the experiments were performed using an Intel QuadCore i7-4770 computer with 16 GB of RAM memory. The algorithm sasita is coded in Java (OpenJDK IcedTea) and uses cplex (IBM ILOG AMPL/CPLEX 12.5.1) for solving the MILP models; the constants are fixed as follows: \(\epsilon = 0.5\), \(\overline{\epsilon }= 0.5\), \(U = 1000.0\). The constraints (2) were coded using indicator constraints to avoid numerical instability. The software and all network and input files can be downloaded at http://sasita.gforge.inria.fr/.
Comparison between sasita and Eker et al.'s approach
We start by calling attention to the fact that the comparison with the method of Eker et al. was difficult due to the fact that it is not publicly available. We also were not able to obtain it upon request. We therefore implemented a version of sasita that enumerates all minimal MD-SPS using the constraints given by Eq. (5). As input we took the metabolic network, the set of sources \(\mathcal {X}\) and the set of targets \(\mathcal {T}\) provided in the supplementary material of Eker et al. [8]. The authors provided also a list of "auxiliary compounds" without which, according to them, their model does not work. No auxiliary compound appears in the minimal precursor sets that are enumerated by Eker et al. It is not clear how these compounds are handled in their approach. If we treat such auxiliary compounds as ordinary ones, we are not able to enumerate a single MD-SPS with sasita. If we add a source-pool reaction for each one of the auxiliary compounds, we obtain the minimal MD-SPS X = {CCO-PERI-BAC@SULFATE}. Eker et al. find 787 solutions and all of them contain Sulfate. So the minimal solution X we found is in fact a subset of all their solutions.
We provide in the Additional file 1 a list of reactions F that form a MD-stoichiometric factory from X to \(\mathcal {T}\), the flux values in F, and the stoichiometric matrix restricted to the reactions in F. Furthermore, we show that all substrates of the reactions in F and the target set \(\mathcal {T}\) are produced in a positive amount using the reactions in F. Hence, the minimal MD-SPS fulfils the properties of a precursor set according to the machinery-duplicating model [8]. Such minimal MD-SPS is not found by Eker et al. probably because they do some preprocessing on the network that is not described in their paper and that we were not able to obtain upon request.
Comparison between sasita and combinatorial approach
We ran both approaches, i.e. sasita and a combinatorial approach (called combi) where minimal \(SPSs\) are obtained from combinations of minimal topological factories in the many-to-one network, on several instances. Our objective was to analyse the differences in the running times between both approaches, so we set cplex into single thread mode for sasita. Table 1 shows clearly that the MILP approach is more efficient than the combinatorial one. The networks of S. muelleri, C. ruddii and B. aphidicola were obtained from metexplore, filtering out ubiquitous metabolites and pairs of co-factors. We obtained the E. coli core model from http://www.systemsbiology.ucsd.edu/InSilicoOrganisms/Ecoli/EcoliSBML. As sources we considered all compounds that are not produced by a reaction or those that are produced by reversible reactions only. For the E. coli strains, we used the same networks from Monk et al. [16] and considered as sources the compounds from Table 3.
We set a time limit to the combinatorial approach of 2 h. sasita is by far more efficient on genome-scale networks where the combinatorial approach did not finish within the time limit. To be able to show an example where both approaches finish and sasita outperforms combi, we removed all compounds from the E. coli core network if they are consumed and produced by more than ten reactions. This is the case for M_atp_c, M_nad_c, M_nadh_c, M_h2o_c, M_h_e, M_h_c. The resulting network is denoted by a superscript a. Notice that the number of reactions remains the same because we remove only the above-mentioned compounds from the reactions. The difference in the time spent to solve the problem is remarkable. It takes less than 1 s with sasita and 42 s with combi.
Table 1 Our MILP approach (sasita) versus the combinatorial one
Enumerating minimal precursor sets in genome-scale metabolic networks
In this case, we based our experiments on the work of Monk et al. [16] who investigated the pan and core metabolic capabilities of 55 Escherichia coli and Shigella strains based on genome-scale reconstructions of their metabolism. By core is meant the elements shared by all strains and by pan the union of the elements from all strains. As concerns the latter in particular, the authors found the pan to be enriched in alternate carbon metabolic pathways. In order to determine the functional differences among the strains, the authors computed by flux balance analysis (FBA) the growth phenotypes of 385 nutrients (henceforth called the test metabolites/compounds), each considered individually as a source of carbon, nitrogen, phosphorus and sulfur, aerobically and anaerobically. To that purpose, an in silico minimal medium that contains a sole carbon, nitrogen, phosphorus and sulfur source was defined. The authors then replaced the sole carbon source by each of the 385 test metabolites one at a time. Whether or not these new media constituted a growth condition was tested by FBA. The procedure was repeated for each source in the minimal media, namely for nitrogen, phosphorus and sulfur, as well as for each strain. The resulting metabolic phenotypes indicated strain-specific adaptation to nutritional environments.
Our first goal was to validate our method: we compared the results obtained with sasita to the ones in Monk et al. [16]. We enumerated and compared the minimal precursor sets allowing for biomass production of the E. coli strains, which included commensals as well as both intestinal and extraintestinal pathogens. We used for this the genome-scale metabolic models from Monk et al. [16]. The strains were E. coli str. K-12 MG1655 (Commensal), E. coli O157:H7 str. Sakai (Enterohemorrhagic E. coli, EHEC), E. coli O157:H7 EDL933 (EHEC), and E. coli CFT073 (Uropathogenic E. coli, UPEC). The same 385 compounds tested in [16] were given as part of the sources for different runs of sasita. Since Monk et al. [16] were not interested in minimal solutions, we wanted to check whether our solutions were subsets of their solutions.
The second goal was to explore some solutions that were only found by sasita in order to illustrate one application of our method. These solutions contain more than one of the test metabolites, after excluding sources from the minimal media (i.e. carbon, nitrogen, phosphorus and sulfur). Such solutions were explored as concerns strain-specific growth and their relation to niches and to pathotypes.
For almost all solutions found by Monk et al. [16], sasita was capable of correctly finding at least one corresponding minimal subset. There is a small number of solutions (7) found by Monk et al. [16] and not by sasita. In Table 2, we show the amounts of solutions found and not found for each strain. We confirmed through FBA that there is indeed no feasible flux for those solutions (this is further discussed below). In the remainder of this section, we explain how we realised our experiments and we present our results in more detail.
Table 2 Differences between solutions found by Monk et al. [16] and by sasita
Two experiments were conducted. In both cases, oxygen was always available and we used as target an artificial compound that is added as an extra product of the core biomass reaction, with stoichiometry of 1.0. Also, the minimal media for E. coli CFT073 contained tryptophan, its auxotrophy. We now give a general description of each experiment. The exact list of compounds for each experiment as well as all networks can be found in the sasita website.
In Experiment 1, we tested growth on minimal media of each strain as defined by Monk et al. [16] by enumerating the minimal precursor sets using as sources only the compounds from such minimal media. In Table 3, we present the list of compounds considered as sources for each strain, for this experiment. For each strain we found two solutions, one aerobic and another anaerobic as expected. This shows that, theoretically, the so-called "minimal media" are in fact not minimal as a whole (i.e. they are minimal in terms of carbon, sulfur, nitrogen and phosphorus sources) and that the considered strains can grow from a proper subset of that set of compounds.
Table 3 Minimal media compounds for each E. coli strain
In order to check the ability of each strain to grow on the 385 test compounds as sources of carbon, nitrogen, phosphorus and sulfur, we ran Experiment 2. For each network we considered as input source set a subset of the minimal media compounds plus a subset of the 385 test metabolites.
For subsets of the minimal media compounds, we considered the set of minimal media compounds minus one of the following: glucose, ammonium, phosphate or sulfate respectively. Since we were removing a compound from the minimal media, we included only the test metabolites that could replace the removed one (if we removed glucose, we considered only the set of test compounds that have carbon in their composition and so on). This was done because considering all the 385 test metabolites together leads to a combinatorial explosion of the number of solutions that are unpractical to enumerate with sasita. In one case, namely when we remove glucose, the set of test metabolites to include was also too big and we needed to split it further in two smaller sets. As a side effect, this split of the input compounds can lead to a loss of some solutions, namely those containing compounds that are in different input sets. We thus may lose some solutions that have more than one minimal media compound replaced by two or more test metabolites. However, our split guarantees that at least all the sets considered in Monk et al. [16] are possible combinations of our input compounds because the authors replace glucose, ammonium, phosphate or sulfate from the minimal media with only one of the test metabolites.
First goal: Comparison with Monk et al. We found 837 minimal precursor sets for E. coli CFT073 and between 11.164 and 13.732 for the other strains (Table 4).
Table 4 Number of solutions found for each E. coli strain
This difference is remarkable but not surprising: E. coli CFT073 has a tryptophan auxotrophy, and tryptophan itself can be a source of carbon and nitrogen. Since we search for minimal precursor sets and tryptophan is always a source, there is no need for any extra source of carbon and nitrogen, thus reducing the number of solutions for this strain.
All solutions found by sasita are either a subset of the given minimal media or a subset of the minimal media plus one or more test metabolites. The number of solutions where at least one of the four sources (carbon, nitrogen, phosphorus and sulfur) is replaced by only one compound among the 385 test metabolites is presented in Table 5.
Table 5 Number of solutions with one test metabolite
These solutions contain one test metabolite plus a subset of the minimal media in which the test metabolite replaces one, or more of the following compounds: glucose, ammonium, phosphate and sulfate. They therefore correspond to minimal sets of the solutions found by Monk et al. [16].
Other differences found in the comparison of our results with those from [16] are presented and discussed below. Some such differences arise for E. coli CFT073 for which, as mentioned, tryptophan can always be a source of carbon and nitrogen. Since we find a minimal solution without any test metabolite and without glucose and ammonium, but with tryptophan, it is a minimal subset of all solutions from [16] considering the replacement of glucose or ammonium by a test metabolite. Furthermore, a few minimal precursor sets for which Monk et al. [16] found no growth are present among our solutions because of the different conditions we allowed in our test, namely some sources were available at bigger amounts and the compounds were allowed to accumulate. There were as well seven solutions for which Monk et al. [16] found a positive flux and we did not (see Table 2). Those solutions are for the K-12 strain. Among these solutions, 6 are related with the compounds 4-Hydroxy-l-threonine and Oxaloacetate from the exchange subsystem, and in fact there are no reactions in the network that use those compounds to produce anything. The remaining solution is the one using Thiosulfate as source of sulfur, and we confirmed no growth by FBA for this condition. There is one last difference, for the solution with the test metabolite Fe(III) dicitrate which allows growth as a carbon source in aerobic and anaerobic conditions for E. coli K-12. We explicitly found only the anaerobic solution. Since the aerobic one can be seen as a superset of the anaerobic, it is not a minimal precursor set. This does not happen in the other test metabolites owing to different iron oxidation states in each solution. We thus found different aerobic and anaerobic solutions when we enumerated the minimal precursor sets.
In conclusion, we obtained almost all of the solutions found by Monk et al. [16], showing that the nutrient sources of alternate catabolic pathways are part of the minimal precursor sets that allow for biomass production in most of the tested E. coli strains.
Second goal: Original results using sasita
The remaining solutions go beyond the analyses performed by Monk et al. [16] because they contain two or more test metabolites (Table 6).
Table 6 Solutions with more than one test metabolite
Most of these solutions actually have two or three test metabolites (4.961 and 7.801 respectively). In both cases, more than 60% of those solutions are aerobic.
Most solutions in which the two or three test metabolites replace all four sources from the minimal media (glucose, ammonium, phosphate, sulfate) are found in all the three strains, E. coli EDL993, E. coli K-12 and E. coli Sakai. Moreover, there are some minimal precursor sets specific to E. coli EDL993 and E. coli Sakai which are both EHEC, and others specific to E. coli K-12 which is commensal (Table 7). The latter are specific to pathotypes and probably indicate adaptations to nutritional environments.
Table 7 Solutions with two or three test metabolites
From these results, the pairs of test metabolites in the six solutions specific to E. coli EDL993 and E. coli Sakai are: N-Acetyl-d-galactosamine 1-phosphate with butanesulfonate, ethanesulfonate or taurine, respectively; all aerobic and each one in two solutions with different iron states (line in italics in Table 7). N-Acetyl-d-galactosamine 1-phosphate was shown to give extraintestinal pathogenic strains of E. coli a catabolic advantage when compared to commensals, supporting growth in 100% of the cases compared to 67%, respectively [16]. This compound supported growth as a sole carbon source in aerobic and anaerobic conditions for extra and intracellular pathogens (E. coli CTF073 together with tryptophan, E. coli EDL993 and E. coli Sakai), however not for the commensal strain E. coli K-12. Furthermore, the enterohemorrhagic strains E. coli EDL933 and E. coli Sakai were shown to occupy the same niche in the streptomycin-treated mouse intestine [17] while E. coli EDL933 was shown not to colonise the same niche and does not use the same sugars as carbon source as the commensal E. coli K-12 [18, 19]. The three solutions detailed above and the solutions presented in Table 7 therefore represent metabolic capabilities that are specific to the pathogenic strains analysed here when compared to the commensal strain E. coli K-12, in agreement with the niches occupied by such strains.
These results suggest that sasita can depict pathotype and niche-specific metabolic capabilities which allow broad in silico studies of strains or species interactions. For instance, an extension of the analysis presented in this paper to a larger dataset of E. coli strains including both pathogenic and commensal biotypes could help predict in silico sets of commensal strains that would prevent the colonisation of pathogens due to a consumption by the native microbiota of the nutrients required by the pathogen (see the experimental study of mutant phenotypes in Maltby et al. [20]).
We examined the relationship between topological and stoichiometric precursor sets. We highlighted that stoichiometric precursor sets can be obtained from combinations of minimal topological factories in the many-to-one network. However, this does not lead to an efficient method. We then presented sasita, an efficient algorithm for the exhaustive enumeration of minimal precursor sets for a given target that takes into account stoichiometry. To the best of our knowledge, there exists only one previous approach for this problem due to Eker et al. [8] who proposed two different constraint models, steady-state and machinery-duplicating. However, in their computations, the authors use only the latter, that requires a strictly positive net production of the intermediate compounds on the path from the sources to the target. This model may exclude solutions as we showed (Fig. 1).
In our experiments, we enumerated and compared the minimal precursor sets of nutrient sources of alternate catabolic pathways allowing for biomass production of some Escherichia coli strains, comprising commensal and both intestinal and extraintestinal pathogens, using genome-scale metabolic models. We compared our results to those of Monk et al. [16] in order to have a guideline on part of the solutions we generated, since our approach is different from the one that the authors used, and our results go beyond such comparison. We found metabolic capabilities that distinguish the strains compared in their ability to catabolise nutrients, and such were specific to pathotypes and niches of E. coli strains.
Our method can therefore be used in a wide variety of applications in order to study minimal growth conditions as well as strains and/or species interactions based on their catabolic abilities and their nutritional niches. One valuable application in this context would be to predict patterns of colonisation of commensal and pathogenic E. coli strains in the intestine.
Our method can furthermore be used to refine a metabolic network. If growth of an organism is observed for a defined medium in the laboratory, but no minimal precursor set is a subset of such medium, then either the metabolic network lacks reactions, e.g. export reactions, or the biomass function is not well formulated.
The execution of all experiments of the comparison with Monk et al. [16] took altogether around 5 days. The execution times ranged from 20 s to 12 h. Running the experiments in parallel, one could such retrieve the results after less than a day. We cannot claim to be more efficient than Eker et al. [8] as their software is not available for testing. However, we are guaranteed to enumerate all minimal \(SPSs\) and MD-SPS.
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RA and MW designed and implemented the algorithms, RA, MW and CCK performed the tests and analyzed the results, RA, MW, VA, AMS, PVM, LS, MFS wrote and revised the theorems and proofs, MFS supervised the work. All authors contributed to the writing of the paper. All authors read and approved the final manuscript.
RA acknowledges CNPq/Brasil for the financial support and VA for the support of Fondecyt 1140631, CIRIC-INRIA Chile and FONDAP Center for Genome Regulation. CCK and MW were recipients of a grant from the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No [247073]10 SISYPHE.
Erable team, INRIA Grenoble Rhône-Alpes, 655 Avenue de l'Europe, 38330, Montbonnot Saint-Martin, France
, Martin Wannagat
, Cecilia C. Klein
, Alberto Marchetti-Spaccamela
, Leen Stougie
& Marie-France Sagot
UMR CNRS 5558, LBBE, "Biométrie et Biologie évolutive", Université Lyon 1, 43 bd du 11 Novembre 1918, 69622, Villeurbanne, France
Center for Mathematical Modeling (UMI 2807 CNRS), University of Chile, Beauchef 851, 837 0456, Santiago de Chile, Chile
Vicente Acuña
Sapienza University of Rome, Via Ariosto 25, 00185, Rome, Italy
Alberto Marchetti-Spaccamela
Applied Research Department, Tecsinapse, São Paulo, Brazil
Paulo V. Milreu
CWI, Science Park 123, 1098 XG, Amsterdam, The Netherlands
Leen Stougie
Vrije Universiteit Amsterdam, De Boelelaan 1105, 1081 HV, Amsterdam, The Netherlands
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Correspondence to Marie-France Sagot.
Ricardo Andrade and Martin Wannagat contributed equally to this work
Additional file 1. A list of reactions F that form a MD-stoichiometric factory, the stoichiometric matrix restricted to the reactions in F, one possible flux vector v, and the vector of the net production of all compounds.
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doi: 10.3934/jimo.2021021
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On the optimal control problems with characteristic time control constraints
Changjun Yu , Shuxuan Su , and Yanqin Bai
Shanghai University, Shanghai, 200444, China
* Corresponding author: Shuxuan Su
Received June 2020 Revised December 2020 Early access January 2021
Fund Project: This work is supported by National Natural Science Foundation of China(NSFC), Grant No.11871039 and Science and Technology Commission of Shanghai Municipality(STCSM), Grant No. 20JC1413900
Figure(7) / Table(2)
In this paper, we consider a class of optimal control problems with control constraints on a set of characteristic time instants. By applying the control parameterization technique, these constraints are imposed on the subdomains that contain the characteristic time points. The values of the control functions as well as the lengths for their corresponding subdomains become decision variables. Time-scaling transformation is an effective technique to optimize the length of each subdomain in a new time horizon. However, the characteristic time instants in the original time horizon become variable time instants in the new time horizon, and hence the control constraints imposed on these characteristic time points are difficult to be formulated in the new time horizon. We propose a surrogate condition and show that the characteristic time control constraints will be satisfied once the surrogate condition holds. Moreover, this surrogate condition is easy to formulate in the new time horizon. The resulting approximate problem can be readily solved by many existing computational methods for solving constrained optimal control problems. Finally, we conclude this paper by solving two examples.
Keywords: Optimal control, characteristic time, control constraints, control parameterization, time-scaling transformation.
Mathematics Subject Classification: Primary: 90C30, 90-08, 34H05.
Citation: Changjun Yu, Shuxuan Su, Yanqin Bai. On the optimal control problems with characteristic time control constraints. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021021
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Figure 1. Time-scaling function for pre-fixed terminal time
Figure 2. Optimal control for Problem 1 with $ p = 9 $.
Figure 3. Optimal state trajectories for Problem 1 obtained using the two different methods with $ p = 9 $.
Figure 4. Optimal controls for Problem 2 with $ p = 5 $
Figure 6. Optimal controls for Problem 2 with p = 7
Figure 7. Optimal state trajectories for Problem 2 with different partition numbers
Table 1. Optimal costs for Problem 1 obtained using the two different methods
Knots $ G_0^* $
Traditional control parameterization New time-scaling transformation
$ p=5 $ $ 110.8 $ $ 1.9531 $
$ p=7 $ $ 42.34 $ $ 1.6671 $
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Table 2. Optimal costs for Problem 2 using different lower bounds on each $ \theta_l $ with different partition $ p $
Lower bounds on $ \theta_l\ (l=1, \ldots, p) $ $ J_0^* $
$ p=5 $ $ p=6 $ $ p=7 $
$ \geq 0.05 $ $ 12.189 $ $ 8.113 $ $ 4.572 $
$ \geq 0.1 $ $ 12.347 $ $ 8.258 $ $ 4.673 $
$ \geq 1 $ $ 14.453 $ $ 10.924 $ $ 7.000 $
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Changjun Yu Shuxuan Su Yanqin Bai | CommonCrawl |
Journal of Health, Population and Nutrition
Research article | Open | Published: 26 February 2018
Effects of home delivery on colostrum avoidance practices in North Wollo zone, an urban setting, Ethiopia: a cross sectional study
Nigus Bililign Yimer1 &
Misgan Legesse Liben2
Journal of Health, Population and Nutritionvolume 37, Article number: 4 (2018) | Download Citation
Colostrum is the first liquid that is produced in the first few days after delivery. It is the perfect first food for newborns which is considered as an infant's first immunization. Despite of this fact colostrum is discarded as unclean and bad for the infant's health. This study aimed to investigate the prevalence and the factors associated with colostrum avoidance in Woldia, Kobo and Lalibela town administrations of North Wollo zone.
A quantitative community based cross sectional study was employed in March 2015 on 810 mothers of children aged less than 24 months. Descriptive statistics, binary and multivariable logistic regression analysis were employed to identify the factors associated with colostrum avoidance. Variables with a p-value < 0.05 in the multivariable model were identified as predictors of colostrum avoidance practices.
Colostrum was discarded by 12.0% (95%CI: 10.0–14.0%) of mothers of children aged less than 24 months. In multivariable logistic regression analysis late initiation of breastfeeding [AOR (95% CI) =2.03 (1.18, 3.49)], prelacteal feeding [AOR (95% CI) =3.38 (1.83, 6.24)], mothers not living with their husband [AOR (95% CI) = 2.24 (1.22, 4.12)] and delivering the index child at home [AOR (95% CI) =2.92 (1.521, 5.59)] were independent positive predictors of colostrum avoidance practices.
The foundation of any nutrition package for the prevention of childhood malnutrition is the promotion of an optimal breastfeeding practices, including colostrum feeding, in the community. Therefore, promoting institutional delivery, early initiation of breastfeeding and creating awareness on the dangers of prelacteal feeding and the advantages of colostrum feeding are recommended interventions to reduce colostrum avoidance practices in the study area.
World Health Organization (WHO) and united nation international children's emergency fund (UNICEF) recommend exclusive breastfeeding for children up to 6 months of age, and to nourish them with appropriate complementary foods and continued breastfeeding from 6 months until 2 years or beyond [1]. Ethiopia also adopts this recommendation [2, 3]. Exclusive breastfeeding is an infant's breast milk consumption without supplementation of any type of foods or drinks, except for vitamins, minerals and necessary medications up to the age of 6 months [1, 2, 4].
Colostrum is the first liquid that is produced in the first few days after delivery [4]. Colostrum flows in very small amounts that suit the infant's very small stomach and the immature kidneys that cannot handle large amount of fluid. Compared to matured milk, colostrum is slightly yellow, more viscous, and thicker. It contains growth factors and protective proteins (immune factors) as well as all other nutrients the newborn needs to survive. Colostrum is an infants' first immunization against many bacteria and viruses. Colostrum is also a laxative which helps the baby to pass meconium (the first sticky black stool). Colostrum is therefore the perfect first food for newborns [1]. Despite of this fact colostrum is discarded as unclean and bad for the infant's health [5,6,7].
Colostrum avoidance is failure to feed infants with the first, thick and yellowish milk that is produced in the first few days after birth [1]. It has a significant association with increased odds of malnutrition among children aged less than 5 years. In India infants who fed on colostrum were less likely to be stunted and wasted compared to children who were deprived of colostrum [8]. In West Gojjam children deprived of colostrum were more likely to be stunted compared to children who had fed on colostrum [9]. Children who fed on colostrum were less likely to be malnourished compared to children who deprived of colostrum in Nigeria [10]. Colostrum avoidance has also a negative association with optimal breastfeeding practices [6, 11, 12].
Prelacteal feeding is a common practice in some cultures over colostrum feeding [7, 13,14,15,16,17,18]. Prelacteal feeds are foods and/or drinks other than human milk, given to newborns before breastfeeding initiation usually on the first 3 days of life. These traditional feedings restrict suckling, making breastfeeding more challenging to establish [1, 4, 19]. Hence, assessing the factors associated with colostrum avoidance is essential in maintaining optimal breastfeeding practices. In addition limited studies are conducted in Ethiopia to describe colostrum avoidance practices. This study aimed to investigate the prevalence and factors associated with colostrum avoidance in Woldia, Kobo and Lalibela town administrations of North Wollo zone.
Study setting and participants
North Wollo zone is one of the ten zones (the smallest administrative units next to region in Ethiopia) of the Amhara Regional State. It is bordered on the south by South Wollo Zone, on the west by South Gondar, on the north by Wag Hemra, on the northeast by Tigray Region, and on the east by Afar Region. The towns in North Wollo zone include Lalibela, Woldia and Kobo. Based on the 2007 Census conducted by the central statistical agency of Ethiopia, this Zone has a total population of 1,500,303, of whom 752,895 are men and 747,408 women. The largest ethnic group reported in North Wollo zone is the Amhara (99.38%) and Amharic is spoken as a first language by 99.28%.
This study was conducted in Woldia, Kobo and Lalibela towns of North Wollo zone in March 2015. Woldia is the capital of North Wollo zone. It is located 521 km north of Addis Ababa and about 360 km south of the capital of Amhara regional state, Bahirdar. This town has eight kebeles (the smallest administrative units next to district in Ethiopia). Based on the 2007 national census conducted by the central statistical agency of Ethiopia, this town has a total population of 46,139, of whom 23,000 are men and 23,139 women. Kobo town is located in the northeast corner of North Wollo zone. Kobo and Lalibela town administrations consist of five and two kebeles, respectively. The majority of the inhabitants in the three towns (Woldia, Kobo and Lalibela) were Amhara and followers of Ethiopian orthodox Christians.
A quantitative community based cross sectional study was employed to survey 810 mothers of children aged less than 24 months. The sample size was determined using a single population proportion formula:
$$ \mathrm{n}=\mathrm{D}\left[\frac{{\left(\mathrm{Z}\frac{\upalpha}{2}\right)}^2\mathrm{p}\left(1-\mathrm{p}\right)}{{\mathrm{d}}^2}\right] $$
Where n = required sample size, Z = critical value for normal distribution at 95% confidence level (1.96), P = prevalence of colostrum avoidance (39.8%) [20], d = 0.05 (5% margin of error), D = 2 (design effect), and an estimated non-response rate of 10%.
Sampling procedure and technique
First the three towns were selected purposely since these towns are the first three top ranked town administrations (Woldia, Kobo and Lalibela) in North Wollo Zone. Out of the fifteen kebeles in the three towns, eight kebeles were randomly selected. Presurvey was done before the actual period of data collection to know which households have the targeted mother-child pairs. As a result, there were 6013 households having the targeted mother-child pairs in the selected eight Kebeles. Then, the sample size was proportionally allocated to each selected kebeles. At the time of survey, from each household unit one eligible mother who had a biological child aged less than 24 months was selected. If there were more than one mother with children aged less than 24 months in one household unit, one mother with the youngest child was selected. From mothers who had two children aged less than 24 months, the youngest child was selected. If mothers had twin children aged less than 24 months, one child was selected by lottery. Non-biological and mothers who are unable to communicate were excluded from the study.
Study variables
In this study the outcome variable was colostrum avoidance among mothers of children aged less than 24 months. Colostrum avoidance is failure to feed infants with the first, thick and yellowish milk that is produced in the first 3 days after birth [1]. Discarding colostrum was coded as "1" while colostrum feeding was coded as "0" for regression analysis. The independent variables were maternal characteristics (age, parity, educational status, religion, ethnicity, marital status, occupation), maternal health service and obstetric variables (antenatal care visit, place of delivery, mode of delivery, postnatal care visit), child's sex, breastfeeding initiation, breastfeeding counseling during antenatal care visits and prelacteal feeding. Prelacteal feeding was understood as providing foods and/or drinks other than human milk for the infant before breastfeeding initiation [1].
Data collection procedure and quality control
Data were collected using a pre-tested, structured and interviewer administered questionnaire. The questionnaire was prepared first in English and translated to Amharic (local language), then back to English to check for consistency. The Amharic version of the questionnaire was used to collect the data. The data were collected by six diploma midwives. The data collectors and the supervisors (three BSc nurses) were trained for 2 days by the investigators on the study instrument, consent form, how to interview and data collection procedures.
Then questionnaire was pretested on mother-child pairs in two kebeles which were not included in the research. The pretest was done to ensure clarity, wordings, logical sequence and skip patterns of the questions. Then the pretest amendments on the questionnaire were made accordingly. The supervisors had checked the day to day activity of data collectors regarding the completion of questionnaires, clarity of responses and proper coding of the responses.
Data processing and analysis
The data were checked for completeness and inconsistencies. It was also cleaned, coded and entered into EpiData version 3.02, then exported to the SPSS 20.0 statistical package for analysis. Descriptive statistics were used to show the prevalence of colostrum avoidance practices and socio-demographic characteristics.
Binary logistic regression analysis was performed. The crude odds ratio (COR) with 95% confidence interval was estimated to assess the association between each independent variables and the outcome variable, and to select candidate variables for the multivariable logistic regression analysis. Variables found statistically significant at p-value < 0.25 [21] during binary logistic regression analysis were included in the multivariable logistic regression model.
The Hosmer-Lemeshow goodness-of-fit with enter procedure was used to test for model fitness. Adjusted odds ratio (AOR) with 95% confidence interval was estimated to assess the strength of the association, and a p-value < 0.05 was used to declare the statistical significance in the multivariable analysis. Variables with p-value < 0.05 in the multivariable logistic regression analysis were considered as significant and independent predictors of colostrum avoidance.
The study was approved by institutional research ethics review committee (IRERC) of Woldia University. An official letter was written from research and development office of Woldia University to Woldia, Kobo and Lalibela town's administration office. Then permission and support letter was written to each selected kebeles. Informed verbal consent was taken from the participants before the interview. The participants were also assured about the confidentiality of the information they provided.
Socio-demographic characteristics of the study participants
A total of 782 mother-child pairs were included in the study, yielding a response rate of 96.5%. The mean (±SD) age of mothers was 27.03 (±SD 5.48) years and ranged from 15 to 48 years. Majority of the mothers (72.9%) had attended formal education and 76.4% were in the age group of 20–34 years (Table 1).
Table 1 Socio-demographic characteristics of mothers of children aged less than 24 months (n = 782) in North Wollo zone, Northeastern Ethiopia, 2015
Maternal health service utilization
Ninety four percent of mothers had attended antenatal care visits. Of these only 41.6% were counseled about breastfeeding. About 90% of respondents delivered in health institution. Nearly 54% of mothers attended postnatal care visits (Table 2).
Table 2 Distribution of mothers based on maternal health service utilization (n = 782) in North Wollo zone, Northeastern Ethiopia, 2015
Breast feeding practices
About 98 % (98.3%) of mothers had ever breastfed their index child. Of those who had ever breastfed, 601(78.2%) mothers initiated breast feeding within 1 h of birth. Colostrum was discarded by 12.0% (95% CI: 10.0–14.0%) of mothers of children aged less than 24 months (Table 3).
Table 3 Breastfeeding patterns (n = 782) in North Wollo zone, Northeastern Ethiopia, 2015
Factors associated with colostrum avoidance
Binary logistic regression showed that breastfeeding initiation time, attending antenatal care visits, prelacteal feeding, marital status, parity and place of delivery were associated with colostrum avoidance.
In multivariable logistic regression analysis late initiation of breastfeeding, prelacteal feeding, mothers who live without their husband and delivering the index child at home remained significant as independent positive predictors of colostrum avoidance among mothers of children aged less than 24 months. Mothers who initiated breastfeeding after 1 h of delivery [AOR (95% CI) =2.03 (1.18, 3.49)] were more likely to discard colostrum compared to mothers who initiated breastfeeding within 1 h after delivery. Mothers who practiced prelacteal feeding [AOR (95% CI) =3.38 (1.83, 6.24)] were more likely to avoid colostrum compared to mothers who avoid prelacteal feeds. Compared to mothers who live with their husband, mothers who live without their husband were more likely to discard colostrum [AOR (95% CI) = 2.24 (1.22, 4.11)]. Mothers who delivered the index child at home were 2.9 times [AOR (95% CI) =2.92 (1.52, 5.59)] more likely to discard colostrum compared to mothers who gave birth at health institution (Table 4).
Table 4 Binary and multivariable logistic regression analysis showing factors associated with colostrum avoidance among mothers of children aged less than 24 months in North Wollo zone, Northeastern Ethiopia, 2015
Proper breastfeeding practices are essential for the growth, development and survival of children. Despite this fact, colostrum is discarded in different parts of the globe since it is thought as yellowish-dirty milk [7, 13,14,15,16,17, 22]. This study showed that colostrum avoidance was practiced by 12.0% (95% CI: 10.00–14.00%) of mothers of children aged less than 24 months which is higher than the findings reported at Kersa district of Ethiopia (8.5%) [12]. Similar findings were reported at Raya Kobo (13.5%) [7] and Arbaminch zuria (11.0%) [23]. However, the prevalence of colostrum avoidance in this study was lower compared to the national prevalence in Ethiopia (39.8%) [20], Bahir Dar city (16.7%) [24], Jimma Arjo district (27.5%) [6], Goba district (35%) [13] and India (15.2%) [25]. This difference might be due to socio-cultural factors that affect colostrum feeding.
Mothers who initiated breastfeeding after 1 h of delivery were more likely to discard colostrum than mothers who initiated breastfeeding within 1 h after delivery. This is in line with the findings at Raya Kobo district [7]. This could be because of the fact that those mothers who initiated breastfeeding lately would have more time for infant feeding malpractices like colostrum avoidance. The reverse might be also correct. When mothers tend to discard colostrum, they might take more time to discard colostrum and initiate breastfeeding later [11].
Prelacteal feeding is positively associated with colostrum avoidance. In the current study, mothers who practiced prelacteal feeding were more likely to throw away colostrum compared to their counterparts. In Egypt mothers considered colostrum as bad to their baby and majority of mothers introduced prelacteal feeds in the first feed [26]. In support of this idea in Ethiopia, newborns fed on prelacteal foods before breastfeeding initiation. This is because prelacteal foods were believed to decrease infant mortality and morbidity [7, 27]. But colostrum is discarded since it was considered as expired yellowish-dirty milk that could cause abdominal cramp [7, 23, 27].
Maternal marital status is associated with colostrum avoidance in the current study area. Compared to mothers who live with their husband, mothers who live without their husband were more likely to discard colostrum. Similarly household head mothers were more likely to discard colostrum compared with mothers who were not household heads [7]. This might be due to the fact that mothers who live without their husband might lack the helpful advices on the importance of colostrum feeding.
Mothers who delivered the index child at home were 2.9 times more likely to discard colostrum compared to mothers who gave birth at health institution. Similarly in northeastern Ethiopia mothers who delivered the index child at home were 2.6 times more likely to discard colostrums as compared with mothers who gave birth at health institutions [7]. This could be grandmothers [7, 23, 28] and traditional birth attendants are the influential individuals for colostrum avoidance [7, 23]. Therefore, delivering at home might prevent women from immediate colostrum feeding in the face of these potentially influential individuals. On the other hand mothers who delivered in health institution might have knowledge provided by health workers which can improve attitude of mothers towards colostrum feeding.
Colostrum was discarded by 12.0% of mothers of children aged less than 24 months. Colostrum avoidance was more common among mothers who initiated breastfeeding lately, practiced prelacteal feeding, live without their husband and delivered the index child at home. However, the foundation of any nutrition package for the prevention of childhood malnutrition is the promotion of an optimal breastfeeding practices, including colostrum feeding, in the community. Therefore, promoting institutional delivery, early initiation of breastfeeding and creating awareness on the dangers of prelacteal feeding and the advantages of colostrum feeding are recommended interventions to reduce colostrum avoidance.
WHO/UNICEF. Baby-friendly hospital initiative: revised, updated and expanded for integrated care. Geneva: World Health Organization; 2009. http://www.who.int/nutrition/topics/bfhi/en/index.html
Federal Ministry of Health (FMOH) of Ethiopia (2004). National strategy for infant and young child feeding. Available at http://motherchildnutritionorg/nutrition-protection-promotion/pdf/mcn-nationalstrategy-for-infant-and-young-child-feeding-pdf.
FMOH. National strategy for child survival in Ethiopia. Addis Ababa: FMOH; 2005.
Central Statistical Agency (CSA) Ethiopia. Demographic and health survey 2011. Addis Ababa, Ethiopia and Calverton, Maryland, USA: CSA and ORC Macro; 2012.
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Tamiru D, Belachew T, Loha E, et al. Sub-optimal breastfeeding of infants during the first six months and associated factors in rural communities of Jimma Arjo Woreda, Southwest Ethiopia. BMC Public Health. 2012;12:363.
Legesse M, Demena M, Mesfin F, Haile D. Factors associated with Colostrum avoidance among mothers of children aged less than 24 months in Raya kobo district, North-eastern Ethiopia: community-based cross-sectional study. J Trop Pediatr. 2015;0:1–7.
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Liben ML. Determinants of early initiation of breastfeeding among mothers: the case of Raya Kobo District, Northeast Ethiopia: a cross-sectional study. Int J Nutrition Food Sci. 2015;4(3):289–94.
Egata G, Berhane Y, Worku A. Predictors of non-exclusive breastfeeding at 6 months among rural mothers in East Ethiopia. Int Breastfeed J. 2013;8:8.
Setegn T, Gerbaba M, Belachew T. Determinants of timely initiation of breastfeeding among mothers in Goba Woreda, south East Ethiopia: a cross sectional study. BioMed Central Public Health. 2011;11:217.
Tamiru D, Aragu D, Belachew T. Survey on the introduction of complementary foods to infants within the first six months and associated factors in rural communities of Jimma Arjo. Int J Nutrition Food Sci. 2013;2(2):77–84.
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Raval D, Jankar DV, Singh MPA. Study of breast feeding practices among infants living in slums of Bhavnagar city, Gujarat, India. Health Line. 2011;2(2):78–83.
Umar AS, Oche MO. Breastfeeding and weaning practices in an urban slum, north western Nigeria. Int J Tropical Dis Health. 2013;3(2):114–25.
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Sapna PS, Ameya HA, Rooma PS, Parmar A, Rashid K, Narayan A. Prevalence of exclusive breast feeding and its correlates in an urban slum in western India. IeJSME. 2009;3(2):14–8.
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Njai M, Dixey R. A study investigating infant and young child feeding practices in Foni Kansala district, western region, Gambia. J Clin Med Res. 2013;5(6):71–9.
Tamiru D, Mohammed S. Maternal knowledge of optimal breastfeeding practices and AssociatedFactors in rural communities of Arba Minch Zuria. Int J Nutrition Food Sci. 2013;2(3):122–9.
Seid AM, Yesuf ME, Koye DN. Prevalence of Exclusive Breastfeeding Practices and associated factors among mothers in Bahir Dar city, Northwest Ethiopia. Int Breastfeed J. 2013; 8(14).
Prateek SB, Saurabh RS. Breastfeeding practices and factors associated with it: a cross sectional study among tribal women in Khardi primary health Centre, thane, India. Int J Public Health Res. 2012;2(1):115–21.
Abdel HE, Doaa MA. Newborn first feed and Prelacteal feeds in Mansoura, Egypt. Biomed Res Int. 2014;2014:258470. Available at http://dx.doi.org/10.1155/2014/258470
Legesse M, Demena M, Mesfin F, Haile D. Prelacteal feeding practices and associated factors among mothers of children aged less than 24 months in Raya kobo district, north eastern Ethiopia: a cross sectional study. Int Breastfeed J. 2014;9:189.
Haider R, Rasheed S, Sanghvi TG, Hassan N, Pachon H, Islam S, Jalal CS. Breastfeeding in infancy: identifying the program-relevant issues in Bangladesh. Int Breastfeed J. 2010;5:21.
Authors thank Woldia University for supporting this study. Authors also thank supervisors, data collectors and study subjects.
Department of Midwifery, Faculty of Health Sciences, Woldia University, Amhara, Ethiopia
Nigus Bililign Yimer
Department of Public Health, College of Medical and Health sciences, Samara University, P.O.Box 132, Afar, Ethiopia
Misgan Legesse Liben
Search for Nigus Bililign Yimer in:
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NB conceived, designed the study and supervised the data collection. ML assisted with the design and conception of the study, and performed the data analysis, interpretation of data and drafted the manuscript. NB critically reviewed the manuscript. All authors read and approved the final manuscript.
Correspondence to Misgan Legesse Liben.
North Wollo | CommonCrawl |
Jan Mandel
Jan Mandel is a Czech-American mathematician. He received his PhD from the faculty of mathematics and physics, Charles University in Prague and was a senior research scientist there. Since 1986, he is professor of mathematics at the University of Colorado Denver. Since 2013, he is senior scientist at the Institute of Computer Science of the Czech Academy of Sciences.
Jan Mandel
Jan Mandel at the Prague Computer Science seminar, May 2015.
OccupationMathematician
Known forBalancing domain decomposition, WRF-Fire
He has worked in the field of multigrid methods and domain decomposition methods. He developed the balancing domain decomposition method and, with coauthors, published the convergence proofs of the FETI, FETI-DP, and BDDC methods, and the proof of the equivalence of the FETI-DP and the BDDC methods. He has been involved in the field of dynamic data driven application systems and data assimilation with applications in wildfire and epidemic modeling. He has contributed to the WRF-Fire software.
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• Jan Mandel at the Mathematics Genealogy Project
• Home page at University of Colorado Denver
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| Wikipedia |
No-Cloning and Uncertainty: Connections or Misconception
In chapter 9 of Scott Aaronson's book "Quantum Computing Since Democritus", he make interesting but peculiar claims relating the no-cloning theorem and the Heisenberg Uncertainty Principle (HUP). Here is his statement:
"On the one hand, if you could measure all properties of a quantum state to unlimited accuracy, then you could produce arbitrarily-accurate clones. On the other hand, if you could copy a state $|\psi\rangle$ an unlimited number of times, then you could learn all its properties to arbitrary accuracy."
In his interpretation, the first sentence proves that no-cloning theorem implies HUP, while the second sentence proves that HUP implies no-cloning theorem. So he is suggesting that the no-cloning theorem is EQUIVALENT to HUP.
However, I cannot feel comfortable about this statement. There are two questions that I have in mind:
To my understanding, HUP is the claim that a quantum state cannot be a simultaneous eigenstate of momentum and position operators. Therefore, suppose I can clone two copies of a quantum state, I could collapse copy 1 to a position eigenstate, and copy 2 to a momentum eigenstate, and never collapse either copy into a simultaneous eigenstate, therefore never violating the HUP. So is Aaronson wrong on his arguments here?
I am a little unsure about the assumptions that go into the proofs of no-cloning theorem and HUP. To my understanding, No-Cloning follows from Linearity and Unitarity of quantum operators, while HUP follows from unitarity and non-commutativity of quantum operators. My question, which might be stupid, is this: Is it possible to derive a generalized HUP from a nonlinear modification of quantum theory (Invent some nonlinear observables that is neither position nor momentum)? If so, I believe that no-cloning theorem would really have nothing to do with HUP whatsoever.
Any comments or insights about the connection or implication of no-cloning and HUP are also welcomed!
quantum-mechanics
heisenberg-uncertainty-principle
non-linear-systems
Zhengyan ShiZhengyan Shi
If you go through the proof of the no-cloning theorem (see e.g. wikipedia) you will notice that only the following properties are used:
Unitarity of the evolution operator
Hilbert space axioms for a composite state $|\phi \rangle | 0 \rangle$ (which imply the Cauchy-Schwartz inequality)
The no-cloning theorem than says that no unitary operator exists that does the transformation $|\phi \rangle | 0 \rangle \rightarrow |\phi \rangle | \phi \rangle$ for arbitrary $|\phi \rangle$.
The Heisenberg uncertainty principle on the other hand is a statement about how the uncertainties for two different operators on the Hilbert space are related, namely that they can not simultaneously be small for non-commuting operators.
But for the no-cloning theorem you didn't even need observables, it is a property of the Hilbert space and the unitary time evolution in Quantum Mechanics alone.
Summary: They are completely different things and not related.
answered Jul 8, 2016 at 15:38
WolpertingerWolpertinger
$\begingroup$ Hi Numrok, thanks for the answer. I have read through the proofs before. I lean toward your conclusion but not the way you arrived at it. Just because HUP and No-Cloning Theorem involve different mathematical objects, we cannot conclude that they are not equivalent. If they are derivable from each other, then they are equivalent! My question is, do HUP and No-Cloning Theorem make the same assumption about quantum mechanical axioms? Or is it that No-Cloning Theorem requires linearity of all quantum operators while HUP doesn't? $\endgroup$
– Zhengyan Shi
$\begingroup$ @ZhengyanShi In Quantum Mechanics we define certain operators on the Hilbert space, which are the observables. We also have the Schrödinger equation which is tells you how things evolve in time and it is unitary. So in QM we have both no-cloning and HUP. But if you want to make a theory with only commuting observables you don't get HUP (you can still have Schrödinger equation) and if you want to make a theory with no unitary time evolution (which doesn't really make sense, but why not), you can do that too. So how would one imply the other? I think my argument in the answer is complete $\endgroup$
– Wolpertinger
$\begingroup$ How about a theory with unitary evolution operators and nonlinear, non-commuting observables? Do we still have HUP in that case? $\endgroup$
$\begingroup$ You probably still have HUP (see e.g. physics.stackexchange.com/questions/247394/… for HUP on unusual operators). I don't see what this has to do with the question though. My point is that you neither have $HUP \Rightarrow No-cloning$ nor $No-Cloning \Rightarrow HUP$, which I thought was the question and which I demonstrated in my answer why it is the case. Of course they both follow from the axioms of Quantum mechanics, but that does not make one imply the other. You need to invoke the other axioms. $\endgroup$
$\begingroup$ @ Zhengyan Shi About the conclusion of Numrok to say they are not related, I may hold my opinion. From a geometrical point of view, the uncertainty relationship is determined by the Kahler structure of the Hilbert space, where the fibre bundle is determined by the unitary operation assumption. So they ARE REALLY closely related. If we eliminate the linear unitary operation assumption, then the geometrical structure will be different and then the uncertainty relation will not hold any more. This is exactly what udrv showed above. May this paper can help (arxiv.org/abs/1503.00238). $\endgroup$
– XXDD
Aaronson's claims are true, but your statement about what he means is not correct.
If cloning were possible, the HUP would still exist but would pose a non-absolute limit on how much you could learn about a single copy of an unknown quantum state- that is, any limit on how much you learned about a state could be attributed to not doing a very good measurement.
On the other hand, with no-cloning the HUP provides a fundamental limit on how much you can learn about a single copy of an unknown quantum state.
So they are certainly not identical. Maybe a better way to think about them is that they act in concert to limit the amount of information one may extract from a general state.
There may be a deeper way of looking at the connection between them. What Numrok says about the structural difference between the two statements is true, but it is also true that it is difficult to modify just one part of quantum mechanics. There is another nice paper by Aaronson that discusses this point (1). I think it is possible that there is a deeper reason that the structure of quantum theories fits together so neatly, which we do not fully understand as of yet.
RococoRococo
$\begingroup$ +1 on this one. My answer is possibly is possibly only scratching the surface. $\endgroup$
$\begingroup$ Thanks for the paper, the abstract seems very interesting! "Act in concert to limit the amount of information" is a cool perspective, although I don't quite understand it yet... Could you clarify your first point a bit more? Why is it that "if cloning were possible, the HUP would... pose a non-absolute limit on how much you could learn about a single copy of an unknown quantum state" ? Are you just saying that you produce arbitrarily many copies, measure them all, and map out the original wavefunctions to arbitrary accuracy? $\endgroup$
$\begingroup$ @ZhengyanShi "Are you just saying that you produce arbitrarily many copies, measure them all, and map out the original wavefunctions to arbitrary accuracy?" Yes, that's all that I (and presumably Aaronson) mean. You can already do this if you can prepare many copies of a given quantum state, of course, but cloning would allow you to do this for any state. $\endgroup$
– Rococo
$\begingroup$ Ok. I think things are much more clear now. Thank you!! $\endgroup$
$\begingroup$ I checked Scott Aaronson's book Quantum Computing Since Democritus, p.126. He did say "I claim that the No-Cloning Theorem basically implies the Uncertainty Principle and vice versa." $\endgroup$
– Hans
Wrote this to address the 2nd question in the OP: "Is it possible to derive a generalized HUP from a nonlinear modification of quantum theory (Invent some nonlinear observables that is neither position nor momentum)?", following the discussion on the topic in comments. Although it is not a clear-cut answer, perhaps it may help.
But first a couple of remarks adding to the other answers:
It is no simple historical coincidence that the HUP was discovered first as essential to the initial development of quantum theory, but not the no-cloning theorem.
There may be many corollaries to the no-cloning theorem, but the main one, which brought it to light in the first place, is that it forbids a faster-than-light communication loophole through an exploitation of entanglement. So regardless of whether the existence of cloning would allow or not some sort of by-pass on the limits set by the HUP, it would definitely raise the much more troubling issue of open conflict with relativity. HUP does nothing of the sort, to the contrary. Hence again, no, HUP cannot be equivalent to the no-cloning theorem.
On the issue of nonlinear observables:
We already know of some useful nonlinear observables, see the entropy $-k_B \text{Tr}(\rho \ln\rho)$ and related entropic UPs, hence the question is appropriate. The real problem is that the issue is quite complex. Even if the Hilbert space structure is left in place, true nonlinear observables would modify the theory drastically and the usual HUP definitely does not apply in all nonlinear cases.
This is because in general nonlinear operators can no longer be characterized in terms of action on bases or even in terms of matrix elements. The simplest example is one that appears precisely in the algebraic proof of the HUP. If $A$ is an arbitrary linear and hermitic operator, define a nonlinear application ${\mathcal \not A}$ through $$ {\mathcal \not A}|\psi\rangle = A|\psi\rangle - \frac{\langle \psi |A|\psi\rangle}{\langle \psi |\psi\rangle} |\psi\rangle $$ Application ${\mathcal \not A}$ is homogeneous, $$ {\mathcal \not A}|a\psi\rangle = a {\mathcal \not A} |\psi\rangle $$ but not linear, $$ {\mathcal \not A}|(a\psi + b\phi)\rangle \neq a {\mathcal \not A} |\psi\rangle + b {\mathcal \not A} |\phi\rangle $$ and therefore has some unusual properties:
Its action on any eigenvector $|\lambda\rangle$ of $A$ vanishes identically: $$ A|\lambda\rangle = \lambda |\lambda\rangle \;\;\;\Rightarrow \;\;\; {\mathcal \not A}|\lambda\rangle = A|\lambda\rangle - \frac{\langle \lambda |A|\lambda\rangle}{\langle \lambda|\lambda\rangle} |\lambda\rangle = 0 $$ An entire basis set is in its kernel, but ${\mathcal \not A}$ is not a null application!
Moreover, its average also vanishes on any $|\psi\rangle$, $$ \langle \psi | {\mathcal \not A}|\psi\rangle = \langle \psi |A|\psi\rangle - \frac{\langle \psi |A|\psi\rangle}{\langle \psi |\psi\rangle} \langle \psi |\psi\rangle = 0 $$
In other words, many of the tools used liberally with linear observables fly out the window even for this modest example.
In particular, the action of observables, and operators at large, can no longer be defined in terms of action on a basis set, but has to be defined for each state vector individually. Unfortunately when basis sets become insufficient for characterization, so do matrix representations. And once this happens, the concepts of hermitian conjugate, self-adjoint observable, and eigenbasis, all loose their celebrated significance.
Suppose though that observable averages would still be given by real diagonal matrix elements $\langle \psi |{\mathcal O}|\psi\rangle \in {\mathbb R}$, a condition that already sets boundaries on the set of acceptable nonlinear applications $\mathcal O$. Then here is a simple example of nonlinear "observables" $A$, $B$ that break the HUP as applied in the usual form $$ \langle \psi |(\Delta A)^2|\psi\rangle \langle \psi |(\Delta B)^2|\psi\rangle \ge |\langle \psi |\frac{1}{2i}[A,B]|\psi\rangle|^2 $$ Let $A$, $B$ be such that $\langle \psi |A|\psi\rangle \in {\mathbb R}$, $\langle \psi |B|\psi\rangle \in {\mathbb R}$ for any $|\psi\rangle$, and in addition such that for some given $|\psi_0\rangle$ $$ A|\psi_0\rangle = 0 \;\;\; \Rightarrow \;\;\; \langle \psi_0 |A|\psi_0 \rangle = 0 $$ $$ B|\psi_0\rangle = a |\psi_0\rangle \neq 0 \;\;\; \Rightarrow \;\;\; \langle \psi_0 |B|\psi_0 \rangle = a = a^* \neq 0 $$ and for any $|\psi_\bot\rangle$, $\langle\psi_\bot|\psi_0\rangle = 0$, $$ A|\psi_\bot \rangle = |\psi_0\rangle \;\;\; \Rightarrow \;\;\; \langle \psi_\bot |A|\psi_\bot \rangle = 0 \\ B|\psi_\bot \rangle = b |\psi_\bot \rangle + c |\psi_0\rangle \;\;\; \Rightarrow \;\;\; \langle \psi_\bot |B|\psi_\bot \rangle = b = b^* \neq 0 \\ A|b\psi_\bot + c\psi_0\rangle = |\psi_\bot\rangle \;\;\; \Rightarrow \;\;\; \langle b\psi_\bot + c \psi_0 |A| b\psi_\bot + c\psi_0 \rangle = b \langle \psi_\bot | \psi_\bot \rangle $$ Then we have that for any $|\psi_\bot\rangle$, $\langle\psi_\bot|\psi_\bot\rangle = 1$, $\langle\psi_\bot|\psi_0\rangle = 0$, $$ \langle \psi_\bot |(\Delta A)^2|\psi_\bot \rangle = \langle\psi_\bot |A^2|\psi_\bot \rangle - \langle\psi_\bot |A|\psi_\bot \rangle^2 = 0 $$ but $$ |\langle \psi |[A,B]|\psi\rangle|^2 = |\langle\psi_\bot |AB|\psi_\bot \rangle - \langle\psi_\bot |BA|\psi_\bot \rangle|^2 = |\langle\psi_\bot |A(b\psi_\bot + c\psi_0)\rangle - \langle\psi_\bot |B|\psi_0 \rangle|^2 = $$ $$ = |\langle\psi_\bot |A(b\psi_\bot + c\psi_0)\rangle - a \langle\psi_\bot |\psi_0 \rangle|^2 = |\langle\psi_\bot |A(b\psi_\bot + c\psi_0)\rangle|^2 = |\langle \psi_\bot |\psi_\bot \rangle|^2 =1 $$ and so $$ \langle \psi |(\Delta A)^2|\psi\rangle \langle \psi |(\Delta B)^2|\psi\rangle 0 \le |\langle \psi |\frac{1}{2i}[A,B]|\psi\rangle|^2 = \frac{1}{4} \;\;\;!! $$
udrvudrv
Heisenberg's uncertainty principle for mean deviation?
Exploiting the Heisenberg Uncertainty Principle as a means to communicate
More Heisenberg Uncertainty Principle (HUP) Clarification
What are the practical consequences of "approximate" quantum cloning with a stimulated emission cloning machine?
Thought experiment about no-cloning theorem and FTL information
Issues with the proof of the no-cloning theorem
The 'Conservation of Information' myth
Could a non-unitary time evolution violate the no-cloning theorem?
Why doesn't this copy circuit violate the no cloning theorem?
How is many world interpretation of quantum mechanics compatible with no cloning theorem?
Why does quantum mechanics "collapse" without the uncertainty principle? | CommonCrawl |
\begin{document}
\begin{center} {\large\bf Unconditional Security Of Quantum Key Distribution Over Arbitrarily Long Distances\footnote{This reprint version contains the same material as the one published in {\it Science} {\bf 283}, 2050--2056 (1999). We also include the refereed supplementary notes (as in {\tt http://www.sciencemag.org/feature/data/984035.shl}) explicitly in the appendix for easy reference.}} \\ ~~~\\ Hoi-Kwong Lo$^1$\footnote{e-mail: [email protected]} and H. F. Chau,$^2$\footnote{e-mail: [email protected]}\\ ~~~\\ $^1$ Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol
BS34 8QZ, UK.\\ $^2$ Department of Physics, University of Hong Kong, Pokfulam Road,
Hong Kong, China\\ ~~\\ (\today) \end{center} \begin{abstract} Quantum key distribution is widely thought to offer unconditional security in communication between two users. Unfortunately, a widely accepted proof of its security in the presence of source, device and channel noises has been missing. This long-standing problem is solved here by showing that, given fault-tolerant quantum computers, quantum key distribution over an arbitrarily long distance of a realistic noisy channel can be made unconditionally secure. The proof is reduced from a noisy quantum scheme to a noiseless quantum scheme and then from a noiseless quantum scheme to a noiseless classical scheme, which can then be tackled by classical probability theory. \end{abstract}
\widetext
The art of secure communication --- cryptography --- has a long history. Before two parties can communicate securely, they often must share a secret random string of numbers (a key) for encryption and decryption. The secrecy of the message depends on the secrecy of the key. A problem in conventional cryptography is the key distribution problem: In classical physics, there is nothing to prevent an eavesdropper from monitoring the key distribution channel passively, without being caught by the legitimate users.
Quantum key distribution (QKD) (1-5) has been proposed as a solution to the problem. The quantum no-cloning theorem states that it is impossible to make an exact copy of an unknown quantum state (6). Thus, it is generally thought that eavesdropping on a quantum channel will almost surely produce detectable disturbances. The two users can, therefore, use part of their quantum signals to test for eavesdropping. Only when the error rates are acceptable will they use the quantum signals to generate a key. Thus, the two users (commonly called Alice and Bob) have the confidence that if an eavesdropper (commonly called Eve) has a nonnegligible amount of information on the final key, she will almost surely be caught, even if she has infinite computing power and access to a quantum computer. Incidentally, several recent experiments have demonstrated the feasibility of QKD over tens of kilometers (7).
``The most important question in quantum cryptography is to determine how secure it really is '' (8, p.16). QKD is widely claimed to provide perfect security. However, this viewpoint has been under renewed scrutiny for two reasons. First, contrary to well-known claims of unconditional security (9), a class of other quantum cryptographic schemes, including so-called quantum bit commitment and quantum one-out-of-two oblivious transfer, has recently been shown to be insecure (10) . Cheaters can defeat these schemes by a subtle application of the well-known Einstein-Podolsky-Rosen (EPR) paradox (11) and by delaying their measurements. These ``no-go'' theorems not only shattered the long-standing belief in the security of those schemes, but they also undermined the confidence in QKD itself. Second, a convincing and rigorous proof of the security of QKD has been missing despite extensive investigations (12-15). Thus, the foundation of quantum cryptography has been shaky. Here, we solve this long-standing problem by proving that, given quantum computers, QKD can be made unconditionally secure over arbitrarily long distances.
A rigorous proof of the security of a QKD scheme requires the explicit construction of a procedure such that, whenever Eve's strategy has a nonnegligible probability of passing the verification test by Alice and Bob, her information on the final key will be exponentially small (16-17). This procedure must be secure and efficient, even when Alice and Bob use imperfect sources and devices and share a noisy quantum channel.
Most analyses on the security of QKD have dealt with single-particle eavesdropping strategies (12), with immediate or delayed measurements, as well as the so-called collective attacks (13), in which Eve brings each signal particle into interaction with a separate probe system but then, after hearing the public discussion between Alice and Bob, measures all probes together. Security against the most general type of attack, the so-called joint attack, has been investigated by Deutsch {\it et al.} and also by Mayers. The discussion by Deutsch {\it et al.} was restricted to the special case of perfect devices (14). It introduced the concept of quantum privacy amplification, based on a process called entanglement purification, which was studied by Bennett, DiVincenzo, Smolin and Wootters (BDSW) (18). Earlier versions of Mayers' proof (15) have not been accepted as definitive. His most recent version of the proof is more detailed and complex (19). He proposes a proof of security of the Bennett and Brassard (BB84) (2) scheme against joint attacks in the presence of detector and channel noise but with an ideal trusted single-photon source. Our current work and work by Mayers (19) are contemporaneous and independent. They differ greatly in their premises, methods and consequences: (i) Mayers' work deals with the standard BB84 QKD protocol (2) for preparation, transmission, and measurements of nonorthogonal states. His approach does not require Alice and Bob to have a quantum computer, although Eve may have one. In contrast, our proof applies to a new QKD protocol, involving fault-tolerant sharing and purification of so-called EPR pairs, and requires that Alice and Bob have quantum computers. (ii) Mayers' work (19) assumes an ideal single-photon or EPR-pair source, thus disallowing a beam-splitter attack. [A testing procedure for an allegedly ideal EPR-pair source from an untrusted vendor has recently been suggested by Mayers and Yao (20).] In contrast, our work allows the reception of untrusted imperfect quantum signals from the channel (21). (iii) Our proof and protocol allow QKD to be securely extended over arbitrarily large distances through a chain of insecure relay stations. A similar extension of BB84 in Mayers' proposed proof would require secure relay stations, to which Eve does not have access. (iv) Our proof is conceptually simpler. (v) Our techniques have widespread applications outside QKD.
Why is a proof of security of QKD so difficult? In a joint attack, Eve treats the whole sequence of quantum signals as a single entity. She couples this entity with her probe and then unitarily evolves the combined system. She forwards a subsystem to Bob and keeps the remaining subsystem for eavesdropping purposes. Eve can use any unitary transformation she likes, and yet, a secure QKD scheme must defeat all of them. Moreover, Eve may attempt to mask her presence by attributing the errors caused by her eavesdropping attack to normal transmission noise. Furthermore, because the particles are now generally entangled with each other, a naive application of classical probability theory may lead to fallacies [See the EPR paradox (11)].
Despite these apparent difficulties,
we show that it is possible to distinguish a malicious Eve from noise. Moreover, it is possible to use classical probability theory to establish the security of QKD.
{\bf Techniques and importance of results.} Assuming that users have access to quantum computers, we show the security of QKD by a reduction in two steps. The central theme of the first step is to reduce the noisy quantum scheme (imperfect devices, noisy channels, storage errors, and so forth) to a noiseless quantum scheme. We do this by combining the ideas of ``quantum repeaters'' (22,23) and fault-tolerant quantum computation (FTQC) (24,25). Although these are existing ideas in the field, we make the nontrivial observation that they can be combined and applied to QKD to distinguish noise from a malicious Eve. In particular, we note that knowing the error syndrome does not help an eavesdropper. Therefore, we can give an eavesdropper full control of the quantum repeater stations without compromising security.
Even in a noiseless quantum scheme, Alice and Bob are required to verify that the particles are untampered by Eve. Things will be easy if one can apply classical arguments to solve this quantum problem at hand. However, as illustrated by the EPR paradox, naive classical arguments often lead to fallacies. The most important technical contribution of this paper is our second theme --- reducing the noiseless quantum verification scheme to a classical one. Finally, we establish the security of the classical verification scheme by classical probability theory. The security of the quantum scheme then follows.
The use of classical arguments in our quantum problem allows us to simplify our discussion greatly. We emphasize that the validity of this usage is highly paradoxical. Classical arguments work in our quantum problem because all the observables $O_i$'s under consideration are diagonal with respect to a single basis $\cal B$. In more detail, let us consider the observable $M$, which represents a complete von Neumann measurement along the same basis $\cal B$. Because all of the $O_i$'s under consideration are diagonal with respect to $\cal B$, $M$ commutes with all the observables $O_i$'s. Therefore, the measurement $M$ along basis $\cal B$ will in no way change the outcome of the subsequent $O_i$'s. Without any loss of generality, one can imagine that such a complete von Neumann measurement $M$ is always performed before the measurement of subsequent $O_i$'s. In other words, the initial state of the quantum system is simply a classical mixture of eigenstates of $M$, and hence, classical arguments carry over to the quantum case. We remark that the $O_i$'s that we consider are coarse-grained observables (observables with degenerate eigenvalues) rather than fine-grained observables (observables with nondegenerate eigenvalues).
{\bf Quantum-computational protocols.} The execution of our secure QKD scheme requires large-scale quantum computers for both error correction and verification. Building such computers is a technological feat that is far beyond our current technology. However, all existing QKD security analyses require some idealization also. In an actual experimental implementation of polarization-coding BB84 (a standard ``prepare-and-measure'' P/M scheme) over a substantial distance (say 40km) of a lossy quantum channel using coherent states, Eve may, in principle, break the system completely by a generalized beam-splitting attack (26). This is so even when the bit error rate of the quantum signals is strictly zero.
Quantum-computational protocols like ours are worthy of analysis for several reasons. First, unlike the usual P/M schemes, they extend the range of secure QKD to arbitrarily long distances even with insecure quantum repeaters. Second, when implemented over a noisy channel without repeaters, it is conceivable that they can tolerate a higher noise level than a standard P/M scheme. Third, a proof of security and the tradeoff between noise and key rate are much easier than those for P/M schemes. Indeed, our scheme provides a conceptually simple and rigorous proof of the security of QKD without the full complexity of a P/M scheme.
{\bf EPR pairs.}
Before we report our QKD scheme in detail, let us first recapitulate the usefulness of an EPR pair, that is, a singlet state ${ 1 \over \sqrt{2}} (|\uparrow
\downarrow\rangle - | \downarrow\uparrow\rangle)$ of a pair of quantum bits (qubits) (27) in QKD. If two members of an EPR pair are measured along any common axis, each member will give a random outcome, and yet, the outcomes of the two members will always be antiparallel. This spooky action at a distance defies any simple classical explanation and is at the core of EPR paradox (11).
Now, suppose two distant users share $R$ EPR pairs. Then, the random outcome of measurement along a common axis generates an $R$-bit key between them. The laws of quantum physics assure that the key is truly random and that Eve cannot have any information on its value. Indeed, the two lemmas in supplementary material (available at www.sciencemag.org/feature/data/984035.shl ) [that is, Supplementary Note 2 in this reprint version] show that, to generate an almost perfectly secure $R$-bit key, Alice and Bob only need to share $R$ EPR pairs of almost perfect fidelity (28). Therefore, all we need for secure QKD is a way for Alice and Bob to share EPR pairs and to verify that, indeed, they are EPR pairs. We focus on these EPR distribution and verification problems. There are two issues that Alice and Bob have to address: noise and Eve.
{\bf Reduction to a noiseless scheme.} One can classify errors in an EPR-pair distribution process into four types. First, the quantum communication channel between Alice and Bob is generally noisy. Second, the EPR source may be imperfect in itself. Third, errors may occur during the storage of quantum information. Fourth, errors may also occur during computation; because elementary gates and measuring devices for quantum computation are generally imperfect, gate errors and measurement errors may arise.
The last three types of errors can be fixed by recently developed quantum error correction (29) and fault-tolerant quantum computation (FTQC) (24,25) techniques. In particular, there is a ``threshold result'' in FTQC: Assuming an independent noise model and that the error rates for each primitive computational gate and for each time step of storage are smaller than some positive threshold values, one can perform arbitrarily long quantum computations with an arbitrarily high fidelity (25). The essence of FTQC is to defeat errors by encoding quantum states in quantum error-correcting codes (QECC) and then performing quantum computation on the encoded states.
{\bf Quantum repeaters.} We must consider the first type of noise --- channel noise. If the quantum communication channel is very noisy (for example, it is very long), we cannot apply the threshold result in FTQC to combat quantum communication errors. Fortunately, the idea of quantum repeaters has been proposed as a much more efficient way of correcting quantum communication errors (23). The idea is summarized as follows .
Given impure EPR pairs shared between two distant observers, they can apply local operations and classical communication to distill a smaller number of higher fidelity EPR pairs in a procedure known as entanglement purification (18). However, for distances much longer than the coherence length of a noisy quantum communication channel, the probability that a quantum state will remain error-free is exponentially small. Therefore, the fidelity of transmission is so low that standard purification methods are not applicable.
Quantum repeaters are essentially simple quantum computers installed throughout a quantum communication channel. They are used to divide the channel into shorter segments, which are then purified separately, before they are connected. The number and locations of quantum repeaters are chosen so that it is possible to create EPR pairs with sufficiently high fidelity between the two ends of each segment. After creating EPR pairs that are shared between the two ends, one applies entanglement purification by using quantum repeaters. This will, at the cost of discarding some pairs, increase the fidelity of the remaining pairs. Afterwards, EPR pairs shared between various segments are connected together by ``quantum teleportation'' (30). Indeed, a highly efficient procedure involving a sequence of entanglement purification and teleportation has been devised that allows the reliable sharing of EPR pairs between two arbitrarily distant locations (23).
Three important remarks are in order. First, with two-way classical communication, quantum repeaters can greatly improve the yield of distillation (18,23) over the standard fault-tolerant circuits. Second, even highly imperfect quantum repeaters can do the job very well: It has been argued (23) that an error rate in the percent level is readily tolerable. Third, a strength of our approach is that, assuming perfectly reliable local quantum operations by Alice and Bob, one can actually calculate the threshold value for tolerable noise between two adjacent quantum repeaters. For example, in the case of a depolarizing channel, it is known that a fidelity of $1/2$ is the threshold value (18).
With quantum repeaters and FTQC, the usual threshold result can be extended to distributed quantum computation over a realistic noisy quantum channel (31). In other words, any distributed quantum algorithm, including the EPR-pairs distribution process, that works in the noiseless case can always be extended to the noisy case.
{\bf Error syndrome contains no useful information for an eavesdropper.} We make the most generous assumption that Eve completely controls the quantum repeaters and the quantum communication channel. Alice and Bob need only trust their own quantum computers and authenticated classical messages from each other. There is a subtlety for us to address. If Eve follows the correct procedure, she will not be caught. However, she does learn about the error syndrome (that is, the pattern of measurement results) generated during a FTQC, which allows the error-correction apparatus to correct a corrupted state to a former value. So, the question is Can the error syndrome tell her anything useful about the state? The answer is``no" because of the following.
Mathematically, each of Alice and Bob's state can be written as a tensor product of the logical qubits (which actually contain the quantum information) and the ancillary qubits (which contain the error syndromes) (32); that is, the wave function
$| \Psi \rangle$ can be written as $ \sum_{i,j} c_{ij} |a_i \rangle_L \otimes |e_j \rangle_A $, where subscripts $L$, $A$, $i$ and $j$ represent the logical qubits, the ancillary qubits, the logical state (in some orthonormal basis), and the error syndrome, respectively, and $c_{ij}$'s are some complex coefficients. (In reality, Alice and Bob's system is generally entangled with Eve's probe. However, this does not change the essential point of our argument.) Because of FTQC, although the state of the ancillary qubits evolves unpredictably over time, the state of the logical qubits will, with a very high fidelity [$1 - O (e^{-\ell})$ for some arbitrarily chosen $\ell> 0$ (33)], follow the desired computation and remain unaffected by the errors. As long as the gate error rate and storage error rate are sufficiently small, the subsequent verification and key generation steps can be thought of as being performed solely on the logical qubits. In other words, the ancillary qubits decouple from the verification step. Accordingly, we shall ignore the ancillary qubits and focus only on the logical qubits.
If there is no appreciable eavesdropping, the logical qubits will represent the desired state. Of course, an honest Eve can learn as much as she likes about the error syndrome. The general theory of QECC tells us that the error syndrome contains absolutely no information about the encoded quantum state (29).
In summary, we have reduced the proof of security of our noisy QKD (or EPR delivery) scheme to that of a noiseless one. Now, we focus on the noiseless scheme.
{\bf The goal of verification.} To make sure that there is no substantial eavesdropping, Alice and Bob must verify that the state of the logical qubits is, indeed, that of $N$ EPR pairs. Without any loss of generality, we can allow Eve to not merely act on the $N$ EPR pairs while they are being shared but to actually prepare them in an arbitrary state of her choosing and then give them to Alice and Bob (14). She claims that they are perfect EPR pairs. Alice and Bob will be happy to sacrifice a small number $m$ of those pairs to verify Eve's claim. If any one of the $m$ tested pairs fails the test, then all the $N$ pairs are discarded. However, if all the $m$ pairs pass the test, the remaining $N-m$ pairs will be accepted as singlets and used to generate the key. The goal of the verification is for Alice and Bob to make sure that Eve has a very small probability of cheating successfully. By cheating successfully, we mean that the $m$ tested pairs pass the verification test and yet the remaining $N-m$ pairs, if given a yes or no test of being $N-m$ singlets, will give ``no'' as an answer (34).
The security of our quantum verification scheme will automatically guarantee the security of the corresponding QKD scheme [refer to (28) for an explicit bound on Eve's information].
We will now consider the security of our quantum verification scheme.
Essentially, what Alice and Bob are trying to do is to distinguish singlets from triplets. Although there is no way for Alice and Bob to do so with certainty using only local operations and classical communication, they can do so with a very high probability.
The goal of a quantum verification scheme is to verify that the state of the $N$ pairs is, in fact, $N$ singlet. A direct testing of a random subset of EPR pairs requires an exponential amount of resources in term of the security parameter $k$, where the probability for Eve to cheat successfully is, at most, $e^{-k}$. Direct testing of a random subset is, therefore, not an efficient verification scheme. To understand this point, suppose Eve cheats by inserting a single nonsinglet among the $N$ pairs and only $m$ random pairs are tested by Alice and Bob. There is a probability $N-m \over N$ for this nonsinglet to remain untested. Consequently, Eve has at least a probability $N-m \over N$ of cheating successfully. To prevent this from happening, it is necessary for $m$ to be of order $N$. Even when $m$ equals $N-1$, the probability for Eve to cheat successfully is still at least $1/N$. For this to be exponentially small in $k$, the number of photons transmitted $N$, must be exponentially large in $k$.
A much more efficient way of verifying a quantum state exists. It is due to the random-hashing idea by BDSW (18). (BDSW proposed it for error correction, but here we use it for verification.) It is in the same spirit of a classical random-hashing scheme which we will now describe.
{\bf A classical verification scheme.} Imagine a game in which Eve locks an $N$-bit string $x$ in a box and Alice and Bob are allowed to ask a small number $m < N$ of ``fair'' questions about it, which Eve must answer truthfully. A fair question is a yes or no question whose answer is "yes" for exactly half of all $N$-bit strings. Thus, Is the first bit 1? and Are the first and third bits equal? are fair questions, but Are all the bits 1's? is unfair. If all of Eve's answers are consistent with the assumption that the string $x$ is all 1's, Alice and Bob must ``accept'' the string. Otherwise, Alice and Bob must ``reject'' the string. Finally, Eve opens the box and shows Alice and Bob the string to prove that she has answered faithfully. Eve wins if the string is not all 1's and yet Alice and Bob have accepted the string. [Alice and Bob win if the string is not all 1's and they have rejected the string. If the string is, in fact, all 1's, the game is a draw.]
If Alice and Bob ask only single-digit questions of the form Is the $k$th bit a 1? then Eve has a good chance of winning by choosing a string with a single $0$ at a random location. However, if Alice and Bob instead ask Eve about the parities of random subsets of the bits, they quite likely catch any string that is not all 1s.
For example, if the unknown string is $x= 1101$ and Alice and Bob choose a subset consisting of the second and third bits (this can be represented conveniently by an index string $s = 0110$), the parity $x \cdot s$ is $1$. This test reveals that $x$ is not all 1's, since an all-1's four-bit string would have had parity $0$ on this subset. More generally, the parity of a subset $s$ of the bits in a string $x$ is the inner product, or modulo-2 sum of the bit-wise AND of strings $x$ and $s$, and it is denoted by $x \cdot s$. [In this example, $x \cdot s = 1 \cdot 0 + 1 \cdot 1 + 0 \cdot 1 + 1 \cdot 0 = 1 \pmod 2$.] The probability that two different strings give the same answer for $m$ iterations of random parity check is no more than $2^{-m}$ (18). Thus, by checking only a few subset parities (say $20$), Alice and Bob can reduce their chance of accepting an $x$ that is not all 1s to less than one in a million.
Eve must not know the index strings beforehand. Otherwise, she could always cheat successfully, in a similar way as a smuggler who knows beforehand which of the several bags a customs inspector will open in an airport. Indeed, because the string has $N$ bits and there are only $m$ constraints (generated by $m$ rounds of parity verification), there are clearly exponentially many ( namely, $2^{N-m}$), strings that will pass the test. However, because Eve does not know the index strings beforehand and because the index strings are chosen randomly, Eve effectively has to put her bet on a single string without prior knowledge. We see from the last paragraph that any string $x \not= 11 \cdots 1$ chosen by Eve has only an exponentially small probability ($2^{-m}$) of passing the verification test.
{\bf Our quantum verification scheme.} Now, we construct an efficient quantum verification scheme that is similar to the classical verification scheme that we have just described. Consider the so-called Bell basis, $\Psi^{\pm} $ and $\Phi^{\pm}$, where \begin{equation} \Psi^{\pm}= { 1 \over \sqrt{2}}
( | \uparrow \downarrow \rangle \pm | \downarrow \uparrow \rangle) \end{equation} and \begin{equation} \Phi^{\pm}= { 1 \over \sqrt{2}}
( | \uparrow \uparrow \rangle \pm | \downarrow \downarrow \rangle). \end{equation} With the convention in (18), Bell basis vectors are represented by two classical bits \begin{eqnarray} \Phi^+ & =& \tilde{0}\tilde{0}, \nonumber \\ \Psi^+ & =& \tilde{0}\tilde{1}, \nonumber \\ \Phi^- & =& \tilde{1}\tilde{0}, \nonumber \\ \Psi^- & =& \tilde{1}\tilde{1}. \label{bellstate} \end{eqnarray} \noindent (Because Bell basis vectors are maximally entangled, one should never think of them as direct product states.) A complete basis for $N$-ordered pairs of qubits (what we shall call $N$-bell basis) consists of products of Bell basis vectors, each of which is described by a $2N$-bit string. In the absence of an eavesdropper, Alice and Bob share $N$ singlets, whose state is described by a $2N$-bit string of $\tilde{1}$'s,
$| \tilde{1}\tilde{1} \cdots \tilde{1} \rangle$.
What happens when there is an eavesdropper? Recall that we allow Eve to not merely act on the $N$ EPR pairs while they are being shared, but to actually prepare them in an arbitrary state of her choosing and then give them to Alice and Bob. The pairs may be entangled among themselves as well as with a probe in Eve's hands (14). A system described by any mixed state can be equivalently described by a pure state of a larger system consisting of the original system and an ancilla (10,14). As discussed by Deutsch {\it et al.} (14), by considering the larger system instead, we shall, without a loss of generality, consider that Eve prepares a pure state \begin{equation}
| u \rangle = \sum_{i_1, i_2 , \cdots, i_N} \sum_j
\alpha_{i_1, i_2 , \cdots, i_N,j} | i_1, i_2 , \cdots, i_N \rangle
\otimes | j \rangle , \label{two0} \end{equation} where $i_k$ denotes the state of the $k$th pair, which runs from $\tilde{0} \tilde{0}$ to $\tilde{1}\tilde{1}$, $\alpha_{i_1, i_2 , \cdots, i_N,j} $'s are some complex coefficients, and the
$| j \rangle$ values form an orthonormal basis for the ancilla. Each state $| u \rangle$ represents a particular cheating strategy chosen by Eve.
The goal of a quantum verification scheme is to verify that the string describing the state of the $N$ pairs is, in fact, all $\tilde{1}$'s. We now construct an efficient quantum verification scheme based on the quantum random-hashing idea by BDSW (18). BDSW showed that one can compute the parity of any subset of the $2N$-bit string by using local operations and classical communication only (35). The parity is ``collected'' into a single destination pair; it is determined by the outcomes of measurements performed on that pair, which has to be discarded afterward. More specifically, the parity is found by noting whether the measurement outcomes on the two members of the destination pair are parallel or antiparallel.
If Eve prepares a classical mixture of products of Bell states, it is not too difficult to show that classical arguments apply directly to the quantum verification problem and Eve's probability of cheating successfully is negligible [see (36) for details].
{\bf Why do classical arguments work for a quantum problem?} However, instead of preparing a classical mixture of products of Bell states, in the most general eavesdropping strategy as shown in Eq.4, Eve prepares a general state, which is entangled with her probe. The big question is Can Eve prepare a more general state to enhance her probability of cheating successfully? The crux of our paper is the following claim: If Eve prepares a general state to cheat in the BDSW random-hashing verification scheme, her probability of cheating successfully will be exactly the same as in the situation when she premeasures that state along the $N$-Bell basis before handing it over to Alice and Bob. In other words, a general state offers no advantage over a mixture of products of Bell states. With this quantum to classical reduction result, (36) applies to any eavesdropping strategy. This proves that Eve's probability of cheating successfully is negligible and our QKD scheme is secure against all possible attacks.
{\bf Proof of our claim.} Consider the following observables on a state
$| u \rangle$ of $N$ pairs of qubits shared between Alice and Bob. We define these observables by their actions on the
$2^{2N}$ $N$-Bell states, which form a complete basis. Let $W$, defined by $W |w \rangle =
w | w \rangle$, be the observable that gives the $2N$-bit string representing the state $w$ in BDSW notation. For any index string $s$, let $Q_s$, defined by
$Q_s |w \rangle = ( s \cdot w ) |w \rangle$, be the observable that gives the parity of the subset $s$ of the bits. Finally, let
$R = | \tilde{1}\tilde{1} \cdots \tilde{1} \rangle
\langle \tilde{1}\tilde{1} \cdots \tilde{1} |$ be the projector onto a state of $N$ singlets. All the above operators refer to a single basis (namely, the $N$-Bell basis). Because all the observables ($R$, $W$ and $Q_s$) are simultaneously diagonalizable with respect to the $N$-Bell basis,
$R$ and all the $Q_s$ values commute with $W$. Therefore, neither the value of $R$ nor any of the $Q_s$ values are affected by a prior measurement of $W$. In other words, for any state $| u \rangle$ that Eve might have supplied, neither the sequence of subset parities measured in the verification stage nor the result of the final hypothetical measurement of $R$ would have been affected if Eve had pre-measured $| u \rangle$ in Bell basis (that is, made a measurement of $W$) before handing the state to Alice and Bob.
Incidentally, the fact that a premeasurement does not change the outcomes of some subsequent measurements is highly reminiscent of work by Griffiths and Niu (37).
{\bf Subtleties in our proof.} The following example illustrates the computation of parities and the subtleties involved. Suppose Alice and Bob share three pairs of qubits. With the procedure specified in BDSW, the computation of the parity of the first subset (for example, $s_1= 001101$) can be done by the circuit diagram shown in Fig.~1A. The parity is collected into a single pair and is determined by whether that pair gives a parallel or antiparallel outcome when both members are measured along the $z$ axis.
The computation itself, up to phases, performs a permutation on the space of all $2N$-bit strings. After the computation, the measured pair is dropped from consideration, and only two pairs remain. The computation of the parity of the second subset (for example, $s_2= 1001$) by the BDSW procedure is shown in Fig.~1B. After the computation, another pair is measured and dropped from consideration. Therefore, only a single pair out of the original three is left after the computation of the two parities (for $s_1$ and $s_2$). A simple unitary description (10) of the overall computation is that it maps, up to phases, the state
$| \tilde{1} \tilde{1} \tilde{1} \tilde{1} \tilde{1} \tilde{1} \rangle$ to
$| \tilde{1} \tilde{0} \tilde{1} \tilde{1} \tilde{1} \tilde{1} \rangle$. Suppose also that, on passing the verification test, Alice and Bob generate their secret key by measuring the remaining pair along the $z$ axis, with an ``up'' for Alice's result meaning ``0'' and a ``down'' meaning ``1''.
A number of subtleties deserve careful discussions. First, as in the classical case, the choice of subsets can be announced only after Alice and Bob receive all of their quantum particles. So long as Eve does not know the subsets beforehand, her probability of cheating successfully is exponentially small (see supplementary material, available at www. sciencemag.org/feature/data/984035.shl) [that is, Supplementary Note 4 in this reprint version].
Second, during the computation of the parities of subsets, the state of the $N$ pairs of qubits is transformed by a unitary transformation $U_{s_1, s_2, \cdots, s_m}$, which depends on the subsets $s_i$. But would that unitary transformation $U_{s_1, s_2, \cdots, s_m}$
somehow spoil our reduction argument from a quantum to classical verification? Fortunately, the answer is ``no". Despite the apparent complexity of the parity computation procedure, the bottom-line answer that Alice and Bob obtain is simply the parities (that is, the eigenvalues of the operators $Q_s$'s of their choice). Therefore, the verification test proves that, for any general cheating strategy by Eve that passes the test with a probability of at least $2^{-r}$, the conditional fidelity of the state before the parity computation as $N$ singlets, $|\tilde{1} \tilde{1} \cdots \tilde{1} \rangle$, is $1 - O (2^{-(m-r)})$. Consequently, the state after the parity computation will, with the same fidelity, be
$U_{s_1, s_2, \cdots, s_m} |\tilde{1} \tilde{1} \cdots \tilde{1} \rangle$.
Third, in our quantum verification procedure, Alice and Bob have to disclose all their measurement outcomes in a public channel. For each measured pair, there are four possible outcomes, ``$\uparrow \downarrow$'', ``$\downarrow \uparrow$'', ``$\uparrow \uparrow$'' and ``$\downarrow \downarrow$'', thus resulting in two bits of information. This is more than the one-bit (0 or 1) parity information. Now, the question is Can Eve somehow benefit from this additional information? The answer is ``no" (this discussion is available at www. sciencemag.org/feature/data/984035.shl) [that is, Supplementary Note 5 in this reprint version]. Finally, the issue of a quantum Trojan horse attack is addressed in (21). This completes our proof of security of QKD.
{\bf Discussion.} An important idea behind our quantum to classical reduction is that a quantum mechanical experiment has a classical interpretation whenever observables that refer to only one basis (the $N$-Bell basis in our case) are considered. The fine-grained measurement operators by Alice and Bob along the three random bases do not commute with the Bell-basis projection operators. However, Alice and Bob base their decision on whether to accept the alleged singlets not on those fine-grained measurement results but on the coarse-grained (parallel or antiparallel) ones. Those coarse-grained operators all commute with a complete von Neumann measurement along the Bell basis (38).
Our quantum to classical reduction technique is a powerful tool of widespread applications. It guarantees that one can apply standard results in the classical world (such as probability theory and statistics theory) to the original quantum problem without leading to fallacies. In effect, this means the extension of classical statistical theory to quantum mechanics, resulting in a quantum statistical theory. To illustrate this point, we give two other examples of applications of our quantum to classical reduction result . (i) Suppose two distant observers share $N$ pairs of qubits, and estimate the number of singlets in those $N$ pairs. By the number of singlets, we mean the expected number of ``yes'' answers if a singlet or triplet measurement was made on each pair individually. (ii) Under the assumption that signal carries are perfect single photons, put a probabilistic bound on an eavesdropper's information in BB84 as a function of the error rates of the sampled photons. (These examples are discussed at www. sciencemag.org/feature/data/984035.shl.) [That is, Supplementary Note 6 in this reprint version.]
The second example gives us a quantitative statement on the trade-off between information gain and disturbance (39). This is a strong result to a notoriously difficult problem because (i) the bound applies not merely to a strategy in which Eve couples a probe to each signal particle but to any information extraction strategy that is consistent with quantum mechanics and (ii) the bound can be derived by a random sampling of a small subset. In other words, a concrete experimental random-sampling procedure (rather than an abstract mathematical equation with little physical meaning) is presented here (40).
Finally, let us return to QKD itself. Although we have focused on the case when Alice and Bob receive allegedly good EPR pairs from Eve, our proof of security of QKD also applies to the case when Alice sends qubits (rather than halves of EPR pairs) to Bob. Consider the following situation. Alice prepares $N$ EPR pairs in her laboratory. She then chooses the subsets for parity determination beforehand and performs all the computations and measurements on her halves of the $N$ EPR pairs in her own laboratory before sending out the other halves to Bob. After Alice's measurements, the subsystem that she sends to Bob is in a pure state; that is, qubits rather than halves of EPR pairs are sent to Bob. However, because Alice's operation is local, it must commute with Eve's eavesdropping operator. Therefore, this qubit-based scheme must be as secure as the original EPR-based scheme. (Just as in the EPR-based case, it is of the utmost importance for Alice to withhold information on the choice of subsets for the parity determination until Bob acknowledges the receipt of quantum transmission. Otherwise, Eve can cheat easily.)
\begin{references} \bibitem{Wiesner} The idea of quantum cryptography was first proposed by S. Wiesner around 1970 but remained unpublished until 1983 [ S. Wiesner, {\it SIGACT News} {\bf 15} (no. 1), 78 (1983)]. Wiesner proposed quantum money and multiplexing channel (essentially one-out-of-two oblivious transfer) but not QKD per se. \bibitem{BB84} C. H. Bennett and G. Brassard, in {\it Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing} (IEEE Press, New York, 1984), pp.175-179. \bibitem{Ekert} A. K. Ekert, {\it\prl} {\bf 67}, 661 (1991). \bibitem{book} H.-K. Lo, S. Popescu, and T. Spiller {\it Introduction to Quantum Computation and Information} (World Scientific,
Singapore, 1998). \bibitem{Loreview} For a review on quantum cryptography, see, for example, H.-K. Lo, in (4), p.~76-119. \bibitem{nocloning} W. K. Wootters and W. Zurek, {\it Nature} {\bf 299},
802 (1982); D. Dieks, {\it Phys. Lett. A} {\bf 92}, 271 (1982). \bibitem{QKD_Expt} P. D. Townsend, {\it Electron.\ Lett.\ } {\bf 30},
809 (1994);
A. Muller, H. Zbinden, N. Gisin, {\it Europhys.\ Lett.\ } {\bf 33},
335 (1996); R. J. Hughes, G. G. Luther, G. L. Morgan, C. G. Peterson,
C. Simmons, in {\it Advances in Cryptology: Proceedings of CRYPTO'96}, vol. 1109 of {\it Lecture Notes in Computer Science,} N. Kobiltz, Ed. (Springer-Verlag, Berlin, 1996),
pp. 329-342. For a review, see, for example, H. Zbinden, in (4),
p.~120-142. \bibitem{25} G. Brassard and C. Cr\'{e}peau, {\it SIGACT News} {\bf 27} (no. 3), 13 (1996). \bibitem{BCJL} G. Brassard, C. Cr\'{e}peau, R. Jozsa, D. Langlois,
in {\it Proceedings of the 34th annual IEEE Symposium on the Foundation of Computer Science} (IEEE Computer Society Press, Los Alamitos, CA, 1993), pp. 362-371, and references cited therein. \bibitem{bit_comm} For the impossibility of bit commitment, see the following:
D. Mayers, Los Alamos e-Print archive (available at http://xxx.lanl.gov/abs/quant-ph/9603015);
H.-K. Lo and H. F. Chau, {\it\prl} {\bf 78}, 3410 (1997);
D. Mayers, {\it ibid.,} p. 3414;
H.-K. Lo and H. F. Chau, {\it Physica\ D}, {\bf 120}, 177 (1998).
For the impossibility of other schemes, including one-out-of-two oblivious transfer, see
H.-K. Lo, {\it\pra} {\bf 56}, 1154 (1997).
For a review, see, for example,
H. F. Chau and H.-K. Lo, {\it Fort.\ de\ Phys.\ } {\bf 46}, 507 (1998)
; (5).
\bibitem{Bell} J. S. Bell, {\it Physics\ } {\bf 1}, 195 (1964); [reprinted in {\it Quantum Theory and Measurement,} J. A. Wheeler and W. H. Zurek, Eds. (Princeton Univ. Press, Princeton, 1983), pp.403-408]; A. Einstein,
B. Podolsky, N. Rosen, {\it Phys.\ Rev.\ } {\bf 47}, 777 (1935) [reprinted in {\it Quantum Theory and Measurement,} J. A Wheeler and W. H. Zurek, Eds. (Princeton Univ. Press, Princeton, 1983), pp.~138-141].
\bibitem{ssingle} C. H. Bennett, F. Bessette, G. Brassard, L. Salvail,
J. Smolin, {\it J.\ Cryptol.\ } {\bf 5}, 3 (1992);
N. L{\rm \"u}tkenhaus, {\it\pra} {\bf 54}, 97 (1996);
C. H. Bennett, T. Mor, J. A. Smolin, {\it ibid.,} p. 2675;
C. A. Fuchs, N. Gisin, R. B. Griffiths, C.-S. Niu, A. Peres,
{\it ibid.,} p. 1163 (1997); R. B. Griffiths and C.-S. Niu,
{\it ibid.,} p.1173; N. L{\rm \"u}tkenhaus
and S. M. Barnett, Los Alamos e-Print archive (available at http://xxx.lanl.gov/abs/quant-ph/9711033).
\bibitem{scollect} E. Biham and T. Mor, {\it\prl} {\bf 78}, 2256 (1997);
E. Biham, M. Boyer, G. Brassard, J. van de Graaf, T. Mor,
Los Alamos e-Print archive (available at http://xxx.lanl.gov/abs/quant-ph/9801022).
\bibitem{sdeutsch} D. Deutsch {\it et al.}, {\it\prl} {\bf 77}, 2818 (1996); D. Deutsch {\it et al.}, {\it ibid.} {\bf 80}, 2022 (1998). \bibitem{smayers} D. Mayers, in {\it Advances in Cryptology: Proceedings
of CRYPTO'95}, vol. 963 of {\it Lecture Notes in Computer Science,} D. Coppersmith, Ed. (Springer-Verlag,
Berlin, 1995), pp. 124-135; in
{\it Advances in Cryptology: Proceedings
of CRYPTO'96}, vol. 1109 of {\it Lecture Notes in Computer Science,} N. Koblitz, Ed. (Springer-Verlag,
Berlin, 1996), pp. 343-357. \bibitem{problem} In some applications, a nonnegligible amount of information leakage to Eve can be disastrous. The following example is due to J. Smolin (41). If a key is used in a so-called one-time pad to encrypt a president's order which ends with the password for launching a nuclear missile, an adversary who is aware of the structure of the message will, in principle, be able to steal the password.
\bibitem{negli} The goal of making Eve's expected information small conditional only on passing the test, is generally unattainable. One must include the proviso:
for any eavesdropping strategy with a nonnegligible chance of success. \bibitem{BDSW} C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, W. K.
Wootters, {\it\pra} {\bf 54}, 3824 (1996).
\bibitem{Mayers} D. Mayers, Los Alamos e-Print archive (available at http://xxx.lanl.gov/abs/quant-ph/9802025), version~4.
\bibitem{Mayao} D. Mayers and A. Yao, in {\it Proceedings of 39th Annual Symposium on Foundations of Computer Science} (IEEE Computer Society Press, Los Alamitos, CA, 1998), pp. 509-515 [also available at Los Alamos e-Print archive (available at http://xxx.lanl.gov/abs/quant-ph/9809039)].
\bibitem{remark} A big worry in cryptography is the Trojan horse attack. Any untrusted material received from an open channel poses serious security risks. As J. Smolin has often remarked (41), it is even conceivable that a robot is hidden in the received material and that it pops out to find and disclose secrets to adversaries. It is not just that the Trojan horse might leak information once it is in Bob or Alice's laboratory. One might think that this problem could be eliminated by simply shielding the laboratory very well or that such shielding is , in fact, assumed anyway in cryptographic protocols. The real problem is that the Trojan horse pretends to be real EPR pairs when Alice and Bob do their testing but behaves differently when they generate key, thus causing them to leak the information themselves. This worry is not unfounded because it is notoriously difficult to prepare almost perfect EPR pairs. [See (20) for a related discussion.] Real quantum systems often contain other degrees of freedom which are ignored in quantum computation. One might wonder if Eve could perform a quantum Trojan horse attack by hiding robots in (the hidden Hilbert space dimensions of) the quantum systems received by Alice and Bob. This would certainly make a rigorous proof of security of QKD based on imperfect sources impossible. Our answer is the following proposition. {\it Proposition~1.}As long as there is no security risk for Alice and Bob to receive untrusted classical messages, quantum Trojan horse attack can be foiled. { \it Remark.} Before we present our proof, notice that the assumption that there is no security risk in receiving classical messages is most reasonable because Eve can always send classical messages to Alice and Bob in a ``man-in-the-middle'' attack during a classical authentication process. If Alice and Bob could not afford to receive any untrusted classical message, the whole enterprise of cryptography would be hopeless. { \it Proof:} Instead of receiving any untrusted quantum system directly from an open quantum channel, a user (say Bob) demands that the state of the system must be converted into classical messages via teleportation (30) right at his doorstep. More concretely, Bob prepares trusted EPR pairs in his laboratory and sends one member of each pair to his untrusted representative Robert, who is working in an insecure area just outside his laboratory, when the untrusted quantum data (potentially a Trojan horse) is waiting. Robert teleports the nominal state of the untrusted system (that is, the state in its nonclandestine variables) into Bob's laboratory. In other words, Bob conveys the untrusted quantum state into his laboratory by means of trusted EPR pairs and untrusted classical messages. Now assuming that there is no security risk in receiving classical messages, Bob can safely receive those classical messages and use them to reconstruct the original quantum state. Teleportation provides an exact counting of the effective dimensions of the Hilbert space because each qubit requires two classical bits to teleport. Therefore, there is no hidden Hilbert space to worry about in the reconstructed quantum system. This conclusion is valid even if the original EPR pairs prepared by Bob do contain hidden dimensions.
\bibitem{relay1} S. J. van Enk, J. I. Cirac, P. Zoller, {\it\prl} {\bf 78},
4293 (1997); J. Preskill, {\it Proc.\ R.\ Soc.\ London\ A\ } {\bf 454},
385 (1998); J. I. Cirac, A. K. Ekert, S. F. Huelga, C. Macchiavello,
Los Alamos e-Print archive (available at http://xxx.lanl.gov/abs/quant-ph/9803017). \bibitem{relay2} H.-J. Briegel, W. D\"{u}r, S. J. van Enk,
J. I. Cirac, P. Zoller, {\it Philos. Trans. R. Soc. London Ser. A } {\bf 356}, 1713 (1998);
H.-J. Briegel, W. D\"{u}r, J. I. Cirac, P. Zoller, {\it\prl}
{\bf 81}, 5932 (1998);
W. D\"{u}r, H.-J. Briegel, J. I. Cirac, P. Zoller, {\it\pra} {\bf 59}, 169 (1999).
\bibitem{Shor} P. W. Shor, in {\it Proceedings of the 37th Symposium on Foundations of
Computer Science} (IEEE Computer Society Press, Los Alamitos, CA, 1996), pp. 56-65. \bibitem{ftqc} A. Yu. Kitaev, {\it Russ. Math. Surv.} {\bf 52}, 1191 (1997);
D. Aharonov and M. Ben-Or, in {\it Proceedings
of the 29th Annual ACM Symposium on the Theory of Computing}, (ACM Press, New York, 1998),
pp. 176-188;
E. Knill, R. Laflamme, W. Zurek, {\it Science\ } {\bf 279}, 342 (1998);
for a review, see, for example, J. Preskill, in (4), pp. 213-269.
\bibitem{attack} For instance, in the study of standard P/M schemes such as BB84 (2), one often assumes that the signal carriers are perfect single photons. Unfortunately, producing almost perfect single-photon pulses is beyond current technology, and dim coherent light pulses with a Poisson distribution in the number of photons are often used instead. The attenuation of an optical fiber is also large (say 0.35~dB/km), and detector efficiencies are far from perfect. Therefore, rather surprisingly, in an actual experimental implementation of polarization-coding BB84 over a significant distance (say 40km), Eve may, in principle, break the system by a generalized beamsplitting attack. The key point is that, many of the signals contain more than one photons and as such Eve is allowed to make copies (details are available at http://xxx.lanl.gov/abs/quant-ph/984035.shl) [that is, Supplementary Note 1 in this reprint version]. For short-distance applications, the relevance of such an attack remains an important subject for future investigations. In summary, standard theoretical security analyses on BB84 do not apply to most real-life experimental systems to date.
\bibitem{qubits} A qubit is simply a
two-level quantum system. It plays the role of a fundamental unit of quantum information,
just like a bit in classical information. \bibitem{Note1} Alice and Bob share $R$ EPR pairs and they
generate a key by measuring these pairs along any common axis. If the fidelity of their pairs is high (that is,
$\langle R\mbox{~singlets} | \rho | R \mbox{~singlets} \rangle > 1- 2^{-k}$ for a sufficiently large $k$), then Eve's information on the final key will be bounded by $2^{-c} + \mbox{O} (2^{-2k})$ where $c = k -\log_2 ( 2R +k + { 1 \over \log_e 2} ) $. In other words, Eve's information (more precisely, mutual information with the final key) is exponentially small as a function of $k$. This result follows directly from two lemmas (see discussion, available at http://xxx.lanl.gov/abs/quant-ph/984035.shl) [that is, Supplementary Note 2 in this reprint version].
\bibitem{qee} P. W. Shor, {\it\pra} {\bf 52}, 2493 (1995);
A. M. Steane, {\it\prl} {\bf 77}, 793 (1996).
\bibitem{tele} C. H. Bennett {\it et al.}, {\it \prl} {\bf 70}, 1895 (1993).
\bibitem{Realistic} Here, we assume that the error rate per unit
length varies smoothly along the channel. For example, the errors for different parts of the channel are almost independent. \bibitem{Peres} The decomposition of the quantum state into the tensor product of the logical qubits and ancillary qubits is a mathematical one. In the actual physical system, the state of the logical qubits is delocalized among all physical qubits. Such a delocalization is necessary for both error correction and fault-tolerant computation. See, for example, A. Peres, Los Alamos e-Print archive (available at http://xxx.lanl.gov/abs/quant-ph/9609015). \bibitem{scaling} Efficient quantum error correcting schemes exist for reducing the error rate to an exponentially small amount (see discussion, available at http://xxx.lanl.gov/abs/quant-ph/984035.shl) [that is, Supplementary Note 3 in this reprint version].
\bibitem{Newnote} Such an ``$(N-m)$-singlets-or-not'' measurement can be performed if Alice and Bob bring the two halves of each EPR pair together to perform a measurement along a Bell basis. This is a very subtle point because such a Bell measurement is not actually performed and, indeed, could not be performed without bringing the two halves together. Successful cheating thus means that the actual verification test is passed, but a hypothetical second test of bringing the remaining pairs back into the same laboratory and measuring them in Bell basis would fail (that is, some of the remaining $N-m$ pairs are shown to be nonsinglets upon Bell measurements).
\bibitem{Note2} This is a surprising result because Bell basis vectors are highly entangled and yet only local operations and classical communication are allowed here. The local operations needed are simply single-qubit operations and bilocal exclusive OR. More specifically, the three types of operations used are (i) unilateral rotations by $\pi$ rad, corresponding to $\sigma_x$, $\sigma_y$ and $\sigma_z$; (ii) bilateral rotations by $\pi/2$ rad; and (iii) bilateral application of the two-bit quantum exclusive OR (or controlled NOT). These basic operations plus local measurements and classical communication allow Alice and Bob to correct quantum errors using the one-way random-hashing scheme by BDSW. See (18) for details.
\bibitem{appendix} We can safely use classical probability theory to derive an explicit bound on Eve's information for any eavesdropping strategy that passes the verification test with a probability at least $2^{-r}$ for some parameter $r> 0$. See also (17). Here we work in the approximation of reliable local quantum operations by Alice and Bob. However, this assumption can be relaxed without changing our essential conclusion. If all the original $N$ pairs are singlets, the remaining $N-m$ pairs must be singlets. Instead of computing the fidelity for the remaining $N-m$ pairs to be $N-m$ singlets, let us compute the fidelity for the original $N$ pairs to be $N$ singlets. This will give us a good enough bound on the fidelity. With any cheating strategy against the quantum verification scheme by Eve, let $p_1$ be the total probability for the state of the $N$ pairs to be $N$ singlets under the measurement along the Bell basis. The case of $N$ singlets, which happens with a probability $p_1$, will automatically pass the verification test. This case is perfectly fine and secure. What about the other case? Upon a random-hashing verification scheme that sacrifices $m$ pairs, the other case (which happens with probability $1- p_1$) will pass a $m$-round random-hashing verification test with a conditional probability of, at most $2^{-m}$. Therefore, the probability that a strategy passes the verification test is given by \begin{equation} P({\rm passing}) \leq p_1 + 2^{-m} ( 1 -p_1) \leq p_1+ 2^{-m}. \label{passing1} \end{equation} Eve would be most interested in a cheating strategy that passes the test with a nonnegligible probability( say at least $2^{-r}$ where we assume that $ 0 < r \ll m$). Therefore, we demand that the probability \begin{equation} P({\rm passing}) \geq 2^{-r} . \label{hell} \end{equation} Combining Eq.\ (\ref{passing1}) and (\ref{hell}), we find that \begin{eqnarray} p_1+ 2^{-m} &\geq& 2^{-r} \nonumber \\ p_1 &\geq& 2^{-r} [ 1 - 2^{-(m-r)}] . \label{conp1} \end{eqnarray} Conditional on passing the verification test, the fidelity of the $N$ pairs as singlets is given by \begin{equation} F'\geq { p_1 \over p_1+ 2^{-m} } \geq { 2^{-r} [ 1 - 2^{-(m-r)}] \over 2^{-r} [ 1 - 2^{-(m-r)}] + 2^{-m}} = [ 1 - 2^{-(m-r)}] \label{heaven} \end{equation} where Eq.\ (\ref{conp1}) and the fact that $p_1 \over p_1+ 2^{-m} $ is an increasing function of $p_1$ have been used. By choosing a value of $m$ that is substantially larger than $r$, the conditional fidelity can be made very close to $1$. Therefore, given any parameter $r$, one can increase the conditional fidelity in Eq.\ (\ref{heaven}) by increasing the number $m$ of random parities computed. In summary, consider any eavesdropping strategy that passes an $m$-round random-hashing verification scheme with a probability at least $2^{-r}$ (where $m \gg r > 0$). From Eq.~(\ref{heaven}, upon passing the test, the conditional fidelity of the $N$ pairs as $N$ singlets is at least $1 - 2^{- (m-r)}$. From (28), this implies that Eve's information is exponentially small in $m-r$, more precisely, $2^{-c} + O (2^{-2(m-r)})$, where $c= m-r - \log_2 [2 (N-m) + m -r + { 1 \over \log_e 2}]$.
\bibitem{Semi} R. B. Griffiths and C.-S. Niu, {\it\prl} {\bf 76},
3228 (1996). \bibitem{notenew} Our classical argument applies to the $N$-Bell basis, whose basis vectors are highly entangled. It is perhaps surprising at first that the coarse-grained probabilities of a quantum mechanical experiment involving only local operations and classical communication can have a classical interpretation with respect to such a highly non-local basis. Put in another way, given the lesson from the EPR paradox, it is perhaps surprising that classical arguments can still be used to demonstrate that two distantly separated quantum subsystems are, in fact, highly quantum (that is, highly entangled).
\bibitem{tradeoff} C. A. Fuchs, {\it Fortschr.\ Phys.\ } {\bf 46}, 535 (1998) and references cited therein.
\bibitem{incident} Incidentally, our result also proves the security of quantum money proposed by Wiesner (1). Indeed, the proof for our second example can be used to derive a probabilistic bound on the entropy of the combined system consisting of the quantum banknote and the bank. Consequently, any double-spending strategy will almost surely fail in the verification step (as in BB84) done by the bank because this entropy will no longer be close to zero. \bibitem{Smolin} J. Smolin, personal communication. \bibitem{Ack} H.-K. Lo particularly thanks A. Ekert for pressing him
to investigate the security of QKD. We thank
numerous colleagues, including
C. H. Bennett, G. Brassard, I. Chuang, D. P. DiVincenzo, C. A. Fuchs, N.
Gisin,
D. Gottesman, E. Knill,
D. W. C. Leung, N. L\"{u}tkenhaus, D. Mayers, S. Popescu, J. Preskill,
J. Smolin, T. Spiller, A. Steane,
and A. C.-C. Yao for invaluable conversations and
suggestions. Many helpful suggestions from an anonymous referee are
gratefully acknowledged.
H. F. Chau is supported by Hong Kong Government
RGC grant HKU~7095/97P.
\end{references}
\begin{figure}
\caption{A sample one-way hashing protocol used to determine $s_1 \cdot x_1$
and $s_2 \cdot x_2$ for an unknown three-Bell state. Following the
convention of (18), $B_x$ and
$B_y$ denote bilateral rotation of $\pi/2$ along the $x$ and $y$ axis,
respectively; $\sigma_x$ denotes a unilateral rotation of $\pi$ along the
$x$ axis; and the symbol $\bullet\!\mbox{---}\!\oplus$ denotes a bilateral
controlled NOT operation. $\mathit{M}$ denotes a bilateral measurement.
}
\label{F:1}
\end{figure}
{\bf Appendix: Refereed Supplementary Materials} \par
\noindent {\em Supplementary Note 1:} \par
The beamsplitter attack has been discussed by C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin in Ref.~\cite{ssingle}. Here we show that, even in the case of zero bit error rate (BER) in BB84, a generalized version of beamsplitter attack can break BB84 completely provided that the loss due to the quantum channel between Alice and Bob is sufficiently large. For ease of discussion, we consider a generalized beamsplitter attack, which has been brought to our attention by N. L\"{u}tkenhaus. Eve measures the photon number observable $N$ of each pulse. If $N$ is less than two, she keeps the signal herself and Bob receives nothing. If $N$ is two or larger, she uses an ideal beamsplitter to take one photon out. She then resends the remaining photons to Bob via a {\it superior} channel so as to ensure that Bob still gets the same bit rate. [This is possible provided that the loss of the quantum channel between Alice and Bob is large enough.] Eve stores her photons and waits for the announcement of the polarization bases. As far as polarization is concerned, Eve has an exact copy of Bob's signals. Therefore, by measuring her own photons along the correct bases in BB84, she learns the polarizations of Bob's received photons and hence the key.
\par
\noindent {\em Supplementary Note 2:}
\par
{\noindent \it Lemma~1}: (High fidelity implies low entropy)
If $ \langle R \mbox{~singlets} | \rho | R \mbox{~singlets} \rangle > 1-\delta$ where $\delta \ll 1$, then the von Neumann entropy $S(\rho) < - ( 1 - \delta) \log_2 ( 1 - \delta) - \delta \log_2 {\delta \over (2^{2R} -1)}$. \par
\noindent
{\noindent \it Proof:} If $ \langle R \mbox{~singlets} | \rho | R \mbox{~singlets} \rangle > 1- \delta$, then the largest eigenvalue of the density matrix $\rho$ must be larger than $ 1- \delta$, the entropy of $\rho$ is, therefore, bounded above by that of $\rho_0 = {\rm diag}~\{1- \delta, {\delta \over (2^{2R} -1)}, {\delta \over (2^{2R} -1)}, \cdots ,{\delta \over (2^{2R} -1)} \}$. That is, $\rho_0$ is diagonal with a large entry $1- \delta$ and with the remaining probability $\delta$ equally distributed between the remaining $2^{2R} -1$ possibilities.
Q.E.D. \par
{\noindent \it Lemma~2}: (Entropy is a bound to mutual information.) Given any pure state $\phi_{AB}$ of a system consisting of two subsystems $A$ and $B$, and any generalized measurements $X$ and $Y$ on $A$ and $B$ respectively, the entropy of each subsystem $S( \rho_A)$
(where $\rho_A = {\rm Tr}_B | \phi_{AB} \rangle \langle \phi_{AB} |$) is an upper bound to the amount of mutual information between $X$ and $Y$. \par
{\noindent \it Proof:} This is a corollary to Holevo's theorem (See A. S. Holevo,
{\it Probl.\ Inf.\ Transm.\ (USSR)\ } {\bf 9}, 117 (1973).)
Q.E.D. \par\noindent
\par
\noindent {\em Supplementary Note 3:}
\par
For instance, one can simply choose a block code of length $k$ (say $k=7$) that can correct a single error to encode a single qubit repeatedly. In the first level of this concatenated coding scheme, the qubit is mapped into $k$ qubits. In the second level of the coding, each of the $k$ qubits is individually encoded into $k$ qubits, resulting in $k^2$ qubits altogether, so on and so forth. Let us assume that the errors for various elementary gates are independent and of order $\epsilon$. After the first level of encoding, all single errors can be corrected. Therefore, the effective error rate is of the order $\epsilon^2$. More generally, the effective error rate of the $(L+1)$-th level is related to that of the $L$-th level by $\epsilon^{(L+1)} \sim ( \epsilon^{(L)})^2$. Consequently, one expects that $\epsilon^{(L)} \sim \epsilon_0 ( { \epsilon \over \epsilon_0 })^{2^L} $ where $\epsilon_0$ is some threshold value of the error rate. (See Eq.\ (36) of Preskill in Ref.~\cite{ftqc}.) Therefore, the error rate becomes exponentially suppressed whenever a level is added.
\par
\noindent {\em Supplementary Note 4:} \par
On the contrary, if Eve were to know the subsets beforehand, then similar to our discussion on classical random hashing, Eve could alway cheat successfully: In the above example, suppose Eve is told beforehand the subsets $s_1$ and $s_2$ and that Alice and Bob will generate their key by measuring their pair along $z$-axis. Then, Eve can cheat by finding out which cheating final state will pass the verification test and then following its evolution backwards in time to work out the initial state: Suppose Eve would like the value of the key to be ``0''. She observes that the final state
$|\tilde{1}\tilde{0}\tilde{1}\tilde{1}\rangle \otimes |\uparrow_z \downarrow_z \rangle = \frac{1}{\sqrt{2}}
|\tilde{1}\tilde{0}\tilde{1}\tilde{1}\rangle\otimes
(|\tilde{1}\tilde{1}\rangle + |\tilde{0}\tilde{1}\rangle)$ will achieve her goal. Evolving it backwards in time, Eve finds that she needs to prepare the initial state
$\frac{1}{\sqrt{2}} |\tilde{1}\tilde{1}\tilde{1}\tilde{1}\rangle\otimes
(|\tilde{1}\tilde{1}\rangle - |\tilde{0}\tilde{1}\rangle)$.
\par
\noindent {\em Supplementary Note 5:} \par
The key point to note is that, starting with an $N$-singlet state
$|\tilde{1} \tilde{1} \cdots \tilde{1} \rangle$, the parity computation will simply evolve it into another $N$-Bell state, up to a phase \cite{BDSW}. Notice that the $N$ pairs in such a final state are {\it not} entangled with one another. Therefore, if we consider the untested $N-m$ pairs, they should be in a {\it pure} state. In fact, they will be in an $(N-m)$-Bell state, described by the density matrix, ${\rm Tr}_{\rm tested}
U_{s_1, s_2, \cdots, s_m} |\tilde{1} \tilde{1} \cdots \tilde{1} \rangle \langle \tilde{1} \tilde{1} \cdots \tilde{1}
| U_{s_1, s_2, \cdots, s_m}^{\dagger}$.
Recall that, for any {\it effective} eavesdropping strategy (that is, one that passes the verification test with a probability $\geq 2^{-r}$), the fidelity of the initial state as $N$-singlets is very close to $1$. By unitarity, the fidelity of the final state as
$U_{s_1, s_2, \cdots, s_m} |\tilde{1} \tilde{1} \cdots \tilde{1} \rangle$ is also close to $1$. Since fidelity does not decrease under tracing [See R. Jozsa, {\it J.\ Mod.\ Opt.\ } {\bf 41}, 2315 (1994) for this property.], the fidelity of the subsystem of the untested pairs as ${\rm Tr}_{\rm tested}
U_{s_1, s_2, \cdots, s_m} |\tilde{1} \tilde{1} \cdots \tilde{1} \rangle
\langle \tilde{1} \tilde{1} \cdots \tilde{1} | U_{s_1, s_2, \cdots, s_m}^{\dagger}$ is very close to $1$.
As ${\rm Tr}_{\rm tested}
U_{s_1, s_2, \cdots, s_m} |\tilde{1} \tilde{1} \cdots \tilde{1} \rangle
\langle \tilde{1} \tilde{1} \cdots \tilde{1} | U_{s_1, s_2, \cdots, s_m}^{\dagger}$ is a pure state, we can apply the argument in note~\cite{Note1} to prove that the von Neumann entropy of those $N-m$ untested pairs is very close to $0$ and, hence, that the mutual information between those $N-m$ pairs and the external universe (Eve plus tested pairs plus anything else) is exponentially small. Since the tested pairs are just part of the external universe, measurements on them do not really help.
\par
\noindent {\em Supplementary Note 6:} \par
Here we give the key arguments for the reduction result in the two examples rigorously, but leave out the irrelevant detailed calculations based on classical statistical theory. In the first example, consider $N$ pairs of qubits shared between Alice and Bob. Those pairs can be entangled with each other and also with the external universe, for example, an ancilla prepared by Eve. Suppose Alice and Bob would like to estimate the number of singlets. i.e., the expected number of ``yes'' answers if a singlet-or-not measurement were performed on each pair individually. This number can, in principle, be determined by Bell measurements if they bring the two halves together. However, Alice and Bob would like to perform local measurements and classical communication only. We argue that they can, nonetheless, estimate it accurately by the following method. They {\it randomly} pick $m$ pairs and measure the two members of each pair along an axis chosen randomly from $x$, $y$ and $z$ axes. Then they publicly announce their outcomes.
{\it Claim}: If $k$ of the outcomes are anti-parallel, then the estimated fraction of singlets in the $N$ pairs is $(3k-m)/2m$. Furthermore, confidence levels can be deduced from classical statistical theory.
{\it Proof}: Consider, for each pair, the projection operators $P^i_{\parallel}$ and $P^i_{{\rm anti}-\parallel}$ for the two coarse-grained outcomes (parallel and antiparallel) of the measurement performed on the $i$th pair. It is straight-forward to express these projection operators as linear combinations of projection operators along a single basis, namely, $N$-Bell basis.
Let us consider the operator $M_B$ which represents the action of a measurement along $N$-Bell basis. Since $M_B$, $P^i_{\parallel}$ and $P^i_{{\rm anti}-\parallel}$ all refer to a single basis ($N$-Bell basis), they clearly commute with each other. Thus, a pre-measurement $M_B$ by Eve along $N$-Bell basis will in no way change the outcome for $P^i_{\parallel}$ and $P^i_{\parallel}$. Therefore, we may as well consider the case when such a pre-measurement is performed and the problem is classical to begin with.
Random sampling is a powerful technique in classical statistical theory. The Central Limit Theorem shows that the asymptotic or large-sample distribution of the mean of a random sample from {\it any} finite population with finite variance is normal. See, for example, B. W. Lindgren, {\it Statistical Theory} (3rd. ed., Collier Macmillan, London, 1976). Therefore, for a sufficiently large $m$, we can simply use a normal distribution to estimate the original mean and establish the confidence level.
There is a minor subtlety. For each pair, only one of three random measurements is performed. Since everything is now classical and is unrelated to the key points of this paper, we shall skip the details here and refer the readers to, for example, W. G. Cochran, {\it Sampling Techniques} (3rd ed.,Wiley, New York, 1977), chap.~5.
Q.E.D.
This result is profound because it allows one to apply classical sampling theory and central limit theorem to a quantum problem at hand.
For the second example, one can establish a rigorous probabilistic bound on the eavesdropper's mutual information $I_{\rm Eve}$ on the final key by random sampling (i.e., by sampling a random subset of photons, computing the error rates in the two bases and then applying classical probability theory to estimate the real error rates, etc). We assume that the signal carriers in BB84 are perfect single photons. Notice that Alice could use EPR pairs to prepare photons in BB84. Let us consider a pre-measurement $M_B$ along $N$-Bell basis by Eve and the coarse-grained projectors $P^i_{\parallel}$ and $P^i_{{\rm anti}-\parallel}$. As before, they refer to a single basis and, thus, commute with each other. Therefore, a pre-measurement $M_B$ will in no way change those coarse-grained outcomes.
Alice and Bob pick $m$ photons randomly from those that are both transmitted and received along the {\em same} basis and publicly compare their polarizations. Alice and Bob can then work out the ``typical subspace'' that is likely to give such error rates and compute its dimensions. The logarithm of the number of dimensions of the typical subspace will be a very good probabilistic bound on the eavesdropper's information. [The probability amplitude on the ``atypical subspace'' will also contribute to the entropy, but this contribution can be made to be negligible in comparison. In more detail, Suppose there are $N$ EPR pairs and the squared amplitude on the atypical subspace is at most $\epsilon$. Then the atypical space's contribution $S_a$ to the entropy is no more than $S_a = - \epsilon \log_2 { \epsilon \over 2^{2N}} \approx
2N \epsilon $. By making sure that $\epsilon$ is much less than $1$, $S_a$ is much less than $N$. Since the contribution of the typical space to the entropy is supposed to be linear in $N$ in a noisy channel, clearly $S_a$ is negligible in comparison.]
\end{document} | arXiv |
Genetic evidence of tri-genealogy hypothesis on the origin of ethnic minorities in Yunnan
Zhaoqing Yang1 na1,
Hao Chen2 na1,
Yan Lu3,
Yang Gao4,
Hao Sun1,
Jiucun Wang3,4,
Li Jin3,4,
Jiayou Chu1 &
Shuhua Xu ORCID: orcid.org/0000-0002-1975-10023,4,5,6
BMC Biology volume 20, Article number: 166 (2022) Cite this article
Yunnan is located in Southwest China and consists of great cultural, linguistic, and genetic diversity. However, the genomic diversity of ethnic minorities in Yunnan is largely under-investigated. To gain insights into population history and local adaptation of Yunnan minorities, we analyzed 242 whole-exome sequencing data with high coverage (~ 100–150 ×) of Yunnan minorities representing Achang, Jingpo, Dai, and Deang, who were linguistically assumed to be derived from three ancient lineages (the tri-genealogy hypothesis), i.e., Di-Qiang, Bai-Yue, and Bai-Pu.
Yunnan minorities show considerable genetic differences. Di-Qiang populations likely migrated from the Tibetan area about 6700 years ago. Genetic divergence between Bai-Yue and Di-Qiang was estimated to be 7000 years, and that between Bai-Yue and Bai-Pu was estimated to be 5500 years. Bai-Pu is relatively isolated, but gene flow from surrounding Di-Qiang and Bai-Yue populations was also found. Furthermore, we identified genetic variants that are differentiated within Yunnan minorities possibly due to the living circumstances and habits. Notably, we found that adaptive variants related to malaria and glucose metabolism suggest the adaptation to thalassemia and G6PD deficiency resulting from malaria resistance in the Dai population.
We provided genetic evidence of the tri-genealogy hypothesis as well as new insights into the genetic history and local adaptation of the Yunnan minorities.
Located in Southwest China, Yunnan is a territory with a diversified ecological environment, a complex terrain of which over 90% is covered by mountains and hills, and it borders multiple countries in mainland Southeast Asia (Fig. 1a), which have given birth to rich human genetic and cultural diversity [1, 2]. With various ethnic minorities living in the same region, in general, population admixture can lead to frequent genetic exchanges among different ethnic groups [3]. Nonetheless, due to the complexity of mountainous landforms, Yunnan serves as a natural barrier for different ethnic groups to be isolated. As a result, these minorities are considered to have retained their cultural traditions and genetic patterns along with their histories. However, the genetic origin and history of most minorities living in Yunnan are still opaque. A comparison of the genetic backgrounds and evolutionary histories of Yunnan minorities is required to uncover the human genetic diversity [4].
Sample information and PCA of Global Panel C. a Geographic location of samples in Global Panel C. The circle color of each population on map corresponds to the dot color of PC plots in b. b M.Yunnan.West and corresponding populations with close affinities in the PCA of Global Panel C, using a total number of 17,101 SNPs in 709 individuals
Archeological records have shown that humans settled in Yunnan in the late Paleolithic period about 10,000 years ago (ya), and the Neolithic culture in Yunnan developed prosperously with rice culture around 5000 ya [5, 6]. In addition, a splendid Bronze Age civilization of Yunnan around 4000–3500 ya was derived from the Haimenkou Relic Site. The unearthed bronzes are distinguished from those from the Central Plains based on their advanced craftsmanship and strong ethnic characteristics [5]. These findings provide robust evidence that indigenous people living in Yunnan were distinct from inland people in the past. However, ethnological and linguistic studies have proposed the tri-genealogy hypothesis, which states that most ethnic minorities living in Yunnan have three different origins and histories. Generally speaking, the present-day diverse minorities in Yunnan are mainly derived from three ancient lineages, i.e., Bai-Yue, Bai-Pu, and Di-Qiang [7]. The Bai-Yue and Bai-Pu were southern indigenous groups speaking Tai-Kadai (also known as Kra-Dai) and Austroasiatic languages, respectively [7,8,9]. The Di-Qiang are northern immigrants speaking Tibeto-Burman languages who migrated from the Upper-Middle Yellow River Basin through the Tibetan-Yi corridor to Northwestern Yunnan because of the expansion of the Qin dynasty around 2700 ya [1, 8, 10,11,12,13,14,15]. Furthermore, previous Y-chromosomal analyses have indicated that the present Yunnan minorities who speak Tibeto-Burman languages are the result of admixture between Neolithic immigrants and local populations based on the relatively high frequency of the NRY haplogroups D-M174 and O-M95 [13, 16, 17]. The geographic distribution of minorities in Yunnan belonging to these three lineages also shows a certain pattern: (i) Di-Qiang mainly live in Northwestern Yunnan, (ii) Bai-Yue in Southwestern Yunnan and near the border between Eastern Yunnan and the Guangxi Zhuang Autonomous Region, and (iii) Bai-Pu in Western Yunnan, which borders with Myanmar (Additional file 1: Fig. S1). Coincidentally, minorities living in Western Yunnan are included in these three ancient lineages (Additional file 1: Fig. S1) with different historical cultures (Additional file 2), making it an appropriate region to investigate different Yunnan ethnic minorities.
Exome sequencing technology is a popular approach to generating high-coverage data to discover associations between coding variants and related complex traits [18, 19]. In recent years, large-scale exome sequencing studies have provided new insights into the associations between genetic variants and the risk for certain diseases [20,21,22,23]. However, it is impossible to fully cover variations with large effects profiled from underrepresented populations by investigations with large-scale data of well-studied populations. These variations, which are associated with biomedical traits, could be influenced by local demographic histories and adaptive processes, some of which benefit from the conservation of isolated circumstances [24,25,26]. Due to the isolation and the small size of the population, however, little is known about the genetic history and adaptive evolution of most Yunnan minorities. Investigation of these issues could shed light on: (i) the effects of functional variations in protein-coding genes and (ii) the evolutionary adaptations that have shaped the genomes of Yunnan minorities.
To elucidate the population structure, demographic history, and adaptive evolution of Yunnan ethnic minorities in detail, in the present study, we sequenced exomes of four minorities living in Western Yunnan (M.Yunnan.West), including Achang (ACH), Dai (DAI), Deang (DEA), and Jingpo (JIP). These minorities live at different altitudes in the same region: JIP live on hilltops above 1500 m; ACH and DEA live on hillsides at about 1000–1500 m, and DAI live in lowlands below 1000 m [27] (Additional file 1: Table S2 and Additional file 3: Table S1). Based on the tri-genealogy hypothesis, these populations belong to three lineages with distinctive cultures and histories (Additional file 2): ACH and JIP belong to Di-Qiang, DAI belong to Bai-Yue, and DEA belong to Bai-Pu [1]. To the best of our knowledge, we have conducted the first comprehensive and systematic analyses to explore the population history and evolutionary adaptation of these minorities. Moreover, we discuss the rationality of the tri-genealogy hypothesis based on our genetic evidence.
Differentiated genetic affinities of Yunnan minorities
Principal component analysis (PCA) in a global context showed that M.Yunnan.West was located between clusters of East Asians and Southeast Asians (Additional file 1: Fig. S3). After removing other worldwide populations while retaining the East Asians and Southeast Asians, the PCA results showed that M.Yunnan.West was not clustered together (Additional file 1: Fig. S3). To further understand the fine-scale genetic affinities of M.Yunnan.West, we selected Han Chinese (HAN) from both South and North China, minorities in South China (M.South), highlander minorities in China (M.Highland), and mainland Southeast Asians (MSEA) as the surrounding populations of M.Yunnan.West (Fig. 1a), and then compared their relationships. As a result, M.Yunnan.West was separated into three distinct clusters in this panel (Fig. 1b and Additional file 1: Fig. S4a): ACH and JIP were clustered closer to the Tibetan (TBN) and other M.Highland, DEA was clustered with the Burmese, and DAI was clustered with the Lahu and Vietnamese. The pattern of the three clusters remained the same when only comparing M.Yunnan.West (Additional file 1: Fig. S4b). We also performed PCA of M.Yunnan.West and their related populations. The results suggest that the different M.Yunnan.West can be distinguished with their related populations in the PC plot (Additional file 1: Fig. S5). The DAI was widely scattered compared with the Vietnamese, probably because of the substructure of DAI in Western Yunnan in our dataset and Southern Yunnan in the Human Genome Diversity Project (HGDP) [28, 29] dataset. We confirmed this substructure by comparing DAI from Western Yunnan in our study and from Southern Yunnan in the 1000 Genomes Project (KGP) [30] and HGDP datasets (Additional file 1: Fig. S6). Overall, these results indicate the genetic components of M.Yunnan.West are differentiated, despite living in the same regions.
We also used the unbiased fixation index (FST) (Additional file 1: Fig. S7) and the outgroup f3 statistics (Additional file 1: Fig. S8) to examine the genetic relationship of M.Yunnan.West and surrounding populations. The overall genetic makeup of JIP was closest to that of ACH (FST = 0.006), followed by surrounding Tibeto-Burman populations such as Burmese, Yi, and Naxi (FST = 0.008–0.009). Similarly, ACH was closest to JIP, followed by Tibeto-Burman populations, including Burmese, Yi, and Tujia (FST = 0.009–0.01). In contrast, the genetic makeup of DAI was closer to those of MSEA and M.South, such as Vietnamese (FST = 0.002), Tujia (FST = 0.005), and Cambodian (FST = 0.006). DEA had close affinities with surrounding populations such as Burmese (FST = 0.007), DAI (FST = 0.009), and JIP (FST = 0.01). The phylogenetic tree constructed based on the pairwise FST also portrayed a similar pattern as PCA: ACH and JIP were located in the clade of M.Highland, and DAI and DEA were located in the clade of MSEA, while DAI was close to the Vietnamese and DEA was close to the Burmese. The results of our outgroup f3 statistical analysis also confirmed our finding that the overall genetic relationship of M.Yunnan.West from three ancient lineages is different from each other.
Migration and admixture scenarios
Global ancestry inference with ADMIXTURE [31] based on the Global Panel B and C revealed the ancestral makeup of M.Yunnan.West (Fig. 2a and Additional file 1: Fig. S9 and Fig. S10). There were most likely six ancestral populations (K = 6) for M.Yunnan.West in Global Panel B based on the estimation of the cross-validation (CV) error (Additional file 1: Fig. S11). The result of ADMIXTURE under the Global Panel B indicates that ACH and JIP mainly shared their ancestral makeup with M.Highland (ancestral component colored as yellow, 71.66% ± 15.58% and 75.18% ± 13.94%, respectively), DEA shared the majority of ancestral makeup with M.Highland (ancestral component colored as yellow, 55.01% ± 9.06%) and Southeast Asians (ancestral component colored as red, 30.15% ± 5.69%), and DAI mainly shared their ancestral makeup with lowland East Asians (ancestral component colored as green, 41.53% ± 6.74%) and Southeast Asians (ancestral component colored as red, 29.41% ± 4.06%), suggesting the different genetic origins and admixture histories of M.Yunnan.West populations. In addition, the DEA-specific component was observed at K = 9, and the specific component of ACH and JIP was observed at K = 10 and further split into two components when K = 12 (Additional file 1: Fig. S9), indicating these three populations are more isolated compared to DAI. Similar results were supported by the ADMIXTURE inference under the Global Panel C (Additional file 1: Fig. S10). These findings were also confirmed by the estimation of the run of homozygosity (ROH) (Additional file 1: Fig. S12) and identity-by-descent (IBD) sharing (Additional file 1: Fig. S13) under the NGS Panel, illustrating that ACH, DEA, and JIP have a greater number of long ROHs and more IBD segments within populations compared to DAI. One possible explanation could be that more admixture events occurred in DAI due to living in the lowlands with other populations.
Global ancestry inference and migration signals of M.Yunnan.West. a ADMIXTURE result of Global Panel B at K = 6, using a total number of 28,462 SNPs in 600 samples. Population categories are labeled on the left. ACH, Achang; DAI, Dai; DEA, Deang; JIP, Jingpo; M.North, minorities in North China; M.South, minorities in South China; M.Highland highland minorities in China; MSEA, mainland Southeast Asians; ISEA, island Southeast Asians. b The model used for D statistic estimations. Positive and negative D values indicate the excess allele sharing with population 1 (pop1) labeled in red and with population 2 (pop2) labeled in blue, respectively. An absolute Z-score greater than 3 is generally accepted as a strong signal of gene flow. c Potential gene introgression of M.Yunnan.West estimated by D statistics. Different pairwise M.Yunnan.West population combinations were used as possible populations (pop1 and pop2) under the gene introgression from the assumed ancestor populations in Global Panel C. D value above and below 0 for an ancestor population is assumed to have a closer genetic affinity to pop1 and pop2, respectively. An absolute Z-score above 3 for an ancestor population is presented in the upper right and lower left, indicating possible gene introgression into pop1 labeled in red and pop2 labeled in blue, respectively
Furthermore, we found that the proportion of M.Highland ancestry was correlated with the living altitude of populations under both the Global Panel B and Global Panel C (R = 0.85 and P = 1.65 × 10−5, and R = 0.75 and P = 5.57 × 10−4, respectively) (Additional file 1: Fig. S14a). We also found the M.Highland ancestry was correlated with the longitude of population settlement (R = − 0.56 and P = 1.82 × 10−2 for Global Panel B, and R = 0.74 and P = 6.55 × 10−4 for Global Panel C), instead of the latitude (R = 0.5 and P = 4.23 × 10−2 for Global Panel B, and R = 0.2 and P = 0.45 for Global Panel C) (Additional file 1: Fig. S14a). These results indicate that populations living in Western China at higher altitudes are more likely to have a higher ancestral component of M.Highland. To test the major contribution of geographical factors, we further performed an analysis of partial correlations for altitude and longitude by using one as a variable and controlling another one as covariable. We found the significance of altitude (P = 2.3 × 10−2 for Global Panel B and P = 2.71 × 10−2 for Global Panel C) was higher than that of latitude (P = 0.23 for Global Panel B and P = 4.49 × 10−2 for Global Panel C) (Additional file 1: Fig. S14b), suggesting the higher contribution of altitude in M.Highland ancestry. In addition, the M.Highland ancestry and Southeast Asian (SEA) ancestry also showed differences among language groups of M.Yunnan.West under both the Global Panel B (Welch's ANOVA P = 7.04 × 10−52 and P = 8.76 × 10−46, respectively) or C (Welch's ANOVA P = 7.31 × 10−29 and P = 1.02 × 10−40, respectively) (Additional file 1: Fig. S14c). A similar result was obtained when surrounding populations were added to the corresponding language groups (Additional file 1: Fig. S14d). Overall, these findings indicate that the different living altitudes and languages could be the causes of the distinctive admixture patterns, which further lead to the genetic differences among M.Yunnan.West.
To further infer the potential admixture of M.Yunnan.West, we applied the D statistics to detect the gene flow signals. In the D statistic models, positive D values indicate more allele sharing with the first population and negative D values indicate more shared alleles with the second population; absolute Z-scores greater than 3 are generally accepted as a strong signal of gene flow (Fig. 2b). As a result, a significant gene flow signal from TBN was detected in ACH and JIP, followed by other M.Highland, possibly illustrating the migration from populations living in the Tibetan area to ACH and JIP. Besides, DAI showed multiple signals of gene flow from surrounding populations, including MSEA, M.South, and HAN, suggesting that frequent admixture events occurred in DAI (Fig. 2c). In contrast, few signals of gene flow were detected in DEA, which proves DEA is the most isolated population among M.Yunnan.West (Fig. 2c). In addition, we found no gene flow signal when the two target populations were ACH and JIP due to the high similarity of their genetics (Fig. 2c). However, the negative D values of M.Highland indicated JIP has a slightly closer genetic affinity with highlander minorities than ACH, which is consistent with the PCA results. We conducted the same analysis based on the f3 statistics and obtained similar results (Additional file 1: Fig. S15).
At last, we applied dadi [32] to analyze the migrations of M.Yunnan.West using the data of the NGS Panel. Population joint site frequency spectrum (SFS) based on putatively neutral sites was used to infer population migration history (Additional file 1: Fig. S16 and Fig. S17). We estimated the pairwise migration rate and direction among populations using the symmetrical migration model (SMM) and asymmetrical migration model (AMM) (Additional file 1: Fig. S16a). Based on the results of the SMM (Additional file 1: Fig. S16b and Additional file 4: Table S3), we found that ACH had a high migration rate with JIP, and JIP had a high migration rate with TBN. Further, the DAI had a high migration rate with HAN, and DEA had a high migration rate with M.Yunnan.West in the same region, including AHC and DAI. We also found that the populations living at higher altitudes were less likely to have migration events with HAN. To further detect the migration direction, we applied AMM (Additional file 1: Fig. S16c and Additional file 4: Table S3). The ratio of migration rate between TBN and M.Yunnan.West revealed that the direction of migration events is from TBN to M.Yunnan.West. In addition, the migration rates of DAI and HAN were approximately equal, indicating the unbiased migration between these two populations. These results were supported by the analyses conducted by TreeMix [33] (Additional file 1: Fig. S18).
Demographic history of Yunnan minorities
Data from the NGS Panel were used to infer demographic population history. We utilized IBDNe [34] to infer the recent demographic history of M.Yunnan.West (Fig. 3a). We found that populations who have an M.Highland component of > 50%, including ACH, DEA, JIP, and TBN, all showed a bottleneck around 15 generations ago. Besides, HAN and DAI depicted larger effective population sizes (Ne) than other populations, which is consistent with previous studies reporting large Ne values for HAN and DAI among East Asians [35, 36]. We also estimated the Ne using the LD-decay approach (Additional file 1: Fig. S19a), which allows us to infer Ne more than 200 generations ago. The results based on the LD-decay method also showed that the Ne values of HAN and DAI were consistently larger than those of other populations, indicating the consistent expansion of these two populations.
Recent effective population size and integrated demography model of M.Yunnan.West. a Recent demographic histories inferred by IBDNe, using populations from the NGS Panel. b Integrated demography model describing the population history of M.Yunnan.West. Population divergence and effective population size were estimated by the 5-population model of dadi. Detected population migrations based on the results of D statistics, dadi, and TreeMix are portrayed as bold dotted arrows
We further estimated the pairwise divergence time using dadi (Additional file 1: Fig. S16d and Additional file 4: Table S3) based on the comparison of the expected and observed SFS (Additional file 1: Fig. S17). We applied different demography models and calculated the log-likelihood for each model to determine the best-fit divergence time. As a result, we inferred that ACH and JIP had the latest divergence time (3900 ya, 95% confidence interval [CI]: 3400–4200 ya), and JIP had a later divergence time with TBN (6700 ya, 95% CI: 6000–7100 ya) than other populations. The divergence between DAI and the M.Highland ancestry-enriched populations, ACH and JIP, was 7000 ya (95% CI: 6400–8200 ya) and 6900 ya (95% CI: 5500–7400 ya), respectively, which is earlier than that of DEA and HAN. The divergences of DEA from populations in the same region, ACH, DAI, and JIP, are 7100 ya (95% CI: 6200–8700 ya), 5500 ya (95% CI: 4200–6600 ya), and 6400 ya (95% CI: 5700–7500 ya), respectively, being more recent than those from other populations. The estimated divergence time was also consistent with the results inferred by the LD-decay method (Additional file 1: Fig. S19b). Although the values estimated by the LD-decay method were lower than those calculated with dadi, the overall relationships among the populations were consistent with those suggested by dadi.
After modeling with pairwise 2-population models, we hypothesized a model topology based on the pairwise best-fit divergence time inferred from 2-population models in dadi, and utilized a hypothesis-testing framework of 3-population models (see the "Methods" section) to confirm this topology was the best-fit one of the four M.Yunnan.West populations and two reference populations (Additional file 5: Table S4) (see the "Methods" section). The model topology was also recovered by the inferred maximum likelihood tree of the TreeMix (Additional file 1: Fig. S18). After the confirmation of model topology, we estimated the Ne and population divergence of the five populations except the outgroup TBN using the 5-population model in dadi (Additional file 1: Table S5). Taken together, we propose an integrated model to describe the population history of M.Yunnan.West (Fig. 3b). As the model shows, the Tibeto-Burman speaking populations ACH and JIP are affected by the gene flow from the highland minority TBN. The DAI is genetically close to DEA and there is a mutual gene flow between DAI and HAN. The DEA is affected by gene flow from the populations with M.Highland ancestry (ACH and JIP) and populations with SEA ancestry (DAI).
Novel variants identified in Yunnan minorities
Based on the results of ROH and IBD sharing, we assumed that ACH, DAI, DEA, and JIP show differences in genetic diversity and novel variants. Using HAN for comparison, we first calculated the average nucleotide differences (\(\overline{\pi }\)) of each M.Yunnan.West population and found that DEA showed the highest value, followed by ACH, JIP, and DAI (Additional file 1: Fig. S20a). We then calculated novel variants that were defined as variants not presented in the dbSNP v154 [37] and Exome Aggregation Consortium (ExAC) [38] datasets, at the population level and the individual level. We found that most novel variants were singletons in these populations, especially in the reference population HAN, due to the large sample size (Additional file 1: Fig. S20b). After removing singletons, we observed that ACH, DEA, and JIP harbored a significant proportion of novel variants that were not represented in public databases, and DEA showed the greatest number of novel variants, followed by ACH and JIP (Additional file 1: Fig. S20b). A similar result was obtained at the individual level, irrespective of whether singletons were included (Additional file 1: Fig. S20c).
Among M.Yunnan.West populations, we observed that DEA showed the largest average number of population-specific novel variants no matter removing singletons or not, followed by ACH, JIP, and DAI (Additional file 1: Fig. S21a). To further investigate the variant type of these novel variants, we annotated variant consequences using the Ensembl Variant Effect Predictor (VEP) [39] and counted the number of each annotation category (Additional file 1: Table S6). We defined variants with a high impact classification and missense variants whose SIFT [40] and PolyPhen [41] scores in VEP both predicted that they are harmful as loss-of-function (LOF) variants (Additional file 1: Table S6 and Additional file 6: Table S7). We then found that while ACH, DEA, and JIP showed a higher number of LOF novel variants than DAI (Additional file 6: Table S7), probably because most novel variants of DAI were reported in the KGP and HGDP datasets, the proportion of LOF variants was highest in DAI (Additional file 1: Fig. S21b), indicating a higher genetic burden in DAI.
Shared and divergent adaptation in Yunnan minorities
To detect the shared signals of adaptive evolution among M.Yunnan.West populations, we calculated the Population Branch Statistic (PBS) for each gene [42], using HAN and CEU as the second and third populations, respectively. This allowed the detection of genes that are likely under selection in all of the M.Yunnan.West populations but not in HAN. To estimate the significance of PBS values, we simulated PBS values under the demographic model inferred from this study using CEU as an outgroup and compared the generated data with observed data. We combined different M.Yunnan.West populations into one population and calculated the PBS values over the 99th percentile, and then we compared the distributions of different population combinations (Fig. 4a). We found that the populations that shared a higher ancestral component also showed higher PBS values. For example, a combination of M.Highland ancestry containing ACH, DEA, and JIP showed a higher PBS value than other three-population combinations. This indicates that M.Yunnan.West populations that share a higher ancestral component tend to share more adaptive signals. We also calculated the PBS values of genes for each population and selected genes with a P value below 0.01 as the extreme adaptive signals (Fig. 4b). DAI showed a greater number of extremely significant signals than the other three populations (Fig. 4c), which suggests that the number of genes that are likely under selection in DAI but not in HAN is higher than for the other three populations. Moreover, we found that FAM185BP, FAM74A3, and TMEM121 were extremely significant selection signals for all M.Yunnan.West populations. TMEM121 expression levels are often determined in skin biopsies, and TMEM121 has been reported to play a role in endothelial cells and chronic inflammatory disorders [43]. In addition, two genes related to the major histocompatibility complex (MHC) in humans, HLA-K and HCG4B, were identified as a significant signal in ACH, DEA, and JIP. The gene SLC24A5, which is involved in skin pigmentation, also showed a significant signal in ACH, DAI, and JIP. Other genes shared between the three populations were GTF2H4 in ACH, DAI, and DEA and FAM115C in DAI, DEA, and JIP.
Shared adaptive signals among M.Yunnan.West. a PBS distribution of genes over the 99th percentile, using different combinations of M.Yunnan.West populations. Box plot outliers were removed. Blue boxes indicate population combinations sharing the apparent same ancestral makeup. b Shared adaptive signals with extreme significance (simulation P value less than 0.01) among M.Yunnan.West populations. Only signals shared by at least two populations are labeled. c Venn plot representing the overlaps of extremely significant shared adaptive genes in four M.Yunnan.West populations
To investigate genes under selection specific to one M.Yunnan.West population, we calculated PBS values using each non-target M.Yunnan.West population as the second population and HAN as the third population [25]. Following the same process above, genes with a P value less than 0.01 were selected as extreme adaptive genes for each M.Yunnan.West populations (Fig. 5). Most of these genes also showed evident signatures of natural selection as estimated by cross-population extended haplotype homozygosity (XP-EHH), using HAN as the reference population (Additional file 1: Fig. S22).
Differential adaptive signals among M.Yunnan.West. Differential adaptive genes with extreme significance (P value less than 0.01) were scanned by PBS in a ACH, b DAI, c DEA, and d JIP. Different colors indicate that M.Yunnan.West populations were used as the second population and HAN was used as the third population in each calculation. Only signals detected as differential genes in at least two populations are labeled
We further identified and annotated variants with PBS values greater than 0.1 as major contributing variants for these differential adaptive genes (Additional file 7: Table S8). Regarding DAI, a population that prefers bitterness as a special eating habit, we identified the gene TAS2R30, which plays a role in the perception of the bitter taste [44, 45]. Derived allele frequencies (DAFs) of missense variants for TAS2R30 were all 0% in DAI but 10%–40% in other M.Yunnan.West populations. Other genes with major contributing missense variants in DAI included ANKRD36C, involved in spermatogenesis, PCNXL4, and GOLGA6L1, involved in immunity, and TMEM121, which plays a role in human skin biopsies. In addition, we identified C4orf17 genes in DEA. This gene is highly related to alcohol metabolism. MTTP, which is involved in lipid metabolism, is affected by alcohol exposure [46] and was reported to be associated with alcoholic fatty liver in the Korean population [47]. The remaining genes of DEA with major contributing missense variants were RAMP3, involved in the regulation of the vascular system, and TNFAIP3, involved in immunity. In JIP, we identified the SYNC gene, which is involved in the formation of muscle fiber and consistently shows a high expression level in muscle tissue in the Genotype-Tissue Expression (GTEx) v8 dataset [48] (Additional file 1: Fig. S23). Other genes with contributing missense variants in JIP were PARS2, associated with mitochondrial disease [49], and HLA-DRA, which plays a key role in the human immune system.
To confirm the divergent adaptation of M.Yunnan.West populations is related to biomedical pathways, we took the intersection of genes with a strong adaptive signal (P < 0.05) in all three comparisons for each target M.Yunnan.West population as the gene sets for enrichment analysis (Additional file 8: Table S9). Functional enrichment and protein − protein interaction (PPI) network analyses were performed using metascape [50]. If − log10(P value) > 2, a functional category was considered significant. As a result, 11 functional groups were recognized for ACH, 20 for DAI, 19 for DEA, and 11 for JIP (Additional file 1: Fig. S24 and Additional file 9: Table S10); 2 PPIs were identified for ACH, 4 for DAI, 2 for DEA, and 1 for JIP (Additional file 1: Fig. S25). Strikingly, some of these functional categories are possibly associated with the habits and adaptations to living circumstances of different M.Yunnan.West populations, such as categories associated with mitochondrial alteration (GO: 0070125 and GO: 0010821) in ACH, categories associated with malaria (KEGG: hsa05144), glucose metabolism (GO: 0042149 and GO: 0010907), and taste transduction (KEGG: hsa04742) in DAI, and a pathway related to alcohol (GO: 0097305 and GO: 0097306) in DEA.
Genetic evidence of the tri-genealogy hypothesis
In this study, we unraveled the genetic components of M.Yunnan.West populations and found they are distinctly differentiated and genetically divided into three groups. Living in the same region, the three groups live at different altitudes and speak different languages. Our observations confirmed that these three groups have distinctive genetic patterns. We propose that the complex landforms and different language families could be the reason behind the lack of genetic exchange and the maintenance of persistent genetic differentiation (Additional file 1: Fig. S14). These differentiations could be attributed to the different migration and regional adaptations along with the population history and have been further validated in our study. Our observation, to some extent, is in agreement with the tri-genealogy hypothesis, which is mainly based on linguistic studies. ACH and JIP belong to the Di-Qiang lineage, showing a genetically close affinity with highland minorities speaking the Tibeto-Burman language. The DAI belongs to the Bai-Yue lineage speaking the Tai-Kadai language and had close genetic affinities with MSEA and lowland East Asians. DEA belongs to the Bai-Pu lineage and an Austroasiatic language, appearing to be related to both M.Highland and MSEA. This tri-genealogy hypothesis has been widely used in studies of Yunnan minorities and in some ways makes sense since it is generally assumed that populations in the same language families tend to show similar genetic patterns because of migration and genetic drift [51].
However, this hypothesis is based on ethnic and linguistic records, and many limitations were shown in the context of the rapid development of human genetics. Previous studies analyzed the NRY haplogroup of Yunnan minorities of these three ancient lineages and found that they have certain typical characteristics but could not be clearly distinguished from each other [2]. The Lahu is an ethnic minority speaking a Tibeto-Burman language and is thus classified as belonging to the Di-Qiang lineage, but in the present study, we found that the Lahu showed little genetic affinity with the highland minorities, and similar to the DAI, the Lahu were more related to M.South and Southeast Asians (Additional file 1: Fig. S9). One possible reason for this discrepancy could be that the Lahu do not live in Northwestern Yunnan like most other Di-Qiang minorities, but in Southwestern Yunnan, which is dominated by minorities of Bai-Yue and Bai-Pu lineages. Languages can change easily when a small population is merged with a large one [2]. We propose that the formation of different Yunnan minorities is influenced by many other factors, such as geographical location, migration, and admixture history, besides the language family, making the tri-genealogy hypothesis outdated in some cases. More studies are also needed to further investigate the complex population history and genetic affinity of the diverse ethnic minorities in Yunnan.
Distinct demographic history of Yunnan minorities
Based on the historical records, ACH and JIP are descendants of the ancient Di-Qiang. They migrated from Southeastern Tibet along the Tibetan-Yi corridor and the ancient Tea-Horse Road [8, 51]. In the present study, we observed ACH and JIP shared a high ancestral component with M.Highland from the result of ADMIXTURE (Fig. 2a). In addition, a previous study using Y-chromosomal data found that JIP harbors a high frequency of the NRY haplogroup D-M174, which is prevalent in M.Highland [17]. These results suggest that ACH and JIP are originated from the Tibeto-Burman speaking groups living in the highlands. We also detected the gene flow from M.Highland led by the TBN to ACH and JIP, using the D statistics (Fig. 2c), dadi (Additional file 1: Fig. S16), and TreeMix (Additional file 1: Fig. S18). Our observations confirmed that the migration from the Tibetan area to Western Yunnan resulted in the present-day ACH and JIP populations. However, ACH, DEA, and JIP that contained the M.Highland ancestry of > 50% all show a recent bottleneck around 15 generations (Fig. 3a). One explanation for the bottleneck could be the chaos caused by the drastic struggle among competing Buddhists in the Tibetan area around the sixteenth century in the Ming dynasty [52].
The Bai-Yue populations are widely distributed in South China and mainland Southeast Asia, and the DAI belonging to the Bai-Yue populations has been used as one of the representative minorities of China in public human genome datasets such as KGP [30] and HGDP [28]. In the present study, we found that the DAI shared a higher proportion of ancestral components with M.South and MSEA compared with other M.Yunnan.West, suggesting the DAI is originated from the ancient Bai-Yue populations. The previous studies showed that DAI has the larger Ne value among East Asians [35, 36]. Our observations based on IBDNe suggest that DAI experienced a different demographic history than the other three M.Yunnan.West populations, which connotes that DAI did not experience population bottlenecks in recent generations (Fig. 3a). In addition, based on the ADMIXTURE (Fig. 2a, Additional file 1: Fig. S9, and Additional file 1: Fig. S10), ROH (Additional file 1: Fig. S12), and IBD sharing (Additional file 1: Fig. S13) results, we found that DAI was less isolated than ACH, DEA, and JIP. The results of D statistical analyses also indicate that DAI received more gene flow than the other three M.Yunnan.West populations, mainly from MSEA, M.South, and HAN (Fig. 2c). Since the implementation of the Tusi System in the Yuan Dynasty, most minorities in ancient Southwest China were under the control of DAI, resulting in the DAI having a higher social status among these southwestern minorities. This may also be explained by our observations that the DAI is more frequently admixed with surrounding populations and undergo continuous population expansion.
As an ethnic minority belonging to the Bai-Pu lineage that was initially called the Ailao tribe, the DEA is recognized as an indigenous population living in Western Yunnan. Our observation based on the ADMIXTURE and D statistics suggests that the DEA was the most isolated one of the M.Yunnan.West populations, suggesting the DEA has a different genetic origin from surrounding population groups. In addition, the different genetic origins with little expansion and migration in DEA possibly resulted in the high number of novel variants compared with other M.Yunnan.West populations. With the development of the Ailao tribe of Bai-Pu lineage, the Ailao state was gradually established under the rule of DAI expanding from the South. In addition, the Di-Qiang lineage from the North also reached the Ailao state to escape the war. As a result, the Bai-Pu lineage was probably influenced by both Di-Qiang and Bai-Yue lineages during history. Our results confirm that DEA shares the ancestral components with M.Highland and MSEA, which are the representative ancestral components of the Di-Qiang and Bai-Yue lineages, respectively.
Local adaptation of M.Yunnan.West populations
We first searched for shared signatures of selection in M.Yunnan.West populations by scanning genes using PBS (Fig. 4c). We identified three genes with extreme significance, i.e., FAM185BP, FAM74A3, and TMEM121. FAM185BP and FAM74A3 are long non-coding RNA genes that have not been fully characterized. TMEM121 is used in skin biopsies and plays a role in endothelial cells and chronic inflammatory disorders [43]. Another study also indicated that TMEM121 plays a role in immunity related to skin diseases such as psoriasis [53]. With respect to extremely significant adaptive signals shared in three populations, we identified HLA-K and HCG4B in ACH, DEA, and JIP, SLC24A5 in ACH, DAI, and JIP, GTF2H4 in ACH, DAI, and DEA, and FAM115C in DAI, DEA, and JIP. Among these genes, HLA-K, HCG4B, and GTF2H4, belonging to the MHC gene family, and SLC24A5, involved in skin pigmentation, were previously reported as candidate genes favored by natural selection in human populations [54, 55]. In addition, most of these genes, including HLA-K, HCG4B, GTF2H4, and FAM115C, play key roles in the regulation of the human immune system. For example, FAM115C is related to cancer cell migration [56, 57], and GTF2H4 plays a key role in DNA repair and has been reported to be associated with the risk for human papillomavirus (HPV) resistance in Costa Ricans [58]. Concerning the divergent adaptations, the DAI showed more differential adaptive variants than other M.Yunnan.West populations (Figs. 4c and 5). We identified the differential genes specific to each M.Yunnan.West population (Additional file 7: Table S8). Furthermore, we performed enrichment analysis for differential genes specific to each M.Yunnan.West population, using the intersection of adaptive variants strongly differentiated from each of the other three M.Yunnan.West populations (Additional file 8: Table S9). Notably, results of differential adaptive genes and functional enrichment categories illustrated specific adaptive evolution possibly related to the living environment and habits of each M.Yunnan.West population (Additional file 9: Table S10).
As a multifactorial disorder, hypertension has been reported to be associated with mitochondrial alterations [59]. In our study, we profiled PARS2, which is associated with mitochondrial disease, as an extremely differential gene in JIP. The functional category of metabolic disorders of biological oxidation enzymes (R-HSA-5579029) was also enriched in the JIP differential gene set. Additionally, functional categories related to mitochondrial alterations, including mitochondrial translational elongation (GO: 0070125) and regulation of mitochondrion organization (GO: 0010821), were enriched in the differential gene set of ACH. The ancient Di-Qiang populations were living in areas with high altitudes like Tibetan areas. The habit of living in the high altitude of the local environment was kept in ACH and JIP after they migrated to Western Yunnan, while such continuity of residence was lacking in the ancestors of DAI and DEA. A previous study indicated that hypertension is more prevalent in Yunnan minorities of Di-Qiang lineages, such as Bai and Tibetan [60]. Therefore, we speculate that the differences in living habits contribute to the detection of hypertension-related selection signatures within M.Yunnan.West. These results suggest that ACH and JIP, as descendants of the Di-Qiang lineage living in the highlands, are likely under adaptive evolution of hypertension.
The ancestor of DAI is one of the indigenous populations living in South China in the past. The present-day DAI people mainly live in the lowlands of Southwestern Yunnan, which is permanently hot and humid and where malaria was prevalent in the past [61]. These special living circumstances give rise to the specific adaptive evolution of DAI. On the one hand, the hot and humid environment drives the DAI to prefer bitterness as their eating habit because the ancient DAI believed that bitterness could dispel dampness and detoxify the body. Among the extremely significant differential adaptive genes for DAI, TAS2R30 is highly associated with the perception of bitter taste. The functional pathway of taste transduction (KEGG: hsa04742) was also enriched in the DAI gene set of strong differential adaptive signals. A previous study indicated that the genetic diversity of the human TAS2R gene family is higher than the genome-wide average due to the elevated rates of non-synonymous substitution [62]. In the present study, most of the major contributing variants identified in TAS2R30 were non-synonymous, and DAFs of these missense variants were all 0% in DAI but 10–40% in other M.Yunnan.West populations. With regard to other M.Yunnan.West that have no traditional habit of eating bitter foods, sensitive bitterness perception would safeguard them from eating toxic substances [63]. Conversely, the TAS2R under selection in DAI may be explained by their low sensitivity to the bitter taste.
Malaria in Yunnan has been documented since 225 AD. It has been a persistent epidemic disease in Yunnan until the 1950s [64]. The border and low-altitude areas of Yunnan are the main malaria-endemic areas [65] and highly overlap with the living areas of DAI. In this study, we identified genes related to the malaria resistance pathway (KEGG: hsa05144), including CCL2, CD40, HBA1, and HBA2 as the differential genes in DAI, indicating different selective pressure posed by malaria on the DAI compared to other M.Yunnan.West. In addition, we found that HBA1 and HBA2, genes involved in malaria resistance by the sickle cell and thalassemia traits [66,67,68], which also explains the high prevalence and specific pattern of thalassemia in the present-day DAI [69, 70]. Similarly, glucose-6-phosphate dehydrogenase (G6PD) deficiency is one of the common enzymopathies affecting people living in regions where malaria is endemic, as a result of natural selection against genes that are associated with susceptibility to malaria [71,72,73]. In our results, functional categories related to the response to glucose, including cellular response to glucose starvation (GO: 004214) and positive regulation of glucose metabolic process (GO: 0010907), were enriched in the differential gene set of DAI, which could be a consequence of G6PD deficiency. The G6PD deficiency can cause the blockade of the pentose phosphate pathway and the accumulation of its substrate, glucose-6-phosphate, to compensate for the glucose metabolic process and further reduce the sensitivity of cells to glucose starvation [74]. At the same time, the G6PD deficiency can lead to a decrease in the production of NADPH, which will increase the level of oxidation of erythrocyte and further result in hemolytic anemia [75, 76]. Thus, these enriched terms from the differential gene set of DAI suggest the DAI might be under the adaptive process to the G6PD deficiency due to the malaria prevalence. Although ACH, DEA, and JIP have also been reported to suffer from malaria, their susceptibility and severity of malaria were lower than that of the DAI since they live in high-altitude areas with a relatively cold and dry environment. Accordingly, the onset and prevalence of thalassemia and G6PD deficiency in these populations could also be different from that of the DAI [70, 77, 78].
As one of the genes involved in divergent adaptation in DEA, C4orf17 is the nearest flanking gene of ADHs, which are the well-known genes related to alcohol metabolism. Previous large-scale genome-wide association studies revealed that C4orf17 is significantly associated with alcohol consumption at the gene level [79]. Moreover, another differential gene, MTTP, which encodes a triglyceride transporter, showed a strong signature of selection in Southeast Asians in a previous study [80]. As the near downstream gene of C4orf17, MTTP has been suggested to be related to alcoholic fatty liver disease [47], possibly because alcohol exposure could increase triglyceride and cholesterol levels [46]. In addition, functional categories involved in the response to alcohol (GO: 0097305 and GO: 0097306) were also enriched from the differential gene set of DEA. The ACH, JIP, and DAI are all considered to be the populations with prosperous liquor cultures, while the ancient DEA had neither winemaking technologies nor liquor culture. This indicates that DEA was less able to undergo the adaptive process of the improvement of alcohol consumption, which possibly explains that selection signals related to alcohol metabolism were differentiated in DEA from the other three M.Yunnan.West populations.
Limitations of this study
In this study, we analyzed the genetic structure, population history, and local adaptations of 4 minority groups in Western Yunnan, and most of the results were based on the analyses of the target region of whole-exome sequencing (WES) data. Since our studied populations are different genetic backgrounds and WES data were less used in population genetic studies, we incorporated diverse populations from both whole-genome sequencing (WGS) and genotyping array data like references and designed different dataset panels based on the different analysis purposes. For example, we included both sequencing data and genotyping array data in analyses of population structure, but only used sequencing data in analyses of population history and local adaptation due to the ascertainment bias of genotyping array data [81]. Such study design with different datasets is effective in eliminating the potential bias stemming from multiple data resources.
The rare variants and unascertained common variants can be identified from WES data, which enables inferences of demographic history based on the SFS [82]. However, due to the lack of non-coding regions that include a great number of common variants, WES data are not as powerful as the WGS data for inference of population history. For example, applying MSMC [83] to infer long-term Ne was only supported by the WGS data. In addition, genetic inference from WES data may be subject to selective pressures since exons contain a substantial proportion of the functional variants [82]. Although some data processing such as only including evolutionary neutral sites as we used in this study (see the "Methods" section), were used to reduce the effect of background selection in CDS regions [25, 84], the analysis results based on the exome target region could inevitably be biased from the genome-wide level that may be closer to the real population history. Thus, due to the limitations of WES data, there would inevitably be potential biases in our results compared to the genome-wide level.
In this study, we characterized genetic structures, population histories, and local adaptations of Yunnan minorities. We found that Yunnan minorities from three ancient lineages, i.e., Di-Qiang, Bai-Yue, and Bai-Pu, show sufficient genetic differences. We modeled the population history of Yunnan minorities from three ancient lineages and provide genetic evidence for the tri-genealogy hypothesis. Di-Qiang populations are related to highland minorities and likely migrated from the Tibetan area about 6700 years ago. The divergence time between Bai-Yue and Di-Qiang was estimated to be 7000 years, and that between Bai-Yue and Bai-Pu was estimated to be 5500 years. Bai-Pu is relatively isolated with few expansions, but evidence of gene flow from surrounding Di-Qiang and Bai-Yue populations was also found. Adaptive variants possibly associated with the living circumstances and habits of the Yunnan minorities were identified. A few functional mitochondrial alterations and TAS2R30 might be associated with a higher incidence of hypertension in Di-Qiang populations. Adaptive variants related to malaria and glucose metabolism were identified in the Dai population and indicate the adaptation to thalassemia and G6PD deficiency resulting from malaria resistance, while selection on PARS2 is likely related to the perception of bitterness. C4orf17 and MTTP in Deang are associated with alcohol metabolism and the potential adaptation to the alcohol response.
Sample collection, exome sequencing, and SNP-calling
Epstein-Barr virus immortalized B lymphoblastoid cell lines (LCLs) from 242 native ethnic individuals, including 65 Achang (ACH), 52 Dai (DAI), 65 Deang (DEA), and 60 Jingpo (JIP) living in Mangshi of Dehong Autonomous Prefecture, Yunnan Province (Fig. 1a and Additional file 3: Table S1), were obtained from the Immortalize Cell Bank of Chinese Ethnics Groups hosted in the Institute of Medical Biology, CAMS. Whole exome sequencing (WES) data with high target coverage (100 × –150 ×) for 150 bp paired-end reads was carried out on the Illumina Hiseq X10 platform (iGeneTech, Ltd., Beijing, China).
Trimmomatic v0.4.0 [85] was applied for raw FASTQ data trimming using the recommended default parameters. Reads of each sample were mapped to the human reference genome (GRCh37) using BWA-MEM v0.7.10 [86]. We executed duplicate mark and base quality recalibration using GATK v3.8 [87]. Variants calling was performed through the HaplotypeCaller module of GATK based on the GVCF mode.
For comparison, WES data of 300 Han Chinese (HAN) residing in different regions of China from the HuaBiao Project [88, 89] were also collected (Additional file 1: Table S2). We performed a joint variant calling of M.Yunnan.West with HAN samples as well as 33 whole-genome sequenced (WGS) Tibetan (TBN) samples with high coverage collected from Lu et al. [90, 91] and implemented strict quality control through VQSR. We filtered the raw variant calling file into the target region of 62,984,579 base pairs. As a result, bi-allelic single-nucleotide variants with high quality were retained for downstream analyses.
Data compilation
To analyze the genetic variation of M.Yunnan.West in a broader context, we collected the global populations from the Human Genome Diversity Project (HGDP) dataset in the WGS version [29, 92]. Since Yunnan province is close to mainland Southeast Asia and there are much fewer Southeast Asians in HGDP dataset, we also collected genotype data including 196 Southeast Asians from Mörseburg et al. [93, 94]. We combined our joint-calling dataset, HGDP dataset, and 196 Southeast Asians as the Global Panel. There are 46,845 SNPs retained in this dataset after filtering SNPs with a missing rate higher than 10%. To solve specific problems under different contexts, we dissect Global Panel into different subsets. Global Panel A contained the whole global population of the Global Panel, Global Panel B contained the East Asian and Southeast Asian included in Global Panel, and Global Panel C only included M.Yunnan.West and their surrounding populations (M.South, M.Highland, MSEA, and HAN) (Additional file 3: Table S1). Besides, we also collected the altitude information of populations in Global Panel C based on the information of longitude and altitude using Google Earth (https://earth.google.com) (Additional file 1: Table S2). The maps used in this study, including GS(2016)1609, GS(2016)1667, and GS(2020)4618, were obtained from a standard map service (http://bzdt.ch.mnr.gov.cn) approved by the Ministry of National Resources of the People's Republic of China.
The Global Panel shows limitations for insufficient genetic information due to the lack of joint-calling and the merging of genotype data. To conduct more comprehensive analyses, including estimating effective population size (Ne), population divergence time, and PBS calculation, we used our joint-calling dataset as NGS Panel. This panel is informative and contains 274,634 SNPs, which enables analyses requiring sequencing data input.
Population structure and genetic affinity
To avoid the bias caused by a close genetic relationship, we identified the relatedness of M.Yunnan.West samples using the KING v2.1.2 [95] and excluded samples within the second-degree relationship for subsequent population structure analyses (Additional file 1: Fig. S2). We used the dataset from Global Panel and performed a series of principal component analyses (PCA) at the individual level using the SNPRelate v1.16.0 [96]. We restricted every single dataset in the target region and selected bi-allelic autosomal SNPs with a missing rate of less than 0.05 using the BCFTools v1.6 [97]. PLINK v1.9 [98] was applied to perform LD-pruning of SNPs using a window of 1,000 base pairs advanced by 100 SNPs at a time and an r-squared coefficient of 0.2 for the combined dataset. After quality control, 17,101 SNPs were left for subsequent analyses.
Global ancestry inference under the Global Panel B and Global Panel C was performed by ADMIXTURE v1.3.0 [31] to dissect the ancestral makeup of each individual. The input data for admixture analysis were prepared using the same method as for the PCA. To avoid the bias caused by differences in sample sizes, we randomly selected 40 samples for populations with larger sample sizes.
Population differentiation was estimated following Weir and Cockerham's FST [99] using SNPRelate v1.16.0. To investigate the relationship between M.Yunnan.West and their surrounding populations, we calculated pairwise FST values among populations in Global Panel C and constructed a phylogenetic tree based on the pairwise FST results.
We also applied the outgroup f3 statistics [100] to examine the relationship between each M.Yunnan.West population and other populations under the Global Panel B. The program qp3pop implemented in ADMIXTOOLS v7.0.2 [101] was applied to calculate the outgroup f3 statistics in the form of f3(Yoruba; X, Y), where X represents the different M.Yunnan.West populations and Y represents other populations under the Global Panel B. The output Z-score was used to measure the genetic affinity between populations.
Estimation of genetic diversity
To measure the consanguinity of the populations in the NGS Panel, we used BCFTools v1.6 to estimate ROH based on the Hidden Markov Model (HMM) approach [102]. We used the -G option and set the argument as 30 to account for GT errors. We classified ROH into three classes: short ROH is less than 1 Mb, medium ROH is from 1 to 5 Mb, and long ROH is larger than 5 Mb. Then, we calculated the total length of each type of ROH for each individual and compared ROH length at the population level.
To estimate the IBD-sharing within and between populations, we first used the Beagle v5.2 [103] to phase the data of the NGS Panel. Based on the phased data, The IBD segments were estimated using the hap-IBD [104], and short gaps in the IBD segments were removed.
Novel variants were defined as those not present in dbsnp v154 [37] and ExAC [38] and were annotated using Ensembl VEP v94 [39]. To rule out singletons mainly caused by sample size, we primarily focused on variants excluding singletons. Variants with a SIFT score lower than 0.05 are considered deleterious and variants with a PholyPhen score higher than 0.446 are considered damaging. LOF variants were defined as variants with a high impact classification or missense variants whose SIFT [40] and PolyPhen [41] scores in VEP both predicted that the variants were harmful.
Population history inference
To detect the gene flow of M.Yunnan.West, we used qpDstat in ADMIXTOOLS v7.0.2 [101] to calculate D statistics by assuming two populations from M.Yunnan.West to be potentially admixed and including the third population from the surrounding populations under Global Panel C as the ancestral population. The Yoruba population from the HGDP dataset was used as the outgroup population. As a result, the tree-like relationship ((X, Y), Z), Yoruba) was used for the detection of gene introgression, where X and Y are target populations to be potentially admixed, and Z is the surrounding population assumed to be the ancestor. Similarly, we also applied qp3pop in ADMIXTOOLS to calculate the f3 statistics.
Recent demographic histories were estimated by IBDNe v23Apr20 [34]. IBDNe was used to estimate estimated the change in effective population size from 1 to 60 generations ago. IBD segments were inferred by hap-IBD using the phased data of the NGS Panel. We set IBD segments shorter than the 4 cM that were ignored in IBDNe.
Divergence times and migration rates were inferred using the dadi v2.1.0 [32] based on the SFS. To avoid the bias caused by the coding sequence regions, we annotated our data using the VEP v94 [39] and selected intergenic, synonymous, and intronic sites from the target region as the neutral sites for analysis. As a result, 125,101 SNPs were used to construct pairwise unfolded joint SFS. Ancestral states of variants were annotated using the Enredo-Pecan-Ortheus (EPO) 6-way primate alignment [30]. We used the misidentification function in dadi to model the proportion of variants with a misidentified ancestral state. To determine the optimal divergence time and migrations, we ran optimization to infer the best-fit 2-population model parameters from the three given models (Additional file 1: Fig. S16) based on the comparison of the expected and observed SFS in dadi (Additional file 1: Fig. S17). We also utilized a 3-step strategy using 3-population models to confirm the topology inferred via pairwise divergence times in 2-population models and the TreeMix as the best-fit topology in dadi (Additional file 5: Table S4). First, we fixed the TBN as an outgroup population and used different 2-population combinations from the other five populations to test the best-fit topology in each combination. There are 20 different TBN-pop1-pop2 combinations with 3 topologies in each combination (each of the three populations as an outgroup), and we confirmed the topology that TBN is outgroup was the best-fit in all TBN-pop1-pop2 tests. Second, in the remaining five populations, we fixed ACH and JIP and used each of the other three populations (DAI, DEA, and HAN) as the third population to test the best-fit topology in each combination. We found the topology that the third population is the outgroup was the best fit with maximum likelihood in each test. Finally, we used the remaining 3 populations (DAI, DEA, and HAN) to test their topology of them and found that the topology that DEA is the outgroup of DAI and HAN, was the best-fit one. Since dadi only supports at most 5 taxa in model construction, after the confirmation of the model topology, we ran the 5-population model using populations except the outgroup TBN to estimate the demographic parameters (Additional file 1: Table S5). For each run of dadi, we repeated the optimization process using the parameters from the last round to seed a subsequent round of model fitting, which improves the log-likelihood values and generally converges in the final round. The CIs of demographic parameters in 2-population and 5-population models were estimated by 500 Bootstrapped SFS generated from 100 800-kb blocks containing the target region. Each 800-kb SFS was randomly sampled with replacement and estimated using the same analysis approach described above.
TreeMix v1.1.3 [33] was used to confirm the tree topology of the proposed model and infer migration events. We combined the NGS Panel dataset with YRI from KGP [30, 105] and used PLINK v1.9 [98] to filter SNPs with a missing rate of less than 0.01. To infer the tree topology, we set the YRI as the outgroup and ran 500 bootstraps in TreeMix with no migration. We then applied SumTrees from the DendroPy [106] o construct a consensus tree based on the 500-bootstrapped trees. To infer migrations, we used the consensus tree as the previously generated tree and ran TreeMix with migrations from 1 to 4.
We also used LD-decay to estimate the effective population size and divergence time of populations [35] based on the NGS Panel. We calculated \({r}_{LD}^{2}\) of each pair of SNPs with a genetic distance less than 0.25 cM in each population. Recombination distances were assigned using PLINK v1.9 [98] based on the genetic map of HapMap [107]. Effective population sizes of \(t\) generations ago were estimated for each population in each recombination distance category as: \({N}_{e}=[(1/{r}_{LD}^{2})-2]*(1/4c)\), where \(\mathrm{c}\) denotes the recombination distance. We adjusted \({r}_{LD}^{2}\) as \({r}_{LD}^{2}-(1/n)\), where \(n\) is the sample size prior to the calculation of \({N}_{e}\). Divergence times were estimated by \({2N}_{e}{F}_{ST}\), where \({N}_{e}\) is the average of the harmonic means of the relevant recombination distance categories. We used distance categories from 0.01 to 0.25 cM (corresponding to 200 to 5000 generations ago) to estimate \({N}_{e}\) values of target populations.
Identification of adaptive signals
PBS was applied to detect signals of adaptive evolution in our study. We only included sites with depth above 50 × and a missing rate of less than 5% for PBS calculation. The PBS is defined as:
$${PBS}_{A}=\frac{{T}_{AB}+{T}_{AC}-{T}_{BC}}{2}$$
where \(T=-\mathrm{log}(1-{F}_{ST})\), A is the concerned population under selection, and B and C are populations used as references. In calculations, only sites that were polymorphic in at least one of the three populations were considered.
To calculate the significance of PBS values, we performed neutral simulations with MSMS v3.2 [108], based on the demographic model inferred in this study. We assumed the divergence time between CEU and East Asians is 2000 generations. Genes with 1–100 SNPs were simulated based on the random sample from all human genes. We then subsampled one million simulations for each number of SNPs per gene (from 1 to 69 SNPs) or using 5-bin categories (from 70 to 100 SNPs). The P value of each PBS was defined as the proportion of observed PBS values that were higher than one million simulated PBS values.
To estimate the shared adaptive genes under natural selection, we used HAN in our NGS Panel and CEU from the KGP dataset as the second and third populations for PBS calculation, respectively. We calculated PBS using each M.Yunnan.West population as the target population and focused on the genes with extreme significance (P < 0.01) in at least two populations as a shared adaptive signal. Besides, we also used different combinations of M.Yunnan.West population as the target population to examine the relationship between ancestry sharing and PBS distribution.
Divergent adaptation based on PBS was estimated for each M.Yunnan.West population, assuming each of the other three M.Yunnan.West populations as the second population and HAN as the third population. A total of 12 combinations were generated for PBS calculation. Similarly, a gene with extreme significance (P < 0.01) in at least two other M.Yunnan.West populations were regarded as a differential adaptive signal.
To validate the differential adaptive genes scanned by the PBS approach, we also estimated cross-population extended haplotype homozygosity (XP-EHH) of extremely significant genes under selection (p < 0.01) for each M.Yunnan.West, using selscan v1.2.0 [109] and regarding HAN as the reference population.
Functional analyses of adaptive variants
VEP v94 [39] was used to annotate differential adaptive genes with extreme significance (P < 0.01). Conservation scores of each variant were calculated by the –sift and –polyphen options in VEP. We also calculated the PBS value of each variant of these genes under selection and listed variants with higher PBS values (> 0.1) as highly differentiated SNPs for each gene.
The gene set was used for functional enrichment of each M.Yunnan.West population was defined as the intersection of strong differential adaptive genes (P < 0.05) with the other three M.Yunnan.West populations. Enrichment analysis and PPI network analysis were performed by metascape [50] (https://metascape.org), which incorporates popular ontologies for functional enrichment. Functional categories with − log10(P value) > 2 are displayed as enriched terms across input gene sets. Similar functional categories were classified into one group, and the category with the highest − log(P value) is shown in the enrichment plot.
The data underlying this article are available in the National Omics Data Encyclopedia (NODE) at https://www.biosino.org/node and can be accessed with accession number OEP002587. Requests for access to data may be directed to [email protected]. All data analyzed during this study are included in this article and its supplementary information files.
ACH:
AMM:
Asymmetrical migration model
Coding sequence
CEU:
Utah residents with Northern and Western European ancestry
DAI:
DAF:
Derived allele frequency
DEA:
Deang
ExAC:
Exome Aggregation Consortium
G6PD:
Glucose-6-phosphate dehydrogenase
GTEx:
Genotype-Tissue Expression
HGDP:
Human Genome Diversity Project
IBD:
Identity-by-descendant
ISEA:
Island Southeast Asians
JIP:
Jingpo
Kyoto Encyclopedia of Genes and Genomes
KGP:
1000 Genomes Project
LOF:
Loss-of-function
M.Highland:
Highland minorities in China
M.North:
Minorities in North China
M.South:
Minorities in South China
M.Yunnan.West:
Minorities in Western Yunnan
MSEA:
Mainland Southeast Asians
PBS:
Population branch statistic
PCA:
PPI:
Protein-protein interaction
ROH:
Run of homozygosity
SEA:
SFS:
Site frequency spectrum
SMM:
Symmetrical migration model
SNP:
Single nucleotide polymorphism
TBN:
VEP:
WES:
Whole-exome sequencing
Whole-genome sequencing
XP-EHH:
Cross-population extended haplotype homozygosity
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Sukumaran J, Holder MT. DendroPy: a Python library for phylogenetic computing. Bioinformatics. 2010;26(12):1569–71.
International HapMap Consortium. A second generation human haplotype map of over 3.1 million SNPs. Nature. 2007;449(7164):851–61.
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We are grateful to all volunteers who donated their peripheral blood for the establishment of the Immortalize Cell Bank of Chinese Ethnic Groups, and all participants who contributed to this study.
This study was supported by the National Natural Science Foundation of China (NSFC) grant (32030020 and 31961130380), the Strategic Priority Research Program (XDPB17, XDB38000000) of the Chinese Academy of Sciences (CAS), the Yunnan Applied Basic Research Program (2016FA048), the Yunnan Leading Medical Scientist Training Program (L-2018003), and the Shanghai Municipal Science and Technology Major Project (2017SHZDZX01). The funders had no role in the study design, data collection, analysis, decision to publish, or preparation of the manuscript.
Zhaoqing Yang and Hao Chen contributed equally to this work.
Department of Medical Genetics, Institute of Medical Biology, Chinese Academy of Medical Sciences, Kunming, 650118, China
Zhaoqing Yang, Hao Sun & Jiayou Chu
Key Laboratory of Computational Biology, Shanghai Institute of Nutrition and Health, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Shanghai, 200031, China
Hao Chen
State Key Laboratory of Genetic Engineering, Collaborative Innovation Center for Genetics and Development, Center for Evolutionary Biology, School of Life Sciences, Fudan University, Shanghai, 200438, China
Yan Lu, Jiucun Wang, Li Jin & Shuhua Xu
Human Phenome Institute, Zhangjiang Fudan International Innovation Center, and Ministry of Education Key Laboratory of Contemporary Anthropology, Fudan University, Shanghai, 201203, China
Yang Gao, Jiucun Wang, Li Jin & Shuhua Xu
Department of Liver Surgery and Transplantation Liver Cancer Institute, Zhongshan Hospital, Fudan University, Shanghai, 200032, China
Shuhua Xu
Center for Excellence in Animal Evolution and Genetics, Chinese Academy of Sciences, Kunming, 650223, China
Zhaoqing Yang
Yan Lu
Yang Gao
Hao Sun
Jiucun Wang
Li Jin
Jiayou Chu
S.X. and J.C. conceived the study, S.X. supervised the project. Z.Y. and H.S. collected the LCLs samples, performed the extraction of the genomic DNA, and generated the genome data. Y.G. and H.C. did variant calling. H.C. and Z.Y. performed population genetic analyses. Y.L. and Y.G. contributed to the collection of ethnic cultural materials. L.J. and J.W. contributed the data on Han Chinese. H.C. drafted the manuscript. S.X. revised the manuscript. The authors have read and approved the manuscript.
Correspondence to Jiayou Chu or Shuhua Xu.
All procedures performed in studies involving human participants were approved by the Biomedical Research Ethics Committee of Shanghai Institutes for Biological Sciences (No. ER-SIBS-261408), and in accordance with the 1964 Helsinki declaration, its later amendments or comparable ethical standards. Informed consent was obtained from all individual participants included in the study. The personal identifiers of all samples, if any existed, were stripped off before sequencing and analysis.
Fig. S1. Linguistic distribution of three ancient lineages in Yunnan. Fig. S2. Relatedness among samples of each M.Yunnan.West. Fig. S3. Principal component analysis (PCA) under the global context. Fig. S4. PCA of Global Panel C and M.Yunnan.West. Fig. S5. PCA of M.Yunnan.West and their related populations. Fig. S6. PCA of DAI from different datasets. Fig. S7. Population differentiation measured by FST under the Global Panel C. Fig. S8. Outgroup f3 statistics of each M.Yunnan.West under the Global Panel B. Fig. S9. Unsupervised ADMIXTURE analysis from K = 2 to K = 12 under the Global Panel B. Fig. S10. Unsupervised ADMIXTURE analysis from K = 2 to K = 7 under the Global Panel C. Fig. S11. Cross-validation (CV) error of ADMIXTURE analysis. Fig. S12. Run of homozygosity (ROH) of the populations in the NGS Panel. Fig. S13. Identity-by-descendant (IBD) sharing of the populations in the NGS Panel. Fig. S14. Correlation of ancestral component with altitude and language family. Fig. S15. Potential gene introgression in M.Yunnan.West estimated by f3 statistics. Fig. S16. 2-population dadi models and estimated demographic parameters. Fig. S17. Observed and expected site frequency spectrum (SFS) for three 2-population models constructed in dadi. Fig. S18. Admixture trees estimated using TreeMix based on the NGS Panel. Fig. S19. Population demography estimated by LD-decay approach under the NGS Panel. Fig. S20. Genetic diversity measured by nucleotide differences and novel variants. Fig. S21. Novel variants identified in M.Yunnan.West. Fig. S22. Adaptive signals with extreme significance validated by cross-population extended haplotype homozygosity (XP-EHH). Fig. S23. Gene expression of SYNC in GTEx dataset. Fig. S24. Functional categories enriched from differential gene set in each of M.Yunnan.West. Fig. S25. Protein-protein interaction (PPI) identified from differential gene set in each of M.Yunnan.West. Table S2. Information of the populations in Global Panel C. Table S5. Demographic parameters of 5-population model with 95% CI estimated by dadi. Table S6. Number of novel variants of annotation categories defined by VEP.
Introduction of studied ethnic minorities.
Table S1. Information of samples used in this study.
Table S3. Demographic parameters of 2-population models with 95% CI estimated by dadi. Pairwise estimations among the populations in Global Panel C were performed using three given models constructed in dadi.
Table S4. Topology test for demography model using 3-population models in dadi.
Table S7. List of loss-of-function (LOF) novel variants of M.Yunnan.West populations. LOF novel variants for each M.Yunnan.West were defined as a variant with a high impact classification or a missense variant whose SIFT score is lower than 0.05 and PholyPhen score is higher than 0.446.
Table S8. Annotation of major contributing SNPs for adaptive genes with extreme significance. Differential adaptive genes with extreme significance (P < 0.01) with major contributing SNPs are shown in the table. Major contributing SNPs were defined as SNPs with PBS values larger than 0.1 in at least one of the other three M.Yunnan.West populations.
Table S9. Differential gene set of each M.Yunnan.West population used for enrichment analysis
Table S10. Functional categories enriched from differential gene sets using metascape. Functional categories with −log10(P-value) > 2 are displayed as enriched terms across input gene sets. Similar functional categories were classified into one group. Differential gene set for each M.Yunnan.West population in Additional file 8: Table S9 was used as an input gene set for functional enrichment.
Yang, Z., Chen, H., Lu, Y. et al. Genetic evidence of tri-genealogy hypothesis on the origin of ethnic minorities in Yunnan. BMC Biol 20, 166 (2022). https://doi.org/10.1186/s12915-022-01367-3
Population history
Tri-genealogy hypothesis
Local adaptation | CommonCrawl |
Is Schatten p-norm a monotone ideal norm?
Let $T$ be bounded operators between Hilbert Spaces and define the Schatten p-norm $(p \geq 1)$ \begin{equation*} \sigma_p(T) = \left( \sum_{n=1}^\infty a_n(T)^p \right)^{1/p}, \end{equation*} where $a_n(T)$ are the singular values of $T$.
Suppose that $T \in \mathcal{K}(H_1,H_2)$, $S \in \mathcal{K}(H_1,H_3)$, and \begin{equation*} ||Tx|| \leq ||Sx|| \quad \text{for all} \ x \in H_1, \end{equation*} Does it follow that $ \sigma_p(T) \leq \sigma_p(S)$?
The case $p = 2$ it's clear, because the 2-Schatten class are the Hilbert-Schmidt operators and \begin{equation*} \sigma_2(T)^2 = \sum_{n=1}^\infty ||Te_n||^2 \leq \sum_{n=1}^\infty ||Se_n||^2 = \sigma_2(S)^2, \end{equation*} where $(e_n)$ is an orthonormal basis of $H_1$.
What can we say if $p \neq 2$?
operator-theory hilbert-spaces norm
Javier GonzálezJavier González
Yes the $p$-Schatten norm is monotone. This can be seen from the following fact: A mapping $T:H_1\rightarrow H_2$ is of $p$-Schatten class if and only if$$\{||T\psi_j||_{H_2}\}_{j=1}^{\infty}\in\ell^p$$ for all($2\leq p<\infty$) orthonormal bases/for some ($0<p<2$) orthonormal basis $\{\psi_j\}_j$ of $H_1$ and the $p$-Schatten norm is obtained by maximizing($2\leq p<\infty$) or minimizing($0<p<2$)the expression $$\left(\sum_{j=1}^{\infty}||T\psi_j||^p_{H_2}\right)^{\frac{1}{p}}$$ where the maximum or minimum is taken over all orthonormal bases $\{\psi_j\}_j$ of $H_1$. I proved this fact in my master thesis:
https://www.fernuni-hagen.de/analysis/download/diplomarbeit_melech.pdf
You find the proof in chapter 6 (p.36). Since: $$\left(\sum_{j=1}^{\infty}||T\psi_j||^p_{H_2}\right)^{\frac{1}{p}}\leq \left(\sum_{j=1}^{\infty}||S\psi_j||^p_{H_3}\right)^{\frac{1}{p}}$$ this proves the monotonicity of the $p$-Schatten norm for $0<p<\infty $.
Peter MelechPeter Melech
$\begingroup$ Ok, I found the proof of maximizing for $2 \leq p < \infty$. The case $ 0 < p < 2$ is equivalent? $\endgroup$ – Javier González Feb 28 '18 at 17:28
$\begingroup$ Yes the minimum is obtained by the orthonormal basis consisting of eigenvectors of $T^*T$ in this case $\endgroup$ – Peter Melech Feb 28 '18 at 17:33
$\begingroup$ $\sum_{j=1}^{\infty}||T\psi_j||^p=\sum_{j=1}^{\infty}(T\psi_j,T\psi_j)^{\frac{p}{2}}=\sum_{j=1}^{\infty}(T^*T\psi_j,\psi_j)^{\frac{p}{2}}=\sigma_p(T)^p$ for this basis. This is actually sufficient for what You want to show. In both cases!! $\endgroup$ – Peter Melech Feb 28 '18 at 17:40
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Relation between Schatten-$p$-norm and $l^p$ norm of operator matrix | CommonCrawl |
\begin{document}
\draft \author{V.B. Svetovoy and M.V. Lokhanin} \address{{\small Department of Physics, Yaroslavl State University,} \\ {\small Sovetskaya 14, Yaroslavl 150000, Russia}} \title{Comment on the temperature dependence of the Casimir force.} \maketitle
\begin{abstract} Linear in temperature correction to the Casimir force is discussed. The correction is important for small separations between bodies tested in the recent experiments and disappears in the case of perfect conductors. \end{abstract} \pacs{12.20 Ds, 03.70.+k}
The Casimir force \cite{Casimir} has been measured with high precision in recent experiments \cite{Lam1,MR,RLM}. Also there are plans \cite{Long,Fisch} to look for very weak hypotetical forces where the Casimir force is the main background. All this makes the precise evaluation of the Casimir force an important problem. Here we will discuss a particular problem concerning the temperature dependence of the force between macroscopic bodies made of nonideal metals.
For perfect conductors the temperature correction has been found many years ago \cite{Meh,Brown,Schw} and it is small for small separations between bodies $a\ll c\hbar /kT$ or equivalently for low temperature. For a sphere above a disk the leading term behaves as $\left( T/T_{eff}\right) ^3$, where $T_{eff}=\hbar c/2a$. This result follows from a general expression for the Casimir force given by Lifshitz \cite{Lif,LP} modified for the case of sphere-disk geometry with the proximity force theorem \cite{PFT}:
\begin{equation} \label{shpl}F(a)=-\frac{kTR}{c^2}{\sum\limits_{n=0}^\infty {}}^{\prime }\zeta _n^2\int\limits_1^\infty dpp\ln \left[ \left( 1-G_1e^{-2p\zeta _na/c}\right) \left( 1-G_2e^{-2p\zeta _na/c}\right) \right] , \end{equation}
\noindent where $R$ is the sphere radius,
$$ G_1=\left( \frac{p-s}{p+s}\right) ^2,\quad G_2=\left( \frac{\varepsilon \left( i\zeta _n\right) p-s}{\varepsilon \left( i\zeta _n\right) p+s}\right) ^2,\quad $$
\begin{equation} \label{defin1}s=\sqrt{\varepsilon \left( i\zeta _n\right) -1+p^2},\quad \zeta _n=\frac{2\pi nkT}\hbar , \end{equation}
\noindent $\varepsilon \left( i\zeta _n\right) $ is the dielectric function of the used material at imaginary frequencies. The prime over the sum sign indicates that the first term $n=0$ has to be taken with the coefficient $ 1/2 $.
For small temperature the sum in (\ref{shpl}) can be replaced by the integral over $\zeta $ and the resulting force does not depend on the temperature at all. In general, the replacement is true with the precision $ \sim T/T_{eff}$. In condition of the atomic force microscope experiments \cite{MR,RLM} the smallest separation was $0.1\ \mu m$ and the replacement error can be as large as 3\%. It exceeds the experimental errors $\sim $1\% and, therefore, the finite temperature effect has to be taken into account. We define the temperature correction $\Delta _TF$ as difference between forces written as the sum over $n$ and as the integral instead of this sum.
Special care needs to treat the first term $n=0$ in Eq.(\ref{shpl}). The formal reason is that $\zeta _n^2$ becomes zero but the integral over $p$ diverges. The physical reason is that this term corresponds to the static limit when for metallic bodies $\varepsilon \rightarrow \infty $. This means that any parameter characterizing the dielectric function of a metal cannot appear in the $n=0$ term in contrast with a dielectric for which it will depend on the static permittivity of the material. In the $\varepsilon \rightarrow \infty $ limit the functions $G_{1,2}$ become $G_1=G_2=1$. The formal problem is overcome by introducing the integration over a new variable $x=2p\zeta _na/c$ and after that one can take $\zeta _n=0$ for the $ n=0$ term. Transformed in this way Eq.(\ref{shpl}) will be
\begin{equation} \label{base}F(a)=\frac{kTR}{4a^2}\left\{ \zeta \left( 3\right) -{ \sum\limits_{n=1}^\infty {}}\int\limits_{x_n}^\infty dxx\ln \left[ \left( 1-G_1e^{-x}\right) \left( 1-G_2e^{-x}\right) \right] \right\} , \end{equation}
\noindent where $\zeta \left( m\right) $ is the zeta-function and
\begin{equation} \label{xn}x_n=\frac{2\zeta _na}c. \end{equation}
\noindent Here the first term is linear in temperature and it corresponds to the $n=0$ term in (\ref{shpl}).
The sum in (\ref{base}) as a function of temperature contains a piece linear in $T$ which exactly cancels for ideal metals the first term giving the well known result
\begin{equation} \label{FT}F_T(a)=F_0(a)\left[ 1+\frac{45\zeta \left( 3\right) }{\pi ^3} \left( \frac T{T_{eff}}\right) ^3-\left( \frac T{T_{eff}}\right) ^4\right] , \end{equation}
\noindent where $F_0(a)=\pi ^3\hbar cR/(360a^3)$ is the bare Casimir force between sphere and plate. (\ref{FT}) is written in the small temperature limit when corrections to $F_0(a)$ are very small.
If we are using the dielectric function of a real metal, the cancellation of the first term in (\ref{base}) can be incomplete and the linear in $T$ contribution can survive. That was noted first in \cite{SL}, where Eq.(\ref {base}) was used for numerical calculation of the Casimir force. It was found that for the experiments \cite{MR,RLM} the temperature correction at the smallest separation is $4\ pN$ against the experimental errors $2\ pN$. This conclusion has been criticized in Ref.\cite{BGKM}, where the linear correction was not found. In this connection we would like to clarify here difference in the approaches.
The $n=0$ term was discussed in \cite{BGKM} on the right basis but for actual calculations the following expression has been used
\begin{equation} \label{alt}F(a)=-\frac{kTR}{4a^2}{\sum\limits_{n=0}^\infty {}} ^{\prime}\int\limits_{x_n}^\infty dxx\ln \left[ \left( 1-G_1e^{-x}\right) \left( 1-G_2e^{-x}\right) \right] , \end{equation}
\noindent where for $n=0$ the function $G_1\neq 1$. It is clear from the expression for the force in the high temperature limit, where only the $n=0$ term survives (Eq.(16) in \cite{BGKM})
\begin{equation} \label{hT}F(a)=\frac{kT}{4a^2}R\zeta \left( 3\right) \left( 1-\frac{2c}{ a\omega _p}\right) . \end{equation}
\noindent Here $\omega _p$ is the plasma frequency of the used metal. The parameter $\omega _p$ in this equation shows that the special prescription for the $n=0$ term has not been done. The dielectric function was described by the plasma model where it is
\begin{equation} \label{plasma}\varepsilon \left( i\zeta \right) =1+\frac{\omega _p^2}{\zeta ^2}. \end{equation}
\noindent In the high temperature limit only low frequency fluctuations are important and in this range metals can be much better described by the Drude dielectric function
\begin{equation} \label{Drude}\varepsilon \left( i\zeta \right) =1+\frac{\omega _p^2}{\zeta \left( \zeta +\omega _\tau \right) }, \end{equation}
\noindent where $\omega _\tau $ is the relaxation frequency. However, if we use (\ref{Drude}) to find the classical limit with the help of (\ref{alt}), the result will be wrong, namely, two times smaller than the well known limit $kTR\zeta \left( 3\right) /4a^2$. Eq.(\ref{base}) does not suffer from this problem.
The authors \cite{BGKM} convincingly demonstrated that for low temperatures Eq.(\ref{alt}) does not give the correction linear in $T$ and the leading correction is only $\left( T/T_{eff}\right) ^3$. One can use this result to extract the linear term from the sum in Eq.(\ref{base}) explicitly. The difference between (\ref{base}) and (\ref{alt}) gives the correction we are looking for if one neglects the higher order terms in $T/T_{eff}$
\begin{equation} \label{delT}\Delta _TF=\frac{kTR}{4a^2}\left\{ \zeta \left( 3\right) +\frac 1 2{}\int\limits_0^\infty dxx\ln \left[ \left( 1-G_1e^{-x}\right) \left( 1-G_2e^{-x}\right) \right] \right\} . \end{equation}
\noindent The integral here is the linear term contained in the sum in (\ref {base}) and, of course, it can depend on the material parameters because the summation is going over nonzero frequencies $\zeta _n$. On the other hand, since this integral appeared as the $n=0$ term in (\ref{alt}), we should take the functions $G_{1,2}$ at $x_n=0$. In this limit $G_2=1$ but $G_1\neq 1 $. Using then the relation $\int_0^\infty dxx\ln \left( 1-e^{-x}\right) =-\zeta \left( 3\right) $ one finds the final expression for the correction linear in $T$:
\begin{equation} \label{delTf}\Delta _TF=\frac{kTR}{8a^2}\left[ \zeta \left( 3\right) +\int\limits_0^\infty dxx\ln \left( 1-G_1e^{-x}\right) \right] , \end{equation}
\noindent where
$$ G_1=\left( \frac{x-\sqrt{x^2+\alpha ^{-2}}}{x+\sqrt{x^2+\alpha ^{-2}}} \right) ^2,\qquad \alpha =\frac c{2a\omega _p}. $$
\noindent Let us stress that (\ref{delTf}) is true only for the plasma model. When $\omega _p\rightarrow \infty $ the correction disappears as it should be. Expansion in powers of $\alpha $ gives
\begin{equation} \label{exp}\Delta _TF=\frac{kTR}{8a^2}\zeta \left( 3\right) \cdot 8\alpha \left( 1-3\alpha +O\left( \alpha ^2\right) \right) . \end{equation}
\noindent For $\omega _p=2\cdot 10^{16}\ s^{-1}$ and $a=0.1\ \mu m$ one gets $\Delta _TF\approx 2.5\ pN$ using (\ref{delTf}) or calculating directly with the help of (\ref{base}) and $2.9\ pN$ using (\ref{exp}). The correction increases further if we will use the Drude dielectric function (\ref{Drude} ). In this case it has to be evaluated numerically using (\ref{base}) and ( \ref{shpl}) with the integral instead of the sum. The relaxation frequency $ \omega _\tau $ influences mostly on the integral since it changes low frequency behavior of the itegrand. For typical value $\omega _\tau =5\cdot 10^{13}\ s^{-1}$ we found $\Delta _TF\approx 4.0\ pN$. It cannot be compared directly with the value given in \cite{SL} because layered body cover has been considered there but it is clear that the calculations here give the same order of magnitude for $\Delta _TF$.
In conclusion, we have considered the linear in temperature correction to the Casimir force at low temperatures or equivalently at small separations. Special care has to be taken to get the contribution of the fluctuations in the static limit ($n=0\ $term). This contribution is canceled for ideal mirrors but cancellation is incomplete for real metals. The right treatment of the $n=0$ term allowed to use the Drude dielectric function for metals which is more appropriate at low frequencies than the function in the plasma model.
\begin{references} \bibitem{Casimir} H.B.G. Casimir, Koninkl. Ned. Adak. Wetenschap. Proc. {\bf 51}, 793 (1948).
\bibitem{Lam1} S.K. Lamoreaux, Phys. Rev. Lett. {\bf 78} , 5 (1997); {\bf 81 } , 5475 (1998).
\bibitem{MR} U. Mohideen and A. Roy, Phys. Rev. Lett. {\bf 81} , 4549 (1998). physics/9805032
\bibitem{RLM} A. Roy, C. -Y. Lin, and U. Mohideen, Phys. Rev. {\bf D 60}, 111101 (1999). quant-ph/9906062
\bibitem{Long} J.C. Long, H.W. Chan, and J.C. Price, Nucl. Phys. {\bf B 539} , 23 (1999). hep-ph/9805217
\bibitem{Fisch} E. Fischbach and C. Talmadge, {\it The Search for Non-Newtonian Gravity} (AIP Press/Springer-Verlag, New York, 1999).
\bibitem{Meh} J. Mehra, Physica {\bf 37}, 145 (1967).
\bibitem{Brown} L.S. Brown and G.J. Maclay, Phys. Rev. {\bf 184}, 1272 (1969).
\bibitem{Schw} J. Schwinger, L.L. DeRaad, Jr., and K.A. Milton, Ann. Phys. (N.Y.) {\bf 115}, 1 (1978).
\bibitem{Lif} E.M. Lifshitz, Sov. Phys. JETP {\bf 2}, 73 (1956).
\bibitem{LP} E.M. Lifshitz and L.P. Pitaevskii, {\it Statistical Physics, Part 2}, (Pergamon Press, Oxford, 1980).
\bibitem{PFT} J.Blocki, J. Randrup, W.J. Swiatecki, and C.F. Tsang, Ann. Phys. (N.Y.) {\bf 105},427 (1977).
\bibitem{SL} V.B. Svetovoy and M.V. Lokhanin, e-print quant-ph/0001010.
\bibitem{BGKM} M. Bordag, B. Geyer, G.L. Klimchitskaya, and V.M. Mostepanenko, e-print quant-ph/0003021. \end{references}
\end{document} | arXiv |
# Lectures on Spectral Graph Theory
Fan R. K. Chung<br>Author address:<br>University of Pennsylvania, Philadelphia, Pennsylvania 19104<br>E-mail address: [email protected]
## CHAPTER 1
## Eigenvalues and the Laplacian of a graph
### Introduction
Spectral graph theory has a long history. In the early days, matrix theory and linear algebra were used to analyze adjacency matrices of graphs. Algebraic methods are especially effective in treating graphs which are regular and symmetric. Sometimes, certain eigenvalues have been referred to as the "algebraic connectivity" of a graph [126]. There is a large literature on algebraic aspects of spectral graph theory, well documented in several surveys and books, such as Biggs [25], Cvetković, Doob and Sachs [90, 91], and Seidel [224].
In the past ten years, many developments in spectral graph theory have often had a geometric flavor. For example, the explicit constructions of expander graphs, due to Lubotzky-Phillips-Sarnak [193] and Margulis [195], are based on eigenvalues and isoperimetric properties of graphs. The discrete analogue of the Cheeger inequality has been heavily utilized in the study of random walks and rapidly mixing Markov chains [224]. New spectral techniques have emerged and they are powerful and well-suited for dealing with general graphs. In a way, spectral graph theory has entered a new era.
Just as astronomers study stellar spectra to determine the make-up of distant stars, one of the main goals in graph theory is to deduce the principal properties and structure of a graph from its graph spectrum (or from a short list of easily computable invariants). The spectral approach for general graphs is a step in this direction. We will see that eigenvalues are closely related to almost all major invariants of a graph, linking one extremal property to another. There is no question that eigenvalues play a central role in our fundamental understanding of graphs.
The study of graph eigenvalues realizes increasingly rich connections with many other areas of mathematics. A particularly important development is the interaction between spectral graph theory and differential geometry. There is an interesting analogy between spectral Riemannian geometry and spectral graph theory. The concepts and methods of spectral geometry bring useful tools and crucial insights to the study of graph eigenvalues, which in turn lead to new directions and results in spectral geometry. Algebraic spectral methods are also very useful, especially for extremal examples and constructions. In this book, we take a broad approach with emphasis on the geometric aspects of graph eigenvalues, while including the algebraic aspects as well. The reader is not required to have special background in geometry, since this book is almost entirely graph-theoretic. From the start, spectral graph theory has had applications to chemistry [27]. Eigenvalues were associated with the stability of molecules. Also, graph spectra arise naturally in various problems of theoretical physics and quantum mechanics, for example, in minimizing energies of Hamiltonian systems. The recent progress on expander graphs and eigenvalues was initiated by problems in communication networks. The development of rapidly mixing Markov chains has intertwined with advances in randomized approximation algorithms. Applications of graph eigenvalues occur in numerous areas and in different guises. However, the underlying mathematics of spectral graph theory through all its connections to the pure and applied, the continuous and discrete, can be viewed as a single unified subject. It is this aspect that we intend to cover in this book.
### The Laplacian and eigenvalues
Before we start to define eigenvalues, some explanations are in order. The eigenvalues we consider throughout this book are not exactly the same as those in Biggs [25] or Cvetković, Doob and Sachs [90]. Basically, the eigenvalues are defined here in a general and "normalized" form. Although this might look a little complicated at first, our eigenvalues relate well to other graph invariants for general graphs in a way that other definitions (such as the eigenvalues of adjacency matrices) often fail to do. The advantages of this definition are perhaps due to the fact that it is consistent with the eigenvalues in spectral geometry and in stochastic processes. Many results which were only known for regular graphs can be generalized to all graphs. Consequently, this provides a coherent treatment for a general graph. For definitions and standard graph-theoretic terminology, the reader is referred to [31].
In a graph $G$, let $d_{v}$ denote the degree of the vertex $v$. We first define the Laplacian for graphs without loops and multiple edges (the general weighted case with loops will be treated in Section 1.4). To begin, we consider the matrix $L$, defined as follows:
$$
L(u, v)=\left\{\begin{array}{cl}
d_{v} & \text { if } u=v \\
-1 & \text { if } u \text { and } v \text { are adjacent } \\
0 & \text { otherwise }
\end{array}\right.
$$
Let $T$ denote the diagonal matrix with the $(v, v)$-th entry having value $d_{v}$. The Laplacian of $G$ is defined to be the matrix
$$
\mathcal{L}(u, v)=\left\{\begin{array}{cl}
1 & \text { if } u=v \text { and } d_{v} \neq 0 \\
-\frac{1}{\sqrt{d_{u} d_{v}}} & \text { if } u \text { and } v \text { are adjacent } \\
0 & \text { otherwise. }
\end{array}\right.
$$
We can write
$$
\mathcal{L}=T^{-1 / 2} L T^{-1 / 2}
$$
with the convention $T^{-1}(v, v)=0$ for $d_{v}=0$. We say $v$ is an isolated vertex if $d_{v}=0$. A graph is said to be nontrivial if it contains at least one edge. $\mathcal{L}$ can be viewed as an operator on the space of functions $g: V(G) \rightarrow \mathbb{R}$ which satisfies
$$
\mathcal{L} g(u)=\frac{1}{\sqrt{d_{u}}} \sum_{\substack{v \\ u \sim v}}\left(\frac{g(u)}{\sqrt{d_{u}}}-\frac{g(v)}{\sqrt{d_{v}}}\right)
$$
When $G$ is $k$-regular, it is easy to see that
$$
\mathcal{L}=I-\frac{1}{k} A
$$
where $A$ is the adjacency matrix of $G$, (i. e., $A(x, y)=1$ if $x$ is adjacent to $y$, and 0 otherwise, ) and $I$ is an identity matrix. All matrices here are $n \times n$ where $n$ is the number of vertices in $G$.
For a general graph, we have
$$
\begin{aligned}
\mathcal{L} & =T^{-1 / 2} L T^{-1 / 2} \\
& =I-T^{-1 / 2} A T^{-1 / 2} .
\end{aligned}
$$
We note that $\mathcal{L}$ can be written as
$$
\mathcal{L}=S S^{*},
$$
where $S$ is the matrix whose rows are indexed by the vertices and whose columns are indexed by the edges of $G$ such that each column corresponding to an edge $e=\{u, v\}$ has an entry $1 / \sqrt{d_{u}}$ in the row corresponding to $u$, an entry $-1 / \sqrt{d_{v}}$ in the row corresponding to $v$, and has zero entries elsewhere. (As it turns out, the choice of signs can be arbitrary as long as one is positive and the other is negative.) Also, $S^{*}$ denotes the transpose of $S$.
For readers who are familiar with terminology in homology theory, we remark that $S$ can be viewed as a "boundary operator" mapping "1-chains" defined on edges (denoted by $C_{1}$ ) of a graph to "0-chains" defined on vertices (denoted by $C_{0}$ ). Then, $S^{*}$ is the corresponding "coboundary operator" and we have
$$
C_{1} \underset{S^{*}}{\stackrel{S}{\rightleftarrows}} C_{0}
$$
Since $\mathcal{L}$ is symmetric, its eigenvalues are all real and non-negative. We can use the variational characterizations of those eigenvalues in terms of the Rayleigh quotient of $\mathcal{L}$ (see, e.g. [162]). Let $g$ denote an arbitrary function which assigns to each vertex $v$ of $G$ a real value $g(v)$. We can view $g$ as a column vector. Then
$$
\begin{aligned}
\frac{\langle g, \mathcal{L} g\rangle}{\langle g, g\rangle} & =\frac{\left\langle g, T^{-1 / 2} L T^{-1 / 2} g\right\rangle}{\langle g, g\rangle} \\
& =\frac{\langle f, L f\rangle}{\left\langle T^{1 / 2} f, T^{1 / 2} f\right\rangle} \\
& =\frac{\sum_{u \sim v}(f(u)-f(v))^{2}}{\sum_{v} f(v)^{2} d_{v}}
\end{aligned}
$$
where $g=T^{1 / 2} f$ and $\sum_{u \sim v}$ denotes the sum over all unordered pairs $\{u, v\}$ for which $u$ and $v$ are adjacent. Here $\langle f, g\rangle=\sum_{x} f(x) g(x)$ denotes the standard inner product in $\mathbb{R}^{n}$. The sum $\sum_{u \sim v}(f(u)-f(v))^{2}$ is sometimes called the Dirichlet sum of $G$ and the ratio on the left-hand side of (1.1) is often called the Rayleigh quotient. (We note that we can also use the inner product $\langle f, g\rangle=\sum \overline{f(x)} g(x)$ for complex-valued functions.)
From equation (1.1), we see that all eigenvalues are non-negative. In fact, we can easily deduce from equation (1.1) that 0 is an eigenvalue of $\mathcal{L}$. We denote the eigenvalues of $\mathcal{L}$ by $0=\lambda_{0} \leq \lambda_{1} \leq \cdots \leq \lambda_{n-1}$. The set of the $\lambda_{i}$ 's is usually called the spectrum of $\mathcal{L}$ (or the spectrum of the associated graph G.) Let $\mathbf{1}$ denote the constant function which assumes the value 1 on each vertex. Then $T^{1 / 2} \mathbf{1}$ is an eigenfunction of $\mathcal{L}$ with eigenvalue 0 . Furthermore,
$$
\lambda_{G}=\lambda_{1}=\inf _{f \perp T \mathbf{1}} \frac{\sum_{u \sim v}(f(u)-f(v))^{2}}{\sum_{v} f(v)^{2} d_{v}} .
$$
The corresponding eigenfunction is $g=T^{1 / 2} f$ as in (1.1). It is sometimes convenient to consider the nontrivial function $f$ achieving (1.2), in which case we call $f$ a harmonic eigenfunction of $\mathcal{L}$.
The above formulation for $\lambda_{G}$ corresponds in a natural way to the eigenvalues of the Laplace-Beltrami operator for Riemannian manifolds:
$$
\lambda_{M}=\inf \frac{\int_{M}|\nabla f|^{2}}{\int_{M}|f|^{2}}
$$
where $f$ ranges over functions satisfying
$$
\int_{M} f=0 .
$$
We remark that the corresponding measure here for each edge is 1 although in the general case for weighted graphs the measure for an edge is associated with the edge weight (see Section 1.4.) The measure for each vertex is the degree of the vertex. A more general notion of vertex weights will be considered in Section 2.5. We note that (1.2) has several different formulations:
$$
\begin{aligned}
\lambda_{1} & =\inf _{f} \sup _{t} \frac{\sum_{u \sim v}(f(u)-f(v))^{2}}{\sum_{v}(f(v)-t)^{2} d_{v}} \\
& =\inf _{f} \frac{\sum_{u \sim v}(f(u)-f(v))^{2}}{\sum_{v}(f(v)-\bar{f})^{2} d_{v}},
\end{aligned}
$$
where
$$
\bar{f}=\frac{\sum_{v} f(v) d_{v}}{\operatorname{vol} G},
$$
and vol $G$ denotes the volume of the graph $G$, given by
$$
\operatorname{vol} G=\sum_{v} d_{v}
$$
By substituting for $\bar{f}$ and using the fact that $N \sum_{i=1}^{N}\left(a_{i}-a\right)^{2}=\sum_{i<j}\left(a_{i}-a_{j}\right)^{2}$ for $a=\sum_{i=1}^{N} a_{i} / N$, we have the following expression (which generalizes the one in $[126])$ :
$$
\lambda_{1}=\operatorname{vol} G \inf _{f} \frac{\sum_{u \sim v}(f(u)-f(v))^{2}}{\sum_{u, v}(f(u)-f(v))^{2} d_{u} d_{v}},
$$
where $\sum_{u, v}$ denotes the sum over all unordered pairs of vertices $u, v$ in $G$. We can characterize the other eigenvalues of $\mathcal{L}$ in terms of the Rayleigh quotient. The largest eigenvalue satisfies:
$$
\lambda_{n-1}=\sup _{f} \frac{\sum_{u \sim v}(f(u)-f(v))^{2}}{\sum_{v} f^{2}(v) d_{v}} .
$$
For a general $k$, we have
$$
\begin{aligned}
\lambda_{k} & =\inf _{f} \sup _{g \in P_{k-1}} \frac{\sum_{u \sim v}(f(u)-f(v))^{2}}{\sum_{v}(f(v)-g(v))^{2} d_{v}} \\
& =\inf _{f \perp T P_{k-1}} \frac{\sum_{u \sim v}(f(u)-f(v))^{2}}{\sum_{v} f(v)^{2} d_{v}}
\end{aligned}
$$
where $P_{i}$ is the subspace generated by the harmonic eigenfunctions corresponding to $\lambda_{i}$, for $i \leq k-1$.
The different formulations for eigenvalues given above are useful in different settings and they will be used in later chapters. Here are some examples of special graphs and their eigenvalues.
EXAmple 1.1. For the complete graph $K_{n}$ on $n$ vertices, the eigenvalues are 0 and $n /(n-1)$ (with multiplicity $n-1$ ).
EXAmPle 1.2. For the complete bipartite graph $K_{m, n}$ on $m+n$ vertices, the eigenvalues are 0,1 (with multiplicity $m+n-2$ ), and 2 .
EXAmple 1.3. For the star $S_{n}$ on $n$ vertices, the eigenvalues are 0,1 (with multiplicity $n-2$ ), and 2 .
Example 1.4. For the path $P_{n}$ on $n$ vertices, the eigenvalues are $1-\cos \frac{\pi k}{n-1}$ for $k=0,1, \cdots, n-1$.
Example 1.5 . For the cycle $C_{n}$ on $n$ vertices, the eigenvalues are $1-\cos \frac{2 \pi k}{n}$ for $k=0, \cdots, n-1$.
EXAmple 1.6. For the $n$-cube $Q_{n}$ on $2^{n}$ vertices, the eigenvalues are $\frac{2 k}{n}$ (with multiplicity $\left(\begin{array}{l}n \\ k\end{array}\right)$ ) for $k=0, \cdots, n$.
More examples can be found in Chapter 6 on explicit constructions.
### Basic facts about the spectrum of a graph
Roughly speaking, half of the main problems of spectral theory lie in deriving bounds on the distributions of eigenvalues. The other half concern the impact and consequences of the eigenvalue bounds as well as their applications. In this section, we start with a few basic facts about eigenvalues. Some simple upper bounds and lower bounds are stated. For example, we will see that the eigenvalues of any graph lie between 0 and 2 . The problem of narrowing the range of the eigenvalues for special classes of graphs offers an open-ended challenge. Numerous questions can be asked either in terms of other graph invariants or under further assumptions imposed on the graphs. Some of these will be discussed in subsequent chapters.
Lemma 1.7. For a graph $G$ on $n$ vertices, we have
(i):
$$
\sum_{i} \lambda_{i} \leq n
$$
with equality holding if and only if $G$ has no isolated vertices.
(ii): For $n \geq 2$,
$$
\lambda_{1} \leq \frac{n}{n-1}
$$
with equality holding if and only if $G$ is the complete graph on $n$ vertices. Also, for a graph $G$ without isolated vertices, we have
$$
\lambda_{n-1} \geq \frac{n}{n-1} .
$$
(iii): For a graph which is not a complete graph, we have $\lambda_{1} \leq 1$.
(iv): If $G$ is connected, then $\lambda_{1}>0$. If $\lambda_{i}=0$ and $\lambda_{i+1} \neq 0$, then $G$ has exactly $i+1$ connected components.
(v): For all $i \leq n-1$, we have
$$
\lambda_{i} \leq 2
$$
with $\lambda_{n-1}=2$ if and only if a connected component of $G$ is bipartite and nontrivial.
(vi): The spectrum of a graph is the union of the spectra of its connected components.
Proof. (i) follows from considering the trace of $\mathcal{L}$.
The inequalities in (ii) follow from (i) and $\lambda_{0}=0$.
Suppose $G$ contains two nonadjacent vertices $a$ and $b$, and consider
$$
f_{1}(v)=\left\{\begin{aligned}
d_{b} & \text { if } v=a, \\
-d_{a} & \text { if } v=b, \\
0 & \text { if } v \neq a, b
\end{aligned}\right.
$$
(iii) then follows from (1.2).
If $G$ is connected, the eigenvalue 0 has multiplicity 1 since any harmonic eigenfunction with eigenvalue 0 assumes the same value at each vertex. Thus, (iv) follows from the fact that the union of two disjoint graphs has as its spectrum the union of the spectra of the original graphs.
(v) follows from equation (1.6) and the fact that
$$
(f(x)-f(y))^{2} \leq 2\left(f^{2}(x)+f^{2}(y)\right) .
$$
Therefore
$$
\lambda_{i} \leq \sup _{f} \frac{\sum_{x \sim y}(f(x)-f(y))^{2}}{\sum_{x} f^{2}(x) d_{x}} \leq 2 .
$$
Equality holds for $i=n-1$ when $f(x)=-f(y)$ for every edge $\{x, y\}$ in $G$. Therefore, since $f \neq 0, G$ has a bipartite connected component. On the other hand, if $G$ has a connected component which is bipartite, we can choose the function $f$ so as to make $\lambda_{n-1}=2$.
(vi) follows from the definition.
For bipartite graphs, the following slightly stronger result holds:
LEMMA 1.8. The following statements are equivalent:
(i): $G$ is bipartite.
(ii): $G$ has $i+1$ connected components and $\lambda_{n-j}=2$ for $1 \leq j \leq i$.
(iii): For each $\lambda_{i}$, the value $2-\lambda_{i}$ is also an eigenvalue of $G$. Proof. It suffices to consider a connected graph. Suppose $G$ is bipartite graph with vertex set consisting of two parts $A$ and $B$. For any harmonic eigenfunction $f$ with eigenvalue $\lambda$, we consider the function $g$
$$
g(x)=\left\{\begin{array}{cc}
f(x) & \text { if } x \in A \\
-f(x) & \text { if } x \in B
\end{array}\right.
$$
It is easy to check that $g$ is a harmonic eigenfunction with eigenvalue $2-\lambda$.
For a connected graph, we can immediately improve the lower bound of $\lambda_{1}$ in Lemma 1.7. For two vertices $u$ and $v$, the distance between $u$ and $v$ is the number of edges in a shortest path joining $u$ and $v$. The diameter of a graph is the maximum distance between any two vertices of $G$. Here we will give a simple eigenvalue lower bound in terms of the diameter of a graph. More discussion on the relationship between eigenvalues and diameter will be given in Chapter 3 .
Lemma 1.9. For a connected graph $G$ with diameter $D$, we have
$$
\lambda_{1} \geq \frac{1}{D \text { vol } G}
$$
Proof. Suppose $f$ is a harmonic eigenfunction achieving $\lambda_{1}$ in (1.2). Let $v_{0}$ denote a vertex with $\left|f\left(v_{0}\right)\right|=\max _{v}|f(v)|$. Since $\sum_{v} f(v)=0$, there exists a vertex $u_{0}$ satisfying $f\left(u_{0}\right) f\left(v_{0}\right)<0$. Let $P$ denote a shortest path in $G$ joining $u_{0}$ and $v_{0}$. Then by $(1.2)$ we have
$$
\begin{aligned}
\lambda_{1} & =\frac{\sum_{x \sim y}(f(x)-f(y))^{2}}{\sum_{x} f^{2}(x) d_{x}} \\
& \geq \frac{\sum_{\{x, y\} \in P}(f(x)-f(y))^{2}}{\operatorname{vol} G f^{2}\left(v_{0}\right)} \\
& \geq \frac{\frac{1}{D}\left(f\left(v_{0}\right)-f\left(u_{0}\right)\right)^{2}}{\operatorname{vol} G f^{2}\left(v_{0}\right)} \\
& \geq \frac{1}{D \operatorname{vol} G}
\end{aligned}
$$
by using the Cauchy-Schwarz inequality.
LeMma 1.10. Let $f$ denote a harmonic eigenfunction achieving $\lambda_{G}$ in (1.2). Then, for any vertex $x \in V$, we have
$$
\frac{1}{d_{x}} \sum_{\substack{y \\ y \sim x}}(f(x)-f(y))=\lambda_{G} f(x) .
$$
Proof. We use a variational argument. For a fixed $x_{0} \in V$, we consider $f_{\epsilon}$ such that
$$
f_{\epsilon}(y)= \begin{cases}f\left(x_{0}\right)+\frac{\epsilon}{d_{x_{0}}} \epsilon & \text { if } y=x_{0}, \\ f(y)-\frac{\epsilon}{\operatorname{vol} G-d_{x_{0}}} & \text { otherwise. }\end{cases}
$$
We have
$$
\begin{aligned}
& \frac{\sum_{\substack{x, y \in V \\
x \sim y}}\left(f_{\epsilon}(x)-f_{\epsilon}(y)\right)^{2}}{\sum_{x \in V} f_{\epsilon}^{2}(x) d_{x}} \\
& =\frac{\sum_{\substack{x, y \in V \\
x \sim y}}(f(x)-f(y))^{2}+\sum_{\substack{y \\
y \sim x_{0}}} \frac{2 \epsilon\left(f\left(x_{0}\right)-f(y)\right)}{d_{x_{0}}}-\sum_{\substack{y \\
y \neq x_{0}}} \sum_{\substack{y^{\prime} \\
y \sim y^{\prime}}} \frac{2 \epsilon\left(f(y)-f\left(y^{\prime}\right)\right)}{\operatorname{vol} G-d_{x_{0}}}}{\sum_{x \in V} f^{2}(x) d_{x}+2 \epsilon f\left(x_{0}\right)-\frac{2 \epsilon}{\operatorname{vol} G-d_{x_{0}}} \sum_{y \neq x_{0}} f(y) d_{y}} \\
& +O\left(\epsilon^{2}\right) \\
& =\frac{\sum_{\substack{x, y \in V \\
x \sim y}}(f(x)-f(y))^{2}+\frac{2 \epsilon \sum_{\substack{y \\
y \sim x_{0}}}\left(f\left(x_{0}\right)-f(y)\right)}{d_{x_{0}}}+\frac{2 \epsilon \sum_{\substack{y \\
y \sim x_{0}}}\left(f\left(x_{0}\right)-f(y)\right)}{\operatorname{vol} G-d_{x_{0}}}}{\sum_{x \in V} f^{2}(x) d_{x}+2 \epsilon f\left(x_{0}\right)+\frac{2 \epsilon f\left(x_{0}\right) d_{x_{0}}}{\operatorname{vol} G-d_{x_{0}}}} \\
& +O\left(\epsilon^{2}\right)
\end{aligned}
$$
since $\sum_{x \in V} f(x) d_{x}=0$, and $\sum_{y} \sum_{y^{\prime}}\left(f(y)-f\left(y^{\prime}\right)\right)=0$. The definition in (1.2) implies that
$$
\frac{\sum_{\substack{x, y \in V \\ x \sim y}}\left(f_{\epsilon}(x)-f_{\epsilon}(y)\right)^{2}}{\sum_{x \in V} f_{\epsilon}^{2}(x) d_{x}} \geq \frac{\sum_{\substack{x, y \in V \\ x \sim y}}(f(x)-f(y))^{2}}{\sum_{x \in V} f^{2}(x) d_{x}}
$$
If we consider what happens to the Rayleigh quotient for $f_{\epsilon}$ as $\epsilon \rightarrow 0$ from above, or from below, we can conclude that
$$
\frac{1}{d_{x_{0}}} \sum_{\substack{y \\ y \sim x_{0}}}\left(f\left(x_{0}\right)-f(y)\right)=\lambda_{G} f\left(x_{0}\right) .
$$
and the Lemma is proved. One can also prove the statement in Lemma 1.10 by recalling that $f=T^{-1 / 2} g$, where $\mathcal{L} g=\lambda_{G} g$. Then
$$
T^{-1} L f=T^{-1}\left(T^{1 / 2} \mathcal{L} T^{1 / 2}\right)\left(T^{-1 / 2} g\right)=T^{-1 / 2} \lambda_{G} g=\lambda_{G} f,
$$
and examining the entries gives the desired result.
With a little linear algebra, we can improve the bounds on eigenvalues in terms of the degrees of the vertices.
We consider the trace of $(I-\mathcal{L})^{2}$. We have
$$
\begin{aligned}
\operatorname{Tr}(I-\mathcal{L})^{2} & =\sum_{i}\left(1-\lambda_{i}\right)^{2} \\
& \leq 1+(n-1) \bar{\lambda}^{2},
\end{aligned}
$$
where
$$
\bar{\lambda}=\max _{i \neq 0}\left|1-\lambda_{i}\right|
$$
On the other hand,
$$
\begin{aligned}
\operatorname{Tr}(I-\mathcal{L})^{2} & =\operatorname{Tr}\left(T^{-1 / 2} A T^{-1} A T^{-1 / 2}\right) \\
& =\sum_{x, y} \frac{1}{\sqrt{d_{x}}} A(x, y) \frac{1}{d_{y}} A(y, x) \frac{1}{\sqrt{d_{x}}} \\
& =\sum_{x} \frac{1}{d_{x}}-\sum_{x \sim y}\left(\frac{1}{d_{x}}-\frac{1}{d_{y}}\right)^{2},
\end{aligned}
$$
where $A$ is the adjacency matrix. From this, we immediately deduce
Lemma 1.11. For a k-regular graph $G$ on $n$ vertices, we have
$$
\max _{i \neq 0}\left|1-\lambda_{i}\right| \geq \sqrt{\frac{n-k}{(n-1) k}}
$$
This follows from the fact that
$$
\max _{i \neq 0}\left|1-\lambda_{i}\right|^{2} \geq \frac{1}{n-1}\left(\operatorname{tr}(I-\mathcal{L})^{2}-1\right)
$$
Let $d_{H}$ denote the harmonic mean of the $d_{v}$ 's, i.e.,
$$
\frac{1}{d_{H}}=\frac{1}{n} \sum_{v} \frac{1}{d_{v}}
$$
It is tempting to consider generalizing (1.11) with $k$ replaced by $d_{H}$. This, however, is not true as shown by the following example due to Elizabeth Wilmer.
EXAmple 1.12. Consider the $m$-petal graph on $n=2 m+1$ vertices, $v_{0}, v_{1}, \cdots$, $v_{2 m}$ with edges $\left\{v_{0}, v_{i}\right\}$ and $\left\{v_{2 i-1}, v_{2 i}\right\}$, for $i \geq 1$. This graph has eigenvalues $0,1 / 2$ (with multiplicity $m-1$ ), and $3 / 2$ (with multiplicity $m+1$ ). So we have $\max _{i \neq 0}\left|1-\lambda_{i}\right|=1 / 2$. However,
as $m \rightarrow \infty$.
$$
\sqrt{\frac{n-d_{H}}{(n-1) d_{H}}}=\sqrt{\frac{m-1 / 2}{2 m}} \rightarrow \frac{1}{\sqrt{2}}
$$
Still, for a general graph, we can use the fact that
$$
\frac{\sum_{x \sim y}\left(\frac{1}{d_{x}}-\frac{1}{d_{y}}\right)^{2}}{\sum_{x \in V}\left(\frac{1}{d_{x}}-\frac{1}{d_{H}}\right)^{2} d_{x}} \leq \lambda_{n-1} \leq 1+\bar{\lambda} .
$$
Combining (1.9), (1.10) and (1.12), we obtain the following:
Lemma 1.13. For a graph $G$ on $n$ vertices, $\bar{\lambda}=\max _{i \neq 0}\left|1-\lambda_{i}\right|$ satisfies
$$
1+(n-1) \bar{\lambda}^{2} \geq \frac{n}{d_{H}}\left(1-(1+\bar{\lambda})\left(\frac{k}{d_{H}}-1\right)\right),
$$
where $k$ denotes the average degree of $G$.
There are relatively easy ways to improve the upper bound for $\lambda_{1}$. From the characterization in the preceding section, we can choose any function $f: V(G) \rightarrow \mathbb{R}$, and its Rayleigh quotient will serve as an upper bound for $\lambda_{1}$. Here we describe an upper bound for $\lambda_{1}$ (see [204]).
Lemma 1.14. Let $G$ be a graph with diameter $D \geq 4$, and let $k$ denote the maximum degree of $G$. Then
$$
\lambda_{1} \leq 1-2 \frac{\sqrt{k-1}}{k}\left(1-\frac{2}{D}\right)+\frac{2}{D} .
$$
One way to bound eigenvalues from above is to consider "contracting" the graph $G$ into a weighted graph $H$ (which will be defined in the next section). Then the eigenvalues of $G$ can be upper-bounded by the eigenvalues of $H$ or by various upper bounds on them, which might be easier to obtain. We remark that the proof of Lemma 1.14 proceeds by basically contracting the graph into a weighted path. We will prove Lemma 1.14 in the next section.
We note that Lemma 1.14 gives a proof (see [5]) that for any fixed $k$ and for any infinite family of regular graphs with degree $k$,
$$
\lim \sup \lambda_{1} \leq 1-2 \frac{\sqrt{k-1}}{k} \text {. }
$$
This bound is the best possible since it is sharp for the Ramanujan graphs (which will be discussed in Chapter ??). We note that the cleaner version of $\lambda_{1} \leq$ $1-2 \sqrt{k-1} / k$ is not true for certain graphs (e.g., 4-cycles or complete bipartite graphs). This example also illustrates that the assumption in Lemma 1.14 concerning $D \geq 4$ is essential.
### Eigenvalues of weighted graphs
Before defining weighted graphs, we will say a few words about two different approaches for giving definitions. We could have started from the very beginning with weighted graphs, from which simple graphs arise as a special case in which the weights are 0 or 1 . However, the unique characteristics and special strength of graph theory is its ability to deal with the $\{0,1\}$-problems arising in many natural situations. The clean formulation of a simple graph has conceptual advantages. Furthermore, as we shall see, all definitions and subsequent theorems for simple graphs can usually be easily carried out for weighted graphs. A weighted undirected graph $G$ (possibly with loops) has associated with it a weight function $w: V \times V \rightarrow$ $\mathbb{R}$ satisfying
$$
w(u, v)=w(v, u)
$$
and
$$
w(u, v) \geq 0 .
$$
We note that if $\{u, v\} \notin E(G)$, then $w(u, v)=0$. Unweighted graphs are just the special case where all the weights are 0 or 1 .
In the present context, the degree $d_{v}$ of a vertex $v$ is defined to be:
$$
\begin{gathered}
d_{v}=\sum_{u} w(u, v), \\
\operatorname{vol} G=\sum_{v} d_{v} .
\end{gathered}
$$
We generalize the definitions of previous sections, so that
$$
L(u, v)= \begin{cases}d_{v}-w(v, v) & \text { if } u=v \\ -w(u, v) & \text { if } u \text { and } v \text { are adjacent } \\ 0 & \text { otherwise }\end{cases}
$$
In particular, for a function $f: V \rightarrow \mathbb{R}$, we have
$$
L f(x)=\sum_{\substack{y \\ x \sim y}}(f(x)-f(y)) w(x, y) .
$$
Let $T$ denote the diagonal matrix with the $(v, v)$-th entry having value $d_{v}$. The Laplacian of $G$ is defined to be
$$
\mathcal{L}=T^{-1 / 2} L T^{-1 / 2}
$$
In other words, we have
$$
\mathcal{L}(u, v)= \begin{cases}1-\frac{w(v, v)}{d_{v}} & \text { if } u=v, \text { and } d_{v} \neq 0 \\ -\frac{w(u, v)}{\sqrt{d_{u} d_{v}}} & \text { if } u \text { and } v \text { are adjacent } \\ 0 & \text { otherwise }\end{cases}
$$
We can still use the same characterizations for the eigenvalues of the generalized versions of $\mathcal{L}$. For example,
$$
\begin{aligned}
\lambda_{G}:=\lambda_{1} & =\inf _{g \perp T^{1 / 2} 1} \frac{\langle g, \mathcal{L} g\rangle}{\langle g, g\rangle} \\
& =\inf _{\substack{f \\
\sum f(x) d_{x}=0}} \frac{\sum_{x \in V} f(x) L f(x)}{\sum_{x \in V} f^{2}(x) d_{x}} \\
& =\inf _{\substack{f \\
\sum f(x) d_{x}=0}} \frac{\sum_{x \sim y}(f(x)-f(y))^{2} w(x, y)}{\sum_{x \in V} f^{2}(x) d_{x}} .
\end{aligned}
$$
A contraction of a graph $G$ is formed by identifying two distinct vertices, say $u$ and $v$, into a single vertex $v^{*}$. The weights of edges incident to $v^{*}$ are defined as follows:
$$
\begin{aligned}
w\left(x, v^{*}\right) & =w(x, u)+w(x, v), \\
w\left(v^{*}, v^{*}\right) & =w(u, u)+w(v, v)+2 w(u, v) .
\end{aligned}
$$
LemMa 1.15. If $H$ is formed by contractions from a graph $G$, then
$$
\lambda_{G} \leq \lambda_{H}
$$
The proof follows from the fact that an eigenfunction which achieves $\lambda_{H}$ for $H$ can be lifted to a function defined on $V(G)$ such that all vertices in $G$ that contract to the same vertex in $H$ share the same value.
Now we return to Lemma 1.14.
SKETCHED PROOF OF LEMMA 1.14:
Let $u$ and $v$ denote two vertices that are at distance $D \geq 2 t+2$ in $G$. We contract $G$ into a path $H$ with $2 t+2$ edges, with vertices $x_{0}, x_{1}, \ldots x_{t}, z, y_{t}, \ldots, y_{2}, y_{1}, y_{0}$ such that vertices at distance $i$ from $u, 0 \leq i \leq t$, are contracted to $x_{i}$, and vertices at distance $j$ from $v, 0 \leq j \leq t$, are contracted to $y_{j}$. The remaining vertices are contracted to $z$. To establish an upper bound for $\lambda_{1}$, it is enough to choose a suitable function $f$, defined as follows:
$$
\begin{aligned}
f\left(x_{i}\right) & =a(k-1)^{-i / 2}, \\
f\left(y_{j}\right) & =b(k-1)^{-j / 2}, \\
f(z) & =0
\end{aligned}
$$
where the constants $a$ and $b$ are chosen so that
$$
\sum_{x} f(x) d_{x}=0
$$
It can be checked that the Rayleigh quotient satisfies
$$
\frac{\sum_{u \sim v}(f(u)-f(v))^{2} w(u, v)}{\sum_{v} f(v)^{2} d_{v}} \leq 1-\frac{2 \sqrt{k-1}}{k}\left(1-\frac{1}{t+1}\right)+\frac{1}{t+1},
$$
since the ratio is maximized when $w\left(x_{i}, x_{i+1}\right)=k(k-1)^{i-1}=w\left(y_{i}, y_{i+1}\right)$. This completes the proof of the lemma.
### Eigenvalues and random walks
In a graph $G$, a walk is just a sequence of vertices $\left(v_{0}, v_{1}, \cdots, v_{s}\right)$ with $\left\{v_{i-1}, v_{i}\right\} \in E(G)$ for all $1 \leq i \leq s$. A random walk is determined by the transition probabilities $P(u, v)=\operatorname{Prob}\left(x_{i+1}=v \mid x_{i}=u\right)$, which are independent of $i$. Clearly, for each vertex $u$,
$$
\sum_{v} P(u, v)=1
$$
For any initial distribution $f: V \rightarrow \mathbb{R}$ with $\sum_{v} f(v)=1$, the distribution after $k$ steps is just $f P^{k}$ (i.e., a matrix multiplication with $f$ viewed as a row vector where $P$ is the matrix of transition probabilities). The random walk is said to be ergodic if there is a unique stationary distribution $\pi(v)$ satisfying
$$
\lim _{s \rightarrow \infty} f P^{s}(v)=\pi(v) .
$$
It is easy to see that necessary conditions for the ergodicity of $P$ are (i) irreducibility, i.e., for any $u, v \in V$, there exists some $s$ such that $P^{s}(u, v)>0$ (ii) aperiodicity, i.e., g.c.d. $\left\{s: P^{s}(u, v)>0\right\}=1$. As it turns out, these are also sufficient conditions. A major problem of interest is to determine the number of steps $s$ required for $P^{s}$ to be close to its stationary distribution, given an arbitrary initial distribution.
We say a random walk is reversible if
$$
\pi(u) P(u, v)=\pi(v) P(v, u) .
$$
An alternative description for a reversible random walk can be given by considering a weighted connected graph with edge weights satisfying
$$
w(u, v)=w(v, u)=\pi(v) P(v, u) / c
$$
where $c$ can be any constant chosen for the purpose of simplifying the values. (For example, we can take $c$ to be the average of $\pi(v) P(v, u)$ over all $(v, u)$ with $P(v, u) \neq 0$, so that the values for $w(v, u)$ are either 0 or 1 for a simple graph.) The random walk on a weighted graph has as its transition probabilities
$$
P(u, v)=\frac{w(u, v)}{d_{u}}
$$
where $d_{u}=\sum_{z} w(u, z)$ is the (weighted) degree of $u$. The two conditions for ergodicity are equivalent to the conditions that the graph be (i) connected and (ii) non-bipartite. From Lemma 1.7, we see that (i) is equivalent to $\lambda_{1}>0$ and (ii) implies $\lambda_{n-1}<2$. As we will see later in (1.15), together (i) and (ii) deduce ergodicity.
We remind the reader that an unweighted graph has $w(u, v)$ equal to either 0 or 1 . The usual random walk on an unweighted graph has transition probability $1 / d_{v}$ of moving from a vertex $v$ to any one of its neighbors. The transition matrix $P$ then satisfies
$$
P(u, v)= \begin{cases}1 / d_{u} & \text { if } u \text { and } v \text { are adjacent } \\ 0 & \text { otherwise }\end{cases}
$$
In other words,
$$
f P(v)=\sum_{\substack{u \\ u \sim v}} \frac{1}{d_{u}} f(u)
$$
for any $f: V(G) \rightarrow \mathbb{R}$.
It is easy to check that
$$
P=T^{-1} A=T^{-1 / 2}(I-\mathcal{L}) T^{1 / 2},
$$
where $A$ is the adjacency matrix.
In a random walk with an associated weighted connected graph $G$, the transition matrix $P$ satisfies
$$
\mathbf{1 T P}=\mathbf{1} T
$$
where $\mathbf{1}$ is the vector with all coordinates 1 . Therefore the stationary distribution is exactly $\pi=1 T /$ vol $G$, We want to show that when $k$ is large enough, for any initial distribution $f: V \rightarrow \mathbb{R}, f P^{k}$ converges to the stationary distribution.
First we consider convergence in the $L_{2}$ (or Euclidean) norm. Suppose we write
$$
f T^{-1 / 2}=\sum_{i} a_{i} \phi_{i}
$$
where $\phi_{i}$ denotes the orthonormal eigenfunction associated with $\lambda_{i}$.
Recall that $\phi_{0}=\mathbf{1} T^{1 / 2} / \sqrt{\operatorname{vol} G}$ and $\|\cdot\|$ denotes the $L_{2}$-norm, so
$$
a_{0}=\frac{\left\langle f T^{-1 / 2}, \mathbf{1} T^{1 / 2}\right\rangle}{\left\|\mathbf{1} T^{1 / 2}\right\|}=\frac{1}{\sqrt{\text { vol } G}}
$$
since $\langle f, \mathbf{1}\rangle=1$. We then have
$$
\begin{aligned}
\left\|f P^{s}-\pi\right\| & =\| f P^{s}-\mathbf{1} T / \text { vol } G \| \\
& =\left\|f P^{s}-a_{0} \phi_{0} T^{1 / 2}\right\| \\
& =\left\|f T^{-1 / 2}(I-\mathcal{L})^{s} T^{1 / 2}-a_{0} \phi_{0} T^{1 / 2}\right\| \\
& =\left\|\sum_{i \neq 0}\left(1-\lambda_{i}\right)^{s} a_{i} \phi_{i} T^{1 / 2}\right\| \\
& \leq\left(1-\lambda^{\prime}\right)^{s} \frac{\max _{x} \sqrt{d_{x}}}{\min _{y} \sqrt{d_{y}}} \\
& \leq e^{-s \lambda^{\prime}} \frac{\max _{x} \sqrt{d_{x}}}{\min _{y} \sqrt{d_{y}}}
\end{aligned}
$$
where
$$
\lambda^{\prime}=\left\{\begin{array}{cl}
\lambda_{1} & \text { if } 1-\lambda_{1} \geq \lambda_{n-1}-1 \\
2-\lambda_{n-1} & \text { otherwise. }
\end{array}\right.
$$
So, after $s \geq 1 / \lambda^{\prime} \log \left(\max _{x} \sqrt{d_{x}} / \epsilon \min _{y} \sqrt{d_{y}}\right)$ steps, the $L_{2}$ distance between $f P^{s}$ and its stationary distribution is at most $\epsilon$.
Although $\lambda^{\prime}$ occurs in the above upper bound for the distance between the stationary distribution and the $s$-step distribution, in fact, only $\lambda_{1}$ is crucial in the following sense. Note that $\lambda^{\prime}$ is either $\lambda_{1}$ or $2-\lambda_{n-1}$. Suppose the latter holds, i.e., $\lambda_{n-1}-1 \geq 1-\lambda_{1}$. We can consider a modified random walk, called the lazy walk, on the graph $G^{\prime}$ formed by adding a loop of weight $d_{v}$ to each vertex $v$. The new graph has Laplacian eigenvalues $\tilde{\lambda}_{k}=\lambda_{k} / 2 \leq 1$, which follows from equation (1.13). Therefore,
$$
1-\tilde{\lambda}_{1} \geq 1-\tilde{\lambda}_{n-1} \geq 0,
$$
and the convergence bound in $L_{2}$ distance in (1.14) for the modified random walk becomes
$$
2 / \lambda_{1} \log \left(\frac{\max _{x} \sqrt{d_{x}}}{\epsilon \min _{y} \sqrt{d_{y}}}\right)
$$
In general, suppose a weighted graph with edge weights $w(u, v)$ has eigenvalues $\lambda_{i}$ with $\lambda_{n-1}-1 \geq 1-\lambda_{1}$. We can then modify the weights by choosing, for some constant $c$,
$$
w^{\prime}(u, v)= \begin{cases}w(v, v)+c d_{v} & \text { if } u=v \\ w(u, v) & \text { otherwise. }\end{cases}
$$
The resulting weighted graph has eigenvalues
$$
\lambda_{k}^{\prime}=\frac{\lambda_{k}}{1+c}=\frac{2 \lambda_{k}}{\lambda_{n-1}+\lambda_{k}}
$$
where
$$
c=\frac{\lambda_{1}+\lambda_{n-1}}{2}-1 \leq \frac{1}{2} .
$$
Then we have
$$
1-\lambda_{1}^{\prime}=\lambda_{n-1}^{\prime}-1=\frac{\lambda_{n-1}-\lambda_{1}}{\lambda_{n-1}+\lambda_{1}} .
$$
Since $c \leq 1 / 2$ and we have $\lambda_{k}^{\prime} \geq 2 \lambda_{k} /\left(2+\lambda_{k}\right) \geq 2 \lambda_{k} / 3$ for $\lambda_{k} \leq 1$. In particular we set
$$
\lambda=\lambda_{1}^{\prime}=\frac{2 \lambda_{1}}{\lambda_{n-1}+\lambda_{1}} \geq \frac{2}{3} \lambda_{1} .
$$
Therefore the modified random walk corresponding to the weight function $w^{\prime}$ has an improved bound for the convergence rate in $L_{2}$ distance:
$$
\frac{1}{\lambda} \log \frac{\max _{x} \sqrt{d_{x}}}{\epsilon \min _{y} \sqrt{d_{y}}} .
$$
We remark that for many applications in sampling, the convergence in $L_{2}$ distance seems to be too weak since it does not require convergence at each vertex. There are several stronger notions of distance several of which we will mention.
A strong notion of convergence that is often used is measured by the relative pointwise distance (see [224]): After $s$ steps, the relative pointwise distance (r.p.d.) of $P$ to the stationary distribution $\pi(x)$ is given by
$$
\Delta(s)=\max _{x, y} \frac{\left|P^{s}(y, x)-\pi(x)\right|}{\pi(x)} .
$$
Let $\psi_{x}$ denote the characteristic function of $x$ defined by:
$$
\psi_{x}(y)= \begin{cases}1 & \text { if } y=x \\ 0 & \text { otherwise. }\end{cases}
$$
Suppose
$$
\begin{aligned}
\psi_{x} T^{1 / 2} & =\sum_{i} \alpha_{i} \phi_{i} \\
\psi_{y} T^{-1 / 2} & =\sum_{i} \beta_{i} \phi_{i} .
\end{aligned}
$$
where $\phi_{i}$ 's denote the eigenfunction of the Laplacian $\mathcal{L}$ of the weighted graph associated with the random walk. In particular,
$$
\begin{aligned}
& \alpha_{0}=\frac{d_{x}}{\sqrt{\operatorname{vol} G}}, \\
& \beta_{0}=\frac{1}{\sqrt{\text { vol } G}} .
\end{aligned}
$$
Let $A^{*}$ denote the transpose of $A$. We have
$$
\begin{aligned}
\Delta(t) & =\max _{x, y} \frac{\left|\psi_{y} P^{t} \psi_{x}^{*}-\pi(x)\right|}{\pi(x)} \\
& =\max _{x, y} \frac{\left|\psi_{y} T^{-1 / 2}(I-\mathcal{L})^{t} T^{1 / 2} \psi_{x}^{*}-\pi(x)\right|}{\pi(x)} \\
& \leq \sum_{x, y} \frac{\left|\left(1-\lambda_{i}\right)^{t} \alpha_{i} \beta_{i}\right|}{d_{x} / \operatorname{vol} G} \\
& \leq \sum^{t} \max _{x, y} \frac{\left|\alpha_{i} \beta_{i}\right|}{d_{x} / \operatorname{vol} G} \\
& =\bar{\lambda}^{t} \max _{x, y} \frac{\left\|\psi_{x} T^{1 / 2}\right\|\left\|\psi_{y} T^{-1 / 2}\right\|}{d_{x} / \operatorname{vol} G} \\
\leq & \bar{\lambda}^{t} \frac{\operatorname{vol} G}{\min _{x, y} \sqrt{d_{x} d_{y}}} \\
\leq & e^{-t(1-\bar{\lambda}) \frac{\operatorname{vol} G}{\min _{x} d_{x}}}
\end{aligned}
$$
where $\bar{\lambda}=\max _{i \neq 0}\left|1-\lambda_{i}\right|$. So if we choose $t$ such that
$$
t \geq \frac{1}{1-\bar{\lambda}} \log \frac{\operatorname{vol} G}{\epsilon \min _{x} d_{x}}
$$
then, after $t$ steps, we have $\Delta(t) \leq \epsilon$.
When $1-\lambda_{1} \neq \bar{\lambda}$, we can improve the above bound by using a lazy walk as described in (1.15). The proof is almost identical to the above calculation except for using the Laplacian of the modified weighted graph associated with the lazy walk. This can be summarized by the following theorem:
THEOREM 1.16. For a weighted graph $G$, we can choose a modified random walk $P$ so that the relative pairwise distance $\Delta(t)$ is bounded above by:
$$
\Delta(t) \leq e^{-t \lambda} \frac{\operatorname{vol} G}{\min _{x} d_{x}} \leq \exp ^{-2 t \lambda_{1} /\left(2+\lambda_{1}\right)} \frac{\operatorname{vol} G}{\min _{x} d_{x}} .
$$
where $\lambda=\lambda_{1}$ if $2 \geq \lambda_{n-1}+\lambda_{1}$ and $\lambda=2 \lambda_{1} /\left(\lambda_{n-1}+\lambda_{1}\right)$ otherwise.
COROllarY 1.17. For a weighted graph $G$, we can choose a modified random walk $P$ so that we have
$$
\Delta(t) \leq e^{-c}
$$
if
$$
t \geq \frac{1}{\lambda} \log \frac{\operatorname{vol} G}{\min _{x} d_{x}}
$$
where $\lambda=\lambda_{1}$ if $2 \geq \lambda_{n-1}+\lambda_{1}$ and $\lambda=2 \lambda_{1} /\left(\lambda_{n-1}+\lambda_{1}\right)$ otherwise.
We remark that for any initial distribution $f: V \rightarrow \mathbb{R}$ with $\langle f, \mathbf{1}\rangle=1$ and $f(x) \geq 0$, we have, for any $x$,
$$
\begin{aligned}
\frac{\left|f P^{s}(x)-\pi(x)\right|}{\pi(x)} & \leq \sum_{y} f(y) \frac{\left|P^{s}(y, x)-\pi(x)\right|}{\pi(x)} \\
& \leq \sum_{y} f(y) \Delta(s) \\
& \leq \Delta(s) .
\end{aligned}
$$
Another notion of distance for measuring convergence is the so-called total variation distance, which is just half of the $L_{1}$ distance:
$$
\begin{aligned}
\Delta_{T V}(s) & =\max _{A \subset V(G)} \max _{y \in V(G)}\left|\sum_{x \in A}\left(P^{s}(y, x)-\pi(x)\right)\right| \\
& =\frac{1}{2} \max _{y \in V(G)} \sum_{x \in V(G)}\left|P^{s}(y, x)-\pi(x)\right|
\end{aligned}
$$
The total variation distance is bounded above by the relative pointwise distance, since
$$
\begin{aligned}
\Delta_{T V}(s) & =\max _{\substack{A \subset V(G) \\
\text { vol } A \leq \frac{\mathrm{volG}}{2}}} \max _{y \in V(G)}\left|\sum_{x \in A}\left(P^{s}(y, x)-\pi(x)\right)\right| \\
& \leq \max _{\substack{A \subset V(G) \\
\operatorname{vol} A \leq \frac{\mathrm{volG}}{2}}} \sum_{x \in A} \pi(x) \Delta(s) \\
& \leq \frac{1}{2} \Delta(s) .
\end{aligned}
$$
Therefore, any convergence bound using relative pointwise distance implies the same convergence bound using total variation distance. There is yet another notion of distance, sometimes called $\chi$-squared distance, denoted by $\Delta^{\prime}(s)$ and defined by:
$$
\begin{aligned}
\Delta^{\prime}(s) & =\max _{y \in V(G)}\left(\sum_{x \in V(G)} \frac{\left(P^{s}(y, x)-\pi(x)\right)^{2}}{\pi(x)}\right)^{1 / 2} \\
& \geq \max _{y \in V(G)} \sum_{x \in V(G)}\left|P^{s}(y, x)-\pi(x)\right| \\
& =2 \Delta_{T V}(s),
\end{aligned}
$$
using the Cauchy-Schwarz inequality. $\Delta^{\prime}(s)$ is also dominated by the relative pointwise distance (which we will mainly use in this book).
$$
\begin{aligned}
\Delta^{\prime}(s) & =\max _{x \in V(G)}\left(\sum_{y \in V(G)} \frac{\left(P^{s}(x, y)-\pi(y)\right)^{2}}{\pi(y)}\right)^{1 / 2} \\
& \leq \max _{x \in V(G)}\left(\sum_{y \in V(G)}(\Delta(s))^{2} \cdot \pi(y)\right)^{\frac{1}{2}} \\
& \leq \Delta(s) .
\end{aligned}
$$
We note that
$$
\begin{aligned}
\Delta^{\prime}(s)^{2} & \geq \sum_{x} \pi(x) \sum_{y} \frac{\left(P^{s}(x, y)-\pi(y)\right)^{2}}{\pi(y)} \\
& =\sum_{x} \psi_{x} T^{1 / 2}\left(P^{s}-I_{0}\right) T^{-1}\left(P^{s}-I_{0}\right) T^{1 / 2} \psi_{x}^{*} \\
& =\sum_{x} \psi_{x}\left((I-\mathcal{L})^{2 s}-I_{0}\right) \psi_{x}^{*},
\end{aligned}
$$
where $I_{0}$ denotes the projection onto the eigenfunction $\phi_{0}, \phi_{i}$ denotes the $i$-th orthonormal eigenfunction of $\mathcal{L}$ and $\psi_{x}$ denotes the characteristic function of $x$. Since
$$
\psi_{x}=\sum_{i} \phi_{i}(x) \phi_{i},
$$
we have
$$
\begin{aligned}
\Delta^{\prime}(s)^{2} & \geq \sum_{x} \psi_{x}\left((I-\mathcal{L})^{2 s}-I_{0}\right) \psi_{x}^{*} \\
& =\sum_{x}\left(\sum_{i} \phi_{i}(x) \phi_{i}\right)\left((I-\mathcal{L})^{2 s}-I_{0}\right)\left(\sum_{i} \phi_{i}(x) \phi_{i}\right)^{*} \\
& =\sum_{x} \sum_{i \neq 0} \phi_{i}^{2}(x)\left(1-\lambda_{i}\right)^{2 s} \\
& =\sum_{i \neq 0} \sum_{x} \phi_{i}^{2}(x)\left(1-\lambda_{i}\right)^{2 s} \\
& =\sum_{i \neq 0}\left(1-\lambda_{i}\right)^{2 s} .
\end{aligned}
$$
Equality in (1.16) holds if, for example, $G$ is vertex-transitive, i.e., there is an automorphism mapping $u$ to $v$ for any two vertices in $G$, (for more discussions, see Chapter 7 on symmetrical graphs). Therefore, we conclude
THEOREM 1.18. Suppose $G$ is a vertex transitive graph. Then a random walk after $s$ steps converges to the uniform distribution under total variation distance or $\chi$-squared distance in a number of steps bounded by the sum of $\left(1-\lambda_{i}\right)^{2 s}$, where $\lambda_{i}$ ranges over the non-trivial eigenvalues of the Laplacian:
$$
\Delta_{T V}(s) \leq \frac{1}{2} \Delta^{\prime}(s)=\frac{1}{2}\left(\sum_{i \neq 0}\left(1-\lambda_{i}\right)^{2 s}\right)^{1 / 2} .
$$
The above theorem is often derived from the Plancherel formula. Here we have employed a direct proof. We remark that for some graphs which are not vertextransitive, a somewhat weaker version of (1.17) can still be used with additional work (see $[\mathbf{8 1}]$ and the remarks in Section 4.6). Here we will use Theorem 1.18 to consider random walks on an $n$-cube.
EXAMPle 1.19. For the $n$-cube $Q_{n}$, our (lazy) random walk (as defined in (1.15)) converges to the uniform distribution under the total variation distance, as estimated as follows: From Example (1.6), the eigenvalues of the $Q_{n}$ are $2 k / n$ of multiplicity $\left(\begin{array}{l}n \\ k\end{array}\right)$ for $k=0, \cdots, n$. The adjusted eigenvalues for the weighted graph corresponding to the lazy walk are $\lambda_{k}^{\prime}=2 \lambda_{k} /\left(\lambda_{n-1}+\lambda_{1}\right)=\lambda_{k} n /(n+1)$. By using Theorem 1.18 (also see $[\mathbf{1 0 4}]$ ), we have
$$
\begin{aligned}
\Delta_{T V}(s) \leq \frac{1}{2} \Delta^{\prime}(s) & \leq \frac{1}{2}\left(\sum_{k=1}^{n}\left(\begin{array}{l}
n \\
k
\end{array}\right)\left(1-\frac{2 k}{n+1}\right)^{2 s}\right)^{1 / 2} \\
& \leq \frac{1}{2}\left(\sum_{k=1}^{n} e^{k \log n-\frac{4 k s}{n+1}}\right)^{1 / 2} \\
& \leq e^{-c}
\end{aligned}
$$
if $s \geq \frac{1}{4} n \log n+c n$.
We can also compute the rate of convergence of the lazy walk under the relative pointwise distance. Suppose we denote vertices of $Q_{n}$ by subsets of an $n$-set $\{1,2, \cdots, n\}$. The orthonormal eigenfunctions are $\phi_{S}$ for $S \subset\{1,2, \cdots, n\}$ where
$$
\phi_{S}(X)=\frac{(-1)^{|S \cap X|}}{2^{n / 2}}
$$
for any $X \subset\{1,2, \cdots, n\}$. For a vertex indexed by the subset $S$, the characteristic function is denoted by
$$
\psi_{S}(X)= \begin{cases}1 & \text { if } X=S \\ 0 & \text { otherwise }\end{cases}
$$
Clearly,
$$
\psi_{X}=\sum_{S} \frac{(-1)^{|S \cap X|}}{2^{n / 2}} \phi_{S}
$$
Therefore,
$$
\begin{aligned}
\frac{\left|P^{s}(X, Y)-\pi(Y)\right|}{\pi(Y)} & =\left|2^{n} \psi_{X} P^{s} \psi_{Y}^{*}-1\right| \\
& \leq\left|2^{n} \psi_{X} P^{s} \psi_{X}^{*}-1\right| \\
& =\sum_{S \neq \emptyset}\left(1-\frac{2|S|}{n+1}\right)^{s} \\
& =\sum_{k=1}^{n}\left(\begin{array}{l}
n \\
k
\end{array}\right)\left(1-\frac{2 k}{n+1}\right)^{s}
\end{aligned}
$$
This implies
$$
\begin{aligned}
\Delta(s) & =\sum_{k=1}^{n}\left(\begin{array}{l}
n \\
k
\end{array}\right)\left(1-\frac{2 k}{n+1}\right)^{s} \\
& \leq \sum_{k=1}^{n} e^{k \log n-\frac{2 k s}{n+1}} \\
& \leq e^{-c}
\end{aligned}
$$
if
$$
s \geq \frac{n \log n}{2}+c n
$$
So, the rate of convergence under relative pointwise distance is about twice that under the total variation distance for $Q_{n}$.
In general, $\Delta_{T V}(s), \Delta^{\prime}(s)$ and $\Delta(s)$ can be quite different [81]. Nevertheless, a convergence lower bound for any of these notions of distance (and the $L_{2}$-norm) is $\lambda^{-1}$. This we will leave as an exercise. We remark that Aldous [4] has shown that if $\Delta_{T V}(s) \leq \epsilon$, then $P^{s}(y, x) \geq c_{\epsilon} \pi(x)$ for all vertices $x$, where $c_{\epsilon}$ depends only on $\epsilon$.
## Notes
For an induced subgraph of a graph, we can define the Laplacian with boundary conditions. We will leave the definitions for eigenvalues with Neumann boundary conditions and Dirichlet boundary conditions for Chapter ??. The Laplacian for a directed graph is also very interesting. The Laplacian for a hypergraph has very rich structures. However, in this book we mainly focus on the Laplacian of a graph since the theory on these generalizations and extensions is still being developed.
In some cases, the factor $\log \frac{\mathrm{vol} G}{\min _{x} d_{x}}$ in the upper bound for $\Delta(t)$ can be further reduced. Recently, P. Diaconis and L. Saloff-Coste [100] introduced a discrete version of the logarithmic Sobolev inequalities which can reduce this factor further for certain graphs (for $\Delta^{\prime}(t)$ ). In Chapter 12, we will discuss some advanced techniques for further bounding the convergence rate under the relative pointwise distance.
| Textbooks |
\begin{document}
\begin{abstract} We study the global existence of solutions to semilinear damped wave equations in the scattering case with derivative power-type nonlinearity on (1+3) dimensional nontrapping asymptotically Euclidean manifolds. The main idea is to exploit local energy estimate, together with local existence to convert the parameter $\mu$ to small one. \end{abstract}
\keywords{global existence, damped wave, local energy estimates, nontrapping asymptotically Euclidean}
\subjclass[2010]{35L05, 35L15, 35L71} \maketitle
\section{Introduction}
In this work, we consider the global existence of solutions for the Cauchy problem of the damped semilinear wave equation \begin{align} \label{1.1}
&u_{tt} - \Delta_{\gm}u + \frac{\mu \pt{u}}{(1 + t)^\be} = |\pt{u}|^p, \end{align} with initial data \begin{align} \label{ea2} &u(0,x)=f(x), \pt{u}(0,x)=g(x), \end{align} where $\be > 1$, and $\mu$, $\be \in \R$. Here we consider the nontrapping asymptotically Euclidean manifolds $(\R^{n}, \gm)$, and
$$\Delta_{\gm} = \sum_{i,j=1}^{n}\sqrt{|\gm|}^{-1}\partial_{i}g^{ij}\sqrt{|\gm|}\partial_{j},$$ \begin{align} \label{ea6} &\gm=g_{0}+g_1(r)+g_2(x), \gm ~is ~~nontrapping, \end{align} where $g^{ij}_{0}= \delta^{ij}$, $g_1$ and $g_2$ are of the form $g_{jk}dx^jdx^k$, $\langle x\rangle = \sqrt{1+x^{2}}$, \begin{align} \label{ea7}
&|\nabla_x^a g_{i,jk}|\les_a\langle x\rangle^{-|a|-\rho_i},i=1,2,\rho=\min(\rho_1,\rho_2-1),\rho_1<\rho_2, \end{align} where $\rho_{1}>0, \rho_{2}>1$ are fixed and $(g^{ij}(t,x))$ denotes the inverse matrix of $(g_{ij}(t,x))$. We assume the first perturbation $g_1$ is radial.\par When $\gm = g_{0}$, and $\mu =0$, Glassey made a conjecture that the critical power $p$ for the problem to admit global solutions with small, smooth initial data with compact support is $$p_c:=1 + \frac{2}{n-1}$$ in Glassey \cite{g} (see also Schaeffer \cite{scha} and Rammaha \cite{ram}), where n is the spatial dimension. The conjecture was verified for $n=2,3$ for general data ( Hidano and Tsutaya \cite{hk} and Tzvetkov \cite{nt} independently, as well as the radical case in Sideris \cite{sideris} for $n=3$). Zhou \cite{zhou} obtained the blow up results for $n\geq 4$, when $p\leq p_c$ and Hidano, Wang, Yokoyama \cite{hyw} obtained the global existence for $p > p_{c}$ when $n \geq 4$ under the radically symmetric assumption. For the results on the Glassey conjecture on certain asymptotically flat manifolds ($g_{1}, g_{2} \neq 0$): $n=3$ was proved by Wang \cite{W15} on certain small space time perturbation of the flat metric, as well as the nontrapping asymptotically Euclidean manifolds. Moreover, in Wang \cite{W15} the Glassey conjecture was proved when $n \geq 3$ in radical case on radical asymptotically flat manifolds. See also Wang \cite {W15j} for related results on exterior domain with nontrapping obstacles.\par When $\gm = g_{0}$, $\mu > 0$, for the corresponding linear problem \beeq \label{1.2} \Box u + \mu\frac{u_{t}}{(1+t)^{\beta}} = 0 \eneq we say that the damped term is ``scattering'' when $\beta \in (1, +\infty)$ since the solution behaves like that of wave equation (see, e.g., Wirth \cite{wirth1} \cite{wirth2} \cite{wirth3} for the classifications of (\ref{1.2})). \par When $\gm = g_{0}$ and $\mu > 0$, Lai and Takamura \cite{ltm} proved that (\ref{1.1}) blows up at finite time when $1 <p \leq p_{c}$ and $n \geq 1$. In view of the results of \cite{W15} and the ``scattering" damping term, it is natural to expect (\ref{1.1}) would admit global solution for any $\mu \in \R$ when $n=3$ and $p >2$. In following Theorem, we prove that it is the case. See Theorem \ref{4.1} for a more precise statement. \begin{theorem} Let $n=3$, and assume (\ref{ea6}) (\ref{ea7}). Consider the problem (\ref{1.1}) with $p >2$. Then there exists a global solution $u$ for any initial data which are sufficiently small, decaying and regular. \end{theorem}
Let us describe the strategy of the proof. We basically follow the approach that appeared in \cite{W15} to give the proof. Firstly, by local existence (Lemma \ref{4.8}), we obtain for any $T > 0$ there exists $\ep_{0}$ such that if the norm of initial data less than $\ep_{0}$, the solution could exist up to time $T$. Thus from $T$ time, we can convert the damping term $\frac{\mu}{(1+t)^{\beta}}$ to some $\frac{\tilde{\mu(t)}}{(1+t)^{\tilde{\beta}}}$ with $|\tilde{\mu}|$ small enough and $\tilde{\beta}>1$ and then we mainly exploit local energy estimate with variable coefficient (Lemma \ref{7.4}) appeared in \cite{W15} Lemma 3.5 to get global solution. \begin{remark}
The similar argument works for nonlinearity $c_{1}(u)u^{2}_{t} + c_{2}(u)|\nabla u|^{p}$, where $c_{1}, c_{2}$ are given smooth functions. \end{remark} \begin{remark} When $n \geq 4$ and $g_{2}=0$, the argument in \cite{W15} can be adapted to prove the global existence for $p > p_{c}$ with small radial data. \end{remark} \begin{remark} When $n=2$ and $g_{1}=g_{2}=0$, we can use the similarly argument to show (\ref{1.1}) admits global solution for $p > p_{c}$. \end{remark} \begin{remark} When $n\geq 2$ and $p=p_{c}$, we can obtain the lower bound of existence time of (\ref{1.1}) $T_{\ep} \geq e^{c\ep^{1-p}}$ in a similar way. \end{remark} We close this section by listing some notations. The vector fields to be used will be labeled as \[Y=(Y_1,\dots, Y_{n(n+1)/2}) = \{\nabla_x,\Omega\},\] Here $\Omega$ denotes the generators of spatial rotations: \[\Omega_{ij} = x_i\partial_j - x_j\partial_i,\quad 1\le i<j\le n.\] For a norm $X$ and a nonnegative integer $m$, we shall use the shorthand
\[|Y^{\le m} u| = \sum_{|\mu|\le m} |Y^\mu u|,\quad \|Y^{\le m} u\|_X
= \sum_{|\mu|\le m} \|Y^\mu u\|_X,\]
For fixed $T>0$, the space-time norm $L_T^qL_x^r$ is simply $L_t^q([0,T],L_x^r(\R^n))$. In the case of $T=\infty$, we use $L_t^qL_x^r$ to denote $L_t^q([0,\infty),L_x^r(\R^n))$. As usual, we use $\|\cdot\|_{E_m}$ to denote the energy norm of order $m\ge0$, \begin{align} \label{ea8}
&\|u\|_E=\|u\|_{E_0}=\|\partial u\|_{L_T^\infty L_x^2},\ \|u\|_{E_m}=\sum_{|a|\leq m}\|Y^au\|_E. \end{align}
We will use $\|\cdot\|_{LE}$ to denote the (strong) local energy norm \begin{align} \label{ea9}
&\|u\|_{LE}=\|u\|_E+\|\partial u\|_{l_\infty^{-1/2}(L_t^2L_x^2)}+\|r^{-1}u\|_{l_\infty^{-1/2}(L_t^2L_x^2)},\ \|u\|_{LE_m}=\sum_{|a|\leq m}\|Y^au\|_{LE}, \end{align} where we write
$$\|u\|_{l_{q}^s(A)}=\|(\phi_j(x)u(t,x))\|_{l^{s}_q(A)},$$ for a partition of unity subordinate to the dyadic (spatial) annuli, $\sum_{j\ge 0}\phi_j^2(x)=1$.
\section{Preliminary} In this section, we give some energy type estimates for linear damped wave equation \begin{equation} \label{ea1} u_{tt}-\Delta_\gm{u} + \frac{\mu \pt{u}}{(1 + t)^\be} = F. \end{equation} Moreover, we list the local energy estimate Lemma \ref{7.4} which we shall use later. \begin{lemma} \label{74} Assume (\ref{ea6}) (\ref{ea7}). Let $u \in C^{2}([0,T]\times \R^{n})$ vanishing for large $x$ and satisfy (\ref{ea1}) then we have \begin{equation} \label{1d3}
\|\partial u(t)\|_{L_x^2}\leq C_0\left(\|\partial u(0)\|_{L_x^2}+\int_0^T\|F\|_{L_x^2}dt\right), \end{equation} where $C_{0}$ is constant depends on $\mu$ and $\beta$. \end{lemma} \begin{prf}
This is standard energy type estimate (See, e.g., Sogge \cite{sogge} Proposition 2.1). For the readers' convenience, we give the sketch of proof.\par Let $E^{2}(t) = \frac{1}{2}\int_{\R^{n}}\big((\pa_{t}u)^{2}+g^{jk}\pa_{x_{j}}u\pa_{x_{k}}u\big)\sqrt{|\gm|}dx$, then \begin{align*}
(E^{2})'&= \frac{d}{dt}\frac{1}{2}\int_{\R^{n}}\big((\pa_{t}u)^{2}+g^{jk}\pa_{x_{j}}u\pa_{x_{k}}u\big)\sqrt{|\gm|}dx\\
&=\int_{\R^{n}}(u_{tt}-\Delta_{\gm}u)u_{t}\sqrt{|\gm|}dx\\
&=\int_{\R^{n}}Fu_{t}\sqrt{|\gm|}dx - \int_{\R^{n}}\frac{\mu}{(1+t)^{\beta}}u_{t}^{2}\sqrt{|\gm|}dx\\
&\les \|F\|_{L^{2}}E + \frac{|\mu|}{(1+t)^{\beta}}E^{2}. \end{align*}
Thus $$E' \les \|F\|_{L^{2}}+ \frac{|\mu|}{(1+t)^{\beta}}E, $$ by Gronwall's inequality
$$E(t) \les (E(0) + \|F\|_{L^{2}})e^{\frac{|\mu|}{\beta-1}},~0\leq t \leq T.$$ By the assumption (\ref{ea6}) (\ref{ea7}), there exists a constant $C$ such that
$$\frac{1}{C}\|\pa u\|_{L^{2}} \leq E(t) \leq C\|\pa u \|_{L^{2}},$$ hence (\ref{1d3}) follows.\par \end{prf} \begin{corollary}(Higher order energy estimate) \label{2d2} Under the same assumption of Lemma \ref{74}. If $\beta > 1$, we have \begin{equation*}
\|\partial Y^{\leq2}u(t)\|_{L_x^2}\leq C_1e^{C_1T}\left(\|\partial Y^{\leq2}u(0)\|_{L_x^2}+\int_0^T\|Y^{\leq2}F\|_{L_x^2}dt\right) , \end{equation*} where $C_{1}$ is constant depends on $\mu$ and $\beta$. \end{corollary} \begin{prf} Applying the vector field $Y^{\leq2}$ to both sides of the equation (\ref{ea1}), and note that
$$[\Delta_{\gm}, \nabla^{\mu}\Omega^{\nu}]u = \sum_{\substack{|\tilde{\mu}|+|\tilde{\nu}|\le
|\mu|+|\nu|\\|\tilde{\nu}|\le |\nu|}} \tilde{b}^\alpha_{\tilde{\mu}\tilde{\nu}} \nabla^{\tilde{\mu}} \Omega^{\tilde{\nu}} \nabla_\alpha u.$$
By assumption (\ref{ea6}) (\ref{ea7}) we have $\|\tilde{b}\|_{L^{\infty}} < \infty$. Then we have \begin{equation} \label{2dd}
(\pa_{tt} -\Delta_{\gm})Y^{\leq2}u + \frac{\mu \pt Y^{\leq2}u}{(1 + t)^\be} = \widetilde{F}:=Y^{\leq2}F + \sum_{|\mu|+|\nu|\leq2}[\Delta_{\gm}, \nabla^{\mu}\Omega^{\nu}]u \end{equation} Applying Lemma \ref{74} to (\ref{2dd}) we obtain \begin{align*}
&\|\partial Y^{\leq2}u(t)\|_{L_x^2}\\
\leq &C_0\big(\|\partial Y^{\leq2}u(0)\|_{L_x^2}+\int_0^T\|\widetilde{F}\|_{L_x^2}dt\big) \\
\les &\big(\|\partial Y^{\leq2}u(0)\|_{L_x^2}+\int_0^T\|Y^{\leq2}F\|_{L_x^2}dt +\sum_{|\mu|+|\nu|\leq2}\int_0^T\|[\Delta_{\gm}, \nabla^{\mu}\Omega^{\nu}]u\|_{L_x^2}dt\big)\\
\les&\big(\|\partial Y^{\leq2}u(0)\|_{L_x^2}+\int_0^T\|Y^{\leq2}F\|_{L_x^2}dt\big)+ \int_{0}^{T}\|\tilde{b}\|_{L_{x}^{\infty}}\|\pa Y^{\leq 2}u\|_{L^{2}_{x}}dt \end{align*} Then by Gronwall's inequality, \begin{align*}
\|\partial Y^{\leq2}u(t)\|_{L_x^2}\leq C_1e^{C_1T}\left(\|\partial Y^{\leq2}u(0)\|_{L_x^2}+\int_0^T\|Y^{\leq2}F\|_{L_x^2}dt\right). \end{align*} \end{prf}
\begin{lemma} \label{7.4} Let $n\geq3$ and consider the problem $u_{tt} - \Delta_{\gm}=F$ on the manifold $(\R^{n},\gm)$ satisfying (\ref{ea6}) (\ref{ea7}) with $\rho=min(\rho_1, \rho_2-1)>0$. Then for any positive $\theta>0$, we have the following higher order local energy estimates: \begin{eqnarray}
\|u\|_{LE_k}\les \sum_{|a|\les k}\left(\|\partial Y^au(0)\|_{L_x^2}+\|Y^aF\|_{L_t^1L_x^2+l_2^{\frac12+\theta}L_t^2L_x^2}\right). \end{eqnarray} \end{lemma} \begin{prf}
See Wang \cite{W15} Lemma 3.5. \end{prf}
\section{Local Existence} \begin{lemma}
\label{4.8}
Consider the problem (\ref{1.1}) on the manifold $(\R^{3}, \gm)$ satisfying (\ref{ea6}) (\ref{ea7}). Then there exists small positive constant $\ep_{0}$, such that for any initial data $(f,g) \in H^{1} \times L^{2}$ satisfying $$\|\nabla Y^{\leq2}f\|_{L_x^2}+\|Y^{\leq2}g\|_{L_x^2}=\ep^p \leq \ep_{0},$$ there exist $T_{\ep}>0$ with $\lim_{\ep \rightarrow 0^{+}}T_{\ep} = \infty$. and a unique solution $u$ with
$$\|\partial Y^{\leq2}u(t)\|_{L_t^{\infty}([0,T])L_x^2}\leq2 C_1\ep.$$ \end{lemma} \begin{prf} We define a closed subset of Banach space
$$E_T=\{u;\ \|u\|_{E_T}:=\sup_{0\leq t\leq T}\|\partial Y^{\leq2}u\|_{L_x^2}\leq2 C_1\ep\},$$ and a map $\Ga:v\to u$, satisfying \beeq
\left\{\begin{array}{l}u_{tt} - \Delta_{\gm}u+\frac{\mu\partial_{t} u}{(1+t)^{\be}}=|\pt v|^p,\quad \\ u(0,x) = f,
\pt u(0,x) = g.\end{array}
\right.
\label{3.1} \eneq Then we are reduced to show $\Ga$ is a contraction map from $E_T$ to $E_T$.
Firstly, we will prove $\Ga :E_T\to E_T$.\\ For any $v\in E_T$, by Corollary \ref{2d2}, we have \begin{align*}
\|\Ga v\|_{E_T}&=\sup_{0\leq t\leq T}\|\partial Y^{\leq2}u\|_{L_x^2}\\
&\leq C_1e^{C_1T}\left(\|\partial Y^{\leq2}u(0)\|_{L_x^2}+\int_0^T\|Y^{\leq2}F\|_{L_x^2}dt\right) . \end{align*} Since \begin{eqnarray*}
Y^{\leq2}F(v)=Y^{\leq2}|\pt v|^p\sim|\pt v|^{p-1}Y^{\leq2}\pt v+|\pt v|^{p-2}(Y^{\leq1}\partial v)^2. \end{eqnarray*} For the first term, by H$\ddot{\mathrm{o}}$lder's inequality and Sobolev imbedding $H^{2}(\R^{3}) \hookrightarrow L^{\infty}_{x}(\R^{3})$ we have \begin{eqnarray*}
\||\pt v|^{p-1}Y^{\leq2}\pt v\|_{L_x^2}\leq\|\pt v\|_{L^{\infty}}^{p-1}\|\pa Y^{\leq2}v\|_{L_x^2} \les \|\pa Y^{\leq2}v\|^{p}_{L_x^2}. \end{eqnarray*} Similarly, for the second term, by Sobolev imbedding $H^{1}(\R^{3}) \hookrightarrow L^{4}(\R^{3})$, we have \begin{eqnarray*}
\||\pt v|^{p-2}(Y^{\leq1}\partial v)^2\|_{L_x^2}\leq\|\pt v\|^{p-2}_{L_x^{\infty}}\|Y^{\leq1}\partial v\|_{L_x^4}^2 \les \|\pt v\|^{p-2}_{L_x^{\infty}}\|\pa Y^{\leq 2}v\|^{2}_{L^{2}_{x}} \les \|\pa Y^{\leq2}v\|^{p}_{L_x^2}. \end{eqnarray*} Hence \begin{align*}
\|\Ga v\|_{E_T} &\leq C_1e^{C_1T}(\ep^{p} + C_{2}(2C_{1}\ep)^{p})\\ &\leq 2C_{1}\ep \end{align*} if we take $$T \leq \frac{1}{C_{1}}\ln(\frac{1}{2^{p}C_{1}^{p}C_{2}\ep^{p-1}}).$$ Secondly, we will prove $\Ga :E_T\to E_T$ is contraction. For any $u,\,v\in E_{T}$ with $(\Ga u, u)$, $(\Ga v, v)$ satisfy (\ref{3.1}), then $\Ga (u- v)$ satisfies that \beeq
\left\{\begin{array}{l}(\pa_{tt} -\Delta_{\gm})\Ga (u- v)+\frac{\mu\pt \Ga (u- v)}{(1+t)^{\be}}=|\pt u|^p-|\pt v|^p,\quad \\ \Ga (u-v)(0,x) = 0,
\pt \Ga (u- v)(0,x) = 0.\end{array}
\right.
\label{3.2} \eneq By applying corollary \ref{2d2}, we have \begin{eqnarray*}
\|\partial Y^{\leq2}\Ga (u-v)\|_{L_x^2}
\leq C_1e^{C_1T}\int_0^T\|Y^{\leq2}(|\pt u|^p-|\pt v|^p)\|_{L_x^2}dt. \end{eqnarray*} Since \begin{eqnarray*}
Y^{\leq2}(|\pt u|^p-|\pt v|^p)\sim (|\pt u|^{p-3}+|\pt v|^{p-3})(Y(\pt u+\pt v))^2(\pt u-\pt v)\\
+(|\pt u|^{p-2}+|\pt v|^{p-2})Y^{\leq2}(\pt u+\pt v)Y^{\leq2}(\pt u-\pt v), \end{eqnarray*} by H$\ddot{\mathrm{o}}$lder's inequality and Sobolev inequality, we have \begin{align*}
&\||\pt u|^{p-3}(Y(\pt u+\pt v))^2(\pt u-\pt v)\|_{L_x^2}\\
\les &\|\pt u\|_{L_x^{\infty}}^{p-3}\|Y(\pt u+\pt v))\|_{L_x^4}^2\|\pt u-\pt v\|_{L_x^{\infty}}\\
\les &\|\partial Y^{\leq2}u\|_{L_x^2}^{p-3}\|Y^{\leq2}\partial u+Y^{\leq2}\partial v))\|_{L_x^2}^2\|Y^{\leq2}\partial(u-v)\|_{L_x^2}. \end{align*} Similarly, \begin{eqnarray*}
&&\|(|\pt u|^{p-2}+|\pt v|^{p-2})Y^{\leq2}(\pt u+\pt v)Y^{\leq2}(\pt u-\pt v)\|_{L_x^2}\\
&\les&(\|\pt u\|_{L_x^{\infty}}^{p-2}+\|\pt v\|_{L_x^{\infty}}^{p-2})\|Y^{\leq2}(\pt u+\pt v)\|_{L_x^4}\|Y^{\leq2}(\pt u-\pt v)\|_{L_x^4}\\
&\les&(\|\partial Y^{\leq2} u\|_{L_x^2}^{p-2}+\|\partial Y^{\leq2} v\|_{L_x^2}^{p-2})\|Y^{\leq2}\partial u+Y^{\leq2}\partial v\|_{L_x^2}\|Y^{\leq2}\partial (u-v)\|_{L_x^2}. \end{eqnarray*} Hence \begin{align*}
\|\Ga (u-v)\|_{E_{T}}&=\|\partial Y^{\leq2}(u-v)\|_{L^{\infty}_{t}L_x^2}\\
& \les C_1e^{C_1T}TC_{3}(2C_1\ep)^{p-1}\|u-v\|_{E_T}\\
& \leq \frac12\|u-v\|_{E_T}, \end{align*} if we take $$T\leq \frac{1}{C_{1}}\ln(\frac{1}{2^{p}C_{1}^{p}C_{3}\ep^{p-1}}).$$ Therefore, $\Ga$ is a contraction map from $E_T$ to $E_T$, if we take
$$T=\min\{\frac{1}{C_{1}}\ln(\frac{1}{2^{p}C_{1}^{p}C_{3}\ep^{p-1}}), \frac{1}{C_{1}}\ln(\frac{1}{2^{p}C_{1}^{p}C_{2}\ep^{p-1}})\}.$$ Thus there is a unique solution in $E_{T}$. \end{prf}
\section{Global Existence of equation (\ref{1.1})} \begin{theorem} \label{4.1} Consider the problem (\ref{1.1}) on the manifold $(\R^{3},\gm)$ satisfying (\ref{ea6}) (\ref{ea7}) with $\rho=min(\rho_1, \rho_2-1)>0$. Then there exists small positive constants $\ep_0$ such that for any initial data satisfying \beeq \label{4.3}
\sum_{|a|\leq2}\|\partial Y^au(0)\|_{L_x^2(\R^3)}=\ep^{p} \leq \ep_0,\
\|u(0)\|_{L^2(\R^3)}<\infty\ .
\eneq There is a global solution $u$ with $\|u\|_{LE_2}\les \ep$. \end{theorem} \begin{prf} We rewrite the equation (\ref{1.1}) as \begin{equation} \label{704}
u_{tt} - \Delta_\gm{u} + \frac{\mu}{(1 + t)^{\gamma}}\frac{\pt u}{(1+t)^{\be-\gamma}} = |\pt u|^p, \end{equation} where $0<\gamma < 1$ such that $\tilde{\be} = \be-\gamma>1$. Let $\tilde{\mu}(t)=\frac{\mu}{(1+t)^{\gamma}}$, then (\ref{704}) becomes \beeq \label{705}
u_{tt} - \Delta_\gm{u} + \frac{\tilde{\mu}(t)u_{t}}{(1+t)^{\tilde{\beta}}}= |\pt u|^p, \eneq From the local existence Lemma \ref{4.8}, we know for any $T>0$, there exits $\ep_0>0$, if initial data satisfy (\ref{4.3}) then (\ref{705}) has a unique solution $u$ in $[0,T]$ with \begin{equation*}
\|\partial Y^{\leq2}u\|_{L_t^{\infty}([0,T])L_x^2}\leq2C_1\ep. \end{equation*}
Thus $|\tilde{\mu}(t)|\leq\frac{|\mu|}{(1+T)^{\gamma}}$ can be sufficiently small if we take $T$ sufficiently large. Then from the $T$ time, we are reduced to consider the equation \beeq \begin{cases}
u_{tt} - \Delta_\gm{u}=G(u):=|\pt u|^p-\frac{\tilde{\mu}(t)\pt u(t)}{(1+t)^{\tilde{\be}}}, t \geq T\\ u(T,x) = f, \pt u(T,x) = g. \end{cases} \eneq with \begin{align} \label{706}
&\|Y^{\leq 2}\nabla f\|_{L_x^2} + \|Y^{\leq 2}g\|_{L_x^2}\leq 2C_{1}\ep, \end{align} and
$$\|u(T)\|_{L^{2}} = \|\int_{0}^{T}\pa_{t}u(s)ds + u(0)\|_{L^{2}} \leq T\|\pa u\|_{L^{\infty}_{t}L^{2}_{x}}+\|u(0)\|_{L^{2}} < \infty.$$
Let $G(u) = G_{1}(u) + G_{2}(u) = |u_{t}|^{p} - \frac{\tilde{\mu}(t)\pt u(t)}{(1+t)^{\tilde{\be}}}$ and from now on, we denote $\|u\|_{L_t^pL_x^q}$ as $\|u\|_{L_t^p([T,{\infty}])L_x^q}$. Set $u_0\equiv0$ and define $u_{k+1}$ to be the solution to the equation $$(\pa_{tt} - \Delta_{\gm})u_{k+1}=G(u_k),u_{{k+1}}(T,x)=f(x),\pt u_{{k+1}}(T,x)=g(x).$$
$Boundedness$: By the smallness condition (\ref{706}) on the data, it follows from Lemma \ref{7.4} that there is a universal constant $C_4$ so that \begin{align} \label{ea4}
&\|u_1\|_{LE_2}\leq C_4\ep, \ \|u_{k+1}\|_{LE_2}\leq C_4\ep+C_4\sum_{|a|\leq 2}\|Y^aG(u_k)\|_{L_t^1L_x^2}. \end{align} We shall argue inductively to prove that \begin{align} \label{ea5}
&\|u_{k+1}\|_{LE_2}\leq 2C_4\ep. \end{align} By the above, it suffices to show \begin{align} \label{ea60}
&\sum_{|a|\leq 2}\|Y^aG(u)\|_{L_t^1L_x^2}\leq \ep, \end{align}
for any $u$ with $\|u\|_{LE_2}\leq 2C_4\ep\leq 1.$\\
For the $G_{1} = |u_{t}|^{p}$ part, by (4.6) of \cite{W15} \begin{equation*}
\|Y^{\leq 2}|\pa_{t} u|^p\|_{L_t^1L_x^2}\les \|u\|_{LE_2}^p. \end{equation*} For the $G_{2}(u)$ part, since \begin{align*}
\|\frac{\tilde{\mu}(t)}{(1+t)^{\tilde{\be}}}Y^{\leq 2}\pt u\|_{L_t^1L_x^2}
&\les \|\frac{\tilde{\mu}(t)}{(1+t)^{\tilde{\be}}}\|_{L_t^1}\|Y^{\leq 2}\pt u\|_{L_t^{\infty}L_x^2}\\
&\les\frac{\|\tilde{\mu}(t)\|_{L_t^{\infty}}}{\tilde{\be}-1}\|u\|_{LE_2} \end{align*} In conclusion, we see that there exists a constant $C_{5}$ such that
$$\|Y^{\leq 2}G(u)\|_{L_t^1L_x^2}\leq C_{5}(2C_{4}\ep)^{p} + 2C_{4}\ep C_{5}\frac{\|\tilde{\mu}(t)\|_{L_t^{\infty}}}{\tilde{\be}-1} \leq \ep$$ for $\ep \leq \ep_{0}$ with
$$C_{5}(2C_{4})^{p}\ep_{0}^{p-1} + 2C_{4} C_{5}\frac{\|\tilde{\mu}(t)\|_{L_t^{\infty}}}{\tilde{\be}-1} \leq 1.$$ $Convergence\ of\ the\ sequence\ {u_k}$: We see that
$$\|G(u)-G(v)\|_{L_t^1L_x^2}\leq \|G_1(u)-G_1(v)\|_{L_t^1L_x^2} + \|G_2(u)-G_2(v)\|_{L_t^1L_x^2}.$$ For the first part, by the 1st display on page 7442 of \cite{W15} we have
$$\|G_1(u)-G_1(v)\|_{L_t^1L_x^2} \les (\|u\|_{LE_{2}}+\|v\|_{LE_{2}})^{p-1}\|u-v\|_{LE}.$$ For the second part, since \begin{align*}
\|G_2(u)-G_2(v)\|_{L_t^1L_x^2}&= \|\frac{\tilde{\mu}(t)}{(1+t)^{\tilde{\be}}}(\pt u-\pt v)\|_{L_t^1L_x^2}\\
&\les \|\frac{\tilde{\mu}(t)}{(1+t)^{\tilde{\be}}}\|_{L_t^1L_x^{\infty}}\|\partial(u-v)\|_{L_t^{\infty}L_x^2}\\
&\les \frac{\|\tilde{\mu}(t)\|_{L_t^{\infty}}}{\tilde{\be}-1}\|u-v\|_{LE}. \end{align*} Hence there exists a constant $C_{6}$ such that
$$\|G(u)-G(v)\|_{L_t^1L_x^2} \leq \big(C_{6}(4C_{4}\ep)^{p-1}+ C_{6}\frac{\|\tilde{\mu}(t)\|_{L_t^{\infty}}}{\tilde{\be}-1}\big)\|u-v\|_{LE}\leq \frac{1}{2}\|u-v\|_{LE}$$ for $\ep \leq \ep_{0}$ and $\ep_{0}$ satisfies
$$\big(C_{6}(4C_{4}\ep_{0})^{p-1}+ C_{6}\frac{\|\tilde{\mu}(t)\|_{L_t^{\infty}}}{\tilde{\be}-1}\big)\leq \frac{1}{2}.$$
Together with the uniform boundedness (\ref{ea5}), we find an unique global solution $u \in L^{\infty}_{x}([T,\infty];H^{3})\cap Lip_{t}([T,\infty];H^{2})$ with $\|u\|_{LE_{2}}\leq 2C_{4}\ep$. \end{prf}
\end{document} | arXiv |
Home » Science » Black Hole and Neutron Star Collisions Detected Twice in Ten Days
Black Hole and Neutron Star Collisions Detected Twice in Ten Days
July 21, 2021 by Brian Wang
Two instances of black hole-neutron star collision events have been detected using the Advanced LIGO and Virgo gravitational wave detectors, details of which have been published today in Astrophysical Journal Letters.
Previous gravitational wave detections have spotted black holes colliding, and neutron stars merging, this is the first time that scientists have detected a collision from one of each.
Dr Vivien Raymond, from Cardiff University's Gravity Exploration Institute, said: "After the detections of black holes merging together, and neutron stars merging together, we finally have the final piece of the puzzle: black holes swallowing neutron stars whole. This observation really completes our picture of the densest objects in the universe and their diet."
Gravitational waves are produced when celestial objects collide and the ensuing energy creates ripples in the fabric of space-time which travel all the way to the detectors we have here on Earth.
On 5 January 2020, the Advanced LIGO (ALIGO) detector in Louisiana in the US and the Advanced Virgo detector in Italy observed gravitational waves from this entirely new type of astronomical system.
The detectors picked up the final throes of the death spiral between a neutron star and a black hole as they circled ever closer and merged together.
On 15 January, a second signal was picked up by Virgo and both ALIGO detectors – in Louisiana and Washington state – again coming from the final orbits and smashing together of another neutron star and black hole pair.
Researchers from Cardiff University, who form part of the LIGO Scientific Collaboration, played a crucial role in the data analysis of both events, unpicking the gravitational wave signals and painting a picture of how the extreme collisions played out.
This involved generating millions of possible gravitational waves and matching them to the observed data to determine the properties of the objects that produced the signals in the first place, such as their masses and their location in the sky.
From the data they were able to infer that the first signal, dubbed GW200105, was caused by a 9-solar mass black hole colliding with a 1.9-solar mass neutron star.
Analysis of the second event, GW200115, which was detected just 10 days later, showed that it came from the merger of a 6-solar mass black hole with a 1.5-solar mass neutron star, and that it took place at a slightly larger distance of around 1 billion light-years from Earth.
During its third observing run, the LIGO–Virgo GW detector network observed GW200105 and GW200115, two GW events consistent with NSBH coalescences. Event GW200105 is effectively a single-detector event observed in LIGO Livingston with an S/N of 13.9. It clearly stands apart from all recorded noise transients, but its statistical confidence is difficult to establish.
Astrophysical Journal Letters – Observation of Gravitational Waves from Two Neutron Star–Black Hole Coalescences
We report the observation of gravitational waves from two compact binary coalescences in LIGO's and Virgo's third observing run with properties consistent with neutron star–black hole (NSBH) binaries. The two events are named GW200105_162426 and GW200115_042309, abbreviated as GW200105 and GW200115; the first was observed by LIGO Livingston and Virgo and the second by all three LIGO–Virgo detectors. The source of GW200105 has component masses $8.{9}_{-1.5}^{+1.2}$ and $1.{9}_{-0.2}^{+0.3}\,{M}_{\odot }$, whereas the source of GW200115 has component masses $5.{7}_{-2.1}^{+1.8}$ and $1.{5}_{-0.3}^{+0.7}\,{M}_{\odot }$ (all measurements quoted at the 90% credible level). The probability that the secondary's mass is below the maximal mass of a neutron star is 89%–96% and 87%–98%, respectively, for GW200105 and GW200115, with the ranges arising from different astrophysical assumptions. The source luminosity distances are ${280}_{-110}^{+110}$ and ${300}_{-100}^{+150}\,\mathrm{Mpc}$, respectively. The magnitude of the primary spin of GW200105 is less than 0.23 at the 90% credible level, and its orientation is unconstrained. For GW200115, the primary spin has a negative spin projection onto the orbital angular momentum at 88% probability. We are unable to constrain the spin or tidal deformation of the secondary component for either event. We infer an NSBH merger rate density of ${45}_{-33}^{+75}\,{\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1}$ when assuming that GW200105 and GW200115 are representative of the NSBH population or ${130}_{-69}^{+112}\,{\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1}$ under the assumption of a broader distribution of component masses.
SOURCES- Cardiff, Astrophysical Journal Letters
Written By Brian Wang, Nextbigfuture.com
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Complete linear optical isolation at the microscale with ultralow loss
JunHwan Kim1 na1,
Seunghwi Kim1 na1 &
Gaurav Bahl1
Scientific Reports volume 7, Article number: 1647 (2017) Cite this article
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Microresonators
Nonlinear optics
Low-loss optical isolators and circulators are critical nonreciprocal components for signal routing and protection, but their chip-scale integration is not yet practical using standard photonics foundry processes. The significant challenges that confront integration of magneto-optic nonreciprocal systems on chip have made imperative the exploration of magnet free alternatives. However, none of these approaches have yet demonstrated linear optical isolation with ideal characteristics over a microscale footprint – simultaneously incorporating large contrast with ultralow forward loss – having fundamental compatibility with photonic integration in standard waveguide materials. Here we demonstrate that complete linear optical isolation can be obtained within any dielectric waveguide using only a whispering-gallery microresonator pumped by a single-frequency laser. The isolation originates from a nonreciprocal induced transparency based on a coherent light-sound interaction, with the coupling originating from the traveling-wave Brillouin scattering interaction, that breaks time-reversal symmetry within the waveguide-resonator system. Our result demonstrates that material-agnostic and wavelength-agnostic optical isolation is far more accessible for chip-scale photonics than previously thought.
Ideal optical isolators should exhibit complete linear isolation – where completeness implies perfect transmission one way (i.e. zero forward insertion loss) and zero transmission in the opposite direction – without any mode shifts, frequency shifts, or dependence on input signal power. In practice, isolators should also exhibit a broadband isolation response for robustness and usability across a wide range of applications. To date, the best method for achieving optical isolation with these characteristics has been through Faraday rotation via the magneto-optic response in gyrotropic materials1, 2. Unfortunately, this well-established technique3 has proven challenging to implement in chip-scale photonics due to fabrication complexity, difficulty in locally confining magnetic fields, and significant material losses4,5,6,7.
In light of this challenge, several non-magnetic alternatives for breaking reciprocity3 have been explored both theoretically8,9,10,11,12,13 and experimentally14,15,16,17,18,19. State-of-the-art experimental implementations of these alternatives have succeeded in meeting various metrics of contrast, linearity, and bandwidth, but complete isolation with ultralow forward loss has remained elusive. Nonlinearity-based isolators can be broadband but are fundamentally dependent on input field strength16, 18 and hence do not produce a linear isolation response20. Dynamic modulation8, 10, is a powerful approach that can generate linear isolation over potentially wide bandwidth, but current microscale demonstrations are still constrained by very large forward insertion loss and low contrast15, 17. These limitations could be overcome in macro-scale implementations21. Finally, the use of Brillouin acousto-optic scattering to induce unidirectional optical loss9, 22 is very promising, as it can generate a linear wide-band isolation response. In practice, this technique requires a large product of scattering gain and waveguide length14, which could soon be realized through advancements in on-chip Brillouin gain23,24,25. To date, however, there has been no SBS isolator demonstrated having a sub-millimeter footprint. A comparison of state-of-the-art experimental results on non-magnetic microscale isolation can be found in Table S1 of the Supplement. In this paper we emphasize forward loss, as it is an especially strong motivator for chip-scale photonics where we consistently strive to lower the product of size, weight, and power (SWaP) parameters.
Recently, a fundamentally different path to obtain nonreciprocal optical transport has emerged, by exploiting opto-mechanically induced transparency26,27,28. These nonreciprocal effects are based on destructive optical interference via a non-radiative acoustic coherence within a resonator-waveguide system, and are acousto-optic analogues of electromagnetically induced transparency (EIT). To date, however, only subtle nonreciprocity has been demonstrated by these optomechanical methods, without any expression or demonstration of a path to achieving complete isolation with ultra-low forward loss, both of which are requirements for practical use. We focus our study on the nonreciprocal Brillouin scattering induced transparency (BSIT) mechanism26, in which momentum conservation requirement between photons and phonons helps break time-reversal symmetry for light propagation. More importantly, BSIT uniquely permits two major technical results that we demonstrate in this work. We show theoretically that when operating within the strong acousto-optical coupling regime (aided by the resonant pump) the BSIT system enables theoretically lossless transmission of light in the forward direction in a waveguide, while maintaining complete absorption in the reverse direction – the condition of complete linear isolation. Second, the non-zero momentum of the traveling phonons involved in BSIT permits independent forward/reverse reconfiguration of the isolation effect, in contrast to the zero-momentum mechanical excitations in optomechanically induced transparency28 that can couple forward and reverse propagating light. Experimentally, we demonstrate a device operating very close to the strong coupling regime and capable of generating a record-breaking 78.6 dB of isolation contrast per 1 dB of forward insertion loss within the induced transparency bandwidth. Since the underlying interaction is available in all dielectrics, this isolation effect can in principle be implemented using any waveguide and resonator materials available in photonics foundries.
Achieving Complete Linear Isolation
Qualitative description
Let us first qualitatively discuss how ideal optical isolation can be achieved by means of the BSIT light-sound interaction in dielectric resonators26, 27. We consider a whispering-gallery resonator having two optical modes (ω 1, k1) and (ω 2, k2) that are separated in (ω, k) space by the parameters of a high coherence traveling acoustic mode (Ω, q). This is the requisite phase matching relation for BSIT (Fig. 1a), indicating that phonons enable coupling of the photon modes through photoelastic scattering. We stress here that the two modes should belong to different mode families of the resonator in order to ensure that scattering to other optical modes from the same phonon population is suppressed. When this system is pumped with a strong 'control' field on the lower optical resonance (ω 1, k1), an EIT-like optomechanically induced transparency29, 30 appears within the higher optical resonance (ω 2, k2), due to coherent interference originating from the acousto-optical interaction26, 27.
Achieving optical isolation through non-reciprocal Brillouin scattering induced transparency in a whispering-gallery resonator: (a) The interference of excitation pathways in the BSIT system are described through an energy-level picture (grey boxes), using probe photon number n p and phonon number n m . Absorption of a probe photon into the (ω 2, k2) optical resonance is modeled as an effective transition \(|{n}_{p},{n}_{m}\rangle \to |{n}_{p}+1,{n}_{m}\rangle \). In presence of the control field, the probe photon could scatter to the lower resonance (ω 1, k1) while adding a mechanical excitation in (Ω, q), which is an effective transition to state \(|{n}_{p},{n}_{m}+1\rangle \). However, the coherent anti-Stokes scattering of the control field from this mechanical excitation would generate an interfering excitation pathway for the original state \(|{n}_{p}+1,{n}_{m}\rangle \). This process is analogous to EIT and results in a window of transparency for the forward optical probe, inhibiting the original \(|{n}_{p},{n}_{m}\rangle \to |{n}_{p}+1,{n}_{m}\rangle \) absorption transition. The necessary momentum matching requirement, not visible in the energy diagram, is represented using the dispersion relation (middle) to elucidate the breaking of time-reversal symmetry for the probe signal. (b) We implement this mechanism using a waveguide and a whispering gallery resonator, in which probe signals tuned to either of the (ω 2, ±k2) optical resonances are typically absorbed by the resonator under the critical coupling condition. The presence of a forward control field, however, creates the BSIT interference26, only for forward probe signals and inhibits absorption. Under strong acousto-optical coupling, the waveguide-resonator system is rendered lossless at the original resonance.
A description of this interference can be presented both classically26, or through by a quantum mechanical approach27. Briefly, one can consider signal or 'probe' photons arriving from the waveguide at frequency ω 2 that are on-resonance and being absorbed by the resonator mode (ω 2, k2). When the control field is present in a BSIT phase-matching situation, these probe photons could scatter to (ω 1, k1) causing a mechanical excitation of the system. However, anti-Stokes scattering of the strong control field from this mechanical excitation will generate a phase-coherent optical field that interferes destructively with the original excitation of the mode at (ω 2, k2). The result is a pathway interference that is measured as an induced optical transparency in the waveguide, where no optical or mechanical excitation takes place, and the resonant optical absorption is inhibited (Fig. 1b - top). The strength of this interference is set by the intensity of the control laser. The phase of the mechanically dark mode is instantaneously set by the phases of the control and probe optical fields, and does not require phase coherence between them. It is crucial, however, to note that this transparency in BSIT only appears for probe signals co-propagating with the control laser. Probe light in the counter propagating i.e. time-reversed direction, on the other hand, occupies the high frequency optical mode with parameters (ω 2, −k2). For BSIT to occur in this case, an acoustic mode having parameters (Ω, −(k1 + k2)) would be required for compensating the momentum mismatch between the forward control and backward probe optical modes. However, since such an acoustic mode is not available in the system, no interaction occurs for the counter-propagating probe and the signal is simply absorbed into the resonator (Fig. 1b - bottom).
Classical treatment of the system
The classical field equations for coupled light and sound in this waveguide-resonator system are presented in the Supplement. The transmission coefficient \({\tilde{t}}_{p}\) of the probe laser field can be derived as:
$${\tilde{t}}_{p}=\frac{{s}_{\mathrm{2,}{\rm{out}}}}{{s}_{\mathrm{2,}{\rm{in}}}}=1-\frac{{\kappa }_{{\rm{ex}}}}{({\kappa }_{2}\mathrm{/2}+j{{\rm{\Delta }}}_{2})+{G}^{2}/({{\rm{\Gamma }}}_{B}\mathrm{/2}+j{{\rm{\Delta }}}_{B})}$$
where \({s}_{i,{\rm{in}}}\) and \({s}_{i,{\rm{out}}}\) are the optical driving and output fields in the waveguide (Fig. 1b) at the control (i = 1) and probe (i = 2) frequencies. G is the pump-enhanced Brillouin coupling rate manipulated by the control optical field \({s}_{\mathrm{1,}{\rm{in}}}\) in the waveguide via the relation \(G=|{s}_{\mathrm{1,}{\rm{in}}}\,\beta \,\sqrt{{\kappa }_{{\rm{ex}}}}/({\kappa }_{1}\mathrm{/2}+j{{\rm{\Delta }}}_{1})|\). Here β is the acousto-optic coupling rate, κ i are the loaded optical loss rates, Γ B is the phonon loss rate, and \({\kappa }_{{\rm{ex}}}\) is the coupling rate between the waveguide and resonator. The loaded optical loss rates are defined as \({\kappa }_{i}={\kappa }_{i,{\rm{o}}}+{\kappa }_{ex}\) where \({\kappa }_{i,{\rm{o}}}\) is the loss rate intrinsic to the optical mode. The \({{\rm{\Delta }}}_{i}\) parameters are the field detunings, with subscript B indicating the acoustic field. This response matches the system of optomechanically induced transparency (OMIT)29, 30, with the exception that the pump field is also resonant and the coupling rate β is dependent on momentum matching. As we explain later, the pump resonance in BSIT significantly enhances the maximum coupling rate G achievable in contrast to single-mode OMIT systems.
Equation (1) is key to understanding how an ideal optical isolator can be obtained. First, we examine the case of no acousto-optic coupling G = 0, resulting from either modal mismatch (β = 0) or zero applied control laser power (\({s}_{\mathrm{1,}{\rm{in}}}=0\)). In this case Eq. (1) exhibits a well-known Lorentzian shaped transmission dip implying that the probe optical field in the waveguide is simply absorbed by the resonator31. Critical coupling between resonator and waveguide is enabled when \({\kappa }_{{\rm{ex}}}={\kappa }_{\mathrm{2,}{\rm{o}}}\) and results in complete absorption of the probe light from the waveguide at resonance (\({{\rm{\Delta }}}_{2}=0\)). With critical coupling in place, let us now introduce the effects of the acousto-optic coupling. For very large acousto-optic interaction strength, i.e. \(G\to \infty \), Eq. (1) indicates that we recover perfect transmission \({|{\tilde{t}}_{p}|}^{2}=1\) even when the waveguide and resonator are critically coupled. Forward propagating probe light in the waveguide, co-propagating with the control laser, can thus transmit perfectly with no absorption at resonance in the ideal case. At the same time, we have no Brillouin coupling (\(\beta =0\)) for counter-propagating control and probe optical fields due to the momentum mismatch as indicated previously. This implies that, for a counter-propagating probe, the system remains in the critical coupling region resulting in complete absorption. Since forward probe signals transmit with zero absorption, and backward probe signals are completely absorbed (Fig. 1b), this system is an ideal linear isolator at the transparency resonance.
A more practically accessible case is \(G\ge {\kappa }_{2}\), also known as the strong coupling regime32, where the induced transparency grows to the width of the optical mode. Strong coupling can be reached for high coherence phonon modes (small Γ B ) with large acousto-optic coupling β and large control driving field \({s}_{\mathrm{1,}{\rm{in}}}\). The evolution of the optical transparency and isolation contrast with increasing coupling G is illustrated in Fig. 2. In the weak coupling regime (\(G\ll {\kappa }_{2}\)) the isolation contrast is defined roughly by the linewidth of the phonon mode. As G increases the transparency window broadens until eventually reaching the strong coupling regime where the isolation contrast bandwidth reaches a maximum equaling the optical loss rate κ 2, as long as the acoustic frequency is higher than this value. Thus, the isolation bandwidth can be improved to the several GHz range if a higher frequency acoustic mode is used33, in conjunction with a low-Q (high κ 2) optical mode, and the reduction in coupling is compensated by other means34. In this regime, we also achieve the desired ultra-low forward insertion loss. Such large transparency can also be interpreted as the splitting of the optical mode35. The absence or minimization of forward loss necessarily implies linear optical response at frequency ω 2 without any nonlinearity or mode conversion.
Evolution of the transparency and isolation contrast as a function of pump-enhanced Brillouin coupling G. In the weak coupling regime (\(G\ll \kappa \)), the transparency linewidth and contrast bandwidth are defined by the acoustic linewidth Γ B 26. As coupling G increases, the isolation contrast improves, bandwidth is expanded and the optical mode with transparency appears as a splitted mode. In the strong coupling regime, the isolation bandwidth is independent of the acoustic mode and is instead defined by optical mode linewidth κ only. The dashed lines indicate the perfect transmission baseline (left) and zero isolation contrast (right) respectively.
Demonstration of high-contrast ultralow-loss isolation
We experimentally demonstrate ultra-low loss optical isolation (Fig. 3) in the waveguide-resonator system by probing optical transmission through the waveguide in the forward and backward directions simultaneously (see Methods). A resonator of diameter 170 μm is used to guarantee the natural existence of multiple triplets of acoustic and optical modes that satisfy the phase-matching condition for BSIT.
Experimental observation of extremely low insertion loss linear optical isolation. (a) Probe power transmission coefficient \({|{\tilde{t}}_{p}|}^{2}\) is measured in the forward direction through the waveguide near the (ω 2, k2) mode, with fixed 66 μW pump power dropped into the (ω 1, k1) mode. The forward probe power transmission coefficient through the waveguide shows only 1.44 dB insertion loss within the transparency. The phonon mode frequency is 145 MHz. (b) The (ω 2, −k2) optical mode measured by the backward probe does not exhibit the induced transparency, resulting in conventional absorption of the probe signal by the resonator. (c) The optical isolation contrast is evaluated as the difference between forward and backward power transmission coefficients. Here we calculate 14.4 dB peak contrast with a −3 dB bandwidth of 90 kHz. Isolation exists over 470 kHz.
The requisite BSIT phase-matching is first experimentally verified by strongly driving the (ω 2, k2) optical mode and observing spontaneous and stimulated Stokes Brillouin scattering into the lower mode (ω 1, k1) in the forward direction33. Subsequently, we drive the (ω 1, k1) optical mode with a strong control laser (<1 mW) and use a weak co-propagating probe laser to measure the power transmission spectrum across the high frequency optical mode (ω 2, k2) revealing the induced transparency window. The control laser detuning and power are adjusted in order to maximize the power transmission within the transparency peak. Experimental measurements of the probe power transmission \({|{\tilde{t}}_{p}|}^{2}\) in both forward and backward directions are presented in Fig. 3. To show optical isolation, the same measurement is taken in the forward and backward directions while the constant control driving field \({s}_{\mathrm{1,}{\rm{in}}}\) is supplied in the forward direction only. In this experiment the two selected optical modes of the resonator have linewidth \({\kappa }_{1}\approx {\kappa }_{2}\approx 4.1\) MHz, and are spaced approximately 145 MHz apart. They are coupled by means of a 145 MHz acoustic mode of intrinsic linewidth ΓB ≈ 12 kHz. Through finite element simulations, we estimate that the acoustic mode corresponds to a first order Rayleigh surface acoustic exictation having an azimuthal order of M = 24. At a diameter of 170 μm, this translates to an acoustic momentum of q = 0.28 μm −1 and ensures breaking of interaction symmetry for co-propagating and counter-propagating probe fields (Fig. 1a).
As seen in Fig. 3a the system exhibits very low forward insertion loss (1.44 dB) at the peak of induced transparency region for 66 μW control laser power absorbed to the resonator (power launched in fiber is 680 μW). This corresponds to an experimentally calculated pump-enhanced Brillouin coupling of \(G\approx {\kappa }_{2}/12\). At this point, the acoustic mode has an effective linewidth of 80.4 kHz due to Brillouin cooling36. Simultaneous measurement of backward probe power transmission (Fig. 3b) shows only the absorption spectrum of the unperturbed (ω 2, −k2) optical mode, generating a power transmission loss of 15.8 dB in the waveguide. Subtraction of the forward and backward measurements provides a measure of the optical isolation contrast, which is 14.4 dB here with ~90 kHz full width at half maximum (Fig. 3c).
Since the forward insertion loss is very low (zero in the ideal theoretical case), the isolation contrast is primarily determined by the proximity of the waveguide-resonator coupling to the critical coupling condition, which if achieved would yield infinite isolation contrast. Achieving critical coupling \({\kappa }_{{\rm{ex}}}={\kappa }_{\mathrm{2,}{\rm{o}}}\) in non-integrated waveguide-microsphere systems is very challenging due to multimode waveguiding in the taper, thermal drifts during the experiment, and vibrational or mechanical stability issues. Previously, up to 26 dB of signal extinction has been experimentally demonstrated in a fiber taper-microsphere system37. In the future, ideal isolation may be approached if the waveguide and resonator are integrated on-chip, since most mechanical issues can be eliminated and the interacting modes can be designed precisely. Alternatively, applications that require high contrast may employ multiple isolators in series with minimal penalty due to the extremely low insertion loss in this system. It is thus appropriate to compare performance of different isolators by referencing the achieved contrast to 1 dB forward loss. The data shown in Fig. 3 indicates this figure of merit of approximately 10 \({{\rm{dB}}}_{{\rm{isolation}}}/{{\rm{dB}}}_{{\rm{loss}}}\) (units preserved for clarity, indicating 14.4 dB constrast vs 1.44 dB forward loss).
Theory indicates that much lower forward insertion loss can be obtained if much higher coupling rate G is arranged, either by lowering the loss rates of the optical modes, or by using higher control laser power. Fortunately, a special feature of two-mode systems such as BSIT38 is the resonant enhancement of the intracavity pump photons in mode ω 1, which enables much easier access to the strong coupling regime. Nonreciprocity based on single-mode OMIT28 does not possess this feature and it is thus impractical to expand the isolation bandwidth and reach the ultra-low loss regime. Making use of this resonant enhancement, in Fig. 4a we show a system nearly reaching the strong coupling regime with \(G\approx \kappa /3\), exhibiting only 0.14 dB forward insertion loss (96.8% transmission) and isolation contrast estimated at 11 dB. Here, 235 μW control power is coupled to the resonator (700 μW launched in fiber). The unmodified optical mode absorption can be easily observed by detuning the control laser such that the interference is generated outside the optical mode (Fig. 4b). This result indicates that the strong coupling regime is also within the reach of this silica waveguide-resonator system38. The isolation figure of merit (referenced to 1 dB insertion loss) for the Fig. 4 result is quantified at 78.6 \({{\rm{dB}}}_{{\rm{isolation}}}/{{\rm{dB}}}_{{\rm{loss}}}\). This compares extremely well to commercial fiber-optic Faraday isolators whose figures of merit typically range between 60–100 \({{\rm{dB}}}_{{\rm{isolation}}}/{{\rm{dB}}}_{{\rm{loss}}}\), and far exceeds the capabilities demonstrated till date by any other non-magnetic microscale optical isolation approach. As shown in Supplementary Table S1, our achieved contrast exceeds the next best microscale experimental result in non-magnetic optical isolation19 by nearly 7 orders-of-magnitude (69.5 dB difference, i.e. 78.6 dB vs 9.09 dB) per 1 dB of insertion loss.
Demonstration of ultra-low forward insertion loss with stronger coupling G. (a) Here, we use a triplet of optical and acoustic modes with an optical mode separation or acoustic frequency of 164.8 MHz. Pump-enhanced Brillouin coupling rate G is much higher due to better acousto-optic modal overlap and 235 μW power absorbed into the control mode. This results in \(G\approx \kappa \mathrm{/3}\) causing the forward insertion loss within the transparency to decrease to only 0.14 dB. The isolation bandwidth also increases to approximately 400 kHz. (b) The transparency-free (ω 2, k2) optical mode is observable by detuning the control laser from the (ω 1, k1) optical mode, which also detunes the scattered light.
Independent reconfiguration of optical isolation
Finally, we also demonstrate the optical reconfigurability of the isolation direction by means of independent control lasers that propagate in opposite directions. This is demonstrated through an experiment (Fig. 5) where the the control laser field is sequentially provided in the forward direction only, backward direction only, and in both directions simultaneously. Since the forward and backward directions in a whispering-gallery resonator are nominally decoupled and the phonon mode also has an associated directionality (i.e. momentum), the transparency is independently observed in the directions in which a control laser field is supplied.
Demonstrating reconfigurable optical isolation. Increasing the control laser power in the forward direction, we observe the appearance of the acousto-optical transparency. While transparency is enabled in the forward direction, we can switch on and off the transparency in the backward direction using a separate backward propagating control laser. The red dashed line represents a fit using theoretical model for induced transparency.
Figure 5 shows that when no control field is provided, the anti-Stokes optical mode is a simple Lorentzian shaped dip. However, when the control laser is supplied in the forward direction, a transparency is observed by the forward probe. While this transparency is sustained in the forward direction, we can independently switch on and off the transparency in the backward direction. This is demonstrated by probing the anti-Stokes optical mode in the backward direction with and without a backward control laser, which results in an optical mode with and without transparency respectively. Such reconfigurable transparency has never previously been demonstrated in any other optical or opto-mechanical system.
OMIT-based nonreciprocity28 does not possess the capability of fully independent reconfiguration since both forward and reverse optical signals interact with the same zero-momentum vibrational mode. Thus photon conversion can occur through an optomechanical dark mode39 shared between forward and reverse pumps, i.e. forward (reverse) sources can modify light propagation in the reverse (forward) direction.
Achieving complete linear optical isolation through opto-mechanical interactions that occur in all media, irrespective of crystallinity or amorphicity, material band structure, magnetic bias, or presence of gain, ensures that the technique could be implemented in nearly any photonic foundry process with any optical material. Example systems that could support this isolation approach are released optomechanical resonators with co-integrated waveguides such as those shown in ref. 40. Since the isolation bandwidth demonstrated here is relatively narrow (about 400 kHz), but is wavelength agnostic, this approach must be tailored for particular photonic device applications. However, we must emphasize that the maximum bandwidth of this isolation approach under strong acousto-optical coupling is only limited by the optical mode linewidth κ 2, allowing future improvement in isolation bandwidth to several GHz with the use of low optical Q-factor modes and higher acoustic frequencies. In contrast to all previous works, this induced transparency approach ensures that bidirectional signals are attenuated by default, and only unidirectional transport is enabled when the control optical stimulus is applied. This scheme additionally ensures protection for the system to be isolated in case of failure of the control source, and allows the possibility of dynamic optical shuttering. The absence of magnetic or radiofrequency electromagnetic driving fields make this approach particularly useful for chip-scale cold atom microsystems technologies, for both isolation and shuttering of optical signals, and laser protection without loss.
Waveguide-Resonator System
We experimentally demonstrate ultra-low loss optical isolation by probing light transmission through the waveguide in the forward and backward directions simultaneously. In our experiment, a tapered optical fiber waveguide is fabricated by linear tension drawing of SMF-28 fiber while being heated with a hydrogen flame41, till the point that the tapered waveguide diameter is comparable to the laser wavelength and supports only a single optical mode with significant evanescent field. With adiabatic tapering42 the loss associated with this waveguide can be made as low as 0.003 dB43. We employed a resonator of diameter 170 μm to guarantee the natural existence of multiple triplets of acoustic and optical modes that satisfy the phase-matching condition for BSIT, although smaller resonators may also be used. The microsphere resonator is fabricated by reflow of a single-ended optical fiber taper using an arc discharge. The fiber mode is coupled to the resonator by means of evanescent field overlap with the resonator's whispering gallery modes. The optical coupling rate is controlled using distance with a piezo-nanopositioner.
The experimental setup used for the simultaneous forward and backward measurements is shown in Fig. 6. We employ a 1520–1570 nm tunable external cavity diode laser (ECDL) to generate the control and probe laser fields. This laser source is first split into the forward and backward directions using a 50:50 splitter. Electro-optic modulators (EOM) are employed as variable optical attenuators in dc mode (i.e. by adjusting the bias voltage) for manipulating control laser power in either direction. The probe laser is also derived from the control laser using the same EOMs to generate two sidebands spectrally separated from the control by the modulation frequency ω m . The probe laser frequency \({\omega }_{p}={\omega }_{c}+{\omega }_{m}\) can be swept using ω m relative to the control laser ω c . An erbium-doped fiber amplifier (EDFA) is used after each EOM to independently modify the control laser power, which in turn regulates the pump-enhanced Brillouin coupling rate G in either direction. Fiber polarization controllers (FPC) are used to match the light polarizations of the forward and backward propagating laser fields. Two circulators are placed before and after the resonator to allow simultaneous measurements of the probe transmissions in the forward and backward directions without reconfiguring the experimental setup. We use a total of four photodetectors, two for measuring the forward and backward probes which are used as references (PD1 and PD2 in Fig. 6) and the other two for measuring the forward and backward probe transmissions through the resonator-waveguide system (PD3 and PD4 in Fig. 6). The experiment is performed at room temperature and atmospheric pressure condition.
Experimental setup for simultaneous forward and backward probe transmission measurements is shown. We use a matched set of optical components including the electro-optic modulator (EOM), the erbium-doped fiber amplifier (EDFA), the fiber polarization contrller (FPC), the circulator and the photodetectors (PD) for the forward and backward measurements. Light is coupled to the resonator via tapered waveguide. An electronic network analyzer (NA) performs ratiometric measurements of NA reference and NA input signals (marked in figure) in the forward and backward directions. The electrical spectrum analyzer (ESA) is also used to observe the acoustic phonon mode by measuring the beat note generated by the control laser and Brillouin light scattering by the phonons.
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Funding for this research was provided through the DARPA Cold Atom Microsystems (CAMS) program, and the Air Force Office for Scientific Research (AFOSR) Young Investigator program.
JunHwan Kim and Seunghwi Kim contributed equally to this work.
Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA
JunHwan Kim, Seunghwi Kim & Gaurav Bahl
JunHwan Kim
Seunghwi Kim
Gaurav Bahl
J.K., S.K., and G.B. conceived and designed the experiments. J.K. and S.K. developed the experimental setup and carried out the experiments. All authors analyzed the data and co-wrote the paper. G.B. supervised all aspects of this project.
Correspondence to Gaurav Bahl.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Kim, J., Kim, S. & Bahl, G. Complete linear optical isolation at the microscale with ultralow loss. Sci Rep 7, 1647 (2017). https://doi.org/10.1038/s41598-017-01494-w
Received: 16 January 2017
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The Journal of Basic and Applied Zoology
Effect of dietary Citrus sinensis peel extract on growth performance, digestive enzyme activity, muscle biochemical composition, and metabolic enzyme status of the freshwater fish, Catla catla
M. S. Shabana1,
M. Karthika1 &
V. Ramasubramanian1
The Journal of Basic and Applied Zoology volume 80, Article number: 51 (2019) Cite this article
The present study was made to assess the effects of dietary Citrus sinensis peel extract on the growth performance, digestive enzyme activity, muscle biochemical compositions, profiles of fatty acid and amino acid, and metabolic enzyme status of the freshwater fish Catla catla. The methanolic extract of C. sinensis peel was supplemented with basal diets at 2, 6, and 10 g kg−1 and fed to C. catla for a 45-day experiment period.
Fish fed with the different concentrations of C. sinensis peel-supplemented extract showed significant (P ˂ 0.05) improvement in the survival, growth, muscle biochemical compositions, digestive enzyme activities, and profile of amino acids and fatty acids when compared to control. Among these different concentrations, 6 g kg− 1 C. sinensis-supplemented diet produced a significantly better performance when compared to other concentrations. Similarly, the insignificant (P > 0.05) difference was observed in the metabolic enzyme activities (glutamic oxaloacetic transaminase and glutamic pyruvic transaminase) in all concentrations of C. sinensis peel-supplemented diet-fed fish. It indicates that the supplemented peel extract did not produce any adverse effect on C. catla.
The obtained results suggested that the 6 g kg− 1 of C. sinensis can be supplemented in the diet of C. catla for regulating better survival and growth.
Aquaculture was continuously intensified due to the decrease of wild capture and increased demand for the protein food. Global production of food fish and other aquatic animals from aquaculture reached 170.9 million tons in 2016 (FAO year book, 2016). Among the cultivated fish species, Catla catla is the most important farmed freshwater fish species with a high economic value due to its delicious taste and presence of rich protein and omega-3 fatty acids, which contain lower triglyceride levels which help to reduce inflammation throughout the body and support brain health. The fish production tended to increase during the first two quarters of 2017–2018, and it was estimated as 5.80 million tons (DAHDFF, 2017).
Nutrient composition of feed, such as protein, carbohydrate, lipid, vitamins, and minerals, is the most important factor affecting the health and growth of fish; hence, properly balanced supplemental feeds with a reliable feeding rate can be helpful to enhance survival and growth (Dawood & Koshio, 2016; Dawood, Koshio, & Esteban, 2017). In recent years, plant products (leaf, root, stem, bark, etc.) have been used as a natural immunostimulant instead of antibiotics in aquaculture feed formulations due to their eco-friendly and cost-effective properties compared to synthetic drugs. Fruit peels, such as Musa sapientum, Citrus limon, Artocarpus heterophyllus, Mangifera indica, Hippophae rhamnoides, and Punica granatum, exhibit anti-inflammatory, antitumor, antioxidant, and antimicrobial activities due to the presence of rich flavonoid glycosides, coumarins, β- and ɤ-sitosterols, vitamins, and volatile compounds (Chiba et al., 2003; Gao et al., 2006; Liu, Heying, & Tanumihardjo, 2012). The orange peel is a primary by-product produced by the fruit processing industries, and it accounts approximately 45% of the total bulk (Farhat et al., 2011). Citrus sinensis showed several medicinal properties like anticancer, antidiuretic, immunity enhancer, and tonic to digestion (Grosso et al., 2014). However, to the best of our knowledge, the information about the influence of dietary C. sinensis peel extract on fish is not yet reported so far. Thus, the present study was aimed to assess the effect of C. sinensis peel extract on the survival, growth, digestive enzyme activities, muscle biochemical compositions, profile of amino acids and fatty acids, and metabolic enzymes of the C. catla.
Experimental fish
The freshwater fish C. catla were obtained from the Aliyar dam in Tamil Nadu fisheries development corporation, Pollachi, Coimbatore District, India. Fish were acclimatized to laboratory condition in a large cement tank (6″ × 4″ × 3″) with ground water with an optimal level of physico-chemical characteristics (temperature, 27.33 ± 0.57 °C; dissolved oxygen, 7.23 ± 0.58 mg/L; pH, 7.2 ± 0.1; total dissolved solids, 0.68 ± 0.06 g/L; biological oxygen demand, 18.63 ± 0.35 mg/L; chemical oxygen demand, 67.33 ± 5.03 mg/L; ammonia, 0.4 ± 0.1 mg/L) for 2 weeks. During the acclimatization period, fish was fed with commercial feed thrice (at 06:00 h, 12:00 h, and 18:00 h) per day. Feces and unfed feeds were cleared out daily while renewing the 80% of tank water to maintain the healthy environment.
Preparation of crude extracts of C. sinensis peel
The orange peels (C. sinensis) were collected from various fruits and juice stalls at Coimbatore. The collected peels were washed thrice in distilled water, chopped, and shade dried at room temperature for 2–3 weeks. The dried peels of C. sinensis were grounded into coarse powder for the ease of extraction of active compounds. The powdered plant material (150 g) was wrapped in a filter paper, placed in a Soxhlet apparatus, and extracted with absolute methanol. After extraction, the condensation process was carried out, which condensed the solvent into a liquid form. Finally, it was kept in a water bath for 1 h and a half to eliminate the solvent from the extract and then stored at 4 °C until used for experimentation (Anju, Arun, Sayeed, & Narasimhan, 2011). This process was repeated five times to get an adequate amount of extract for feed formulation.
Feed preparation was made in the laboratory according to Table 1. The ingredients including fishmeal, soybean meal, wheat bran, tapioca flour, eggs, and cod liver oil and vitamin mix were purchased from the local markets. For this diet preparation, the fish meal and soybean meal were served as the protein source, the carbohydrate sources were wheat and tapioca flour, and lipid source was cod liver oil. Also, tapioca flour and egg albumin were taken as binding agents, and vitamin B complex with vitamin C was also added as an essential micronutrient. The above ingredients except egg albumin, cod liver oil, vitamins, and minerals were mixed thoroughly and steam cooked for 20 min at 105 °C. Different concentrations of C. sinensis peel extract (2 g kg− 1, 6 g kg− 1, and 10 g kg− 1) were added along with the heat-sensitive ingredients like vitamin, mineral premix, egg albumin, and cod liver oil to the steam cocked basal diet and mixed well to form a dough. Further, the dough was pelleted using indigenous hand pelletizer (Retro stainless steel, BM brand) and dried at room temperature until the constant weight was reached.
Table 1 Composition of formulated feed along with the Citrus sinensis peel extract
Experimental procedure
Four groups of C. catla were assigned for 45 days of the experiment in triplicate. Three groups were fed with 2, 6, and 10 g kg− 1 C. sinensis peel-supplemented diets. The remaining one group was served as control (fed with "0" concentration of C. sinensis peel-supplemented diet). Each group consisted of 50 fish. The water medium was renewed every day by siphoning method. At the end of the feeding experiment, fish from each treatment were sampled to analyze various parameters.
Assessment of survival, growth, and food index
Survival, growth, weight gain, length gain, specific growth rate, and food index parameters, such as feed intake, feed conversion ratio, and protein efficiency ratio were calculated according to the following equations (Tekinay & Davies, 2001)
$$ {\displaystyle \begin{array}{l}\mathrm{Survival}\ \left(\%\right)=\mathrm{no}.\mathrm{of}\ \mathrm{live}\ \mathrm{fish}/\mathrm{no}.\mathrm{of}\ \mathrm{fish}\ \mathrm{in}\mathrm{troduced}\times 100\\ {}\mathrm{Length}\ \mathrm{gain}\ \left(\mathrm{cm}\right)=\mathrm{final}\ \mathrm{length}\ \left(\mathrm{cm}\right)-\mathrm{initial}\ \mathrm{length}\ \left(\mathrm{cm}\right)\\ {}\mathrm{Weight}\ \mathrm{gain}\ \left(\mathrm{g}\right)=\mathrm{final}\ \mathrm{weight}\ \left(\mathrm{g}\right)-\mathrm{initial}\ \mathrm{weight}\ \left(\mathrm{g}\right)\\ {}\mathrm{Feed}\ \mathrm{in}\mathrm{take}\ \left(\mathrm{g}\ {\mathrm{day}}^{-1}\right)=\mathrm{feed}\ \mathrm{in}\mathrm{take}\ \left(\mathrm{g}\right)/\mathrm{total}\ \mathrm{number}\ \mathrm{of}\ \mathrm{days}\\ {}\mathrm{Specific}\ \mathrm{growth}\ \mathrm{rate}\ \left(\%\right)=\log {w}_2-\log {w}_1/t\times 100\ \left(\mathrm{where}\ {w}_1\ \mathrm{and}\ {w}_2=\mathrm{initial}\ \mathrm{an}\mathrm{d}\ \mathrm{final}\ \mathrm{weight}\ \left(\mathrm{g}\right),\mathrm{and}\ t=\mathrm{duration}\ \mathrm{of}\ \mathrm{an}\ \mathrm{experiment}\ \mathrm{in}\ \mathrm{days}\right)\\ {}\mathrm{Feed}\ \mathrm{conversion}\ \mathrm{ratio}=\mathrm{feed}\ \mathrm{in}\mathrm{take}\ \left(\mathrm{g}\right)/\mathrm{weight}\ \mathrm{gain}\ \left(\mathrm{g}\right)\\ {}\mathrm{Protein}\ \mathrm{efficiency}\ \mathrm{ratio}=\mathrm{weight}\ \mathrm{gain}\ \left(\mathrm{g}\right)/\mathrm{protein}\ \mathrm{in}\mathrm{take}\ \left(\mathrm{g}\right).\end{array}} $$
Assay of digestive enzyme activity and muscle biochemical compositions
Activities of the digestive enzymes (protease, amylase, and lipase) were assayed on the initial and final days of the feeding experiment. Forty fish per treatment (10 fish per tank) were randomly selected; the whole digestive tract and the muscle were taken to analyze the digestive enzyme activity and biochemical compositions. The whole digestive tract was homogenized in ice-cold double-distilled water and centrifuged at 9300g under 4 °C for 20 min. The supernatant was used as crude enzyme source. The casein-hydrolysis method was used to determine the total protease activity (Furne et al., 2005). Amylase activity was determined by the starch-hydrolysis method (Bernfeld, 1955), and the lipase activity was analyzed by the method of Furne et al. (2005). The biochemical constituents of the fish muscle, such as protein (Lowery, Rosebrough, Farr, & Randall, 1951), carbohydrate (Roe, 1954), and total lipid (Folch, Less, & Sloane Stanley, 1956) were estimated by the following standard methods.
Assay of metabolic enzyme activity
The metabolic enzymes, such as glutamic oxaloacetate transaminase (GOT) and glutamic pyruvate transaminase (GPT), were analyzed in the muscle of C. catla according to the method of Reitman and Frankel (1957). Five fish from each individual tank (20 fish per treatment) were collected, and the tissue (100 mg) was homogenized in 0.25 M sucrose and centrifuged at 3300 rpm for 20 min in a high-speed cooling centrifuge at 4 °C. The supernatant was used as the enzyme source. The optical density was taken using a spectrophotometer at 505 nm within 15 min. GOT and GPT activity was expressed as units per liter.
Amino acid profile analysis
High-performance thin-layer chromatographic (HPTLC) method (Hess & Sherma, 2004) was used to analyze the profile of amino acids in the muscle of C. catla (20 fish per treatment, 5 fish per tank) fed with formulated experimental feeds. Standard amino acids like proline, serine, asparagine, glutamine, methionine, aspartic acid, glutamic acid, alanine, valine, phenyl alanine, lysine, glycine, threonine, isoleucine, and tyrosine, arginine, cysteine, histidine, leucine, and tryptophan were also performed in parallel. The peak area of the sample was compared and quantified with standard amino acids. The obtained amino acids were expressed as grams per kilogram of dry weight.
Fatty acid profile analysis
Gas chromatographic and mass spectrometry (GC-MS) method of Martins et al. (2003) was used to analyze the profile of fatty acids using 5 fish per group. Fatty acids were obtained from lipids by saponification. Each fatty acid in the unknown sample was identified based on the retention time and peak area of the standard fatty acids and expressed as %/2 μL methylated fatty acid.
The data were expressed as mean ± S.D. and analyzed by one-way analysis of variance (ANOVA) using SPSS (21.0), followed by Duncan's multiple range test (DMRT) to compare the differences among treatments. Differences were considered significant at P ˂ 0.05.
Survival and nutritional index
Survival, growth, weight gain, feed intake, specific growth rate, and protein efficiency ratio were significantly increased (P ˂ 0.05) in the fish fed with 6 g kg− 1 C. sinensis-supplemented diets when compared to other concentrations of C. sinensis and control diet-fed fish. In context, the feed conversion ratio was found to be significantly decreased in the fish fed with 6 g kg− 1 C. sinensis-supplemented diets when compared with control and other concentrations of C. sinensis-incorporated feed-fed fish group (Table 2).
Table 2 Survival, growth, and food index evaluation of Catla catla fed with C. sinensis-supplemented diets
Activity of digestive enzymes and muscle biochemical compositions
The digestive enzymes such as protease, amylase, and lipase were found to be significantly elevated (P ˂ 0.05) in the fish fed with 2–10 g kg− 1 C. sinensis-supplemented diets when compared to control. However, the differences in these enzymes' activity between 2 and 10 g kg− 1 C. sinensis were insignificant in the case of amylase and lipase activities (Table 3). In the present study, the concentrations of biochemical constituents, such as protein, carbohydrate, and lipid contents were significantly (P < 0.05) increased in fish fed with 6 g kg− 1 of C. sinensis peel-supplemented diets when compared to other concentrations of C. sinensis and control, while lipid content showed insignificant (P > 0.05) difference between 2 and 6 g kg− 1 of C. sinensis-supplemented diet-fed fish when compared to control diet-fed fish. In context, these biochemical compositions were significantly (P < 0.05) decreased in 10 g kg− 1 of C. sinensis peel extract-supplemented diet-fed fish group when compared to 6 g kg− 1 C. sinensis peel extract diet-fed fish group (Table 3).
Table 3 Muscle biochemical composition of Catla catla fed with different concentrations of Citrus sinensis-supplemented diets during the experimental period
Metabolic enzyme status
The metabolic enzymes (GOT and GPT) were insignificantly (P > 0.05) elevated in liver tissue of the fish fed with 2–6 g kg− 1 C. sinensis peel-supplemented diets, whereas fish fed with 10 g kg− 1 C. sinensis peel extracts showed significantly (P < 0.05) better elevation in the GOT and GPT activities (Table 4).
Table 4 Activities of digestive and the metabolic enzymes (U/mg protein) in Catla catla fed with different concentration of Citrus sinensis-supplemented diets
Amino acid profile
Seventeen amino acids were detected in the muscle of C. catla fed with different concentrations of C. sinensis-supplemented diets. Among these, histidine, isoleucine, leucine, lysine, methionine, phenylalanine, threonine, tryptophan, and valine were essential amino acids, and arginine, cysteine, glutamine, glycine, proline, tyrosine, alanine, and aspartic acid were non-essential amino acids. The essential amino acids such as lysine, phenylalanine, threonine, tryptophan, and valine were found to be significantly (P < 0.05) elevated in the fish fed with the different concentration of C. sinensis extract-supplemented diets compared to control, while the non-essential amino acids, such as glutamine, glycine, tyrosine, and aspartic acids, were significantly (P < 0.05) elevated in 2–6 g kg− 1 C. sinensis-supplemented diet fish when compared to the control. However, the insignificant difference was observed between 10 g kg− 1 C. sinensis extract and control diet fed fish in the case of glutamine and tyrosine in (Table 5).
Table 5 Amino acid (g kg− 1) profile of Catla catla fed with C. sinensis-supplemented diets
In the present study, four saturated (myristic acid, palmitic acid, stearic acid, and heptadecanoic acid), two unsaturated (paullinic acid, fumaric acid), two mono-unsaturated (palmitoleic acid, oleic acid), and seven poly unsaturated (linoleic acid, arachidonic acid, eicosatetraenoic acid, docosahexaenoic acid, docosapentaenoic acid, docosatetraenoic acid, and eicosapentaenoic acid) fatty acids were detected in fish muscle through GC-MS analysis. All these fatty acids were found to be insignificantly elevated in the fish fed with the different concentration of C. sinensis when compared to control (Table 6).
Table 6 Fatty acid profile of Catla catla fed with C. sinensis-supplemented diets
Citrus fruit contains a rich source of secondary metabolites like natural flavonoids, polyphenols, steroids, and saponins. Citrus has antimicrobial and antioxidant properties against various microbes like Streptococcus mutans, Lactobacillus acidophilus, Staphylococcus aureus, and Escherichia coli (Mathur et al., 2011). Essential oil obtained from the citrus peel manifest antibacterial activity has also been reported (Upadhyay, Dwivedi, & Ahmad, 2010). Plant-based extracts with antimicrobial and immunostimulant properties have been used as therapeutic and prophylactic agents against fish pathogens in aquaculture industries to maintain an eco-friendly environment (Newaj-Fyzul & Austin, 2015). Further, dietary administration of plant extracts can stimulate the immune response by reducing the pathogen load which leads to better survival and growth of fish culture (Abdel–Tawwab, Ahmad, Seden, & Sakr, 2010; El-Desouky, El-Asely, Shaheen, & Abbass, 2012; Gabriel et al., 2015; Kaleeswaran, Ilavenil, & Ravikumar, 2011). In the present study, the significant improvement in survival, growth rate, length and weight gain, feed intake, specific growth rate, and protein efficiency ratio indicates that the supplementation of 2–6 g kg− 1 C. sinensis peel extract has the ability to promote growth performance and feed intake of C. catla. Previously, Acar et al. (2015) reported that the dietary inclusion of essential oil extract from C. sinensis produced better survival and growth of Oreochromis mossambicus. Plant extract, such as Citrus sinensis, Cynodon dactylon, Aloe vera, Camellia sinensis, Echinacea purpurea, and Allium sativum, -supplemented diet-fed C. catla, Penaeus monodon, Oreochromis niloticus, Carrasius auratus, Lates calcarifer, and Macrobrachium rosenbergii which showed better survival, growth performance, feed intake, specific growth rate, and protein efficiency have been reported (Abdel–Tawwab et al., 2010; Aly & Mohamed, 2010; El-Desouky et al., 2012; Gabriel et al., 2015; Kaleeswaran et al., 2011; Kumar et al., 2013; Yogeeswaran et al., 2012). The significant decreases of survival, growth, feed intake, specific growth rate, and protein efficiency ratio in 10 g kg− 1 C. sinensis feed-fed fish suggest that this concentration might be over dose, which led to the negative impact on the fish. Similar results have been reported in Cyprinus carpio when fed with ethanolic extract of Ocimum basilicum-supplemented diets (Amirkhani & Firouzbakhsh, 2013).
The biochemical compositions, such as protein, carbohydrate, and lipid, are the physiological indicators of fish health, and the nutritive value of fish depends upon their biochemical constituents. In the present study, the significant improvement in muscle biochemical composition (protein, carbohydrate, and lipid) suggests that the synthesis and the storage of the biochemical compositions in C. catla were promoted due to supplementation of C. sinensis extracts in the diet. Similar results have also been reported in tilapia (Oreochromis niloticus) fed on citrus essential oil-supplemented diet (Acar, Kesbic, Yilmaz, Gultepe, & Turker, 2015). Xiaohong et al. (2017) reported the significant increase in the muscle biochemical composition of golden pompanos (Trachinotus auratus) fed with dietary dandelion extracts.
The fish digest the nutrients in the feed with the help of the digestive enzymes, subsequently increasing the feed efficiency (Widanarni & Jusadi, 2015). In the present study, the digestive enzymes (protease, amylase, and lipase) were found to be significantly improved in the C. sinensis-supplemented diets; it indicates that the supplementation of C. sinensis promotes the secretion of these digestive enzymes, which in turn improves the digestion of nutrients, followed by growth of the C. catla. Similarly, administration of Ricinus communis in the diet of black tiger shrimp showed significant improvement in the activity of digestive enzymes. The administration of garlic, ginger, turmeric, and fenugreek into the diets of M. rosenbergii PL which showed an increase in the activities of protease, amylase, and lipase has been reported earlier (Poongodi, Saravana Bhavan, Muralisankar, & Radhakrishnan, 2012).
In the present study, the insignificant elevations in the GOT and GPT in 2–6 g kg− 1 C. sinensis indicate the normal health of liver in fish. In context, the significant alterations of GOT and GPT in the 10 g kg− 1 C. sinensis-supplemented diets suggest some damage in the liver of fish, which leads to poor survival and growth of fish. Previously, the administration of Origanum vulgare extract in the diet of Nile Tilapia (O. niloticus) showed significant alteration in GOT (glutamic oxaloacetic transaminase) and GPT (glutamic pyruvate transaminase) has been reported (El-Araby & EL-Arabey, 2016).
Amino acids are the building blocks of proteins and serve as body builders of an organism. Amino acids are utilized by various cell structures as key components (Anaya & Daniello, 2006). All animals need a constant source of amino acids for tissue protein synthesis and synthesis of other compounds associated with metabolism including hormones, neurotransmitters, purines, and metabolic enzymes (Halver & Hardy, 2002). In the current study, significant improvements in essential and non-essential amino acids in the fish fed with C. sinensis indicate that the supplementation of C. sinensis extract had influence on the synthesis of amino acids which led to better growth and survival of C. catla. Previously, administration of mango seed kernel, banana peel, and papaya peel in the diets of M. rosenbergii which influenced the synthesis of essential and non-essential amino acids have been reported (Aarumugam et al., 2013). Similar results have also been reported in Acipenser ruthenus (juvenile sterlet sturgeon) fed on garlic extracts in the supplemented diet (Lee, Seong, Chang, & Jeong, 2012). Poongodi (2011) reported that the significant increase of essential and non-essential amino acids in the diets of M. rosenbergii PL after the administration of garlic, ginger, turmeric, and fenugreek. Similar results have also been reported in Acipenser ruthenus (juvenile sturgeon) fed on garlic extracts in the supplemented diet (Lee et al., 2012).
Fatty acids play the crucial role in the maintenance of the metabolic and the physiological process which leads to the better growth, survival, and the reproduction of the aquatic organisms. In the present study, the insignificant elevation in the saturated and the unsaturated fatty acids in C. catla fed with C. sinensis extract-supplemented diets suggests that methanolic extracts of C. sinensis did not produce any negative impact on the fatty acid synthesis in the experimental fish C. catla. Similarly, the influence of dietary garlic extracts on fatty acid synthesis has been reported in Acipenser ruthenus and M. rosenbergii (Aarumugam et al., 2013; Lee et al., 2012). Previously, administration of spirulina (Arthrospira platensis) and/ or thyme (Thymus vulgaris) in the diets of Oryctolagus cuniculus (New Zealand white rabbit) showed significant changes in fatty acid contents (Mattioli et al., 2017). Similar results have also been reported in M. rosenbergii PL fed on garlic, ginger, turmeric, and fenugreek extracts in the supplemented diet (Poongodi, 2011).
The result of the present study revealed that the dietary incorporation of 2–6 g kg− 1 methanolic extract of C. sinensis fruit peel significantly improved the survival, growth performance, activities of digestive enzyme, muscle biochemical constituents, and amino acids. The insignificant alteration in metabolic enzyme activity in 2–6 g kg− 1 C. sinensis indicates good health status of the fish. Among these different concentrations of C. sinensis, 6 g kg− 1 produced better performance. Therefore, the present study suggests that the methanolic extract of 6 g kg− 1 C. sinensis peel can be supplemented with the basal diets of C. catla for regulating better culture practice in the aquaculture industry.
The data supporting the conclusions of this work is included within the article. The authors can be contacted for any additional supporting data required by the journal.
DAHDFF:
Department of Animal Husbandry, Dairying & Fisheries Ministry of Agriculture & Farmers Welfare
DMRT:
Duncan's multiple range test
FAO:
GC-MS:
Gas chromatographic and mass spectrometry
Glutamic oxaloacetate transaminase
GPT:
Glutamic pyruvate transaminase
HPTLC:
High-performance thin-layer chromatographic
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The first author would like to thank Bharathiar University for providing the necessary facilities and sincerely acknowledge the constant support, encouragement, and valuable suggestions from those who helped for the completion of this work.
This work did not receive any financial support.
Aquatic Biotechnology and Live Feed Culture Laboratory, Department of Zoology, School of Life Sciences, Bharathiar University, Coimbatore, Tamil Nadu, 641 046, India
M. S. Shabana, M. Karthika & V. Ramasubramanian
M. S. Shabana
M. Karthika
V. Ramasubramanian
MS participated in the design of this research work and performed the collection of samples. MK contributed in the analytical part. MS undertook the characterization studies. MS wrote the manuscript. VR supervised the findings of this work. All authors discussed the result and contributed to the final manuscript. All authors read and approved the final manuscript.
Correspondence to V. Ramasubramanian.
The authors declare that no animal was sacrificed for this study. The collected species was not in the IUCN red list. We declare that we do not need ethical clearance for the present work.
Shabana, M.S., Karthika, M. & Ramasubramanian, V. Effect of dietary Citrus sinensis peel extract on growth performance, digestive enzyme activity, muscle biochemical composition, and metabolic enzyme status of the freshwater fish, Catla catla. JoBAZ 80, 51 (2019). https://doi.org/10.1186/s41936-019-0119-x
Catla catla
Glutamic pyruvic transaminase
Glutamic oxaloacetic transaminase
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Table of prime factors
The tables contain the prime factorization of the natural numbers from 1 to 1000.
When n is a prime number, the prime factorization is just n itself, written in bold below.
The number 1 is called a unit. It has no prime factors and is neither prime nor composite.
Properties
Many properties of a natural number n can be seen or directly computed from the prime factorization of n.
• The multiplicity of a prime factor p of n is the largest exponent m for which pm divides n. The tables show the multiplicity for each prime factor. If no exponent is written then the multiplicity is 1 (since p = p1). The multiplicity of a prime which does not divide n may be called 0 or may be considered undefined.
• Ω(n), the big Omega function, is the number of prime factors of n counted with multiplicity (so it is the sum of all prime factor multiplicities).
• A prime number has Ω(n) = 1. The first: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 (sequence A000040 in the OEIS). There are many special types of prime numbers.
• A composite number has Ω(n) > 1. The first: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21 (sequence A002808 in the OEIS). All numbers above 1 are either prime or composite. 1 is neither.
• A semiprime has Ω(n) = 2 (so it is composite). The first: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34 (sequence A001358 in the OEIS).
• A k-almost prime (for a natural number k) has Ω(n) = k (so it is composite if k > 1).
• An even number has the prime factor 2. The first: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 (sequence A005843 in the OEIS).
• An odd number does not have the prime factor 2. The first: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23 (sequence A005408 in the OEIS). All integers are either even or odd.
• A square has even multiplicity for all prime factors (it is of the form a2 for some a). The first: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 (sequence A000290 in the OEIS).
• A cube has all multiplicities divisible by 3 (it is of the form a3 for some a). The first: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728 (sequence A000578 in the OEIS).
• A perfect power has a common divisor m > 1 for all multiplicities (it is of the form am for some a > 1 and m > 1). The first: 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100 (sequence A001597 in the OEIS). 1 is sometimes included.
• A powerful number (also called squareful) has multiplicity above 1 for all prime factors. The first: 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72 (sequence A001694 in the OEIS).
• A prime power has only one prime factor. The first: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19 (sequence A000961 in the OEIS). 1 is sometimes included.
• An Achilles number is powerful but not a perfect power. The first: 72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968 (sequence A052486 in the OEIS).
• A square-free integer has no prime factor with multiplicity above 1. The first: 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17 (sequence A005117 in the OEIS)). A number where some but not all prime factors have multiplicity above 1 is neither square-free nor squareful.
• The Liouville function λ(n) is 1 if Ω(n) is even, and is -1 if Ω(n) is odd.
• The Möbius function μ(n) is 0 if n is not square-free. Otherwise μ(n) is 1 if Ω(n) is even, and is −1 if Ω(n) is odd.
• A sphenic number has Ω(n) = 3 and is square-free (so it is the product of 3 distinct primes). The first: 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154 (sequence A007304 in the OEIS).
• a0(n) is the sum of primes dividing n, counted with multiplicity. It is an additive function.
• A Ruth-Aaron pair is two consecutive numbers (x, x+1) with a0(x) = a0(x+1). The first (by x value): 5, 8, 15, 77, 125, 714, 948, 1330, 1520, 1862, 2491, 3248 (sequence A039752 in the OEIS), another definition is the same prime only count once, if so, the first (by x value): 5, 24, 49, 77, 104, 153, 369, 492, 714, 1682, 2107, 2299 (sequence A006145 in the OEIS)
• A primorial x# is the product of all primes from 2 to x. The first: 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810 (sequence A002110 in the OEIS). 1# = 1 is sometimes included.
• A factorial x! is the product of all numbers from 1 to x. The first: 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600 (sequence A000142 in the OEIS). 0! = 1 is sometimes included.
• A k-smooth number (for a natural number k) has largest prime factor ≤ k (so it is also j-smooth for any j > k).
• m is smoother than n if the largest prime factor of m is below the largest of n.
• A regular number has no prime factor above 5 (so it is 5-smooth). The first: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16 (sequence A051037 in the OEIS).
• A k-powersmooth number has all pm ≤ k where p is a prime factor with multiplicity m.
• A frugal number has more digits than the number of digits in its prime factorization (when written like below tables with multiplicities above 1 as exponents). The first in decimal: 125, 128, 243, 256, 343, 512, 625, 729, 1024, 1029, 1215, 1250 (sequence A046759 in the OEIS).
• An equidigital number has the same number of digits as its prime factorization. The first in decimal: 1, 2, 3, 5, 7, 10, 11, 13, 14, 15, 16, 17 (sequence A046758 in the OEIS).
• An extravagant number has fewer digits than its prime factorization. The first in decimal: 4, 6, 8, 9, 12, 18, 20, 22, 24, 26, 28, 30 (sequence A046760 in the OEIS).
• An economical number has been defined as a frugal number, but also as a number that is either frugal or equidigital.
• gcd(m, n) (greatest common divisor of m and n) is the product of all prime factors which are both in m and n (with the smallest multiplicity for m and n).
• m and n are coprime (also called relatively prime) if gcd(m, n) = 1 (meaning they have no common prime factor).
• lcm(m, n) (least common multiple of m and n) is the product of all prime factors of m or n (with the largest multiplicity for m or n).
• gcd(m, n) × lcm(m, n) = m × n. Finding the prime factors is often harder than computing gcd and lcm using other algorithms which do not require known prime factorization.
• m is a divisor of n (also called m divides n, or n is divisible by m) if all prime factors of m have at least the same multiplicity in n.
The divisors of n are all products of some or all prime factors of n (including the empty product 1 of no prime factors). The number of divisors can be computed by increasing all multiplicities by 1 and then multiplying them. Divisors and properties related to divisors are shown in table of divisors.
1 to 100
1 − 20
1
22
33
422
55
62·3
77
823
932
102·5
1111
1222·3
1313
142·7
153·5
1624
1717
182·32
1919
2022·5
21 − 40
213·7
222·11
2323
2423·3
2552
262·13
2733
2822·7
2929
302·3·5
3131
3225
333·11
342·17
355·7
3622·32
3737
382·19
393·13
4023·5
41 − 60
4141
422·3·7
4343
4422·11
4532·5
462·23
4747
4824·3
4972
502·52
513·17
5222·13
5353
542·33
555·11
5623·7
573·19
582·29
5959
6022·3·5
61 − 80
6161
622·31
6332·7
6426
655·13
662·3·11
6767
6822·17
693·23
702·5·7
7171
7223·32
7373
742·37
753·52
7622·19
777·11
782·3·13
7979
8024·5
81 − 100
8134
822·41
8383
8422·3·7
855·17
862·43
873·29
8823·11
8989
902·32·5
917·13
9222·23
933·31
942·47
955·19
9625·3
9797
982·72
9932·11
10022·52
101 to 200
101 − 120
101101
1022·3·17
103103
10423·13
1053·5·7
1062·53
107107
10822·33
109109
1102·5·11
1113·37
11224·7
113113
1142·3·19
1155·23
11622·29
11732·13
1182·59
1197·17
12023·3·5
121 − 140
121112
1222·61
1233·41
12422·31
12553
1262·32·7
127127
12827
1293·43
1302·5·13
131131
13222·3·11
1337·19
1342·67
13533·5
13623·17
137137
1382·3·23
139139
14022·5·7
141 − 160
1413·47
1422·71
14311·13
14424·32
1455·29
1462·73
1473·72
14822·37
149149
1502·3·52
151151
15223·19
15332·17
1542·7·11
1555·31
15622·3·13
157157
1582·79
1593·53
16025·5
161 − 180
1617·23
1622·34
163163
16422·41
1653·5·11
1662·83
167167
16823·3·7
169132
1702·5·17
17132·19
17222·43
173173
1742·3·29
17552·7
17624·11
1773·59
1782·89
179179
18022·32·5
181 − 200
181181
1822·7·13
1833·61
18423·23
1855·37
1862·3·31
18711·17
18822·47
18933·7
1902·5·19
191191
19226·3
193193
1942·97
1953·5·13
19622·72
197197
1982·32·11
199199
20023·52
201 to 300
201 − 220
2013·67
2022·101
2037·29
20422·3·17
2055·41
2062·103
20732·23
20824·13
20911·19
2102·3·5·7
211211
21222·53
2133·71
2142·107
2155·43
21623·33
2177·31
2182·109
2193·73
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See also
• Fundamental theorem of arithmetic – Integers have unique prime factorizations
• List of prime numbers – List of prime numbers and notable types of prime numbers
• Table of divisors – numbers divisible by another numberPages displaying wikidata descriptions as a fallback
| Wikipedia |
\begin{document}
\setlength{\arraycolsep}{0.3em}
\title{Online Abstract Dynamic Programming with Contractive Models \thanks{}}
\author{Xiuxian Li and Lihua Xie
\thanks{X. Li is with Department of Control Science and Engineering, College of Electronics and Information Engineering, Institute for Advanced Study, and Shanghai Research Institute for Intelligent Autonomous Systems, Tongji University, Shanghai, China (e-mail: [email protected]).} \thanks{L. Xie is with School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798 (e-mail: [email protected]).}
}
\maketitle
\setcounter{equation}{0} \setcounter{figure}{0} \setcounter{table}{0}
\begin{abstract} This paper addresses the abstract dynamic programming (DP) in the online scenario, where the abstract DP mapping is time-varying, instead of static. In this case, optimal costs and policies at different time instants are not the same in general, and the problem amounts to tracking time-varying optimal costs and policies, which is of interest to many practical problems. It is thus necessary to analyze the performance of classical value iteration (VI) and policy iteration (PI) algorithms in the online case. In doing so, this paper develops and provides the theoretical analysis for several online algorithms, including approximate online VI, online PI, approximate online PI, online optimistic PI, approximate online optimistic PI, and asynchronous online PI and VI algorithms. It is proved that the tracking error bounds for all algorithms critically depend upon the largest difference between any two consecutive abstract mappings. Meanwhile, examples are presented to illustrate the theoretical results. \end{abstract}
\begin{IEEEkeywords} Abstract dynamic programming, online algorithms, contractive mappings, value iteration, policy iteration, optimization. \end{IEEEkeywords}
\section{Introduction}\label{s1}
Dynamic programming (DP) is a powerful tool in handling total cost sequential decision problems, which has been extensively investigated up to now and can find lots of applications in optimal control, Markovian decision problems (MDPs), stochastic shortest path problems (SSP), zero-sum dynamic game, and reinforcement learning, and so on \cite{bertsekas2018abstract,bertsekas2018proper,bertsekas2015value,yang2017hamiltonian,liu2013policy,wei2015value,heydari2014revisiting,song2014adaptive,seiffertt2008hamilton, chang2006policy,ni2015model,bucsoniu2011approximate}. In this paper, the focus is on abstract DP, which provides a unified analysis for DP models by abstracting their substantial structures.
In general, the models for abstract DP are classified into three types. The first is the contractive models, where there exists an abstract mapping that is a contraction over a space consisting of bounded functions defined on the state space, which is first introduced in \cite{denardo1967contraction}. These models have well-behaved analytical and computational properties. The second is the semicontractive models, introduced in \cite{bertsekas2018abstract}, and in this case, the abstract mapping is no longer a contraction over the whole bounded function space. However, in this model, some policies possess a contraction-like property while others do not, and these models can have a good enough theory nearly as in the contractive models when certain conditions hold. The third is the noncontractive models \cite{bertsekas1975monotone,bertsekas1977monotone}, in which the abstract mapping is monotone, instead of contractive. It is known that pathologies emerge in the noncontractive models, leading to that it is difficult to seek effective solutions \cite{bertsekas2017regular}.
There are mainly two fundamental algorithms in abstract DP, i.e., value iteration (VI) and policy iteration (PI), based on which various algorithms have been developed, including approximate VI and PI in finite-state discounted MDP \cite{scherrer2012on}, optimistic PI (or modified PI) \cite{puterman1994markovian}, approximate optimistic PI \cite{canbolat2013approximate}, $\lambda$-PI method \cite{bertsekas1996neuro}, approximate $\lambda$-PI method \cite{thiery2010least}, asynchronous VI \cite{bertsekas1982distributed}, and asynchronous PI \cite{williams1993analysis}. The core of VI and PI is the so-called Bellman's equation, and the key point is to find a fixed point of the corresponding mapping to Bellman's equation.
To date, although there exist numerous works on abstract DP problems as discussed above, most of them are devoted to the case of stationary abstract DP mappings. Nevertheless, in practical problems one often encounters the scenarios where the abstract DP mapping is time-varying itself or caused by the environment's uncertainties, that is, the cost function is time-varying and one usually does not have enough time to perform offline calculation for completely solving the problem at each time step before it goes forward to the next time step. For instance, when tracking a moving target for an unmanned aerial vehicle (UAV), the cost for penalizing the distance between this vehicle and the target is apparently time-varying. To meet the needs of practical applications, such as in reinforcement learning, researchers in optimization, machine learning, and control communities, etc. have put their great effort on online optimization/learning, where the cost functions are time-varying and gradually revealed to the decision maker, that is, the decision maker only knows the information on cost functions at hand until now, without aware of future information. Of pertinent literature along this line are \cite{shahrampour2018distributed,dixit2019online,bliek2016online,li2018distributedon,li2021distri,yi2019distributed,yi2021distri,bernstein2018asynchronous}, to just name a few.
Motivated by the above discussions, this paper aims to study the abstract DP problems with time-varying abstract DP mappings, called {\em online (or running) abstract DP} problems in this paper. To the best of our knowledge, it is the first time to consider the online scenario for abstract DP problems. Of closely relevant work is \cite{bernstein2018asynchronous}, which investigated the fixed point seeking problem for a time-varying sequence of contractive mappings or operators. However, the results in \cite{bernstein2018asynchronous} is unavailable in the context of abstract DP since policy iteration in abstract DP is more complicated than that in \cite{bernstein2018asynchronous}. The contributions of this paper are to develop and analyze online algorithms for online abstract DP, including approximate online VI, online PI, approximate online PI, online optimistic PI, approximate online optimistic PI, and asynchronous online PI and VI algorithms. It is shown that all error bounds for optimal cost tracking are closely related to the differences between consecutive mappings $H_k$ and $H_{k+1}$ for $k\geq 0$.
This paper is organized as follows. Section \ref{s2} formulates the problem, and online PI and VI algorithms in the synchronous case are discussed in Sections \ref{s3} and \ref{s4}, respectively. The asynchronous online PI and VI algorithms are given in Section \ref{s5}, following examples in Section \ref{s6}. Finally, the conclusion is drawn in Section \ref{s7}.
\section{Problem Formulation}\label{s2}
Let $\mathbb{R}$ and $\mathbb{N}$ be the sets of real numbers and nonnegative integers, respectively. Denote by $X$ and $U$ two sets, which can be roughly viewed as the sets of ``states'' and ``controls'', respectively. Given a state $x\in X$, let $U(x)\subset U$ be a subset of $U$, denoting feasible controls at state $x$. Let $\mathcal{M}:=\{\mu: X\to U|~\mu(x)\in U(x),~\forall~x\in X\}$, representing a collection of functions. Similar to DP, a sequence $\{\mu_k\}_{k=0}^\infty$ with $\mu_k\in\mathcal{M}$ for all $k\in\mathbb{N}$ is called a {\em nonstationary policy}, and if all $\mu_k$'s are identical, that is, $\mu_k=\mu$ for some $\mu\in\mathcal{M}$ for all $k\in\mathbb{N}$, then it is called a {\em stationary policy}. To simplify the notation, any single $\mu\in\mathcal{M}$ is also referred to as a {\em policy} when $\{\mu\}$ is a stationary policy.
Denote by $\mathcal{R}(X)$ a set of real-valued functions $J: X\to \mathbb{R}$. In online abstract DP, consider a family of time-varying mappings $H_k: X\times U\times \mathcal{R}(X)\to \mathbb{R}$, where $k\in\mathbb{N}$ is interpreted as time index. The mappings $\{H_k\}_{k=0}^\infty$ are only gradually revealed: at each time $k\in\mathbb{N}$, we only know the mappings before time $k$, but without awareness of future information on $H_l$ for $l\geq k$. Given a time $k\in\mathbb{N}$ and a policy $\mu\in\mathcal{M}$, let us consider the mapping $T_{k,\mu}: \mathcal{R}(X)\to\mathcal{R}(X)$ defined as \begin{align} (T_{k,\mu} J)(x)=H_k(x,\mu(x),J),~~\forall~x\in X,~J\in\mathcal{R}(X) \label{1} \end{align} and also consider a mapping $T_k: \mathcal{R}(X)\to\mathcal{R}(X)$ defined as \begin{align} (T_k J)(x)&=\inf_{u\in U(x)}H_k(x,u,J) \nonumber\\ &=\inf_{\mu\in\mathcal{M}}(T_{k,\mu})(x),~~~\forall~x\in X,~J\in\mathcal{R}(X). \label{2} \end{align}
The objective of online abstract DP is to find a function $J_k^*\in\mathcal{R}(X)$ at each time $k$ such that \begin{align} J_k^*(x)=\inf_{u\in U(x)}H_k(x,u,J_k^*), \label{3} \end{align} i.e., seeking a fixed point of $T_k$ at each time step $k\in\mathbb{N}$, which is typically called {\em Bellman's equation}. Meanwhile, it is desirable to obtain a policy $\mu_k^*\in\mathcal{M}$ such that $T_{k,\mu_k^*}J_k^*=T_kJ_k^*$. That is, $\mu_k^*$ is an {\em optimal policy} corresponding to the {\em optimal cost} $J_k^*$.
The following is an example for illustrating the above problem. \begin{example}[Online Optimal Control]\label{e1} Consider a deterministic discrete-time online optimal control problem, where a nonlinear control system is given as \begin{align} x_{k+1}=f(x_k,u_k),~~~k\in\mathbb{N} \label{4} \end{align} with $x_k\in X$ and $u_k\in U$ being the state and control of the system, respectively.
At each time slot $k\in\mathbb{N}$, there is an objective or cost function $g_k(x,u)$, and the aim is to minimize the total cost incurred by a policy $\pi_k=\{\mu_k,\mu_{k+1},\ldots\}$ over an infinite number of stages with the initial state $x_k$, i.e., \begin{align} \text{minimize}~~J_{\pi_k}(x_k):=\sum_{m=0}^\infty \alpha^m g_k(x_{k+m},\mu_{k+m}), \label{5} \end{align} where $\alpha\in(0,1]$ is a discounted factor.
The optimal cost function is defined by \begin{align} J_k^*(x)=\inf_{\pi_k\in\Pi_k} J_{\pi_k}(x),~~~\forall~x\in X, \label{6} \end{align} where \begin{align}
\Pi_k:=\{\{\mu_k,\mu_{k+1},\ldots\}|~\mu_m\in\mathcal{M},~\forall~m\geq k\}. \label{7} \end{align}
For arbitrary policy $\pi_k=\{\mu_k,\mu_{k+1},\ldots\}$ and writing $\pi_{k+1}=\{\mu_{k+1},\mu_{k+2},\ldots\}$, one can easily rewrite $J_{\pi_k}(x)$ as \begin{align} J_{\pi_k}(x)=g_k(x,\mu_k)+\alpha J_{\pi_{k+1}}(f(x,\mu_k)),~~\forall~x\in X \label{8} \end{align} which leads to that \begin{align} J_k^*(x)&=\inf_{\pi_k=\{\mu_k,\pi_{k+1}\}\in\Pi_k}\Big\{g_k(x,\mu_k)+\alpha J_{\pi_{k+1}}(f(x,\mu_k))\Big\} \nonumber\\ &=\inf_{\mu_k\in\mathcal{M}}\Big\{g_k(x,\mu_k)+\alpha\inf_{\pi_{k+1}\in\Pi_{k+1}} J_{\pi_{k+1}}(f(x,\mu_k))\Big\} \nonumber\\ &=\inf_{\mu_k\in\mathcal{M}}\Big\{g_k(x,\mu_k)+\alpha J_k^*(f(x,\mu_k))\Big\}. \label{9} \end{align} Once defining $H_k(x,u,J)=g_k(x,u)+\alpha J(f(x,u))$, through the above equation, it is easy to see that \begin{align} J_k^*(x)=\inf_{u\in U(x)} H_k(x,u,J_k^*),~~\forall x\in X, \label{10} \end{align} which is exactly consistent with (\ref{3}). As a result, this online optimal control problem can be viewed as an instance of online abstract DP. \end{example}
More examples for abstract DP can be found in \cite{bertsekas2018abstract}, including stochastic Markovian decision problems, finite-state discounted Markovian decision problems, discounted semi-Markov problems, discounted zero-sum dynamic games, minimax problems, and stochastic shortest path problems, etc. It should be noted that online abstract DP will reduce to abstract DP when the mapping $H_k$ is time-invariant.
To proceed, it is necessary to introduce a new space $\mathcal{B}(X)$, composed of functions $J$ on $X$ such that $J(x)/\nu(x)$ is bounded for all $x\in X$, where $\nu: X\to\mathbb{R}$ is a function with $\nu(x)>0$ for all $x\in X$. On the space $\mathcal{B}(X)$, a {\em weighted sup-norm} is defined as \begin{align}
\|J\|=\sup_{x\in X}\frac{|J(x)|}{\nu(x)}. \label{11} \end{align} It has been shown in Appendix B of \cite{bertsekas2018abstract} that $\mathcal{B}(X)$ is a complete normed space with respect to the weighted sup-norm.
At this moment, two important assumptions are listed below.
\begin{assumption}[Monotonicity]\label{a1} For all $k\in\mathbb{N}$ and any $J_1,J_2\in\mathcal{R}(X)$, if $J_1\leq J_2$, then \begin{align} H_k(x,u,J_1)\leq H_k(x,u,J_2),~~~\forall x\in X,~u\in U(x). \label{12} \end{align} \end{assumption}
\begin{assumption}[Uniform Contraction]\label{a2} For all $k\in\mathbb{N}$, $J\in\mathcal{B}(X)$, and $\mu\in\mathcal{M}$, there holds $T_{k,\mu}J,T_k J\in\mathcal{B}(X)$. Moreover, there exists $\alpha_k\in(0,1)$ such that for all $k\in\mathbb{N}$ and $\mu\in\mathcal{M}$ \begin{align}
\|T_{k,\mu}J_1-T_{k,\mu}J_2\|\leq \alpha_k\|J_1-J_2\|,~\forall J_1,J_2\in\mathcal{B}(X) \label{13} \end{align} and $\alpha:=\max_{k\in\mathbb{N}}\alpha_k\in(0,1)$. \end{assumption}
It is noteworthy that the monotonicity assumption holds for almost all relevant DP mappings, and the weighted sup-norm contraction assumption is satisfied for a multitude of important DP models, such as discounted finite-state MDP, and undiscounted finite-state SSP with all policies being proper. More discussions can be found in \cite{bertsekas2018abstract}.
To conclude this section, the following lemma is conducive to the ensuing analysis, which can be found in \cite{bernstein2018asynchronous}. \begin{lemma}\label{l0} For a positive sequence $\{a_k\}$, if there exist $K<1$, $b<1$, and $0<\tau<1$ such that for all $k>K$ \begin{align} a_k\leq b+\tau a_{k-\delta_k}, \nonumber \end{align} for some $\delta_k\in\{1,\ldots,K\}$, then, $\mathop{\lim\sup}_{k\to\infty}a_k\leq \frac{b}{1-\tau}$. \end{lemma}
\section{Synchronous Online Value Iteration}\label{s3}
This section is devoted to online VI algorithms' development and analysis in the synchronous setting.
As seen from (\ref{3}), the goal is to find the fixed point of $T_k$ at each time slot $k\in\mathbb{N}$. To this end, an {\em approximate VI} is proposed as \begin{align} J_{k+1}&=\tilde{T}_k^{m_k} J_k \label{14} \end{align} with any initial condition $J_0\in\mathcal{B}(X)$, where $\tilde{T}_k J_k$ stands for an approximation of $T_k J_k$, satisfying \begin{align}
\|\tilde{T}_k^{m_k}J-T_k^{m_k}J\|\leq e_k,~~\forall J\in\mathcal{B}(X) \label{15} \end{align} with $e:=\max_{k\in\mathbb{N}}e_k<\infty$, and $m_k\geq 1$ is an integer, representing the computational power at time step $k\in\mathbb{N}$. For this online problem, it is of necessity to impose a condition on the switching rate of consecutive optimal costs, that is, there exists a constant $\rho_k\geq 0$ for each $k\in\mathbb{N}$ such that \begin{align}
\|J_k^*-J_{k+1}^*\|\leq \rho_k, \label{16} \end{align} and $\rho:=\max_{k\in\mathbb{N}}\rho_k<\infty$.
It is now ready to present the tacking error bound for the approximate VI (\ref{14}). \begin{theorem}\label{t1} Under Assumption \ref{a2}, there holds for $J_k$ generated by approximate VI (\ref{14}) that \begin{align}
\|J_k-J_k^*\|\leq \alpha^{\sum_{s=0}^{k-1}m_s}\|J_0-J_0^*\|+\frac{\rho+e}{1-\alpha^{m_d}}, \label{17} \end{align} where $m_d:=\min_{k\in\mathbb{N}}m_k \geq 1$. \end{theorem}
\begin{proof} In view of (\ref{14}), it can be obtained that \begin{align}
\|J_{k+1}-J_{k+1}^*\|&=\|\tilde{T}_k^{m_k}J_k-J_{k+1}^*\| \nonumber\\
&\leq \|\tilde{T}_k^{m_k}J_k-T_k^{m_k}J_k\|+\|T_k^{m_k}J_k-J_k^*\| \nonumber\\
&\hspace{0.4cm}+\|J_k^*-J_{k+1}^*\| \nonumber\\
&\leq \alpha_k^{m_k}\|J_k-J_k^*\|+e_k+\rho_k, \nonumber \end{align} where the second inequality has employed Assumption \ref{a2} and (\ref{15})-(\ref{16}). By recursion, one has that \begin{align}
\|J_{k+1}-J_{k+1}^*\|&\leq \prod_{s=0}^k\alpha_s^{m_s}\|J_0-J_0^*\|+\sum_{s=0}^k \alpha_{k:s}(\rho_s+e_s) \nonumber\\
&\hspace{-0.8cm}\leq \alpha^{\sum_{s=0}^k m_s}\|J_0-J_0^*\|+(\rho+e)\sum_{s=0}^k\alpha^{\sum_{r=s+1}^k m_r} \nonumber\\
&\hspace{-0.8cm}\leq \alpha^{\sum_{s=0}^k m_s}\|J_0-J_0^*\|+\frac{\rho+e}{1-\alpha^{m_d}}, \nonumber \end{align} where $\alpha_{k:s}:=\prod_{r=s+1}^k\alpha_r^{m_r}$ when $t=0,1,\ldots,k-1$, and $\alpha_{k:s}:=1$ when $s=k$. This ends the proof. \end{proof}
\begin{remark}\label{r1} It is worth mentioning that a similar online algorithm for finding fixed points of time-varying mappings is addressed in \cite{bernstein2018asynchronous}, which is a special case of Theorem \ref{t1} with $m_k=1$ for all $k\in\mathbb{N}$. It can be observed from (\ref{17}) that $J_k$ will approach to $J_k^*$ with an error bound $(\rho+e)/(1-\alpha^{m_d})$ at an exponential rate as $k$ tends to infinity. \end{remark}
\section{Synchronous Online Policy Iteration}\label{s4}
This section is concerned with the online PI algorithms in the synchronous setup, including exact/approximate online PI and optimistic PI algorithms.
\subsection{Online Policy Iteration}
First, let us consider the exact online PI for solving online abstract DP, for which, given the current policy $\mu_k$ with an initial policy $\mu_0$, the policy update $\mu_{k+1}$ at time step $k+1$ is given as \begin{subequations}\label{18} \begin{align} J_{k,\mu_k}&=T_{k,\mu_k}J_{k,\mu_k},\text{\em (Online policy evaluation)} \label{18a}\\ \hspace{-0.19cm}T_{k,\mu_{k+1}}J_{k,\mu_k}&=T_k J_{k,\mu_k},\text{\em (Online policy improvement).} \label{18b} \end{align} \end{subequations} It is assumed that one can attain the minimum of $H_k(x,u,J_{k,\mu_k})$ over $u\in U(x)$ for all $x\in X$, such that the update $\mu_{k+1}$ at online policy improvement is well defined, and this assumption is always exploited for PI algorithms in this paper. The purpose of online policy evaluation (\ref{18a}) is to calculate $J_{k,\mu_k}$, i.e., to find the fixed point of $T_{k,\mu_k}$, and (\ref{18b}) is leveraged to obtain $\mu_{k+1}$.
To move forward, it is imperative to introduce the following bounds for the online abstract DP: \begin{align}
\|J_{k,\mu}-J_{k+1,\mu}\|&\leq \gamma_{1,k},~~\forall \mu\in U \nonumber\\
\|J_k^*-J_{k+1}^*\|&\leq \gamma_{2,k},~~\forall k\in\mathbb{N} \label{19} \end{align} where $J_{k,\mu}$ is the fixed point of $T_{k,\mu}$ for any $k\in\mathbb{N}$ and $\mu\in \mathcal{M}$, the first inequality indicates to what extent $H_k$ is different from $H_{k+1}$ in the case of the same input, and the second one connotes the switching bound on consecutive optimal costs.
With the above preparations, the main result on online VI (\ref{18}) is given as follows.
\begin{theorem}\label{t2} Under Assumptions \ref{a1} and \ref{a2}, there holds for online VI (\ref{18}) that for all $k\in\mathbb{N}$ \begin{align}
\|J_{k,\mu_k}-J_k^*\|\leq \alpha^k\|J_{0,\mu_0}-J_0^*\|+\frac{\gamma_1+\gamma_2}{1-\alpha}, \label{20} \end{align} where $\gamma_l:=\max_{k\in\mathbb{N}}\gamma_{l,k}$ for $l=1,2$. \end{theorem}
\begin{proof} Invoking (\ref{18}) and the definition of $T_k$, it can be concluded that \begin{align} T_{k,\mu_{k+1}}J_{k,\mu_k}=T_kJ_{k,\mu_k}\leq T_{k,\mu_k}J_{k,\mu_k}=J_{k,\mu_k}, \nonumber \end{align} which, together with Assumption \ref{a1}, follows that \begin{align} T_{k,\mu_{k+1}}^2J_{k,\mu_k}\leq T_{k,\mu_{k+1}}J_{k,\mu_k}=T_kJ_{k,\mu_k}\leq J_{k,\mu_k}. \nonumber \end{align} Performing the above operation iteratively, one can obtain that \begin{align} T_{k,\mu_{k+1}}^mJ_{k,\mu_k}\leq T_{k}J_{k,\mu_k},~~\forall m\geq 1. \nonumber \end{align} By letting $m\to\infty$, it results in \begin{align} J_{k,\mu_{k+1}}\leq T_{k}J_{k,\mu_k}, \nonumber \end{align} which yields by Assumption \ref{a2} that for all $x\in X$ \begin{align} J_{k,\mu_{k+1}}(x)-J_k^*(x)&\leq T_{k}J_{k,\mu_k}(x)-J_k^*(x) \nonumber\\
&\leq\alpha_k\|J_{k,\mu_k}-J_k^*\|\nu(x). \label{pf1} \end{align}
It is known that $J_k^*(x)=\inf_{\mu\in\mathcal{M}}J_{k,\mu}(x)$ for all $x\in X$ and $k\in\mathbb{N}$ by Proposition 2.1.2 in \cite{bertsekas2018abstract}, and $\alpha_k\leq \alpha$. Therefore, one has by (\ref{pf1}) that $J_{k,\mu_{k+1}}(x)\geq J_k^*(x)$ and \begin{align}
\|J_{k,\mu_{k+1}}-J_k^*\|\leq \alpha \|J_{k,\mu_k}-J_k^*\|, \nonumber \end{align} which in combination with (\ref{19}) leads to that \begin{align}
\|J_{k+1,\mu_{k+1}}-J_{k+1}^*\|&\leq \|J_{k,\mu_{k+1}}-J_k^*\|+\|J_k^*-J_{k+1}^*\| \nonumber\\
&\hspace{0.4cm}+\|J_{k+1,\mu_{k+1}}-J_{k,\mu_{k+1}}\| \nonumber\\
&\hspace{-0.3cm}\leq\alpha\|J_{k,\mu_k}-J_k^*\|+\|J_k^*-J_{k+1}^*\|+\gamma_{1,k}, \nonumber \end{align} further implying (\ref{20}) by recursive iterations. This completes the proof. \end{proof}
\begin{remark}\label{r2} Note that unlike the case where $H_k$'s are time-invariant, it generally cannot guarantee the convergence of $\{\mu_k\}_{k=0}^\infty$ generated by online VI (\ref{18}) under arbitrary compactness and continuity conditions, since there exists an error term $(\gamma_1+\gamma_2)/(1-\alpha)$ in the online case. \end{remark}
\subsection{Approximate Online Policy Iteration}
In this subsection, let us consider the online policy iteration through approximations, called {\em approximate online policy iteration}, which generates a sequence of approximate cost functions $\{J_k\}$ and policies $\{\mu_k\}$ satisfying that for all $k\in\mathbb{N}$ \begin{align}
\|J_k-J_{k,\mu_k}\|\leq \delta_{1,k},~~\|T_{k,\mu_{k+1}}J_k-T_kJ_k\|\leq \epsilon_{1,k}, \label{21} \end{align} where $\delta_{1,k},\epsilon_{1,k}\geq 0$ are some constants. Then the following result can be obtained.
\begin{theorem}\label{t3} Under Assumptions \ref{a1} and \ref{a2}, the sequences $\{\mu_k\}$ generated by approximate online PI (\ref{21}) satisfy \begin{align}
\|J_{k,\mu_k}-J_k^*\|\leq \alpha^k\|J_{0,\mu_0}-J_0^*\|+\frac{r_1}{1-\alpha}, \label{22} \end{align} where $r_1:=\gamma_1+\gamma_2+(\epsilon_1+2\alpha\delta_1)/(1-\alpha)$ with $\epsilon_1:=\max_{k\in\mathbb{N}}\epsilon_{1,k}$, $\delta_1:=\max_{k\in\mathbb{N}}\delta_{1,k}$, and $\gamma_1,\gamma_2$ are defined in Theorem \ref{t2}. \end{theorem}
\begin{proof} For each $k\in\mathbb{N}$, in view of Proposition 2.4.4 in \cite{bertsekas2018abstract}, one can obtain that \begin{align}
\|J_{k,\mu_{k+1}}-J_k^*\|\leq \alpha_k\|J_{k,\mu_k}-J_k^*\|+\frac{\epsilon_{1,k}+2\alpha_k\delta_{1,k}}{1-\alpha_k}, \nonumber \end{align} which implies that \begin{align}
\|J_{k+1,\mu_{k+1}}-J_{k+1}^*\|&\leq \|J_{k,\mu_{k+1}}-J_k^*\|+\|J_k^*-J_{k+1}^*\| \nonumber\\
&\hspace{0.4cm}+\|J_{k+1,\mu_{k+1}}-J_{k,\mu_{k+1}}\| \nonumber\\
&\leq \alpha_k\|J_{k,\mu_k}-J_k^*\|+\gamma_{1,k}+\gamma_{2,k} \nonumber\\ &\hspace{0.4cm}+\frac{\epsilon_{1,k}+2\alpha_k\delta_{1,k}}{1-\alpha_k} \nonumber\\
&\leq \alpha\|J_{k,\mu_k}-J_k^*\|+r_1, \nonumber \end{align} where we have used (\ref{19}) in the second inequality and the facts $\alpha_k\leq\alpha,\gamma_{l,k}\leq\gamma_l$ for $k\in\mathbb{N},l=1,2$ in the last inequality. By recursively iterating the above inequality, the conclusion (\ref{22}) can be asserted. \end{proof}
\subsection{Online Optimistic Policy Iteration}
In online PI, the online policy evaluation (\ref{18a}) requires to exactly resolve the fixed point of $T_{k,\mu_k}$, which is usually computationally prohibitive. To alleviate the computational burden, another algorithm, called {\em online optimistic PI (or online ``modified'' PI)}, aims to approximately solve the fixed point of $T_{k,\mu_k}$, delineated as for a given initial cost function $J_0\in\mathcal{B}(X)$ \begin{align} T_{k,\mu_k}J_k=T_kJ_k,~~J_{k+1}=T_{k,\mu_k}^{m_k}J_k, \label{23} \end{align} producing a sequence of $\{\mu_k\}$ and $\{J_k\}$, where $m_k\geq 1$ is an integer for iterating the mapping $T_{k,\mu_k}$ totally $m_k$ times dependent on the computation power at time step $k\in\mathbb{N}$. To analyze (\ref{23}), a metric to measure the consecutive difference between $T_k$ and $T_{k+1}$ is postulated as \begin{align}
\|(T_k-T_{k+1})J\|\leq \eta_{1,k},~~\forall J\in\mathcal{B}(X) \label{24} \end{align} for some constant $\eta_{1,k}\geq 0$ and for all $k\in\mathbb{N}$.
At this stage, it is helpful to introduce a preliminary result on the boundedness of $J_{k+1}$, which is an extension of Lemma 2.5.3 in \cite{bertsekas2018abstract} to the online case considered in this paper.
\begin{lemma}\label{l1} Under Assumptions \ref{a1} and \ref{a2}, if $J_0\geq T_{0}J_0-c\nu$ for some $c\geq 0$, then for all $k\in\mathbb{N}$ \begin{align} T_kJ_k+\frac{\alpha_k}{1-\alpha_k}\lambda_k(c)\nu &\geq J_{k+1} \nonumber\\ &\geq T_{k+1}J_{k+1}-\lambda_{k+1}(c)\nu, \label{25} \end{align} where $\lambda_k(c)$ is defined by \begin{align} \lambda_k(c)=\left\{
\begin{array}{ll}
c, & if~k=0 \\
\sum_{s=0}^{k-1}\eta_{1,s}\prod_{l=s+1}^{k-1}\alpha_l^{m_l}+c\prod_{l=0}^{k-1}\alpha_l^{m_l}, & if~k\geq 1
\end{array}
\right. \nonumber \end{align} with the convention $\prod_{l=s+1}^{k-1}\alpha_l^{m_l}=1$ when $s=k-1$. \end{lemma} \begin{proof} Because of $J_0\geq T_0 J_0-c\nu$, in view of Lemma 2.5.2 in \cite{bertsekas2018abstract} by letting $T=T_0,J=J_0,k=m_0$, and $\mu=\mu_0$, one has \begin{align} T_0J_0\geq T_{\mu_0}^{m_0}J_0-\frac{\alpha_0 c\nu}{1-\alpha_0}=J_1-\frac{\alpha_0\lambda_0(c)\nu}{1-\alpha_0}, \nonumber \end{align} and \begin{align} J_1&=T_{\mu_0}^{m_0}J_0\geq T_0(T_{\mu_0}^{m_0}J_0)-\alpha_0^{m_0}c\nu=T_0J_1-\alpha_0^{m_0}c\nu \nonumber\\ &=T_1J_1+(T_0-T_1)J_1-\alpha_0^{m_0}c\nu \nonumber\\ &\geq T_1J_1-\eta_{1,0} \nu-\alpha_0^{m_0}c\nu \nonumber\\ &=T_1J_1-\lambda_1(c)\nu, \nonumber \end{align} where (\ref{24}) has been employed in the second inequality. Therefore, (\ref{25}) holds when $k=0$.
By induction, it is assumed that (\ref{25}) holds for $k\geq 1$, and then one has $J_k\geq T_kJ_k-\lambda_k(c)\nu$, which in conjunction with Lemma 2.5.2 in \cite{bertsekas2018abstract} with $T=T_k,J=J_k,k=m_k$ and $\mu=\mu_k$ yields that \begin{align} T_kJ_k\geq T_{\mu_k}^{m_k}J_k-\frac{\alpha_k\lambda_k(c)\nu}{1-\alpha_k}=J_{k+1}-\frac{\alpha_k\lambda_k(c)\nu}{1-\alpha_k}, \nonumber \end{align} and \begin{align} J_{k+1}&=T_{\mu_k}^{m_k}J_k\geq T_k(T_{\mu_k}^{m_k}J_k)-\alpha_k^{m_k}\lambda_k(c)\nu \nonumber\\ &=T_kJ_{k+1}-\alpha_k^{m_k}\lambda_k(c)\nu \nonumber\\ &=T_{k+1}J_{k+1}+(T_k-T_{k+1})J_{k+1}-\alpha_k^{m_k}\lambda_k(c)\nu \nonumber\\ &\geq T_{k+1}J_{k+1}-\eta_{1,k}\nu-\alpha_k^{m_k}\lambda_k(c)\nu \nonumber\\ &=T_{k+1}J_{k+1}-\lambda_{k+1}(c)\nu, \nonumber \end{align} where the second inequality has leveraged (\ref{24}). This ends the proof. \end{proof}
It is now ready to provide the error bounds on online optimistic PI (\ref{23}). \begin{theorem}\label{t4} Under Assumptions \ref{a1} and \ref{a2}, let $c\geq 0$ such that $J_0\geq T_0 J_0-c\nu$. Then for all $k\in\mathbb{N}$ \begin{align}
-\frac{\lambda_k(c)\nu}{1-\alpha_k}\leq J_k-J_k^*&\leq \alpha_0^k\|J_0-J_0^*\|\nu+\sum_{l=0}^{k-1}(J_l^*-J_{l+1}^*) \nonumber\\ &\hspace{0.4cm}+\sum_{l=1}^{k-1}(T_l^{k-l}-T_{l-1}^{k-l})J_l+e_k', \label{26} \end{align} where $e_k':=\sum_{l=0}^{k-1}\frac{\alpha_l^{k-l}\lambda_l(c)\nu}{1-\alpha_l}$. \end{theorem} \begin{proof} The proof is motivated by Lemma 2.5.4 in \cite{bertsekas2018abstract}. In light of $J_0\geq T_0J_0-c\nu$ and Lemma \ref{l1}, it can be obtained that \begin{align} J_k\geq T_k J_k-\lambda_k(c)\nu,~~\forall k\in\mathbb{N} \nonumber \end{align} which in conjunction with Lemma 2.5.1(b) in \cite{bertsekas2018abstract} with $W=T_k,J=J_k$ and $k=0$ follows that \begin{align} J_k\geq J_k^*-\frac{\lambda_k(c)\nu}{1-\alpha_k}, \nonumber \end{align} thus ending the proof of (\ref{26}) on the left-hand side.
Now, invoking Lemma \ref{t1}, one has that \begin{align} T_jJ_j\geq T_{j+1}-\frac{\alpha_j\lambda_j(c)\nu}{1-\alpha_j},~~j=0,1,\ldots,k-1 \nonumber \end{align} which, together with Proposition 2.1.3 in \cite{bertsekas2018abstract} with $T_\mu=T_j^{k-j-1}$, leads to that \begin{align} T_{j}^{k-j}J_j\geq T_j^{k-j-1}J_{j+1}-\frac{\alpha_j^{k-j}\lambda_j(c)\nu}{1-\alpha_j}. \nonumber \end{align} Summing the above inequality over $j=0,1,\ldots,k-1$ gives rise to that \begin{align} T_0^kJ_0\geq J_k+\sum_{l=1}^{k-1}(T_{l-1}^{k-l}-T_l^{k-l})J_l-e_k', \nonumber \end{align} which implies that \begin{align} &T_0^kJ_0-J_0^*+\sum_{l=0}^{k-1}(J_l^*-J_{l+1}^*)+J_k^* \nonumber\\ &\hspace{0.4cm}\geq J_k+\sum_{l=1}^{k-1}(T_{l-1}^{k-l}-T_l^{k-l})J_l-e_k'. \nonumber \end{align}
Using $\|T_0^kJ_0-J_0^*\|\leq \alpha_0^k\|J_0-J_0^*\|$ in the above inequality can obtain the right-hand side inequality in (\ref{26}). This completes the proof. \end{proof}
\begin{remark} It can be observed in the right-hand side of (\ref{26}) that the differences among $H_k$ will incur a larger error bound than (approximate) online VI and PI due to the presence of $\sum_{l=0}^{k-1}(J_l^*-J_{l+1}^*)$ and $\sum_{l=1}^{k-1}(T_l^{k-l}-T_{l-1}^{k-l})J_l$, resulting in accumulative errors as $k$ advances. However, due to $\lambda_k(c)\leq\frac{\eta_1}{1-\alpha^{m_d}}+c\alpha^{\sum_{l=0}^{k-1}m_l}$, where $\eta_1:=\max_{k\in\mathbb{N}}\eta_{1,k}$ and $m_d:=\min_{k\in\mathbb{N}}m_k$, the online optimistic PI has a faster convergence rate than $\alpha^k$ from one side, i.e., the left-hand side of (\ref{26}), with rate $\alpha^{\sum_{l=0}^{k-1}m_l}$. This result is consistent with the case where $H_k$'s are time-invariant, see Section 2.5.1 in \cite{bertsekas2018abstract}. \end{remark}
\subsection{Approximate Online Optimistic Policy Iteration}
In this subsection, it is desirable to consider the approximate algorithm for the online optimistic PI, where both operations in (\ref{23}) are approximate. To be specific, {\em approximate online optimistic PI} generates sequences $\mu_k$ and $J_k$ by \begin{subequations} \begin{align}
&\|T_{k,\mu_{k+1}}J_k-T_kJ_k\|\leq \epsilon_k, \label{27a}\\
&\|J_k-T_{k-1,\mu_k}^{m_k}J_{k-1}\|\leq \delta_k,~~\forall k\in\mathbb{N} \label{27b} \end{align}\label{27} \end{subequations} where $\epsilon_k,\delta_k\geq 0$ are some constants. As previously done, it is of help to introduce the metric to measure how different two consecutive $H_k$ and $H_{k+1}$ are, that is, there are constants $\eta_{2,k},\eta_{3,k}\geq 0$ such that for all $k\in\mathbb{N}$ and any $J\in\mathcal{B}(X),\mu\in\mathcal{M}$, and for $j\in\{1,k+1\}$ \begin{align}
\|(T_{k,\mu}^j-T_{k+1,\mu}^j) J\|\leq \eta_{2,k},~~\|J_k^*-J_{k+1}^*\|\leq \eta_{3,k}. \label{29} \end{align}
For example, in Example \ref{e1}, the first inequality in (\ref{29}) when $j=1$ means $\|g_k(x,\mu)-g_{k+1}(x,\mu)\|\leq \eta_{2,k}$ for all $x\in X,\mu\in U(x)$.
It is known from the case where $H_k$'s are time-invariant \cite{bertsekas2018abstract} that a stronger assumption than Assumptions \ref{a1} and \ref{a2} is required, and thus it is also employed here for the online case. \begin{assumption}[Semilinear Monotonic Contraction]\label{a3} For all $k\in\mathbb{N}$, $J\in\mathcal{B}(X)$ and $\mu\in\mathcal{M}$, there holds $T_{k,\mu}J,T_{k}J\in \mathcal{B}(X)$. Moreover, there exists $\alpha_k\in(0,1)$ for each $k\in\mathbb{N}$ such that for all $J_1,J_2\in\mathcal{B}(X),\mu\in\mathcal{M}$ \begin{align} M(T_{k,\mu}J_1-T_{k,\mu}J_2)\leq \alpha_k M(J_1-J_2), \label{30} \end{align} where the mapping $M:\mathcal{B}(X)\to\mathbb{R}$ is defined as $M(y)=\sup_{x\in X}\frac{y(x)}{\nu(x)}$ for a function $y\in\mathcal{B}(X)$. \end{assumption}
With the above at hand, we are now in a position to give the error bound for approximate online optimistic PI. \begin{theorem}\label{t5} Under Assumption \ref{a3}, the sequences $\{\mu_k\}$ generated by (\ref{27}) satisfy \begin{align}
\|J_{\mu_k}-J_k^*\|&\leq \frac{\alpha^{\sum_{l=1}^km_l}}{1-\alpha}M(T_{1,\mu_1}J_0-J_0) \nonumber\\ &\hspace{0.0cm}+\alpha^{k-1}M(T_{1,\mu_1}J_0-J_1^*)+\frac{c_1\beta^{\lceil\frac{k}{2}\rceil}}{1-\beta}+\frac{c_1\beta\alpha^{\lfloor\frac{k}{2}\rfloor}}{1-\alpha} \nonumber\\ &\hspace{0.0cm}+\frac{\alpha^{m_k}\varepsilon_1}{(1-\alpha)(1-\alpha^{m_d})}+\frac{\varepsilon_2}{1-\alpha}, \label{31} \end{align} where $c_1:=\frac{\alpha-\alpha^{m_s}}{1-\alpha}M(T_{1,\mu_1}J_0-J_0)$, $m_s:=\max_{k\in\mathbb{N}}m_k$, $\beta:=\alpha^{m_d}$, $m_d:=\min_{k\in\mathbb{N}}m_k$, $\varepsilon_1:=\epsilon+(1+\alpha)\delta+(2+\alpha)\eta_2$, $\varepsilon_2:=\frac{(\alpha-\alpha^{m_s})\varepsilon_1}{(1-\alpha)(1-\alpha^{m_d})}+\epsilon+\eta_2+\eta_3+\alpha(\delta+\eta_2)$, $\epsilon:=\max_{k\in\mathbb{N}}\epsilon_k$, $\delta:=\max_{k\in\mathbb{N}}\delta_k$, $\alpha:=\max_{k\in\mathbb{N}}\alpha_k$, $\eta_l:=\max_{k\in\mathbb{N}}\eta_{l,k}$ for $l=2,3$, and $\lfloor d\rfloor,\lceil d\rceil$ mean the largest integer not greater than $d$ and smallest integer not less than $d$ for a real number $d$, respectively. \end{theorem} \begin{proof} This proof is adapted from Proposition 2.5.3 in \cite{bertsekas2018abstract}, which is given in the Appendix for the completeness. \end{proof}
\begin{remark}
From (\ref{29}), it can be easily verified that the error bound on $\|J_{k,\mu_k}-J_k^*\|$ in the asymptotic sense is given as \begin{align*}
\mathop{\lim\sup}_{k\to\infty}\|J_{k,\mu_k}-J_k^*\|\leq \frac{\hat{\alpha}\varepsilon_1}{(1-\alpha)(1-\alpha^{m_d})}+\frac{\varepsilon_2}{1-\alpha}, \end{align*} where $\hat{\alpha}:=\alpha^{\mathop{\lim\inf}_{k\to\infty}m_k}$. \end{remark}
\section{Asynchronous Algorithms}\label{s5}
This section aims at further alleviating the computational complexity by taking into account asynchronous algorithms.
\subsection{Asynchronous Approximate Online Value Iteration}
Consider that there are $N$ processors for solving online abstract DP, and partition the state set $X$ into $N$ disjoint nonempty subsets $X_1,\ldots,X_N$. Correspondingly, let us partition $J$ as $J=(J_1,\ldots,J_N)$, where $J_l$ is the restriction of $J$ on $X_l$ for $l\in[N]$ with the notation $[N]:=\{1,\ldots,N\}$. Let $\mathcal{T}_l$ be a subset of iterations, denoting the updating or activation of processor $l\in[N]$. Then the {\em asynchronous approximate online VI} is given as {\small\begin{align} J_{l,k+1}(x)=\left\{
\begin{array}{ll}
\tilde{T}_k^{m_k}(J_{1,\tau_{l1}(k)},\cdots,J_{N,\tau_{lN}(k)})(x), & k\in\mathcal{T}_l,x\in X_l \\
J_{l,k}(x), & k\notin\mathcal{T}_l,x\in X_l
\end{array}
\right. \label{32} \end{align}} where $\tau_{li}(k)\in\{0,1,\ldots,k\}$ with $k-\tau_{li}(k)$ being the communication delay from processor $i\in[N]$ to processor $l$.
In the online case, some conditions on updating frequency and communication delays are listed below. \begin{assumption}[Continuous Updating and Uniformly Bounded Delay]\label{a4} ~~~ \begin{enumerate}
\item There exists an integer $T_a>0$ such that $\mathcal{T}_l\cap[k,k+T_a]\neq\emptyset$ for all $k\in\mathbb{N}$ and $l\in[N]$;
\item There holds $|k-\tau_{ij}(k)|\leq T_d$ for some integer $T_d\geq 0$, for all $k\in\mathbb{N}$ and $i,j\in[N]$. \end{enumerate} \end{assumption}
The first condition in the above assumption means that each processor must update or activate at least once within consecutive $T_a$ time instants, and the second one indicates an upper bound on the communication delays.
With the above preparations, it is ready to develop the error bound on asynchronous approximate online VI. \begin{theorem}\label{t6} Under conditions (\ref{15})-(\ref{16}), Assumption \ref{a2} with $\nu(x)\equiv 1$ and Assumption \ref{a4}, the sequence $\{J_{k}\}$ generated by (\ref{32}) satisfies \begin{align}
\limsup_{k\to\infty}\|J_k-J_k^*\|\leq \frac{\rho(T_a+\alpha^{m_d}T_d)+e}{1-\alpha^{m_d}}, \label{33} \end{align} where $\rho$ and $e$ are defined after (\ref{16}) and $m_d=\min_{k\in\mathbb{N}}m_k$. \end{theorem} \begin{proof} Consider the time step $k+1$ and processor $l$. To simplify the notations, denote by $J_{\tau_l(k)}:=(J_{1,\tau_{l1}(k)},\cdots,J_{N,\tau_{lN}(k)})$. The analysis is divided into two cases: $k\in\mathcal{T}_l$ and $k\notin\mathcal{T}_l$.
If $k\in\mathcal{T}_l$, then one has \begin{align}
&|J_{l,k+1}(x)-J_{k+1}^*(x)| \nonumber\\
&=|\tilde{T}_k^{m_k}(J_{\tau_l(k)})(x)-J_{k+1}^*(x)| \nonumber\\
&\leq |T_k^{m_k}(J_{\tau_l(k)})(x)-J_k^*(x)|+|J_k^*(x)-J_{k+1}^*(x)| \nonumber\\
&\hspace{0.4cm}+|\tilde{T}_k^{m_k}(J_{\tau_l(k)})(x)-T_k^{m_k}(J_{\tau_l(k)})(x)| \nonumber\\
&\leq \alpha^{m_k}\|J_{\tau_l(k)}-J_k^*\|+\rho+e \nonumber\\
&\leq \alpha^{m_k}\big(\|J_{\tau_l(k)}-J_{\tau_l(k)}^*\|+\cdots+\|J_{k-1}^*-J_k^*\|\big)+\rho+e \nonumber\\
&\leq \alpha^{m_k}\|J_{\tau_l(k)}-J_{\tau_l(k)}^*\|+\alpha^{m_k}T_d\rho+\rho+e, \nonumber \end{align} where the second condition in Assumption \ref{a4} has been exploited to obtain the last inequality.
If $k\notin \mathcal{T}_l$, then there must exist an integer $t'\in[k+1-T_a,k)$ such that processor $l$ updates or activates at time slot $t'$. As a result, one can obtain that \begin{align}
&|J_{l,k+1}(x)-J_{k+1}^*(x)| \nonumber\\
&=|J_{l,k}(x)-J_{k+1}^*(x)|=\cdots=|J_{l,t'+1}(x)-J_{k+1}^*(x)| \nonumber\\
&=|\tilde{T}_{t'}^{m_{t'}}J_{\tau_l(t')}(x)-J_{k+1}^*(x)| \nonumber\\
&\leq |T_{t'}^{m_{t'}}J_{\tau_l(t')}(x)-J_{t'}^*(x)|+|J_{t'}^*(x)-J_{k+1}^*(x)| \nonumber\\
&\hspace{0.4cm}+|\tilde{T}_{t'}^{m_{t'}}J_{\tau_l(t')}(x)-T_{t'}^{m_{t'}}J_{\tau_l(t')}(x)| \nonumber\\
&\leq \alpha^{m_{t'}}\|J_{\tau_l(t')}-J_{t'}^*\|+\sum_{i=t'}^k|J_i^*(x)-J_{i+1}^*(x)|+e \nonumber\\
&\leq \alpha^{m_{t'}}\|J_{\tau_l(t')}-J_{\tau_l(t')}^*\|+\alpha^{m_{t'}}T_d\rho+T_a\rho+e, \nonumber \end{align} where the similar technique to the last step of the above inequality has been used to obtain the last inequality.
Combining the above two inequalities yields that \begin{align}
\|J_k-J_k^*\|&\leq \alpha^{m_d}\|J_{k-\tau(k)}-J_{k-\tau(k)}^*\| \nonumber\\ &\hspace{0.4cm}+\rho(T_a+\alpha^{m_d}T_d)+e, \nonumber \end{align} where $\tau(k)\in\{1,\ldots,T_a+T_d\}$. Consequently, in view of Lemma \ref{l0}, the conclusion can be obtained. \end{proof}
\begin{remark} Note that in the case where $H_k$'s are time-invariant \cite{bertsekas2018abstract}, the asynchronous value iteration is anatomized under less conservative conditions than Assumption \ref{a4}, i.e., each set $\mathcal{T}_l$ is infinite for all $l\in[N]$ and $\lim_{k\to\infty}\tau_{li}(k)=\infty$ for all $l,i\in[N]$. However, the analysis under the aforementioned conditions is no longer available to the online case studied in this paper. \end{remark}
\subsection{Asynchronous Online Policy Iteration}
This subsection is to study the asynchronous algorithms for online policy iteration. To do so, let us first review the case of $H_k$'s being time-invariant. It is known that the natural asynchronous version of optimistic PI is not reliable in general, having a possibility of oscillation, and thus two another asynchronous PI algorithms have been proposed in \cite{bertsekas2018abstract}, i.e., an optimistic asynchronous algorithm with randomization and a policy iteration with a uniform fixed point. Usually, the first algorithm has some restrictions, for instance, assuming totally finite policies. In contrast, the second one is more advantageous without such restriction. Hence, the second algorithm is only take into consideration for the online case in this subsection. The idea is to introduce new functions to eliminate the anomaly that $T_k$ and $T_{k,\mu}$ do not have identical fixed points.
To do so, it is necessary to introduce two additional functions \begin{align} V:X\to\mathbb{R}~~\text{and}~~Q:X\times U\to\mathbb{R}, \label{34} \end{align} referred to as a cost function and $Q$-factor as in the DP context, respectively. Meanwhile, for all $k\in\mathbb{N}$, define two functions $F_{k,\mu}(V,Q)$ and $MF_{k,\mu}(V,Q)$ as \begin{align} F_{k,\mu}(V,Q)(x,u)&:=H_k(x,u,\min\{V,Q_\mu\}), \label{35}\\ MF_{k,\mu}(V,Q)(x)&:=\min_{u\in U(x)}F_{k,\mu}(V,Q)(x,u), \label{36} \end{align} where $Q_\mu(x):=Q(x,\mu(x))$ for all $x\in X$.
Now, a new mapping $G_{k,\mu}$ is defined as \begin{align} G_{k,\mu}(V,Q):=(MF_{k,\mu}(V,Q),F_{k,\mu}(V,Q)). \label{37} \end{align} and the norm is defined by \begin{align}
\|(V,Q)\|:=\max\{\|V\|,\|Q\|\}, \label{38} \end{align}
where $\|V\|$ is the weighted sup-norm of $V$, and \begin{align}
\|Q\|:=\sup_{x\in X,u\in U(x)}\frac{|Q(x,u)|}{\nu(x)}. \label{39} \end{align} Some good properties have been shown for $G_{k,\mu}$ in Proposition 2.6.4 in \cite{bertsekas2018abstract}, that is, for each fixed $k\in\mathbb{N}$ under Assumption \ref{a2}, $G_{k,\mu}$ has a unique fixed point $(J_k^*,Q_k^*)$, in which $Q_k^*$ is defined as $Q_k^*(x,u)=H_k(x,u, J_k^*)$ for $x\in X,u\in U(x)$, and $G_{k,\mu}$ is contractive in the sense \begin{align}
&\|G_{k,\mu}(V_1,Q_1)-G_{k,\mu}(V_2,Q_2)\| \nonumber\\
&\hspace{3.0cm}\leq \alpha_k\|(V_1,Q_1)-(V_2,Q_2)\|. \label{40} \end{align}
As in the last subsection, let us consider $N$ processors and divide the set $X$ into $N$ parts as $X_1,\ldots,X_N$, each of which is assigned to a separate processor. Each processor $l\in[N]$ maintains $V_k(x)$, $Q_k(x,u)$, and $\mu_k(x)$ only for $x$ in its local set $X_l$, and enjoy disjoint activation or updating time set $\mathcal{T}_l$ and $\bar{\mathcal{T}}_l$ for all processors $l\in[N]$.
At this position, the {\em asynchronous online PI} is proposed as \begin{enumerate}
\item {\em Online local policy improvement:} If $k\in\mathcal{T}_l$, processor $l$ updates that for all $x\in X_l$ \begin{align} V_{k+1}(x)&=MF_{k,\mu_k}(V_k,Q_k)(x), \nonumber\\ \mu_{k+1}(x)&=\mathop{\arg\min}_{u\in U(x)}H_k(x,u,\min\{V_k,Q_{k,\mu_k}\}), \label{41} \end{align} and $Q_{k+1}(x,u)=Q_k(x,u)$ for all $x\in X_l,u\in U(x)$.
\item {\em Online local policy evaluation:} If $k\in\bar{\mathcal{T}}_l$, processor $l$ updates for all $x\in X_l$ and $u\in U(x)$ \begin{align} Q_{k+1}(x,u)=F_{k,\mu_k}(V_k,Q_k)(x,u), \label{42} \end{align} and $V_{k+1}(x)=V_k(x)$, $\mu_{k+1}(x)=\mu_k(x)$ for all $x\in X_l$. \end{enumerate}
To proceed, the following assumptions are of help for the subsequent analysis.
\begin{assumption}[Bounds on Consecutive Optimal Costs and Updating Frequency]\label{a5} ~~~ \begin{enumerate}
\item There exists a constant $\bar{\rho}_k$ such that $\|(J_k^*,Q_k^*)-(J_{k+1}^*,Q_{k+1}^*)\|\leq \bar{\rho}_k$;
\item There exists an integer $T_a>0$ such that $\mathcal{T}_l\cap [k,k+T_a]\neq \emptyset$ and $\bar{\mathcal{T}}_l\cap[k,k+T_a]\neq \emptyset$ for all $k\in\mathbb{N}$ and $l\in[N]$. \end{enumerate} \end{assumption}
At present, it is ready to establish the following error bound result. \begin{theorem}\label{t7} Under Assumptions \ref{a2} and \ref{a5}, for the sequence $\{(V_k,Q_k)\}$ generated by asynchronous online PI (\ref{41})-(\ref{42}), there holds that \begin{align}
\limsup_{k\to\infty}\|(V_k,Q_k)-(J_k^*,Q_k^*)\|\leq \frac{\bar{\rho}T_a}{1-\alpha}, \label{43} \end{align} where $\bar{\rho}:=\max_{k\in\mathbb{N}}\bar{\rho}_k$ and $\alpha:=\max_{k\in\mathbb{N}}\alpha_k$. \end{theorem} \begin{proof} For any $k>0$, based on Assumption \ref{a5}(1), there must exist two constants $t_1,t_2\in [k+1-T_a,k]$ such that processor $l\in[N]$ performs online local policy improvement and evaluation, respectively. Therefore, it can be concluded that for all $x\in X,u\in U(x)$ \begin{align}
|V_{k+1}(x)-&J_{k+1}^*(x)|=|MF_{t_1,\mu_{t_1}}(V_{t_1},Q_{t_1})(x)-J_{k+1}^*(x)| \nonumber\\
&\leq |MF_{t_1,\mu_{t_1}}(V_{t_1},Q_{t_1})(x)-J_{t_1}^*(x)| \nonumber\\
&\hspace{0.4cm}+\sum_{i=t_1}^k|J_i^*(x)-J_{i+1}^*(x)| \nonumber\\
&\leq |MF_{t_1,\mu_{t_1}}(V_{t_1},Q_{t_1})(x)-J_{t_1}^*(x)|+\bar{\rho}T_a, \nonumber \end{align} and \begin{align}
&|Q_{k+1}(x,u)-Q_{k+1}^*(x,u)| \nonumber\\
&=|F_{t_2,\mu_{t_2}}(V_{t_2},Q_{t_2})(x,u)-Q_{k+1}^*(x,u)| \nonumber\\
&\leq |F_{t_2,\mu_{t_2}}(V_{t_2},Q_{t_2})(x,u)-Q_{t_2}^*(x,u)| \nonumber\\
&\hspace{0.4cm}+\sum_{i=t_2}^k|Q_i^*(x,u)-Q_{i+1}^*(x,u)| \nonumber\\
&\leq |F_{t_2,\mu_{t_2}}(V_{t_2},Q_{t_2})(x,u)-Q_{t_2}^*(x,u)|+\bar{\rho}T_a, \end{align} where Assumption \ref{a5} has been applied to obtain the last inequalities of the above two expressions.
As a consequence, it can be obtained that \begin{align}
&\|(V_{k+1},Q_{k+1})-(J_{k+1}^*,Q_{k+1}^*)\| \nonumber\\
&=\max\{\|V_{k+1}-V_{k+1}^*\|,\|Q_{k+1}-Q_{k+1}^*\|\} \nonumber\\
&\leq \max\{\|MF_{t_1,\mu_{t_1}}(V_{t_1},Q_{t_1})-J_{t_1}^*\|, \nonumber\\
&\hspace{1.3cm}\|F_{t_2,\mu_{t_2}}(V_{t_2},Q_{t_2})-Q_{t_2}^*\|\}+\bar{\rho}T_a \nonumber\\
&\leq \max\{\|G_{t_1,\mu_{t_1}}(V_{t_1},Q_{t_1})-(J_{t_1}^*,Q_{t_1}^*)\|, \nonumber\\
&\hspace{1.3cm}\|G_{t_2,\mu_{t_2}}(V_{t_2},Q_{t_2})-(J_{t_2}^*,Q_{t_2}^*)\|\}+\bar{\rho}T_a, \nonumber \end{align} which implies that there must exist a constant $\tau\in[k-T_a,k-1]$ such that \begin{align}
&\|(V_{k},Q_{k})-(J_{k}^*,Q_{k}^*)\| \nonumber\\
&\leq \|G_{\tau,\mu_{\tau}}(V_{\tau},Q_{\tau})-(J_{\tau}^*,Q_{\tau}^*)\|+\bar{\rho}T_a \nonumber\\
&\leq \alpha\|(V_\tau,Q_\tau)-(J_{\tau}^*,Q_\tau^*)\|+\bar{\rho}T_a. \nonumber \end{align} Invoking Lemma \ref{l0} to the above inequality gives rise to the desired conclusion (\ref{43}), which completes the proof. \end{proof}
\begin{remark} It should be pointed out that communication delays, approximate algorithms, and multiple iterations at single step can be similarly addressed for the asynchronous online PI as previously done in this paper. \end{remark}
It can be observed that it is not necessary to evaluate $Q$ over the entire state space (its value at $\mu_k(x)$ is enough), since the goal is only to calculate $J_k^*$. Consequently, by letting $J_k(x):=Q_k(x,\mu_k(x))$ for all $x\in X$, iterations (\ref{41}) and (\ref{42}) in the asynchronous online PI can, respectively, reduce to \begin{align} J_{k+1}=V_{k+1}(x)&=\min_{u\in U(x)}H_k(x,u,\min\{V_k,J_{k}\}), \nonumber\\ \mu_{k+1}(x)&=\mathop{\arg\min}_{u\in U(x)}H_k(x,u,\min\{V_k,J_{k}\}), \label{44}\\ J_{k+1}(x,u)&=H_k(x,u,\min\{V_k,J_k\})(x,u). \label{45} \end{align}
\section{Examples}\label{s6}
In Example \ref{e1}, an online optimal control problem has been introduced to illustrate the problem formulation for online abstract DP, where $H_k$ is defined by $H_k(x,u,J)=g_k(x,u)+\alpha J(f(x,u))$. It is straightforward to see that $H_k$ satisfies Assumption \ref{a1}, and given $\alpha\in (0,1)$ and the boundedness of $g_k$, Assumption \ref{a2} is also satisfied by $H_k$ with respect to standard unweighted sup-norm, i.e., $\nu\equiv 1$. As a result, the theoretical results in this paper can be applied to the problem in Example \ref{e1}.
\begin{example}[Online Finite-State Discounted MDPs]\label{e2} As another example, consider online finite-state discounted MDPs, which involves a system $x_{k+1}=f(x_k,u_k,w_k),k\in\mathbb{N}$ with finite states, where $x_k\in X$ is the state, $u_k\in U$ is the control, and $w_k\in W$ is a random disturbance with $W$ being countable. Also, the state equation is given in terms of transition probabilities \begin{align}
p_{xy}(u)=Prob(y=f(x,u,w)|x), \label{46} \end{align} for all $x,y\in X$ and $u\in U(x)$. In the meantime, taking into account a cost function $g_k(x,u)$ at each time step $k\in\mathbb{N}$. Then the abstract DP mapping $H_k$ can be written as \begin{align} H_k(x,u,J)=\sum_{y\in X} p_{xy}(u)(g_k(x,u,y)+\alpha J(y)). \label{47} \end{align} It is easy to verify that $H_k$ is monotone, thus satisfying Assumption \ref{a1}. Moreover, if $\alpha\in(0,1)$ and $g_k$ are bounded, then $H_k$ is also contractive with respective to the standard unweighted sup-norm.
As a consequence, the online algorithms in previous sections are applicable to this problem. For instance, asynchronous online PI can be leveraged in which case the function in (\ref{35}) can be explicitly written as \begin{align} F_{k,\mu}(V,Q)(x,u)&=\sum_{y\in X}p_{xy}(u)\Big(g_{k}(x,u,y) \nonumber\\ &\hspace{0.4cm}+\alpha\min\{V(y),Q(y,\mu(y))\}\Big). \label{48} \end{align} \end{example}
Basically, all those problems, which satisfy monotone and contractive assumptions in the stationary case, i.e., $H_k$'s being independent of time, will still meet the two assumptions in the online case.
\section{Conclusion}\label{s7}
This paper has studied the online abstract DP problems, where the abstract mappings are time-varying, leading to that the optimal costs and policies are time-varying as well. It is known that to accurately track time-varying optimal costs and polices is in general impossible in the online case, thus necessitating the investigation on this problem. In this paper, we have developed quite a few algorithms based on classical ones in the static case where $H_k$'s are independent of time, and the tracking error bounds have been provided for these online algorithms, including approximate online VI, online PI, approximate online PI, online optimistic PI, approximate online optimistic PI, and asynchronous online PI and VI algorithms. It has been shown that the largest difference between consecutive abstract mappings $H_k$ and $H_{k+1}$ for $k\in\mathbb{N}$ play a critical part in the tracking error bounds. This paper focuses on the contractive models, as a first step to investigate the abstract DP in the online case, and thereby the future directions can be placed on the online abstract DP with semicontractive and noncontractive models.
\section*{Acknowledgment}
The authors would like to thank Dr. Zhirong Qiu for his helpful suggestions on this paper.
\section*{Appendix}
\noindent{\em The Proof of Theorem \ref{t5}:}
Throughout this proof, for notation ease, let $T_{\mu_k}$ (resp. $J_{\mu_k}$) simply denote $T_{k,\mu_k}$ (resp. $J_{k,\mu_k}$) when having the same time $k$, where $J_{k,\mu}$ means the fixed point of $T_{k,\mu}$, and denote \begin{align} &\underline{J}=J_{k-1}, J=J_k, \mu=\mu_k, \overline{\mu}=\mu_{k+1}, m=m_k, \overline{m}=m_{k+1}, \nonumber\\ &J^*=J_k^*, \overline{J}^*=J_{k+1}^*, s=J_\mu-T_\mu^m\underline{J}, \overline{s}=J_{\overline{\mu}}-T_{\overline{\mu}}^{\overline{m}}J, \nonumber\\ &t=T_\mu^m\underline{J}-J^*, \overline{t}=T_{\overline{\mu}}^{\overline{m}}J-\overline{J}^*, r=T_\mu\underline{J}-\underline{J}, \overline{r}=T_{\overline{\mu}}J-J. \nonumber \end{align} Then, it is easy to see that \begin{align} J_\mu-J^*=s+t. \nonumber \end{align} In what follows, let us develop the bounds on $M(r)$, $M(s)$, and $M(t)$.
First, consider $M(r)$. It can be obtained that {\small\begin{align*} \overline{r}&=T_\mu J-J=(T_{\overline{\mu}}J-T_\mu J)+(T_\mu J-J) \\ &\leq (T_{\overline{\mu}}J-T_k J)+(T_\mu J-T_\mu(T_\mu^m \underline{J})) \\ &\hspace{0.4cm}+(T_\mu^m\underline{J}-J)+(T_\mu^m(T_\mu\underline{J})-T_\mu^m\underline{J}) \\ &\leq (T_{k,\overline{\mu}}J-T_kJ)+(T_{\overline{\mu}} J-T_{k,\overline{\mu}}J)+\alpha M(J-T_\mu^m\underline{J})\nu \\ &\hspace{0.4cm}+(T_{k-1,\mu}^m\underline{J}-J)+(T_\mu^m\underline{J}-T_{k-1,\mu}^m\underline{J})+\alpha^m M(T_\mu\underline{J}-\underline{J})\nu \\ &\leq (\epsilon+\delta)\nu+2\eta_2\nu+\alpha^m M(r)\nu+\alpha M(J-T_{k-1,\mu}^m\underline{J})\nu \\ &\hspace{0.4cm}+\alpha M(T_{k-1,\mu}^m\underline{J}-T_{\mu}^m\underline{J})\nu \\ &\leq (\epsilon+\delta)\nu+(2\eta_2+\alpha\delta+\alpha\eta_2)\nu+\alpha^m M(r)\nu, \end{align*}} where (\ref{29}) has been utilized to obtain the last two inequalities, which implies that \begin{align*} M(\overline{r})\leq \alpha^m M(r)+\varepsilon_1. \end{align*} By defining $M_{r,k}:=M(r)$, one has $M_{r,k+1}=M(\overline{r})$, and thus, by recursively iterating the above inequality, it yields that \begin{align} M_{r,k}&\leq \alpha^{\sum_{l=1}^{k-1}m_l}M_{r,1}+\varepsilon_1\sum_{j=1}^{k-1}\alpha^{\sum_{l=j+1}^{k-1}m_l} \nonumber\\ &\leq \alpha^{\sum_{l=1}^{k-1}m_l}M_{r,1}+\frac{\varepsilon_1}{1-\alpha^{m_d}}, \label{pf2} \end{align} with the convention $\alpha^{\sum_{l=k}^{k-1}m_l}=1$.
Now, consider the bound on $M(s)$. To do so, invoking Proposition 2.1.4(b) in \cite{bertsekas2018abstract} gives rise to \begin{align*} J_\mu\leq \underline{J}+\frac{T_\mu\underline{J}-\underline{J}}{1-\alpha_k}\leq \underline{J}+\frac{T_\mu\underline{J}-\underline{J}}{1-\alpha}, \end{align*} which together Assumption \ref{a3} follows that \begin{align*} s&=J_\mu-T_\mu^m\underline{J}=T_\mu^m J_\mu-T_\mu^m\underline{J}\leq \alpha^m M(J_\mu-\underline{J})\nu \\ &\leq \frac{\alpha^m}{1-\alpha}M(T_\mu\underline{J}-\underline{J})\nu \\ &\leq \frac{\alpha^m}{1-\alpha}M(r)\nu, \end{align*} further implying that \begin{align} M(s)&\leq \frac{\alpha^m}{1-\alpha}M(r) \nonumber\\ &\leq \frac{\alpha^{\sum_{l=1}^k m_l}}{1-\alpha}M_{r,1}+\frac{\varepsilon_1\alpha^m}{(1-\alpha)(1-\alpha^{m_d})}, \label{pf3} \end{align} where (\ref{pf2}) has been used in the last inequality.
In what follows, let us focus on the bound on $M(t)$. Some manipulations with (\ref{29}) lead to that \begin{align*} \overline{t}&=T_{\overline{\mu}}^{\overline{m}}J-J^*+J^*-\overline{J}^* \\ &=(T_{\overline{\mu}}^{\overline{m}}J-T_{\overline{\mu}}^{\overline{m}-1}J)+\cdots+(T_{\overline{\mu}}^{2}J-T_{\overline{\mu}}J) \\ &\hspace{0.4cm}+(T_{\overline{\mu}}J-T_k J)+(T_k J-T_k J^*)+(J^*-\overline{J}^*) \\ &\leq (\alpha^{\overline{m}-1}+\cdots+\alpha)M(T_{\overline{\mu}}J-J)\nu+(T_{\overline{\mu}}J-T_{k,\overline{\mu}}J) \\ &\hspace{0.4cm}+(T_{k,\overline{\mu}}J-T_k J)+(T_k J-T_k J^*)+(J^*-\overline{J}^*) \\ &\leq \frac{\alpha-\alpha^{\overline{m}}}{1-\alpha}M(\overline{r})\nu+(\epsilon+\eta_2+\eta_3)\nu+(T_kJ-T_kJ^*). \end{align*}
Take into account the term $T_kJ-T_kJ^*$ in the last inequality. In view of Assumption \ref{a3} and (\ref{29}), one can obtain that \begin{align*} T_kJ-T_kJ^*&\leq \alpha M(J-J^*)\nu \\ &\leq \alpha [M(J-T_{k-1,\mu}^m \underline{J})+M(T_{k-1,\mu}^m \underline{J}-T_{\mu}^m \underline{J}) \\ &\hspace{0.4cm}+M(T_{\mu}^m \underline{J}-J^*)]\nu \\ &\leq \alpha(\delta+\eta_2)\nu+\alpha M(t)\nu, \end{align*} which in conjunction with the above inequality results in that \begin{align*} \overline{t}&\leq \frac{\alpha-\alpha^{\overline{m}}}{1-\alpha}M(\overline{r})\nu+(\epsilon+\eta_2+\eta_3)\nu \\ &\hspace{0.4cm}+\alpha(\delta+\eta_2)\nu+\alpha M(t)\nu. \end{align*} Hence, in light of (\ref{pf2}), it can be concluded that \begin{align*} M(\overline{t})&\leq \frac{\alpha-\alpha^{\overline{m}}}{1-\alpha}M(\overline{r})+\epsilon+\eta_2+\eta_3+\alpha(\delta+\eta_2)+\alpha M(t) \\ &\leq c_1\beta^k+\varepsilon_2+\alpha M(t), \end{align*} which, after defining $M_{t,k}:=M(t)$, follows that \begin{align*} M_{t,k+1}\leq \alpha M_{t,k}+c_1\beta^k+\varepsilon_2. \end{align*} As a result, it is straightforward to verify that \begin{align} M_{t,k}\leq \alpha^{k-1}M_{t,1}+c_1\sum_{l=0}^{k-2}\alpha^l\beta^{k-l-1}+\frac{\varepsilon_2}{1-\alpha}. \label{pf4} \end{align}
Equipped with the above preparations, making use of (\ref{pf3})-(\ref{pf4}), one has that \begin{align*} M(J_{\mu_k}-J_k^*)&\leq M(s)+M(t) \\ &\leq \frac{\alpha^{\sum_{l=1}^k m_l}}{1-\alpha}M_{r,1}+\frac{\varepsilon_1\alpha^{m_k}}{(1-\alpha)(1-\alpha^{m_d})} \\ &\hspace{0.4cm}+\alpha^{k-1}M_{t,1}+c_1\sum_{l=0}^{k-2}\alpha^l\beta^{k-l-1}+\frac{\varepsilon_2}{1-\alpha}, \end{align*} which, in combination with the fact that $J_{\mu_k}\geq J_k^*$ by Proposition 2.1.2 in \cite{bertsekas2018abstract}, follows that \begin{align*}
\|J_{\mu_k}-J_k^*\|&\leq \frac{\alpha^{\sum_{l=1}^km_l}}{1-\alpha}M_{r,1}+\alpha^{k-1}M_{t,1}+\frac{\varepsilon_2}{1-\alpha} \nonumber\\ &\hspace{0.4cm}+\frac{\alpha^{m_k}\varepsilon_1}{(1-\alpha)(1-\alpha^{m_d})}+c_1\sum_{l=0}^{k-2}\alpha^l\beta^{k-l-1}. \end{align*}
In the last inequality, the term $\sum_{l=0}^{k-2}\alpha^l\beta^{k-l-1}$ can be analyzed as \begin{align*} \sum_{l=0}^{k-2}\alpha^l\beta^{k-l-1}&=\big(\alpha^0\beta^{k-1}+\alpha\beta^{k-2}+\cdots+\alpha^{\lfloor\frac{k}{2}\rfloor-1}\beta^{k-\lfloor\frac{k}{2}\rfloor}\big) \\ &\hspace{0.4cm}+\big(\alpha^{\lfloor\frac{k}{2}\rfloor}\beta^{k-1-\lfloor\frac{k}{2}\rfloor}+\cdots+\alpha^{k-2}\beta\big) \\ &\leq \frac{\beta^{k-\lfloor\frac{k}{2}\rfloor}}{1-\beta}+\frac{\beta\alpha^{\lfloor\frac{k}{2}\rfloor}}{1-\alpha} \\ &= \frac{\beta^{\lceil\frac{k}{2}\rceil}}{1-\beta}+\frac{\beta\alpha^{\lfloor\frac{k}{2}\rfloor}}{1-\alpha}, \end{align*} where the last equality has employed the fact that $k=\lfloor\frac{k}{2}\rfloor+\lceil\frac{k}{2}\rceil$, by substituting which into the last inequality one can obtain the inequality (\ref{31}). This ends the proof.
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\end{document} | arXiv |
Journal of Fluid Mechanics
The shape and motion of gas bub...
Core reader
The shape and motion of gas bubbles in a liquid flowing through a thin annulus
2 Experimental apparatus and procedure
3 Mathematical formulation and numerical model
3.1 Governing equations
3.2 Gap-averaging process
3.3 Curvature and contact angle
3.4 Model set-up and numerical methods
4.1 Experimental observation and numerical simulation
4.2 Effects of gap thickness and pipe diameter
5 Flow mechanisms
6 Discussion and conclusions
Journal of Fluid Mechanics, Volume 855
25 November 2018 , pp. 1017-1039
Q. Lei (a1), Z. Xie (a1) (a2) (a3), D. Pavlidis (a1), P. Salinas (a1), J. Veltin (a4), O. K. Matar (a2), C. C. Pain (a1), A. H. Muggeridge (a1), A. J. Gyllensten (a5) and M. D. Jackson (a1)
1Department of Earth Science and Engineering, Imperial College London, SW7 2AZ, UK
2Department of Chemical Engineering, Imperial College London, SW7 2AZ, UK
3School of Engineering, Cardiff University, CF24 3AA, UK
4TNO, 2628 CK, The Netherlands
5Statoil ASA, 3936, Norway
© 2018 Cambridge University Press
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
DOI: https://doi.org/10.1017/jfm.2018.696
Figure 1. Schematic illustration of the experimental apparatus.
Figure 2. Schematic of (a) the thin annulus and (b) the equivalent Hele-Shaw cell. (c) An illustration showing how the direction of gravitational acceleration changes along the equivalent Hele-Shaw cell to ensure the flow replicates that seen in the actual annulus.
Table 1. Parameters of the experimental apparatus and fluids.
Figure 3. Schematic of (a) the cross-gap meniscus profile governed by the local contact angles, $\unicode[STIX]{x1D703}_{o}$ and $\unicode[STIX]{x1D703}_{i}$ , at the outer and inner walls, respectively, and (b) the moving contact line in the $x$ – $y$ plane with the local advancing/receding state determined by the intersection angle $\unicode[STIX]{x1D719}$ between the velocity vector $\bar{\boldsymbol{u}}$ and the interface normal $\boldsymbol{n}$ .
Figure 4. Interpolation of contact angles around a contact line at the outer or inner wall of the annulus using a third-order polynomial fitting (Antonini et al. 2009); $\unicode[STIX]{x1D703}$ denotes the local contact angle and $\unicode[STIX]{x1D719}$ denotes the intersection angle between the local velocity vector and the interface normal as shown in figure 3.
Figure 5. The 2-D model set-up for numerical simulation.
Figure 6. The formation and evolution of bubbles in the horizontal annulus (top view): (a) generation of the bubble near the inlet (in annulus subsection 1), (b) disconnection of the bubble (in annulus subsection 1), (c) stabilisation of the bubble after slight contraction (in annulus subsection 2) and (d) translation of the steady bubble to the outlet (in annulus subsection 3). Each figure panel includes the experimental observation, the simulation result and the adaptive unstructured mesh used in the simulation.
Figure 7. The formation and evolution of bubbles in the $1.9^{\circ }$ inclined annulus (top view): (a) generation of bubbles near the inlet (in annulus subsection 1), (b) interaction of bubbles, with the trailing bubble catching up with the leading bubble (in annulus subsection 2), (c) coalescence of the trailing and leading bubbles (in annulus subsection 2) and (d) translation of steady bubbles to the outlet (in annulus subsection 3). Each figure panel includes the experimental observation, the simulation result and the adaptive unstructured mesh used in the simulation.
Figure 8. Simulation results showing the coalescence process for two bubbles in the $1.9^{\circ }$ inclined annulus.
Figure 9. The formation and evolution of bubbles in the $4.6^{\circ }$ inclined annulus (top view): (a) generation and interaction of bubbles close to the inlet (in annulus subsection 1), (b) coalescence of the trailing, middle and leading bubbles (in annulus subsection 1), (c) stabilisation of merged larger bubbles (in annulus subsection 2) and (d) translation of steady bubbles to the outlet (in annulus subsection 3). Each figure panel includes the experimental observation, the simulation result and the adaptive unstructured mesh used in the simulation.
Figure 10. Variation of bubble properties with the inclination angle $\unicode[STIX]{x1D6FD}$ : (a) normalised bubble length $l/D$ , (b) normalised bubble cap diameter $d/D$ and (c) normalised bubble terminal velocity $u/v_{i}$ .
Figure 11. Variation of bubble properties with the gap thickness $h$ while the pipe diameter $D$ is fixed to be $0.137$ m: (a) normalised bubble length $l/D$ , (b) normalised bubble cap diameter $d/D$ and (c) normalised bubble terminal velocity $u/v_{i}$ .
Figure 12. Variation of bubble properties with the pipe diameter $D$ while the gap thickness $h$ is fixed to be 0.0035 m: (a) normalised bubble length $l/D$ , (b) normalised bubble cap diameter $d/D$ and (c) normalised bubble terminal velocity $u/v_{i}$ .
Figure 13. Simulation results showing (a) typical steady bubbles in annuli with different inclinations, and their surrounding (b) pressure and (c) velocity fields. To facilitate comparison, the absolute pressure at the frontal point of each bubble is chosen as the reference pressure for that case, so that the pressure contour gives the differential pressure between the absolute pressure and the reference pressure.
Figure 14. (a) The unrolled geometry of a bubble in the $x$ – $y$ plane, and the views of (b) a transversal cross-section in the $y$ – $z$ plane and (c) a longitudinal cross-section in the $x$ – $z$ plane.
Figure 15. Comparison of the predicted terminal velocity $u_{pred}$ from (5.5) and the measured terminal velocity $u_{meas}$ from experiments and simulations for different configurations of pipe diameter $D$ , gap thickness $h$ and inclination angle $\unicode[STIX]{x1D6FD}$ . The velocity values are normalised by the superficial inflow velocity $v_{i}$ .
Figure 16. (a) Variation of the bubble aspect ratio $l/d$ with the inverse Froude number $\hat{Fr}_{\Vert }^{-1}$ . (b) Variation of the bubble Reynolds number $Re$ with the inverse Froude number $\hat{Fr}_{\bot }^{-1}$ . (c) Variation of the gap Reynolds number $Re(h/d)^{2}$ with the confinement ratio $\hat{\unicode[STIX]{x1D716}}$ . Refer to figure 15 for the legend.
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We study the shape and motion of gas bubbles in a liquid flowing through a horizontal or slightly inclined thin annulus. Experimental data show that in the horizontal annulus, bubbles develop a unique 'tadpole-like' shape with a semi-circular cap and a highly stretched tail. As the annulus is inclined, the bubble tail tends to vanish, resulting in a significant decrease of bubble length. To model the bubble evolution, the thin annulus is conceptualised as a 'Hele-Shaw' cell in a curvilinear space. The three-dimensional flow within the cell is represented by a gap-averaged, two-dimensional model, which achieved a close match to the experimental data. The numerical model is further used to investigate the effects of gap thickness and pipe diameter on the bubble behaviour. The mechanism for the semi-circular cap formation is interpreted based on an analogous irrotational flow field around a circular cylinder, based on which a theoretical solution to the bubble velocity is derived. The bubble motion and cap geometry is mainly controlled by the gravitational component perpendicular to the flow direction. The bubble elongation in the horizontal annulus is caused by the buoyancy that moves the bubble to the top of the annulus. However, as the annulus is inclined, the gravitational component parallel to the flow direction becomes important, causing bubble separation at the tail and reduction in bubble length.
Simultaneous flow of gas and liquid through a conduit, over a wide range of flow rates, exhibits a 'slug flow' pattern, which consists of the pseudo-periodic appearance of large bubbles (often called Taylor bubbles) separated by liquid slugs (Fabre & Liné 1992). The shape and motion of Taylor bubbles in tubes have been extensively investigated over the past decades (Davies & Taylor 1950; White & Beardmore 1962; Zukoski 1966; Collins 1967b ; Collins et al. 1978; Reinelt 1987b ; Vanden-Broeck 1984, 1992; Viana et al. 2003; Figueroa-Espinoza & Fabre 2011; Fabre 2016). In contrast, much less effort has been devoted to studying Taylor bubbles in annuli, which are, however, very relevant for many industrial applications such as energy dissipation facilities for chemical/nuclear reactors, casing-tubing annuli for oil production and double-pipe heat exchangers for geothermal systems (Kelessidis & Dukler 1990; Das et al. 1998; Ekberg et al. 1999).
In an annulus, flow is confined between the inner surface of an outer pipe and the outer surface of an inner pipe. If the thickness of the gap between the two surfaces is much smaller than the perimeter of the pipe, the annulus is then similar to a Hele-Shaw cell. The problem of two-phase flow in a Hele-Shaw cell between two parallel flat plates has been the focus of many previous theoretical, experimental and numerical studies (e.g. Taylor & Saffman 1959; Collins 1965a , 1967a ; Maxworthy 1986; Kopf-Sill & Homsy 1988; Roig et al. 2012; Cueto-Felgueroso & Juanes 2014). Large bubbles in a Hele-Shaw cell (the bubble characteristic length, $d$ , is much greater than the gap thickness between the plates, $h$ ) behave very differently from bubbles in an unbounded liquid due to the presence of wall confinement and wettability effects (Thompson 1968; Eck & Siekmann 1978; Saffman & Tanveer 1989; Cueto-Felgueroso & Juanes 2014). The bubble dynamics in a Hele-Shaw cell can be characterised by the bubble Reynolds number $Re=\unicode[STIX]{x1D70C}_{l}u_{\infty }d/\unicode[STIX]{x1D707}_{l}$ ( $u_{\infty }$ is the bubble velocity relative to the ambient liquid having a density $\unicode[STIX]{x1D70C}_{l}$ and a dynamic viscosity $\unicode[STIX]{x1D707}_{l}$ ). In addition, the gap Reynolds number $Re(h/d)^{2}$ is often used to compare the in-plane inertial effect and the cross-gap viscous effect (Thompson 1968; Eck & Siekmann 1978; Bush & Eames 1998; Roig et al. 2012). When $Re(h/d)^{2}\ll 1$ , the flow corresponds to the classical Hele-Shaw flow regime where the inertia is negligible (Saffman & Taylor 1958; Taylor & Saffman 1959; Eck & Siekmann 1978; Maxworthy 1986; Tanveer 1986, 1987; Tanveer & Saffman 1987; Kopf-Sill & Homsy 1988; Meiburg 1989; Saffman & Tanveer 1989; Maruvada & Park 1996; Cueto-Felgueroso & Juanes 2014). When $Re(h/d)^{2}$ is close to or larger than 1, the inertia becomes important, as has been observed for bubbles in inclined or vertical Hele-Shaw cells (Collins 1965a , b ; Grace & Harrison 1967; Lazarek & Littman 1974; Maneri & Zuber 1974; Hills 1975; Vanden-Broeck 1984, 1992; Couet & Strumolo 1987; Bush 1997; Kelley & Wu 1997; Bush & Eames 1998; Huisman, Ern & Roig 2012; Roig et al. 2012; Wang et al. 2014, 2016; Filella, Ern & Roig 2015; Piedra, Ramos & Herrera 2015).
In this paper, we study bubble behaviour in another type of Hele-Shaw cell, i.e. thin annuli. We provide an insight, based on experimental observations, numerical simulations and theoretical analysis, into simultaneous gas–liquid flow through such systems. We focus on the regime where inertia becomes important, i.e. $Re(h/d)^{2}$ is close to or larger than unity. Laboratory experiments are conducted to measure the bubble distribution and evolution in a horizontal or slightly inclined thin annulus (§ 2). A gap-averaged, two-dimensional (2-D) formulation of the Navier–Stokes equations is then derived to represent the three-dimensional (3-D) flow dynamics (§ 3). The numerical model predictions are compared with the experimental data for different inclinations. Further numerical simulations are then performed to elucidate the roles of gap thickness and pipe diameter in bubble formation (§ 4). In addition, a theoretical analysis is presented to interpret the flow mechanisms that govern bubble dynamics in thin annuli (§ 5). Finally, a brief discussion is given and conclusions are drawn (§ 6).
The experimental apparatus consisted of air and water supply systems, an annulus formed between two concentric pipes (comprising three connected subsections and initially filled with water), and a collection tank (figure 1). The parameters of the experimental apparatus and working fluids are as given in table 1. During the experiments, air and dyed water were continuously pumped into a mixer before entering the test section. Visual observation showed the two fluids were well segregated by the time they entered the annulus. Three high-speed cameras were mounted above the annulus subsections for visualisation and recording purposes. The outer pipe was made of Plexiglas to facilitate visualisation, while the inner stainless steel tubing was spray coated to create a white surface which maximised the colour contrast between the air and dyed water. The videos were taken at a frame rate of 17 frames per second with a resolution of $1280\times 720$ pixels, and a colour depth of 24 bits per pixel. The data were imported into MATLAB for post-processing to extract bubble size and velocity data.
As shown in figure 2(a), we define a 3-D curvilinear coordinate system ( $x,y,z$ ) based on the mid-plane of the thin annulus (i.e. the distance of this mid-plane from both the outer and inner walls is $h/2$ ), where $x$ is defined to be along the pipe length ( $0\leqslant x\leqslant L$ ), $y$ is around the perimeter ( $-W/2\leqslant y\leqslant W/2$ with $y=0$ and $y=\pm W/2$ corresponding to the top and bottom of the pipe, respectively), and $z$ is the direction across the gap ( $-h/2\leqslant z\leqslant h/2$ with $z=h/2$ and $z=-h/2$ defining the outer and inner channel walls, respectively). The system is therefore equivalent to a Hele-Shaw cell (viz. figure 2 b) under a distorted gravity field (viz. figure 2 c).
We use the approximation that the flow velocity in the $z$ direction is negligible ( $u_{z}=0$ ) given the very high aspect ratio (i.e. $h/W\ll 1$ ). By eliminating the terms involving $u_{z}$ , the governing equations of the incompressible two-phase, air–water flow are obtained, including the mass and momentum conservation equations given respectively by
(3.1) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0, & \displaystyle\end{eqnarray}$$
(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70C}\left(\frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}\right)=-\unicode[STIX]{x1D735}p+\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D707}[\unicode[STIX]{x1D735}\boldsymbol{u}+(\unicode[STIX]{x1D735}\boldsymbol{u})^{\text{T}}]+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}\left(\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}z}\right)+\unicode[STIX]{x1D70C}\boldsymbol{g}+\unicode[STIX]{x1D70E}\unicode[STIX]{x1D705}\unicode[STIX]{x1D6FF}\boldsymbol{n}, & \displaystyle\end{eqnarray}$$
and an equation for the volume fraction of air given as
(3.3) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}=0. & & \displaystyle\end{eqnarray}$$
Here, $\boldsymbol{u}=[u_{x},u_{y}]$ is the velocity vector, $\unicode[STIX]{x1D735}=[\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x,\unicode[STIX]{x2202}/\unicode[STIX]{x2202}y]$ is the 2-D gradient, $t$ is the time, $\unicode[STIX]{x1D6FC}$ is the volume fraction of air, the bulk density is $\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70C}_{a}+(1-\unicode[STIX]{x1D6FC})\unicode[STIX]{x1D70C}_{w}$ , $p$ is the pressure, the bulk dynamic viscosity is $\unicode[STIX]{x1D707}=\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D707}_{a}+(1-\unicode[STIX]{x1D6FC})\unicode[STIX]{x1D707}_{w}$ , the gravitational acceleration vector is $\boldsymbol{g}=[g\sin \unicode[STIX]{x1D6FD},g\cos \unicode[STIX]{x1D6FD}\sin (2y/d)]$ , $\unicode[STIX]{x1D70E}$ is the surface tension, $\unicode[STIX]{x1D705}$ is the interface curvature, $\unicode[STIX]{x1D6FF}$ is the Dirac delta function and $\boldsymbol{n}$ is the interface outward-pointing unit normal. Surface tension force is treated as a continuous surface force (Brackbill, Kothe & Zemach 1992). Note (3.1)–(3.3) are written in a form following Bush (1997) that eases the derivation of gap-averaged, 2-D equations in the following § 3.2.
We assume the velocity profile in the annulus to be parabolic in the $z$ -direction (Gondret & Rabaud 1997):
(3.4) $$\begin{eqnarray}\displaystyle \boldsymbol{u}=\frac{3}{2}\left[1-\left(\frac{2z}{h}\right)^{2}\right]\bar{\boldsymbol{u}}, & & \displaystyle\end{eqnarray}$$
where $\bar{\boldsymbol{u}}$ is the gap-averaged velocity. This is consistent with our calculation $\unicode[STIX]{x1D70C}_{w}v_{i}(2h)/\unicode[STIX]{x1D707}_{w}=1134$ , which reveals the flow to be laminar (Beavers, Sparrow & Magnuson 1970).
We derive a coupled set of gap-averaged equations by integrating equations (3.1)–(3.3) along the $z$ direction (Roig et al. 2012):
(3.5) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\bar{\boldsymbol{u}}=0, & \displaystyle\end{eqnarray}$$
(3.6) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70C}\left(\frac{\unicode[STIX]{x2202}\bar{\boldsymbol{u}}}{\unicode[STIX]{x2202}t}+\frac{6}{5}\bar{\boldsymbol{u}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\bar{\boldsymbol{u}}\right)=-\unicode[STIX]{x1D735}p+\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D707}[\unicode[STIX]{x1D735}\bar{\boldsymbol{u}}+(\unicode[STIX]{x1D735}\bar{\boldsymbol{u}})^{\text{T}}]-\frac{12\unicode[STIX]{x1D707}}{h^{2}}\bar{\boldsymbol{u}}+\unicode[STIX]{x1D70C}\boldsymbol{g}+\unicode[STIX]{x1D70E}\unicode[STIX]{x1D705}\unicode[STIX]{x1D6FF}\boldsymbol{n}, & \displaystyle\end{eqnarray}$$
(3.7) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x2202}t}+\bar{\boldsymbol{u}}\boldsymbol{\cdot }\unicode[STIX]{x1D6FB}\unicode[STIX]{x1D6FC}=0. & \displaystyle\end{eqnarray}$$
We acknowledge the fact that (3.5)–(3.7) may not represent fully all relevant dynamics, e.g. 3-D effects local to the air–water interface (Oliveira & Meiburg 2011). It is, nonetheless, of interest to determine the extent to which our approximate model can capture as many of the phenomena observed experimentally as possible.
The shape of the air–water interface is affected by the wettability of the channel walls. To incorporate this effect, the curvature in (3.6) may be replaced by (Thompson, Juel & Hazel 2014)
(3.8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}=\unicode[STIX]{x1D705}_{l}+\unicode[STIX]{x1D705}_{t}\approx \unicode[STIX]{x1D705}_{l}+\frac{\cos \unicode[STIX]{x1D703}_{o}+\cos \unicode[STIX]{x1D703}_{i}}{h}, & & \displaystyle\end{eqnarray}$$
where $\unicode[STIX]{x1D705}_{l}$ is the lateral curvature (on the $x$ – $y$ plane) which is estimated using the diffuse-interface approach based on the volume fraction field (Xie et al. 2016), $\unicode[STIX]{x1D705}_{t}$ is the transverse curvature (across the gap) which is estimated from the meniscus profile (figure 3 a), $\unicode[STIX]{x1D703}_{o}$ is the local contact angle at the outer wall and $\unicode[STIX]{x1D703}_{i}$ is that at the inner wall. The local contact angle along the contact line can vary significantly from the advancing value (at the bubble tail) to the receding value (at the bubble front). The experimental measurements by Antonini et al. (2009) showed that the angle can be interpolated using a third-order polynomial function:
(3.9) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}=2\frac{\unicode[STIX]{x1D703}_{min}-\unicode[STIX]{x1D703}_{max}}{\unicode[STIX]{x03C0}^{3}}\unicode[STIX]{x1D719}^{3}-3\frac{\unicode[STIX]{x1D703}_{min}-\unicode[STIX]{x1D703}_{max}}{\unicode[STIX]{x03C0}^{2}}\unicode[STIX]{x1D719}^{2}+\unicode[STIX]{x1D703}_{min}, & & \displaystyle\end{eqnarray}$$
where $\unicode[STIX]{x1D703}$ corresponds to the local contact angle at either the outer or inner wall (i.e. $\unicode[STIX]{x1D703}_{o}$ or $\unicode[STIX]{x1D703}_{i}$ , respectively), $\unicode[STIX]{x1D703}_{max}$ and $\unicode[STIX]{x1D703}_{min}$ denote the advancing and receding contact angles of the corresponding surface, respectively, $\unicode[STIX]{x1D719}$ is the angle between the local velocity vector $\bar{\boldsymbol{u}}$ and the interface normal $\boldsymbol{n}$ (figure 3 b). Figure 4 shows the variation of the local contact angle $\unicode[STIX]{x1D703}$ at the outer and inner walls with respect to $\unicode[STIX]{x1D719}$ . Contact angle hysteresis is promoted by roughness of the channel wall surface (Dussan 1979); we also discuss in § 6 the potential effect of roughness on the presence/absence of thin water films on the channel walls.
A numerical experiment was designed as shown in figure 5. The length and width of the 2-D domain were the pipe length $L$ and perimeter $W$ , respectively. This domain was originally filled with water that had an initial velocity $v_{i}$ along the $x$ direction. The inlet condition was defined by the inflow velocity $v_{i}$ uniformly imposed at the left boundary, such that the air flowed into the domain through the middle part ( $W/5$ wide), corresponding to the inflow rate condition of $q_{a}/(q_{a}+q_{w})=1/5$ (table 1), while the water was injected from the remaining portion of the left boundary. This is consistent with the experimental observation that air and water were well segregated in the mixer before entering the annulus. A free-slip condition was used for the two longitudinal sides of the domain, considering that the air phase is away from the pipe bottom and the flow across the pipe bottom is negligible. A hydrostatic pressure condition was imposed along the outlet to account for the gravity-induced pressure gradient across the channel.
The numerical model was solved based on a mixed control-volume finite-element method, which has been validated against a series of benchmark cases of single rising bubbles, coalescence of two bubbles and droplet impacts (Xie et al. 2014, 2016, 2017). The computational domain was discretised into an unstructured grid of triangular $\text{P}_{1}\text{DG}\text{-}\text{P}_{2}$ elements (linear discontinuous velocity between elements and quadratic continuous pressure between elements) (Cotter et al. 2009). A finite volume discretisation of the mass conservation equation and a linear discontinuous Galerkin discretisation of the momentum equation were used with an adaptive implicit/explicit time stepping scheme (Xie et al. 2016). Within each time step, the equations were iterated upon using a pressure projection method until all equations are simultaneously balanced (Pavlidis et al. 2014; Xie et al. 2014). Interface dynamics were captured through a compressive advection-based volume-of-fluid approach, which used a novel and mathematically rigorous nonlinear Petrov–Galerkin method for maintaining sharp interfaces (Pavlidis et al. 2016). Surface tension was modelled using a diffused-interface formulation that can accurately estimate the lateral curvature based on the volume fraction field (Xie et al. 2016). Anisotropic mesh adaptation (Pain et al. 2001) was used to place a finer mesh around the interface during its evolution as required to capture the dynamics (Xie et al. 2014, 2016). The criteria for mesh refinement, based on the volume fraction field, included an interpolation error bound of 0.1, a minimum element size of $W/60$ and maximum element size of $L/6$ and $W/20$ in the $x$ and $y$ directions, respectively. The 2-D simulation outputs were projected onto the original 3-D Cartesian coordinate system to facilitate visual comparison with experimental data.
Figures 6–9 show the formation of air bubbles in the water flowing through the annulus at different inclinations. Despite the limitations of the gap-averaged 2-D numerical model in representing some 3-D effects such as the detailed transverse shape of the air–water interface and the cross-gap velocity/gravity component, very similar flow behaviour in the lateral directions (i.e. $x$ and $y$ directions) was observed in the laboratory experiments and numerical simulations.
In the horizontal case (figure 6), air was conveyed by long bubbles having a tadpole-like shape with a semi-circular cap and a highly stretched tail connected via an apparent neck. These bubbles, separated by liquid slugs, were forced to flow at the top of the annulus due to buoyancy. The tail of a bubble was initially connected to the inlet (figure 6 a) and was stretched as the bubble advanced until it finally disconnected from the inlet (figure 6 b). The bubble then contracted as a result of surface tension (figure 6 c) and continued to move with an essentially steady shape through the co-current flowing liquid towards the outlet (figure 6 d). When the bubble was separated from the inlet, another bubble began to form, resulting in the appearance of a sequence of similarly shaped/sized bubbles moving along the top of the annulus.
In the $1.9^{\circ }$ inclined case (figure 7), bubbles still exhibited a tadpole-like shape but with a much shorter tail, whilst the cap shape remained similar to that in the horizontal case. The bubble detached quickly from the inlet after its formation (figure 7 a). Due to the constant inflow of air, another bubble formed, disconnected and migrated following the leading one. In this way, a stream of distinct bubbles developed as in the horizontal case. However, if the spacing between two bubbles was sufficiently small, the trailing bubble might catch up (figure 7 b) and merge with its upstream neighbour (figure 7 c). The larger bubble formed might further coalesce with another bubble upstream or downstream. Finally, the bubbles that formed relatively stable shapes were transported to the outlet by the flowing water (figure 7 d).
There appears to be three stages in the coalescence process. (i) Interacting stage (0.0–0.4 s in figure 8). The trailing bubble with a larger velocity caught up with the leading bubble. As the two bubbles became closer, the trailing bubble experienced a marked change in shape, with its cap becoming more pointed with an increased local curvature at the front. The geometry of the leading bubble only changed slightly with the tail becoming shorter and flatter. The changing shape of the two bubbles was caused by the increased pressure gradient between the two bubble interfaces as they got closer together, during which time the liquid between the two bubbles was gradually pushed out. (ii) Merging stage (0.5–0.6 s in figure 8). The cap of the trailing bubble invaded the rear of the leading bubble. The liquid film between the two bubbles was ruptured and drained as the two bubbles united. The merged bubble exhibited a complex, tortuous shape with the cap and tail connected by a narrow neck and a wider body. (iii) Stabilising stage (0.7–1.0 s in figure 8). The merged larger bubble underwent deformation and adjusted itself from a complex, unsteady shape to the stable tadpole-like form.
In the $4.6^{\circ }$ inclined case (figure 9), bubbles became more compact and almost lost the tail structure although the cap shape remained similar to that in the horizontal case. Bubbles separated from the inlet quickly and again formed a train (figure 9 a). The bubbles could merge at an earlier stage by virtue of their smaller spacing and simultaneous coalescence of three bubbles could occur (figure 9 a,b). During the coalescence of three bubbles, the leading bubble remained a semi-circular cap, whereas its rear was flattened; the middle bubble was between the leading and trailing bubbles and showed a sharpened front and a flattened rear; the trailing bubble displayed a narrowed front invading into the back of middle bubble. As merging occurred, the liquid films between bubbles broke and dissipated, such that a united larger bubble was formed with a highly tortuous, unsteady shape. The coalesced larger bubble gradually stabilised itself (figure 9 c) and then migrated to the exit (figure 9 d).
The statistics of the normalised length, cap diameter and terminal velocity of stabilised bubbles in the cases of different inclinations are compared in figure 10. Data from ten fully developed bubbles were collected in the region 2–5 m downstream of the inlet, where the bubbles were not influenced by either inlet or outlet effects. As expected from the discussion above, the bubble length reduces significantly as the inclination of the annulus increases (figure 10 a), due to the deterioration of tail structure. However, the cap diameter, measured based on the curvature of the cap front, only shows a slight increase with the inclination of the annulus (figure 10 b). The terminal velocity of the bubbles also increases slightly with inclination (figure 10 c). An excellent quantitative match with the experimental measurements was achieved by the numerical model. The pseudo-periodic nature of the slug flow and variation of bubble properties as observed in the experiment was also well captured in the simulation (figure 10). Although the inlet condition was stationary, the slug flow pattern varied constantly.
We used the numerical model to further elucidate the effects of gap thickness and pipe diameter on bubbles in thin annuli. First, we set the pipe diameter to be $D=0.137$ m, the air-to-water inflow rate ratio to be $q_{a}/q_{w}=1/4$ and the superficial inflow velocity to be $v_{i}=0.162~\text{m}~\text{s}^{-1}$ . We varied the gap thickness $h$ from 0.001 to 0.005 m, for which the flow remained to be laminar (Beavers et al. 1970) and cross-gap gravity effect was not significant. As shown in figure 11(a), when $h=0.001$ m (i.e. $h/D=0.0073$ ), the bubbles in the horizontal and inclined annuli are very similar, having a slender oval shape with a semi-circular front and a gradually narrowed rear (no discernible neck existed). In the horizontal case, as $h$ becomes larger, the tadpole-like shape becomes more pronounced with a much longer tail and a more distinct neck where the tail joins the cap. If the annulus is inclined, as $h$ increases, the tail tends to vanish and the bubble length decreases significantly. In contrast to the marked variation of bubble length with $h$ , the cap diameter and terminal velocity are much less sensitive to $h$ (figure 11 b,c).
Next, we explored the effect of changing pipe diameter $D=0.05$ –0.4 m (the pipe length $L$ also varied with $D$ according to a fixed ratio $L/D=43.8$ ), keeping $h=0.0035$ m and the same inflow conditions as before. Note that, in figure 12, the bubble length and cap diameter in each case are normalised by the corresponding pipe diameter. As shown in figure 12(a), the ratio of bubble length to the pipe diameter generally does not vary significantly with the change of $D$ . However, in the horizontal annulus with $D=0.4$ m (i.e. $h/D=0.0087$ ), the bubble has a very long, extremely stretched, wavy tail, which does not disconnect from the inlet; the bubble length, therefore, corresponds to the whole pipe length. With the decrease of $D$ (i.e. increase of $h/D$ ), the cap diameter relative to the pipe diameter increases while the bubble velocity decreases (figure 12 b,c).
We use the numerical model to investigate the pressure and velocity fields around the bubbles, which determine the bubble motion, shape and dynamics, and can thus provide clues for interpreting the mechanisms that control these behaviours. Figure 13(a) shows the typical steady bubbles for three different inclinations, and figures 13(b) and 13(c) give their pressure and velocity fields, respectively. To facilitate comparison, the absolute pressure at the foremost point of each bubble is chosen as the reference pressure in each case, so that the pressure contour gives the differential pressure between the absolute pressure and the reference pressure. It can be seen that inside the bubble, the pressure is generally constant due to the very low air density. The pressure in the liquid behind the bubble is higher than that in front of the bubble, driving the bubble to flow towards the outlet. The velocity of air inside the bubble is significantly higher than that of the surrounding liquid.
We explore the mechanisms controlling the bubble shape and motion through a simple analysis. We define two polar coordinate systems: one based on the unrolled 2-D bubble cap geometry in the $x$ – $y$ plane (figure 14 a), and the other based on the transverse cross-section of the pipe in the $y$ – $z$ plane (figure 14 b). By applying Bernoulli's equation to the steady flow around the bubble cap, assumed to be irrotational with the surface tension being neglected, the following relationship is obtained:
(5.1) $$\begin{eqnarray}\displaystyle \frac{u_{\unicode[STIX]{x1D713}}^{2}}{2}=g\left[\frac{D}{2}(1-\cos \unicode[STIX]{x1D6FE})\cos \unicode[STIX]{x1D6FD}+\frac{d}{2}(1-\cos \unicode[STIX]{x1D713})\sin \unicode[STIX]{x1D6FD}\right], & & \displaystyle\end{eqnarray}$$
where $u_{\unicode[STIX]{x1D713}}$ is the relative velocity of the surrounding liquid to the bubble at a point with a polar angle of $\unicode[STIX]{x1D713}$ , and $\unicode[STIX]{x1D6FE}$ is the polar angle in the coordinate system of the transverse cross-section. We assume that flow around the bubble cap is similar to that seen around a circular cylinder to a first approximation (Collins 1967a ). This flow field can be solved using the stream function and velocity potential, with the following relationship obtained (Wilkes 2006):
(5.2) $$\begin{eqnarray}\displaystyle \frac{u_{\unicode[STIX]{x1D713}}^{2}}{u_{\infty }^{2}\sin ^{2}\unicode[STIX]{x1D713}}=4, & & \displaystyle\end{eqnarray}$$
where $u_{\infty }$ is the far-field relative velocity of the flow to the bubble. By combining (5.1), (5.2) and the relation $(d/2)\sin \unicode[STIX]{x1D713}=(D/2)\unicode[STIX]{x1D6FE}$ (see figure 14 a,b), we obtain
(5.3) $$\begin{eqnarray}\displaystyle u_{\infty }^{2}=\left(\frac{1-\cos \unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D6FE}^{2}}\frac{d}{D}\cos \unicode[STIX]{x1D6FD}+\frac{1-\cos \unicode[STIX]{x1D713}}{\sin ^{2}\unicode[STIX]{x1D713}}\sin \unicode[STIX]{x1D6FD}\right)\frac{gd}{4}. & & \displaystyle\end{eqnarray}$$
In the limit of small $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D713}$ , equation (5.3) becomes
(5.4) $$\begin{eqnarray}\displaystyle u_{\infty }\approx \frac{1}{2}\sqrt{\left(\frac{d}{D}\cos \unicode[STIX]{x1D6FD}+\sin \unicode[STIX]{x1D6FD}\right)\frac{gd}{2}}. & & \displaystyle\end{eqnarray}$$
If $r\rightarrow \infty$ , $\unicode[STIX]{x1D6FD}=90^{\circ }$ , then solution (5.4) converges to that for plane bubbles rising through a quiescent liquid between vertical, parallel flat plates, i.e. $0.5(gd/2)^{1/2}$ (Davies & Taylor 1950; Collins 1965a ).
For the simultaneous gas–liquid flow here, the bubble terminal velocity $u$ is calculated as (Griffith & Wallis 1961)
(5.5) $$\begin{eqnarray}\displaystyle u=u_{\infty }+v_{i}\approx \frac{1}{2}\sqrt{\left(\frac{d}{D}\cos \unicode[STIX]{x1D6FD}+\sin \unicode[STIX]{x1D6FD}\right)\frac{gd}{2}}+v_{i}, & & \displaystyle\end{eqnarray}$$
where $v_{i}$ is the superficial inflow velocity (table 1). We substitute the measured diameters of bubble cap from the experimental/numerical results into (5.5) to derive the predicted bubble terminal velocity $u_{pred}$ . The predicted values are compared to the actual velocity $u_{meas}$ measured in the experiment and simulation. It can be seen that our simplified analysis gives a good estimation of the bubble velocity in thin annuli for different inclination angles, pipe diameters and gap thicknesses (figure 15).
We then identify the following dimensionless groups of the annulus system:
(5.6a-c ) $$\begin{eqnarray}\displaystyle \hat{Fr}_{\Vert }^{-1}=\frac{\sqrt{gD\sin \unicode[STIX]{x1D6FD}}}{v_{i}},\quad \hat{Fr}_{\bot }^{-1}=\frac{\sqrt{gD\cos \unicode[STIX]{x1D6FD}}}{v_{i}},\quad \hat{\unicode[STIX]{x1D716}}=\displaystyle \frac{h}{D}, & & \displaystyle\end{eqnarray}$$
where the two inverse Froude numbers, $\hat{Fr}_{\Vert }^{-1}$ and $\hat{Fr}_{\bot }^{-1}$ , are related to the effects of gravity parallel and perpendicular to the flow direction, respectively, and $\hat{\unicode[STIX]{x1D716}}$ characterises the effects of confinement associated with the annulus. We also characterise the bubble shape and motion based on the bubble aspect ratio $l/d$ , the bubble Reynolds number $Re=\unicode[STIX]{x1D70C}_{w}u_{\infty }d/\unicode[STIX]{x1D707}_{w}$ and the gap Reynolds number $Re(h/d)^{2}$ . Here, $u_{\infty }$ is used because the bubble behaviour is governed by the relative motion between the two fluids.
As shown in figure 16(a), as $\hat{Fr}_{\Vert }^{-1}$ increases, the bubble shape becomes less elongated, i.e. $l/d$ approaches unity, since the gravitational component parallel to the flow direction causes an increased pressure at the bubble rear, which tends to separate the bubble and reduce the bubble length. As $\hat{Fr}_{\bot }^{-1}$ increases, $Re$ increases (figure 16 b), indicating inertia becomes more dominant. This is because, for the annuli with small inclinations to the horizontal, the gravitational component perpendicular to the flow direction mainly controls the flow field around the bubble cap, as can be expected from (5.1). As the annulus confinement decreases (i.e. $\hat{\unicode[STIX]{x1D716}}$ increases), the gap Reynolds number $Re(h/d)^{2}$ increases (figure 16 c), due to the reduced viscous effect across the gap, reaching a plateau with increasing $\hat{\unicode[STIX]{x1D716}}$ .
In this paper, we presented an experimental and numerical investigation into the shape and motion of gas bubbles in a liquid flowing through a horizontal or slightly inclined thin annulus. We have focused on the regime where inertia is important (the gap Reynolds number is close to or larger than unity) and the bubble behaviour is governed by the complex interplay of inertia, gravity, viscosity and surface tension. Bubbles in the horizontal annulus developed a unique 'tadpole-like' shape featuring a semi-circular cap and a highly stretched tail. As the annulus was inclined with respect to the horizontal, the length of the bubble decreased. We developed a gap-averaged, 2-D numerical model to represent the 3-D flow dynamics, which achieved a close match to the experimental data for different small inclinations. The numerical model was used to further elucidate the effects of gap thickness and pipe diameter on the bubble evolution in thin annuli. We found that the bubble velocity is strongly correlated to the cap structure, but is independent of the bubble length, as has also been reported for bubbles in tubes (Davies & Taylor 1950; Zukoski 1966; Fabre & Liné 1992; Viana et al. 2003) and between parallel flat plates (Grace & Harrison 1967; Maneri & Zuber 1974; Hills 1975; Couet & Strumolo 1987). We reported that the elongated bubble shape in horizontal annuli is due to the buoyancy which causes the bubbles to spread along the top of the annulus. The gravitational component along the flow direction, which increases as the annulus is inclined, impinges the liquid slug and causes a reduction of the bubble tail. The gravitational component perpendicular to the flow direction controls the bubble motion and the cap structure. These mechanisms produce the unique tadpole-like shape with a sharp tail tip because of the cross-sectional curvature of the annulus channel, in contrast to the bubble shape with a tongue-like rear seen in flows between parallel flat plates (Maneri & Zuber 1974; Couet & Strumolo 1987).
It is remarkable that the 2-D numerical model well captured the bubble evolution and interaction behaviour as observed in the experimental annulus. It is still worth mentioning that some complex 3-D effects were not fully represented in the gap-averaged formulation, such as the detailed transverse shape of the bubble cap and tail within the gap as well as the influence of the cross-gap velocity/gravity component (Oliveira & Meiburg 2011), which require direct three-dimensional but computationally very expensive numerical simulations. Such limitations of the 2-D model may explain the discrepancy of the bubble geometry, e.g. in the region near the bubble neck (figure 6), between our experimental and numerical results. The gap-averaged 2-D model assumed an absence of liquid films between the bubble and channel walls. We examined different mathematical formulations that account for the presence of liquid films (Pitts 1980; McLean & Saffman 1981; Park & Homsy 1984; Reinelt 1987a ; Kopf-Sill & Homsy 1988), which however resulted in a poor prediction of the experimentally observed bubble patterns (the simulated bubbles were much smaller, probably due to an overestimation of the curvature of the air–water interface across the gap). Saffman & Tanveer (1989) also reported that the thin film hypothesis, compared to the contact angle assumption, gave a less consistent prediction of bubble shapes in Hele-Shaw cells. Furthermore, we did not see any evidence of such films in the experiments. The thickness of thin films (if present under no-gravity conditions) is estimated to be $\unicode[STIX]{x1D6FF}/h\approx 0.67(\unicode[STIX]{x1D707}_{w}u_{\infty }/\unicode[STIX]{x1D70E})^{2/3}$ (Park & Homsy 1984; Reinelt 1987a ; Klaseboer, Gupta & Manica 2014), which may however be reduced further in the top region of the annulus under gravity-induced drainage downwards along the peripheral direction (Tso & Sugawara 1990; Paras & Karabelas 1991). Thus, we expect that $\unicode[STIX]{x1D6FF}$ may be close to the scale of the surface roughness of the Plexiglas and stainless steel walls, i.e. around 0.01 mm (Li et al. 2018). We therefore do not expect the films to persist in a stable way. The observed contact angle hysteresis in our study may also be attributed to the presence of pronounced surface roughness effects (Dussan 1979). The phenomenon that air wets the channel wall has also been found between the air bubble and the upper wall of a horizontal/slightly inclined tube (Fabre & Liné 1992) or Hele-Shaw cell (Maneri & Zuber 1974). We suggest that a good approximation of the gap-averaged 2-D model to 3-D flow holds if the cross-gap effects are minor.
We would like to acknowledge Statoil ASA for funding Q.L., M.D.J., C.C.P., O.K.M. and A.H.M., and for granting permission to publish this work. We thank K. Årland from Statoil ASA for project management. O.K.M. and C.C.P. acknowledge the funding provided by the Engineering and Physical Sciences Research Council (EPSRC) through the Programme Grant MEMPHIS (grant EP/K003976/1). D.P. is grateful to the EPSRC (grant EP/M012794/1). P.S. thanks ExxonMobil and the EPSRC (grant EP/R005761/1) for funding. No data were generated in the course of this work. For further information, please contact the corresponding author Q.L. (email: [email protected] or [email protected]), the AMCG Group (http://www.imperial.ac.uk/earth-science/research/research-groups/amcg/), the NORMS group (http://www.imperial.ac.uk/earth-science/research/research-groups/norms) or the Matar Fluids Group (http://www.imperial.ac.uk/matar-fluids-group).
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\begin{document}
\title{A Variant Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic}
\begin{abstract}
On a closed Riemannian surface $(M,\bar g)$ with negative Euler characteristic, we study the problem of finding conformal metrics with prescribed volume $A>0$ and the property that their Gauss curvatures $f_\lambda= f + \lambda$ are given as the sum of a prescribed function $f \in C^\infty(M)$ and an additive constant $\lambda$. Our main tool in this study is a new variant of the prescribed Gauss curvature flow, for which we establish local well-posedness and global compactness results. In contrast to previous work, our approach does not require any sign conditions on $f$. Moreover, we exhibit conditions under which the function $f_\lambda$ is sign changing and the standard prescribed Gauss curvature flow is not applicable. \end{abstract}
\section*{Acknowledgment} This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), project 408275461 (Smoothing and Non-Smoothing via Ricci Flow).\\ We would like to thank Esther Cabezas--Rivas for helpful discussions.
\section{Introduction}
Let $(M,\bar g)$ be a two-dimensional, smooth, closed, connected, oriented Riemann manifold endowed with a smooth background metric $\bar g$. A classical problem raised by Kazdan and Warner in \cite{KazWar74_1} and \cite{KazWar74_2} is the question which smooth functions $f \colon M \to \mathds R$ arise as the Gauss curvature $K_g$ of a conformal metric $g(x)=\mathrm{e}^{2u(x)}\bar g(x)$ on $M$ and to characterise the set of all such metrics.
For a constant function $f$, this prescribed Gauss curvature problem is exactly the statement of the {\em Uniformisation Theorem (see e.g.~\cite{Poi08}, \cite{Koe08}):}\\ {\em There exists a metric $g$ which is pointwise conformal to $\bar g$ and has constant Gauss curvature $K_{g}\equiv\bar K\in\mathds R$.}\\ We now use this statement to assume in the following without loss of generality that the background metric $\bar g$ itself has constant Gauss curvature $K_{\bar g}\equiv \bar K\in\mathds R$. Furthermore we can normalise the volume of $(M,\bar g)$ to one. We recall that the Gauss curvature of a conformal metric $g(x)=\mathrm{e}^{2u(x)}\bar g(x)$ on $M$ is given by the Gauss equation \begin{equation}\label{GaussEquation} K_{g}(x) = \mathrm{e}^{-2u(x)}(-\Delta_{\bar g}u(x) + \bar K). \end{equation} Therefore the problem reduces to the question for which functions $f$ there exists a conformal factor $u$ solving the equation \begin{equation}\label{PCP} -\Delta_{\bar g}u(x) + \bar K = f(x) \mathrm{e}^{2u(x)} \qquad \text{in $M$.} \end{equation} Given a solution $u$, we may integrate \eqref{PCP} with respect to the measure $\mu_{\bar g}$ on $M$ induced by the Riemannian volume form. Using the Gauss--Bonnet Theorem, we then obtain the identity \begin{equation}\label{Condition1} \int_M f(x) d\mu_g(x) = \int_M \bar K d\mu_{\bar g}(x) = \bar K \vol_{\bar g}=\bar K= 2\pi \chi(M), \end{equation} where $d\mu_g(x) = \mathrm{e}^{2u(x)}d\mu_{\bar g}(x)$ is the element of area in the metric $g(x)=\mathrm{e}^{2u(x)}\bar g(x)$. We note that (\ref{Condition1}) immediately yields necessary conditions on $f$ for the solvability of the prescribed Gauss curvature problem. In particular, if $\pm \chi(M)>0$, then $\pm f$ must be positive somewhere. Moreover, if $\chi(M)=0$, then $f$ must change sign or must be identically zero.
In the present paper we focus on the case $\chi(M)<0$, so $M$ is a surface of genus greater than one and $\bar K < 0$. The complementary cases $\chi(M) \ge 0$---i.e., the cases where $M = S^2$ or $M=T$, the $2$-torus---will be discussed briefly at the end of this introduction, and we also refer the reader to \cite{Str05,Str20,BuzSchStr16,Gal15} and the references therein. Multiplying equation \eqref{PCP} with the factor $\mathrm{e}^{-2u}$ and integrating over $M$ with respect to the measure $\mu_{\bar g}$, we get the following necessary condition---already mentioned by Kazdan and Warner in \cite{KazWar74_1}---for the average $\bar f:=\frac{1}{\vol_{\bar g}}\int_M f(x)d\mu_{\bar g}(x)$, with $\vol_{\bar g}:=\int_Md\mu_{\bar g}(x)$: \begin{equation}\label{Condition2} \begin{split} \bar f&=\frac{1}{\vol_{\bar g}}\int_M f(x)d\mu_{\bar g}(x)=\int_M(-\Delta_{\bar g}u(x)+\bar K)\mathrm{e}^{-2u(x)}d\mu_{\bar g}(x)\\
&=\int_M(-2|\nabla_{\bar g}u(x)|^2_{\bar g}+\bar K)\mathrm{e}^{-2u(x)}d\mu_{\bar g}(x)<0. \end{split} \end{equation} This condition is not sufficient. Indeed, it has already been pointed out in \cite[Theorem 10.5]{KazWar74_1} that in the case $\chi(M)<0$ there always exist functions $f \in C^\infty(M)$ with $\bar f< 0$ and the property that \eqref{PCP} has no solution.
We recall that solutions of \eqref{PCP} can be characterised as critical points of the functional \begin{equation}\label{Ef}
E_f:H^1(M,\bar g)\to \mathds R;\quad E_f(u) := \frac12\int_M \left(|\nabla_{\bar g} u(x)|^2_{\bar g} + 2\bar K u(x) - f(x)\mathrm{e}^{2u(x)}\right)
d\mu_{\bar g}(x). \end{equation} Under the assumption $\chi(M)<0$, i.e., $\bar K < 0$, the functional $E_f$ is strictly convex and coercive on $H^1(M,\bar g)$ if $f\le 0$ and $f$ does not vanish identically. Hence, as noted in \cite{DinLiu95}, the functional $E_f$ admits a unique critical point $u_f \in H^1(M,\bar g)$ in this case, which is a strict absolute minimiser of $E_f$ and a (weak) solution of \eqref{PCP}. The situation is more delicate in the case where $f_\lambda=f_0+\lambda$, where $f_0\le0$ is a smooth, nonconstant function on $M$ with $\max_{x\in M}f_0(x)=0$, and $\lambda >0$. In the case where $\lambda>0$ sufficiently small (depending on $f_0$), it was shown in \cite{DinLiu95} and \cite{BorGalStr15} that the corresponding functional $E_{f_\lambda}$ admits a local minimiser $u_\lambda$ and a further critical point $u^\lambda\neq u_\lambda$ of mountain pass type.
These results motivate our present work, where we suggest a new flow approach to the prescribed Gausss curvature problem in the case $\chi(M)<0$. It is important to note here that there is an intrinsic motivation to formulate the static problem in a flow context. Typically, elliptic theories are regarded as the static case of the corresponding parabolic problem; in that sense, many times the better-understood elliptic theory has been a source of intuition to generalise the corresponding results in the parabolic case. Examples of this feedback are minimal surfaces/mean curvature flow, harmonic maps/solutions of the heat equation, and the uniformisation theorem/the two-dimensional normalised Ricci flow.
In this spirit, a flow approach to \eqref{PCP}, the so-called prescribed Gauss curvature flow, was first introduced by Struwe in \cite{Str05} (and \cite{BuzSchStr16}) for the case $M=S^2$ with the standard background metric and a positive function $f \in C^2(M)$. More precisely, he considers a family of metrics $(g(t,\cdot))_{t\ge0}$ which fulfils the initial value problem \begin{align} \partial_tg(t,x)&=2(\alpha(t)f(x)-K_{g(t,\cdot)}(x))g(t,x)\quad\text{in }(0,T)\times M;\\ g(0,x)&= g_0(x)\quad\text{on }\{0\}\times M, \end{align} with \begin{equation}
\label{eq:def-alpha-t} \alpha(t)=\frac{\int_MK_{g(t,\cdot)}(x)d\mu_{g(t,\cdot)}(x)}{\int_M f(x)d\mu_{g(t,\cdot)}(x)}=\frac{2\pi\chi(M)}{\int_Mf(x)d\mu_{g(t,\cdot)}(x)}. \end{equation} This choice of $\alpha(t)$ ensures that the volume of $(M,g(t,\cdot))$ remains constant throughout the deformation, i.e., \[\int_M d\mu_{g(t,\cdot)}(x)=\int_M\mathrm{e}^{2u(t,x)}d\mu_{\bar g}(x)\equiv\vol_{g_0}\quad\text{for all }t\ge0,\] where $g_0$ denotes the initial metric on $M$. Equivalently one may consider the evolution equation for the associated conformal factor $u$ given by $g(t,x)=\mathrm{e}^{2u(t,x)}\bar g(x)$: \begin{align} \partial_tu(t,x)&=\alpha(t) f(x)-K_{g(t,\cdot)}(x)\quad\text{in }(0,T)\times M;\label{PCFnegative}\\ u(0,x)&=u_0(x)\quad\text{on }\{0\}\times M. \end{align} Here the initial value $u_0$ is given by $g_0(x)=\mathrm{e}^{2u_0(x)}\bar g(x)$. The flow associated to this parabolic equation is usually called the prescribed Gauss curvature flow.
With the help of this flow, Struwe \cite{Str05} provided a new proof of a result by Chang and Yang \cite{ChaYan87} on sufficient criteria for a function $f$ to be the Gauss curvature of a metric $g(x)=\mathrm{e}^{2u(x)}g_{S^2}(x)$ on $S^2$. He also proved the sharpness of these criteria.
In the case of surfaces with genus greater than one, i.e., with negative Euler characteristic, the prescribed Gauss curvature flow was used by Ho in \cite{Ho11} to prove that any smooth, strictly negative function on a surface with negative Euler characteristic can be realised as the Gaussian curvature of some metric. More precisely, assuming that $\chi(M)< 0$ and that $f \in C^\infty(M)$ is a strictly negative function, he proves that equation \eqref{PCFnegative} has a solution which is defined for all times and converges to a metric $g_\infty$ with Gaussian curvature $K_{g_\infty}$ satisfying \[K_{g_\infty}(x)=\alpha_\infty f(x)\] for some constant $\alpha_\infty$.
While the prescribed Gauss curvature flow is a higly useful tool in the cases where $f$ is of fixed sign, it cannot be used in the case where $f$ is sign-changing. Indeed, in this case we may have $\int_Mf(x)d\mu_{g(t,\cdot)}(x)=0$ along the flow and then the normalising factor $\alpha(t)$ is not well-defined by (\ref{eq:def-alpha-t}). As a consequence, a long-time solution of (\ref{PCFnegative}) might not exist. In particular, the static existence results of \cite{DinLiu95} and \cite{BorGalStr15} can not be recovered and reinterpreted with the standard prescribed Gauss curvature flow.
In this paper we develop a new flow approach to (\ref{PCP}) in the case $\chi(M)<0$ for general $f \in C^\infty(M)$, which sheds new light on the results in \cite{DinLiu95}, \cite{BorGalStr15} and \cite{Ho11}. The main idea is to replace the multiplicative normalisation in (\ref{PCFnegative}) by an additive normalisation, as will be described in details in the next chapter.
At this point, it should be noted that the normalisation factor $\alpha(t)$ in the prescribed Gauss curvature flow given by (\ref{eq:def-alpha-t}) is also not the appropriate choice in the case of the torus, where, as noted before, $f$ has to change sign or be identically zero in order to arise as the Gauss curvature of a conformal metric. The case of the torus was considered by Struwe in \cite{Str20}, where, in particular, he used to a flow approach to reprove and partially improve a result by Galimberti \cite{Gal15} on the static problem. In this approach, the normalisation in (\ref{eq:def-alpha-t}) is replaced by \begin{equation}
\label{eq:def-alpha-t-torus} \alpha(t)=\frac{\int_M f(x)K_{g(t,\cdot)}(x)d\mu_{g(t,\cdot)}(x)}{\int_Mf^2(x)d\mu_{g(t,\cdot)}(x)}. \end{equation} With this choice, Struwe shows that for any smooth $$ u_0\in C^*:=\left\{u\in H^1(M,\bar g)\mid\int_Mf(x)\mathrm{e}^{2u(x)}d\mu_{\bar g}(x)=0,\:\int_M\mathrm{e}^{2u(x)}d\mu_{\bar g}(x)=1\right\} $$ there exists a unique, global smooth solution $u$ of \eqref{PCFnegative} satisfying $u(t,\cdot)\in C^*$ for all $t>0$. Moreover, $u(t,\cdot)\to u_\infty(\cdot)$ in $H^2(M,\bar g)$ (and smoothly) as $t\to\infty$ suitably, where $u_\infty+c_\infty$ is a smooth solution of \eqref{PCP} for some $c_\infty\in\mathds R$.
In principle, the normalisation (\ref{eq:def-alpha-t-torus}) could also be considered in the case $\chi(M)<0$, but then the flow is not volume-preserving anymore, which results in a failure of uniform estimates for solutions of (\ref{PCFnegative}). Consequently, we were not able to make use of the associated flow in this case.
The paper is organised as follows. In \autoref{SectionProperties} we set up the framework for the new variant of the prescribed Gauss curvature flow with additive normalisation, and we collect basic properties of it. In \autoref{SectionMainResults}, we then present our main result on the long-time existence and convergence of the flow (for suitable times $t_k \to \infty$) to solutions of the corresponding static problem. In particular, our results show how sign changing functions of the form $f_\lambda = f_0 + \lambda$ arise depending on various assumptions on the shape of $f_0$ and on the fixed volume $A$ of $M$ with respect to the metric $g(t)$. Before proving our results on the time-dependent problem, we first derive, in \autoref{sec:stat-minim-probl}, some results on the static problem with volume constraint. Most of these results will then be used in \autoref{sec:proof-main-results}, where the parabolic problem is studied in detail and the main results of the paper are proved. In the appendix, we provide some regularity estimates and a variant of a maximum princple for a class of linear evolution problems with H\"older continuous coefficients.
In the remainder of the paper, we will use the short form $f$, $g(t)$, $u(t)$, $K_{g(t)}$, $\vol_{g(t)}:=\int_M d\mu_{g(t)}=\int_M\mathrm{e}^{2u(t)}d\mu_{\bar g}$, and so on instead of $f(x)$, $g(t,x)$, $u(t,x)$, $K_{g(t,\cdot)}(x)$, $\int_M d\mu_{g(t,\cdot)}(x)=\int_M\mathrm{e}^{2u(t,x)}d\mu_{\bar g}(x)$, et cetera.
\section{A New Flow Approach and Some of its Properties}\label{SectionProperties} Let $f\in C^\infty(M)$ be a smooth function. We consider now the additive rescaled prescribed Gauss curvature flow given by \begin{equation}\label{PCFVol1}
\partial_tu(t)=f-K_{g(t)}-\alpha(t) = f-\mathrm{e}^{-2u(t)}( \Delta_{\bar g}u(t)-\bar K)-\alpha(t) \quad\text{in }(0,T)\times M, \end{equation} where $\alpha(t)$ is chosen such that the volume $\vol_{g(t)}$ of $M$ with respect to $g(t)=\mathrm{e}^{2u(t)}\bar g$ remains constant along the flow, that is, we require the condition \begin{equation}\label{VolPres} \frac12\frac{d}{dt}\vol_{g(t)}=\int_M\partial_t u(t) d\mu_{g(t)}=\int_M(f-K_{g(t)}-\alpha(t))d\mu_{g(t)}=0. \end{equation}
Solving for $\alpha(t)$ then we find \[\alpha(t)=\frac{1}{\vol_{g(t)}}\left(\int_Mf d\mu_{g(t)}-\bar K\right).\]
So, starting with $$ u_0\in \mathcal{C}_{p,A}:=\left\{v\in W^{2,p}(M,\bar g)\mid \int_M\mathrm{e}^{2v}d\mu_{\bar g}=A\right\},\quad p>2, $$ for a given $A>0$, we have \[\vol_{g(t)}=\vol_{g(0)}=\vol_{g_0}=A,\quad\text{for all }t\ge0,\] hence we can define \begin{equation}\label{tildealpha} \alpha_A(t)=\frac{1}{A}\left(\int_M fd\mu_{g(t)}-\bar K\right). \end{equation}
Therefore in the following we consider the flow \begin{align} \partial_tu(t)&=f-K_{g(t)}-\alpha_A(t)\quad\text{in }(0,T)\times M; \label{PCFVolNew1}\\ u(0)&=u_0\in \mathcal{C}_{p,A} \quad\text{on }\{0\}\times M,\label{PCFVolNew2} \end{align} with $\alpha_A(t)$ is chosen like in \eqref{tildealpha}. We can now state some first properties of the flow.
\begin{proposition}\label{Properties1} Let $u$ be a (sufficiently smooth) solution of \eqref{PCFVolNew1}, \eqref{PCFVolNew2}. Then \begin{enumerate} \item the volume $\vol_{g(t)}$ of $(M,g(t))$ is preserved along the flow, i.e., $\vol_{g(t)}\equiv \vol_{g_0}=A$ for all $t\ge0$; \item along this trajectory, we have a uniform bound for $\alpha$ given by \begin{equation}\label{LowerBoundtildealpha}
\alpha(t)\ge\min_{x\in M}f(x)+\frac{|\bar K|}{A}=:\alpha_1>-\infty \end{equation} and \begin{equation}\label{UpperBoundtildealpha}
\alpha(t)\le\max_{x\in M}f(x)+\frac{|\bar K|}{A}=:\alpha_2<\infty; \end{equation} \item the flow is invariant under adding or subtracting a constant $C>0$ to the function $f$; \item and the energy $E_{f}$, defined in \eqref{Ef}, is decreasing in time along the flow, so \[E_f(u(t))\le E_f(u_0)\quad \text{for all }t\ge0.\] \end{enumerate} \end{proposition}
\begin{proof} The first statement directly follows by \eqref{VolPres} and the choice of $\alpha$ in \eqref{tildealpha}.\\ The second one we get since $f$ is smooth and $\vol_{g(t)}=A$.\\ To show the invariance of the flow, let $C>0$ be a constant. We then replace $f$ by $f\pm C$ in \eqref{PCFVolNew1} and see that \[f\pm C-K_{g(t)}-\frac1A\left(\int_M(f\pm C)d\mu_{g(t)}-\bar K\right)=f-K_{g(t)}-\frac1A\left(\int_Mfd\mu_{g(t)}-\bar K\right)=\partial_tu(t).\] So, the flow \eqref{PCFVolNew1} is left unchanged if we replace $f$ by $f\pm C$ for a constant $C>0$.\\ To see that the energy $E_f$ is decreasing along the flow, we use \eqref{VolPres} and get \begin{equation}\label{EnergyEstimate} \begin{split} \frac{d}{dt}E_{f}(u(t))&=\int_M(-\Delta_{\bar g}u(t)+\bar K-f\mathrm{e}^{2u(t)})\partial_tu(t)d\mu_{\bar g}\\ &=\int_M((-\Delta_{\bar g}u(t)+\bar K)\mathrm{e}^{-2u(t)}-f)\mathrm{e}^{2u(t)}\partial_tu(t)d\mu_{\bar g}\\ &=\int_M((-\Delta_{\bar g}u(t)+\bar K)\mathrm{e}^{-2u(t)}-f)\partial_tu(t)d\mu_{g(t)}\\ &=\int_M(K_{g(t)}-f)\partial_tu(t)d\mu_{g(t)}=\int_M(K_{g(t)}-f+\alpha(t))\partial_tu(t)d\mu_{g(t)}\\
&=-\int_M|\partial_tu(t)|^2d\mu_{g(t)}\le0. \end{split} \end{equation} Therefore on an interval $[0,T]$, we have the uniform a-priori bound \begin{equation}\label{APrioriBound}
E_{f}(u(T))+\int_0^T\int_M|\partial_tu(t)|^2d\mu_{g(t)}dt= E_{f}(u(0)) \end{equation} for any $T>0$. \end{proof}
\section{Main Results}\label{SectionMainResults} The following is our first main result. \begin{theorem}\label{ShortTimeExistence} Let $f\in C^\infty(M)$, $p>2$, and $u_0\in \mathcal{C}_{p,A}$ for a given $A>0$. Then the initial value problem \eqref{PCFVolNew1}, \eqref{PCFVolNew2} admits a unique global solution $u \in C([0,\infty);C(M))\cap C([0,\infty);H^1(M,\bar g))\cap C^\infty((0,\infty)\times M)$ satisfying the energy bound $E_{f}(u(t))\le E_{f}(u_0)$ for all $t$.
Moreover, $u$ is uniformly bounded in the sense that \[
\sup_{t>0}\|u(t)\|_{L^\infty(M,\bar g)}<\infty. \] Furthermore, as $t \to \infty$ suitably, $u$ converges to a function $u^\infty$ in $H^2(M,\bar g)$ solving the equation \begin{equation}\label{PCP-f-lambda} -\Delta_{\bar g}u + \bar K = f_\lambda \mathrm{e}^{2u} \qquad \text{in $M$,} \end{equation} where $f_\lambda:= f +\lambda$ with \begin{equation}
\label{eq:def-lambda} \lambda=\frac1A\left(\bar K- \int_M f \mathrm{e}^{2u^\infty}d\mu_{\bar g}\right). \end{equation} In other words, $u^\infty$ induces a metric $g^\infty$ with Gauss curvature $K_{g^\infty}$ satisfying \begin{equation}
\label{eq:def-f-lambda} K_{g^\infty}(x)= f_{\lambda}(x)=f(x)+\lambda\quad\text{for}\quad x\in M. \end{equation} \end{theorem}
\begin{remark} For functions $f<0$, the convergence of the flow \eqref{PCFnegative} is shown in \cite{Ho11}. For the additive rescaled flow \eqref{PCFVolNew1} with initial data \eqref{PCFVolNew2} we get convergence for arbitrary functions $f\in C^\infty(M)$. In general we do not have any information about $\lambda$ and therefore no information about the sign of $f_{\lambda}$ in \autoref{ShortTimeExistence}. On the other hand, more information can be derived for certain functions $f \in C^\infty(M)$ and certain values of $A>0$.
\begin{enumerate}
\item[(i)] In the case where $A \le - \frac{\bar K}{\|f\|_{L^\infty(M,\bar g)}}$, it follows that $$
\lambda= \frac1A\left(\bar K- \int_M f \mathrm{e}^{2u}d\mu_{\bar g}\right) \le \frac{\bar K}{A} + \frac{\|f\|_{L^\infty(M,\bar g)}}{A} \int_M \mathrm{e}^{2u}d\mu_{\bar g}
= \frac{\bar K}{A} + \|f\|_{L^\infty(M,\bar g)} \le 0 $$ for every solution $u \in \mathcal{C}_{2,A}:=\left\{v\in H^2(M,\bar g)\mid\int_M\mathrm{e}^{2v}d\mu_{\bar g}=0\right\}$ of the static problem \eqref{PCP-f-lambda}, and therefore this also applies to $\lambda$ in \autoref{ShortTimeExistence} in this case.
\item[(ii)] The following theorems show that $f_{\lambda}$ in \autoref{ShortTimeExistence} may change sign if
$A > - \frac{\bar K}{\|f\|_{L^\infty(M,\bar g)}}$, so in this case we get a solution of the static problem \eqref{PCP} for sign-changing functions $f\in C^\infty(M)$ by using the additive rescaled prescribed Gauss curvature flow \eqref{PCFVolNew1}. \end{enumerate} \end{remark}
\begin{theorem}
\label{sign-changing}
Let $p>2$. For every $A>0$ and $c> - \frac{\bar K}{A}$ there exists $\varepsilon= \varepsilon(c,A,\bar K)>0$ with the following property.
If $u_0 \equiv \frac{1}{2}\log(A) \in \mathcal{C}_{p,A}$ and $f\in C^\infty(M)$
with $-c \le f \le 0$ and $\|f+c\|_{L^1(M,\bar g)} < \varepsilon$ is chosen in \autoref{ShortTimeExistence}, then the value $\lambda$ defined in \eqref{eq:def-lambda} is positive.
In particular, if $f$ has zeros on $M$, then $f_\lambda$ in \eqref{eq:def-f-lambda} is sign changing. \end{theorem}
Under fairly general assumptions on $f$, we can prove that $\lambda>0$ if $A$ is sufficiently large and $u_0 \in \mathcal{C}_{p,A}$ is chosen suitably.
\begin{theorem} \label{sec:stat-minim-probl-1-cor-2-theorem} Let $f \in C^\infty(M)$ be nonconstant with $\max_{x\in M} f(x) = 0$. Then there exists $\kappa>0$ with the property that for every $A \ge \kappa$ there exists $u_0 \in \mathcal{C}_{p,A}$ such that the value $\lambda$ defined in \eqref{eq:def-lambda} is positive. \end{theorem}
In fact we have even more information on the associated limit $u^\infty$ in this case, see \autoref{sec:stat-minim-probl-1-cor-2} below.
It remains open how large $\lambda$ can be depending on $A$ and $f$. The only upper bound we have is \begin{equation}\label{Conditionlambda1} \lambda<- \int_M f d\mu_{\bar g}, \end{equation} since we must have \[ \bar f_\lambda=\frac{1}{\vol_{\bar g}}\int_M f_\lambda d\mu_{\bar g}=\int_M f d\mu_{\bar g}+\lambda\overset{!}{<}0, \] so that $f_\lambda$ fulfills the necessary condition \eqref{Condition2} provided by Kazdan and Warner in \cite{KazWar74_1}.\\
\section{The static Minimisation Problem with Volume Constraint} \label{sec:stat-minim-probl} To obtain additional information on the limiting function $u^\infty$ and the value $\lambda \in \mathds R$ associated to it by \eqref{eq:def-lambda} and \eqref{eq:def-f-lambda}, we need to consider the associated static setting for the prescribed Gauss curvature problem with the additional condition of prescribed volume.
Before going into the details of this static problem, we recall an important and highly useful estimate. The following lemma (see e.g.~\cite[Corollary 1.7]{Cha04}) is a consequence of the Trudinger's inequality \cite{Tru67} which was improved by Moser in \cite{Mos71} (for more details see e.g.~\cite[Theorem 2.1 and Theorem 2.2]{Str20}):
\begin{lemma}\label{Onofri-ungleichung} For a two-dimensional, closed Riemannian manifold $(M,\bar g)$ there are constants $\eta>0$ and $C_{\text{MT}}>0$ such that \begin{equation}\label{OU}
\int_M\mathrm{e}^{(u-\bar u)}d\mu_{\bar g}\le C_{\text{MT}}\exp\left(\eta\|\nabla_{\bar g}u\|^2_{L^2(M,\bar g)}\right) \end{equation} for all $u\in H^1(M,\bar g)$ where \[\bar u:=\frac{1}{\vol_{\bar g}}\int_Mu\:d\mu_{\bar g}=\int_Mu\:d\mu_{\bar g},\] in view of our assumption that $\vol_{\bar g}=1$. \end{lemma}
As a consequence of \autoref{Onofri-ungleichung}, we have $$ \int_M \mathrm{e}^{p u}d\mu_{\bar g} = \mathrm{e}^{p \bar u} \int_M\mathrm{e}^{(pu-\bar{pu})}d\mu_{\bar g} \le \mathrm{e}^{p \bar u}
C_{\text{MT}}\exp\left(\eta\|\nabla_{\bar g}(pu)\|^2_{L^2(M,\bar g)}\right)< \infty $$ for every $u \in H^1(M,\bar g)$ and $p>0$. Consequently, for a given $A>0$, the set
\begin{equation}\label{NB} \mathcal{C}_{1,A}:= \left\{u\in H^1(M,\bar g)\mid V(u):=\int_M \mathrm{e}^{2u}d\mu_{\bar g}=A\right\} \end{equation} is well defined and coincides with the closure of $\mathcal{C}_{2,A}$ with respect to the $H^1$-norm. We also note that \begin{equation} \label{Jensen-consequence} \bar u\le\frac12\log(A) \qquad \text{for $u\in \mathcal{C}_{1,A}$,} \end{equation} since by Jensen's inequality and our assumption that $\vol_{\bar g}=1$ we have \begin{equation*} 2\bar u=\dashint_M 2 ud\mu_{\bar g}=\int_M 2ud\mu_{\bar g}\le \log\left(\dashint\mathrm{e}^{2u}d\mu_{\bar g}\right)=\log(A)\qquad \text{for $u\in \mathcal{C}_{1,A}$.} \end{equation*}
Furthermore we want to recall the Gagliardo--Nirenberg--Lady\v{z}henskaya interpolation, see e.g.~\cite{CecMon08}.
\begin{lemma}[Gagliardo--Nirenberg--Lady\v{z}henskaya inequality]\label{eindSob} There exists a constant $C_{\text{GNL}}>0$ such that we have for every $\zeta\in H^1(M,\bar g)$ the inequality
\[\|\zeta\|^4_{L^4(M,\bar g)}
\le C_{\text{GNL}}\|\zeta\|^2_{L^2(M,\bar g)}\|\zeta\|^2_{H^1(M,\bar g)}.\] \end{lemma}
Now we enter the details of the static prescribed Gauss curvature problem with volume constraint. In this problem, we wish to find, for given $f \in C^\infty(M)$ and $A>0$, critical points of the restriction of the functional $E_{f}$ defined in \eqref{Ef} to the set $\mathcal{C}_{1,A}$. A critical point $u \in \mathcal{C}_{1,A}$ of this restriction is a solution of \eqref{PCP-f-lambda} for some $\lambda \in \mathds R$, where, here and in the following, we put again $f_\lambda:= f + \lambda \in C^\infty(M)$. In other words, such a critical point induces, similarly as the limit $u^\infty$ in \autoref{ShortTimeExistence}, a metric $g^u$ with Gauss curvature $K_{g^u}$ satisfying $K_{g^u}(x)= f_{\lambda}(x)=f(x)+\lambda.$ The unknown $\lambda \in \mathds R$ arises in this context as a Lagrangian multiplier and is a posteriori characterised again by $$ \lambda=\frac1A\left(\bar K- \int_M f \mathrm{e}^{2u}d\mu_{\bar g}\right). $$
In the study of critical points of the restriction of $E_{f}$ to $\mathcal{C}_{1,A}$, it is natural to consider the minimisation problem first. For this we set $$ m_{f,A} = \inf_{u \in \mathcal{C}_{1,A}}E_f(u). $$ We have the following estimates for $m_{f,A}$: \begin{lemma} \label{m-A-est} Let $f \in C^\infty(M)$, $A>0$. Then we have \begin{equation} \label{eq:m-A-first-estimate} m_{f,A} \le \frac{1}{2} \left(\bar K \log(A) - A\int_{M} f d \mu_{\bar g}\right). \end{equation} Moreover, if $\max f \ge 0$, then we have \begin{equation} \label{eq:m-A-second-estimate} \limsup_{A \to \infty} \frac{m_{f,A}}{A} \le 0. \end{equation} \end{lemma}
\begin{proof} Let $u_0(A)\equiv\frac12\log(A)$, so that $\int_M\mathrm{e}^{2u_0(A)}d\mu_{\bar g}=A$. Hence $u_0(A)$ is the (unique) constant function in $\mathcal{C}_{1,A}$, and \begin{align*}
m_{f,A}&\le E_{f}(u_0(A))=\frac12\int_M(|\nabla_{\bar g}u_0(A)|^2_{\bar g}+2\bar Ku_0(A)-f\mathrm{e}^{2u_0(A)})d\mu_{\bar g}\\ &=\frac12\int_M(\bar K\log(A)-f A)d\mu_{\bar g}\\ &= \frac{1}{2} \left(\bar K \log(A) - A\int_{M} f d \mu_{\bar g}\right). \end{align*} This shows \eqref{eq:m-A-first-estimate}. To show \eqref{eq:m-A-second-estimate}, we let $\varepsilon>0$. Since $f \in C^\infty(M)$ and $\max f \ge 0$ by assumption, there exists an open set $\Omega \subset M$ with $f \ge -\varepsilon$ on $\Omega$. Next, let $\psi \in C^\infty(M)$, $\psi \ge 0$, be a function supported in $\Omega$ and with
$\|\psi\|_{L^\infty(M,\bar g)} = 2$. Consequently, the set $\Omega':= \{x \in M\mid \psi >1\}$ is a nonempty open subset of $\Omega$, and therefore $\mu_{\bar g}(\Omega')>0$.
Next we consider the continuous function $$ h: [0,\infty) \to [0,\infty);\quad h(\tau)= \int_M\mathrm{e}^{2 \tau \psi}d\mu_{\bar g} $$ and we note that $h(0)= \int_M d\mu_{\bar g}=1$, and that $$ h(\tau) \ge \int_{\Omega'} \mathrm{e}^{2 \tau \psi}d\mu_{\bar g} \ge \mathrm{e}^{2\tau} \mu_{\bar g}(\Omega') \qquad \text{for $\tau \ge 0$.} $$ Hence for every $A \ge 1$ there exists \begin{equation}
\label{eq:tau-A} 0 \le \tau_A \le \frac{1}{2}\Bigl(\log(A) -\log (\mu_{\bar g}(\Omega'))\Bigr) \end{equation} with $h(\tau_A)=A$ and therefore $\tau_A \psi \in \mathcal{C}_{1,A}$. Consequently, \begin{align*}
m_{f,A} &\le E_f(\tau_A \psi) = \frac12 \int_M(|\nabla_{\bar g}\tau_A \psi|^2_{\bar g}+2\bar K \tau_A \psi-f\mathrm{e}^{2\tau_A \psi})d\mu_{\bar g}\\
&= \tau_A^2 c_1 - \tau_A c_2 -c_3 - \frac{1}{2} \int_{\Omega} f\mathrm{e}^{2\tau_A \psi}d\mu_{\bar g} \end{align*} with $$
c_1 = \frac{1}{2}\int_M |\nabla_{\bar g}\psi|^2_{\bar g}d\mu_{\bar g},\quad c_2 = - \bar K \int_{M} \psi d\mu_{\bar g} \quad \text{and}\quad c_3 = \frac{1}{2} \int_{M \setminus \Omega} f d\mu_{\bar g}. $$ Since $f \ge -\varepsilon$ on $\Omega$, we thus deduce that $$ m_{f,A} \le \tau_A^2 c_1 -2 \tau_A c_2 +c_3 + \frac{\varepsilon}{2} \int_{\Omega} \mathrm{e}^{2\tau_A \psi}d\mu_{\bar g} \le \tau_A^2 c_1 -2 \tau_A c_2 +c_3 +\frac{\varepsilon A}{2}. $$ Since $\frac{\tau_A}{A} \to 0$ as $A \to \infty$ by \eqref{eq:tau-A}, we conclude that $$ \limsup_{A \to \infty} \frac{m_{f,A}}{A} \le \frac{\varepsilon}{2}. $$ Since $\varepsilon>0$ was chosen arbitrarily, \eqref{eq:m-A-second-estimate} follows. \end{proof}
\begin{lemma} \label{lemma-lagrange-multiplier} Let $f \in C^\infty(M)$ nonconstant with $\max_{x\in M} f(x) = 0$. For every $\varepsilon>0$ there exists $\kappa_0>0$ with the following property. If $A \ge \kappa_0$ and $u \in \mathcal{C}_{1,A}$ is a solution of \begin{equation} \label{eq:PCP-lambda} -\Delta_{\bar g}u + \bar K = (f+\lambda) \mathrm{e}^{2u} \end{equation} for some $\lambda \in \mathds R$ with $E_f(u)< \frac{\varepsilon A}{2}$, then we have $\lambda<\varepsilon$. \end{lemma}
\begin{proof} For given $\varepsilon>0$, we may choose $\kappa_0>0$ sufficiently large so that
$\frac{|\bar K|}{2} \frac{\log(A)}{|A|}< \frac{\varepsilon}{2}$ for $A \ge \kappa_0$.
Now, let $A \ge \kappa_0$, and let $u \in \mathcal{C}_{1,A}$ be a solution of \eqref{eq:PCP-lambda} satisfying $E_f(u)< \frac{\varepsilon A}{2}$. Integrating \eqref{eq:PCP-lambda} over $M$ with respect to $\mu_{\bar g}$ and using that $\vol_{\bar g}(M)=1$ and $\int_{M} \mathrm{e}^{2u}d\mu_{\bar g}=A$, we obtain \begin{align*} \lambda &= \frac{1}{A}\left(\bar K - \int_{M} f \mathrm{e}^{2u}d\mu_{\bar g}\right) \le - \frac{1}{A}\int_{M} f \mathrm{e}^{2u}d\mu_{\bar g} \\
&= \frac{1}{A} \left(E_{f}(u)-\frac12\int_M(|\nabla_{\bar g}u|^2_{\bar g}+2\bar K u)d\mu_{\bar g} \right)\le \frac{1}{A}\left( E_f(u) +|\bar K| \bar u\right)\\
&\le \frac{\varepsilon }{2} + \frac{|\bar K|}{2} \frac{\log(A)}{A} <\varepsilon, \end{align*} as claimed. Here we used \eqref{Jensen-consequence} to estimate $\bar u$. \end{proof}
\begin{proposition} \label{local-minimizers-sequence} Let $f \in C^\infty(M)$ be a nonconstant function with $\max_{x\in M} f(x) =0$. Moreover, let $\lambda_n \to 0^+$ for $n\to\infty$, and let $(u_n)_{n\in\mathds N}$ be a sequence of solutions of \begin{equation} \label{eq1:parameter-dep} -\Delta_{\bar g} u_n + \bar K = (f +\lambda_n)\mathrm{e}^{2u_n} \quad \text{in $M$} \end{equation} which are weakly stable in the sense that \begin{equation} \label{eq:weakly-stable}
\int_{M}(|\nabla_{\bar g} h|_{\bar g}^2 -2 (f+\lambda_n) \mathrm{e}^{2u_n} h^2) d \mu_{\bar g} \ge 0 \quad \text{for all $h \in H^1(M)$.} \end{equation} Then $u_n \to u_0$ in $C^2(M)$, where $u_0$ is the unique solution of \begin{equation} \label{eq:parameter-dep-limit} -\Delta_{\bar g} u_0 + \bar K = f \mathrm{e}^{2 u_0} \qquad \text{in $M$.} \end{equation} \end{proposition}
\begin{proof} We only need to show that \begin{equation} \label{eq:sufficient-bounded} \text{$(u_n)_{n\in\mathds N}$ is bounded in $C^{2,\alpha}(M)$ for some $\alpha>0$.} \end{equation} Indeed, assuming this for the moment, we may complete the argument as follows. Suppose by contradiction that there exists $\varepsilon>0$ and a subsequence, also denoted by $(u_n)_{n\in\mathds N}$, with the property that \begin{equation} \label{eq:sufficient-bounded-contradiction}
\|u_n-u_0\|_{C^2(M)} \ge \varepsilon \qquad \text{for all $n \in \mathds N$.} \end{equation} By \eqref{eq:sufficient-bounded} and the compactness of the embedding $C^{2,\alpha}(M) \hookrightarrow C^2(M)$, we may then pass to a subsequence, still denoted by $(u_n)_{n\in\mathds N}$, with $u_n \to u_*$ in $C^2(M)$ for some $u_* \in C^2(M)$. Passing to the limit in \eqref{eq1:parameter-dep}, we then see that $u_*$ is a solution of \eqref{eq:parameter-dep-limit}, which by uniqueness implies that $u_* = u_0$. This contradicts \eqref{eq:sufficient-bounded-contradiction}, and thus the claim follows.
The proof of \eqref{eq:sufficient-bounded} follows by similar arguments as in \cite[p.~1063 f.]{DinLiu95}. Since the framework is slightly different, we sketch the main steps here for the convenience of the reader. We first note that, by the same argument as in \cite[p.~1063 f.]{DinLiu95}, there exists a constant $C_0>0$ with \begin{equation} \label{eq:proof-lower-bound} u_n \ge -C_0 \qquad \text{for all $n$.} \end{equation}
Since $\{f < 0\}$ is a nonempty open subset of $M$ by assumption, we may fix a nonempty open subdomain $\Omega \subset \subset \{f < 0\}$. By \cite[Appendix]{BorGalStr15}, there exists a constant $C_1>0$ with $$
\|u_n^+\|_{H^1(\Omega,\bar g)} \le C_1 \qquad \text{for all $n$} $$ and therefore \begin{equation} \label{eq:v-n-omega-est} \int_{\Omega}\mathrm{e}^{2u_n}d\mu_{\bar g} \le \int_{\Omega}\mathrm{e}^{2u_n^+}d\mu_{\bar g} \le C_2 \qquad \text{for all $n$} \end{equation} for some $C_2>0$ by the Moser--Trudinger inequality. Next, we consider a nontrivial, nonpositive function $h \in C^\infty_c(\Omega) \subset C^\infty(M)$ and the unique solution $w \in C^\infty(M)$ of the equation $$ -\Delta_{\bar g} w + \bar K = h\mathrm{e}^{2w} \quad \text{in $M$.} $$ Moreover, we let $w_n:= u_n - w$, and we note that $w_n$ satisfies $$ -\Delta_{\bar g} w_n +h\mathrm{e}^{2w} = (f+{\lambda_n})\mathrm{e}^{2u_n} \quad \text{in $M$.} $$ Multiplying this equation by $\mathrm{e}^{2w_n}$ and integrating by parts, we obtain \begin{align}
\int_{M}(f+\lambda_n)\mathrm{e}^{2(u_n+w_n)} d\mu_{\bar g} &= \int_{M}\Bigl(-\Delta_{\bar g} w_n + h \mathrm{e}^{2w} \Bigr)\mathrm{e}^{2w_n} d\mu_{\bar g}= \int_{M} \Bigl(2\mathrm{e}^{2w_n} |\nabla_{\bar g}w_n|_{\bar g}^2 +
h \mathrm{e}^{2(w+w_n)} \Bigr) d\mu_{\bar g} \nonumber\\
&= 2 \int_{M}|\nabla_{\bar g} \mathrm{e}^{w_n}|_{\bar g}^2 d \mu_{\bar g} + \int_{\Omega} h \mathrm{e}^{2u_n} d\mu_{\bar g}. \label{eq:proof-dl-1} \end{align} Moreover, applying \eqref{eq:weakly-stable} to $h= \mathrm{e}^{w_n}$ gives \begin{equation} \label{eq:proof-dl-2}
\int_{M}(f+\lambda_n)\mathrm{e}^{2(u_n+w_n)} d\mu_{\bar g} \le \frac{1}{2} \int_{M}|\nabla_{\bar g} \mathrm{e}^{w_n}|_{\bar g}^2 d \mu_{\bar g}. \end{equation} Combining \eqref{eq:v-n-omega-est}, \eqref{eq:proof-dl-1} and \eqref{eq:proof-dl-2} yields \begin{equation}
\label{eq:inverse-poincare}
\|\nabla_{\bar g} \mathrm{e}^{w_n}\|_{L^2(M,\bar g)}^2 \le -\frac{2}{3} \int_{\Omega} h \mathrm{e}^{2u_n} d\mu_{\bar g}
\le \frac{2}{3}\|h\|_{L^\infty(M,\bar g)}C_2 \quad \text{for all $n$.} \end{equation}
Next we claim that also $\|\mathrm{e}^{w_n}\|_{L^2(M,\bar g)}$ remains uniformly bounded. Suppose by contradiction that \begin{equation}
\label{eq:contradiction-dl}
\|\mathrm{e}^{w_n}\|_{L^2(M,\bar g)} \to \infty \qquad \text{as $n \to \infty$.} \end{equation}
We then set $v_n:= \frac{\mathrm{e}^{w_n}}{\|\mathrm{e}^{w_n}\|_{L^2(M,\bar g)}}$, and we note that \begin{equation}
\label{eq:limit-v-n-1}
\|v_n\|_{L^2(M,\bar g)} = 1 \quad \text{for all $n$}\quad \text{and}\quad \|\nabla_{\bar g} v_n\|_{L^2(M,\bar g)}^2 \to 0 \quad \text{as $n \to \infty$} \end{equation} by \eqref{eq:inverse-poincare}. Consequently, we may pass to a subsequence satisfying $v_n \rightharpoonup v$ in $H^1(M,\bar g)$, where $v$ is a constant function with \begin{equation}
\label{eq:norm-limit}
\|v\|_{L^2(M,\bar g)}=1. \end{equation} However, since $$
\|\mathrm{e}^{w_n}\|_{L^2(\Omega,\bar g)} \le \|\mathrm{e}^{u_n}\|_{L^2(\Omega,\bar g)} \|\mathrm{e}^{-w}\|_{L^\infty(\Omega,\bar g)}
\le \sqrt{C_2} \|\mathrm{e}^{-w}\|_{L^\infty(\Omega,\bar g)}\quad \text{for all $n \in \mathds N$} $$ by \eqref{eq:v-n-omega-est} and therefore $$
\|v\|_{L^2(\Omega,\bar g)} = \lim_{n \to \infty}\|v_n\|_{L^2(\Omega,\bar g)}=
\lim_{n \to \infty}\frac{\|\mathrm{e}^{w_n}\|_{L^2(\Omega,\bar g)}}{\|\mathrm{e}^{w_n}\|_{L^2(M,\bar g)}} = 0 $$ by \eqref{eq:contradiction-dl}, we conclude that the constant function $v$ must vanish identically, contradicting \eqref{eq:norm-limit}.
Consequently, $\|\mathrm{e}^{w_n}\|_{L^2(M,\bar g)}$ remains uniformly bounded, which by \eqref{eq:inverse-poincare} implies that $\mathrm{e}^{w_n}$ remains bounded in $H^1(M,\bar g)$ and therefore in $L^p(M,\bar g)$ for any $p < \infty$. Since $\mathrm{e}^{u_n} \le \|\mathrm{e}^{w}\|_{L^\infty(M,\bar g)} \mathrm{e}^{w_n}$ on $M$ for all $n \in \mathds N$, it thus follows that also $\mathrm{e}^{u_n}$ remains bounded in $L^p(M,\bar g)$ for any $p< \infty$. Moreover, by \eqref{eq:proof-lower-bound}, the same applies to the sequence $u_n$ itself. Therefore, applying successively elliptic $L^p$ and Schauder estimates to \eqref{eq1:parameter-dep}, we deduce \eqref{eq:sufficient-bounded}, as required. \end{proof}
\begin{proposition}
\label{sec:stat-minim-probl-2} Let $f \in C^\infty(M)$ be a nonconstant function with $\max_{x\in M} f(x) =0$. Then there exists $\lambda_\sharp$ and a $C^1$-curve $(-\infty,\lambda_{\sharp}] \to C^2(M); \quad \lambda \mapsto u_\lambda$ with the following properties.
\begin{enumerate}
\item[(i)] If $\lambda \le 0$, then $u_\lambda$ is the unique solution of \begin{equation}
\label{eq:parameter-dep-cor} -\Delta_{\bar g} u + \bar K = f_\lambda \mathrm{e}^{2 u} \quad \text{in $M$} \end{equation} and a global minimum of $E_{f_\lambda}$. \item[(ii)] If $\lambda \in (0,\lambda_\sharp]$, then $u_\lambda$ is the unique weakly stable solution of \eqref{eq:parameter-dep-cor} in the sense of \eqref{eq:weakly-stable}, and it is a local minimum of $E_{f_\lambda}$. \item[(iii)] The curve of functions $\lambda \mapsto u_\lambda$ is pointwisely strictly increasing on $M$, and so the volume function
\begin{equation}
\label{eq:def-volume} (-\infty,\lambda_\sharp] \to [0,\infty);\quad \lambda \mapsto V(\lambda):= \int_{M}\mathrm{e}^{2u_\lambda} d\mu_{\bar g}
\end{equation}
is continuous and strictly increasing. \end{enumerate} \end{proposition}
\begin{proof}
We already know that, for $\lambda \le 0$, the energy $E_{f_\lambda}$ admits a strict global minimiser $u_\lambda$ which depends smoothly on $\lambda$. Moreover, by \cite[Proposition 2.4]{BorGalStr15}, the curve $\lambda \mapsto u_\lambda$ can be extended as a $C^1$-curve to an interval $(-\infty,\lambda_{\sharp}]$ for some $\lambda_{\sharp}>0$. We also know from \cite[Proposition 2.4]{BorGalStr15} that, for $\lambda \in (-\infty,\lambda_{\sharp}]$, the solution $u_\lambda$ is strongly stable in the sense that \begin{equation}
\label{eq:stable}
C_\lambda := \inf_{h \in H^1(M,\bar g)} \frac{1}{\|h\|_{H^1(M,\bar g)}^2}\int_{M}\Bigl(|\nabla_{\bar g} h|_{\bar g}^2 -2 f_\lambda \mathrm{e}^{2u_\lambda} h^2\Bigr) d \mu_{\bar g} >0. \end{equation} Here we note that the function $\lambda \mapsto C_\lambda$ is continuous since $u_\lambda$ depends continuously on $\lambda$ with respect to the $C^2$-norm. Next we prove that, after making $\lambda_\sharp>0$ smaller if necessary, the function $u_\lambda$ is the unique weakly stable solution of \eqref{eq:parameter-dep-cor} for $\lambda \in (0,\lambda_{\sharp}]$. Arguing by contradiction, we assume that there exists a sequence $\lambda_n \to 0^+$ and corresponding weakly stable solutions $(u_n)_{n\in\mathds N}$ of \begin{equation}
\label{eq:parameter-dep} -\Delta_{\bar g} u_n + \bar K = (f + \lambda_n)\mathrm{e}^{2u_n} \quad \text{in $M$} \end{equation} with the property that $u_n \not = u_{\lambda_n}$ for every $n \in \mathds N$. By \autoref{local-minimizers-sequence}, we know that $u_n \to u_0$ in $C^2(M)$. Consequently, $v_n := u_n - u_{\lambda_n} \to 0$ in $C^2(M)$ as $n \to \infty$, whereas the functions $v_n$ solve \begin{equation}
\label{eq:parameter-dep-v-n}
-\Delta_{\bar g} v_n = (f + \lambda_n)\bigl(\mathrm{e}^{2u_n}-\mathrm{e}^{2u_{\lambda_n}}\bigr)=
(f + \lambda_n)\mathrm{e}^{2u_{\lambda_n}} \bigl(\mathrm{e}^{2v_n}-1\bigr)
\quad \text{in $M$} \quad \text{for every $n \in \mathds N$.} \end{equation} Combining this fact with \eqref{eq:stable}, we deduce that \begin{align*}
\|v_n\|_{H^1(M,\bar g)}^2 &\le \frac{1}{C_\lambda} \int_{M}\Bigl(|\nabla_{\bar g} v_n|_{\bar g}^2 -2 (f+\lambda_n) \mathrm{e}^{2u_{\lambda_n}} v_n^2\Bigr)d\mu_{\bar g}\\ &= \frac{1}{C_\lambda} \int_{M}(f+\lambda_n) \mathrm{e}^{2u_{\lambda_n}} \bigl(\mathrm{e}^{2v_n}-1- 2v_n\Bigr)v_nd\mu_{\bar g}. \end{align*}
Since $v_n \to 0$ in $C^2(M)$, there exists a constant $C>0$ with $|(\mathrm{e}^{2v_n}-1- 2v_n)v_n| \le C |v_n|^3$ on $M$ for all $n \in \mathds N$, which then implies with H\"older's inequality and \autoref{eindSob} that \begin{align*}
\|v_n\|_{H^1(M,\bar g)}^2&\le C \|(f+\lambda_n) \mathrm{e}^{2u_{\lambda_n}}\|_{L^\infty(M,\bar g)}\|v_n\|_{L^3(M,\bar g)}^3\\
&\le C\left(\int_M|v_n|^{3\cdot\frac43}d\mu_{\bar g}\right)^{\frac34}= C \|v_n\|_{L^4(M,\bar g)}^3\le C \|v_n\|_{H^1(M,\bar g)}^3 \end{align*} with a constant $C>0$ independent on $M$. This contradicts the fact that $v_n \to 0$ in $H^1(M)$ as $n \to \infty$. The claim thus follows.
It remains to prove that the curve of functions $\lambda \mapsto u_\lambda$ is pointwisely strictly increasing on $M$. This is a consequence of the uniqueness of weakly stable solutions stated in (ii) and the fact that, as noted in \cite{DinLiu95}, if $u_{\lambda_0}$ is a solution for some $\lambda_0 \in (-\infty,\lambda_{\sharp}]$, it is possible to construct, via the method of sub- and supersolutions, for every $\lambda < \lambda_0$, a {\em weakly stable} solution $u_\lambda$ with $u_\lambda < u_{\lambda_0}$ everywhere in $M$. \end{proof}
\begin{corollary}
\label{sec:stat-minim-probl-1-cor-1} Let $f \in C^\infty(M)$ be nonconstant with $\max_{x\in M} f (x)= 0$, and let $\lambda_\sharp>0$ be given as in \autoref{sec:stat-minim-probl-2}. Then there exists $\kappa_1>0$ with the following property.
If $A \ge \kappa_1$ and $u \in \mathcal{C}_{1,A}$ is a solution of
\begin{equation}
\label{eq:PCP-lambda-corollary}
-\Delta_{\bar g}u + \bar K = (f+\lambda) \mathrm{e}^{2u}
\end{equation}
for some $\lambda \in \mathds R$ with $E_f(u)< \frac{\lambda_\sharp A}{2}$, then $0< \lambda < \lambda_\sharp$, and $u$ is not a weakly stable solution of \eqref{eq:PCP-lambda-corollary}, so $u \not = u_\lambda$. \end{corollary}
\begin{proof}
Let $\kappa_0>0$ be given as in \autoref{lemma-lagrange-multiplier} for $\varepsilon = \lambda_\sharp>0$. Moreover, let
$$
\kappa_1:= \max \left \{ \kappa_0, V(u_{\lambda_\sharp})\right\}
$$
with $V$ defined in \eqref{eq:def-volume}. Next, let $u \in \mathcal{C}_{1,A}$ be a solution of \eqref{eq:PCP-lambda-corollary} for some $\lambda \in \mathds R$ with $E_f(u)< \frac{\lambda_\sharp A}{2}$. From \autoref{lemma-lagrange-multiplier}, we then deduce that $0< \lambda < \lambda_\sharp$, and by \autoref{sec:stat-minim-probl-2} (iii) we have $u \not = u_\lambda$. Since $u_\lambda$ is the unique weakly stable solution of \eqref{eq:PCP-lambda-corollary}, it follows that $u$ is not weakly stable. \end{proof}
\begin{corollary}
\label{sec:stat-minim-probl-1-cor-2}
Let $p>2$, $f \in C^\infty(M)$ be nonconstant with $\max_{x\in M} f (x)= 0$, and let $\lambda_\sharp>0$ be given as in \autoref{sec:stat-minim-probl-2}. Then there exists $\kappa>0$ with the property that for every $A \ge \kappa$ the set \[ \tilde{\mathcal{C}}:=\left\{u_0\in\mathcal{C}_{1,A}\cap W^{2,p}(M,\bar g)\mid E_f(u_0)<\frac{\lambda_\sharp A}{2}\right\} \] is nonempty, and for every $u_0\in\tilde{\mathcal{C}}$ the global solution $u \in C([0,\infty);C(M))\cap C([0,\infty);H^1(M,\bar g))\cap C^\infty((0,\infty)\times M)$ of the initial value problem \eqref{PCFVolNew1}, \eqref{PCFVolNew2} converges, as $t \to \infty$ suitably, to a solution $u^\infty$ of the static problem \eqref{eq:PCP-lambda-corollary} for some $\lambda \in (0,\lambda_\sharp)$ which is not weakly stable and hence no local minimiser of $E_{f_\lambda}$. \end{corollary}
\begin{proof}
Let $\kappa_1>0$ be given by \autoref{sec:stat-minim-probl-1-cor-1}. By \eqref{eq:m-A-second-estimate}, there exists $\kappa\ge\kappa_1>0$ with $m_{f,A}<\frac{\lambda_\sharp A}{4}$ for fixed $A>\kappa$. Consequently, there exists $u_0 \in \mathcal{C}_{1,A}\cap W^{2,p}(M,\bar g)$ with $E_f(u_0)< \frac{\lambda_\sharp A}{2}$. By \autoref{ShortTimeExistence}, the global solution $u \in C([0,\infty);C(M))\cap C([0,\infty);H^1(M,\bar g))\cap C^\infty((0,\infty)\times M)$ of the initial value problem \eqref{PCFVolNew1}, \eqref{PCFVolNew2} converges, as $t \to \infty$ suitably, to a solution $u^\infty \in \mathcal{C}_{1,A}$ of the static problem \eqref{eq:PCP-lambda-corollary} for some $\lambda \in \mathds R$, whereas $E_f(u^\infty) \le E_f(u_0) < \frac{\lambda_\sharp A}{2}$. Consequently, $\lambda \in (0,\lambda_\sharp)$ by \autoref{sec:stat-minim-probl-1-cor-1}, and $u^\infty$ is not weakly stable. \end{proof}
\section{Proof of the Main Results} \label{sec:proof-main-results}
\subsection{Notation and Some Regularity Results} In this chapter we summarise different kind of estimates which will be useful later. In the following, for $T>0$ we use the notation \[L^p_tL^r_x:=L^p([0,T];L^r(M,\bar g))\quad\text{and}\quad L^p_tH^q_x:=L^p([0,T];H^q(M,\bar g)).\]
A first regularity result is therefore given by \autoref{eindSob}. \begin{remark}\label{kleineReg} We have
\[\|\theta\|^4_{L^p_tL^4_x}\le C_{\text{GNL}}\|\theta\|^2_{L^p_tL^2_x}\|\theta\|^2_{L^p_tH^1_x}\] for $\theta\in L^p_tH^1_x$ with $p\in[1,\infty]$. \end{remark}
\begin{lemma}[Sobolev inequality]\label{SobolevLemma} There exists a constant $C_S>0$ such that for every $\rho\in L^\infty_t H^1_x$, $T\le1$, we have \begin{equation}\label{Sobolev}
\|\rho\|^2_{L^4_tL^4_x}\le C_S(\|\rho\|^2_{L^\infty_tL^2_x}+\|\nabla_{\bar g}\rho\|^2_{L^2_tL^2_x})<\infty. \end{equation} \end{lemma}
\begin{proof} With \autoref{eindSob} there exists a constant $C_{\text{GNL}}>0$ such that we have for all $T\le1$ \begin{align*}
\|\rho\|^4_{L^4_tL^4_x}&=\int_0^T\|\rho(t)\|^4_{L^4(M,\bar g)}dt \le C_{\text{GNL}}\int_0^T\|\rho(t)\|^2_{L^2(M,\bar g)}\|\rho(t)\|^2_{H^1(M,\bar g)}dt\\
&\le C_{\text{GNL}}\|\rho\|^2_{L^\infty_tL^2_x}\int_0^T(\|\rho(t)\|_{L^2(M,\bar g)}^2+\|\nabla_{\bar g}\rho(t)\|^2_{L^2(M,\bar g)})dt\\
&\le C_{\text{GNL}}\cdot T\:\|\rho\|^4_{L^\infty_tL^2_x}+C_{\text{GNL}}\|\rho\|^2_{L^\infty_tL^2_x}\|\nabla_{\bar g}\rho\|^2_{L^2_tL^2_x}\\
&\le C_{\text{GNL}}\left(\|\rho\|^4_{L^\infty_tL^2_x}+\|\rho\|^2_{L^\infty_tL^2_x}\|\nabla_{\bar g}\rho\|^2_{L^2_tL^2_x}\right). \end{align*} By using Young's inequality we have
\[\|\rho\|_{L^\infty_tL^2_x}\|\nabla_{\bar g}\rho\|_{L^2_tL^2_x}\le \frac12\left(\|\rho\|^2_{L^\infty_tL^2_x}+\|\nabla_{\bar g}\rho\|^2_{L^2_tL^2_x}\right)\] and therefore \begin{align*}
\|\rho\|^2_{L^4_tL^4_x}
&\le C_{\text{GNL}}^{\frac12}\sqrt{\|\rho\|^4_{L^\infty_tL^2_x}+\frac14(\|\rho\|^2_{L^\infty_tL^2_x}+\|\nabla_{\bar g}\rho\|^2_{L^2_tL^2_x})^2}\\
& \le C_{\text{GNL}}^{\frac12} (\|\rho\|^2_{L^\infty_tL^2_x}+\frac12\|\rho\|^2_{L^\infty_tL^2_x}+\frac12\|\nabla_{\bar g}\rho\|^2_{L^2_tL^2_x})\\
&\le \frac32 C_{\text{GNL}}^{\frac12}(\|\rho\|^2_{L^\infty_tL^2_x}+\|\nabla_{\bar g}\rho\|^2_{L^2_tL^2_x})\\
&=: C_S(\|\rho\|^2_{L^\infty_tL^2_x}+\|\nabla_{\bar g}\rho\|^2_{L^2_tL^2_x}). \end{align*} Since $T$ is finite, $\rho\in L^\infty_tH^1_x$ implies that $\rho\in L^p_tH^1_x$ for all $p\in[1,\infty]$ which shows that the upper bound is finite. \end{proof}
Furthermore, since $T<\infty$ and $\vol_{\bar g}=1$, with \autoref{Onofri-ungleichung} we also have for every $p,s\in[1,\infty]$ that $L^q_tL^r_x\subset L^s_tL^p_x$ for $q\ge s$, $r\ge p$.
Since we will often use it in the following, we recall that for $v\in C_tC_x:=C([0,T],C(M))$ we have \begin{equation}\label{Exp1Exp2}
\|1-\mathrm{e}^v\|^2_{L^\infty_tL^\infty_x}\le\mathrm{e}^{2\|v\|_{L^\infty_tL^\infty_x}}\|v\|^2_{L^\infty_tL^\infty_x} \end{equation} since for $x\in\mathds R$ we get with the Taylor expansion \begin{equation}\label{EstimateExp}
|\mathrm{e}^x-1|=|1-\mathrm{e}^x|\le |x|\mathrm{e}^{|x|}. \end{equation}
\begin{lemma}\label{Properties2} With \autoref{Onofri-ungleichung} we get the following statements: \begin{enumerate} \item For a (sufficiently smooth) solution $u$ of \eqref{PCFVolNew1}, \eqref{PCFVolNew2} we have \begin{equation}\label{LowerEstimateBarU}\bar u(t)\ge\frac12\log\left(\frac{A}{C_{\text{up}}}\right)=:m_0(A, E_{f}(u_0), f, C_{\text{MT}},\eta_1), \end{equation}
with $C_{\text{up}}=C_{\text{MT}}\exp(4\eta_1(2E_{f}(u_0)+|\bar K|\log(A)+A\max_{x\in M}f(x)))$ where $\eta_1$ is a number determined by \autoref{Onofri-ungleichung}. So, especially for a solution $u$ of \eqref{PCFVolNew1}, \eqref{PCFVolNew2} we have the uniform bound \begin{equation}\label{UniBarU} m_0\le \bar u(t)\le\frac12\log(A), \end{equation} where we used \eqref{Jensen-consequence} and the volume preserving property to get the upper bound of $\bar u(t)$. \item For a solution $u$ of \eqref{PCFVolNew1}, \eqref{PCFVolNew2} we have for all $p\in\mathds R$ that \begin{equation}\label{Uniformexp} \int_M \mathrm{e}^{2pu(t)}d\mu_{\bar g}\le C_{\text{int}}(A, C_{\text{MT}},E_{f}(u_0),f,\bar K,\eta_1,\eta_2,p), \end{equation} where again, $\eta_1$, $\eta_2$ are numbers determined by \autoref{Onofri-ungleichung}. \item For this part we choose $f=f_0$ where $f_0\le0$ is a nonconstant, smooth function with $\max_{x\in M}f_0(x)=0$. Then there exists a constant $C_{\text{low}}=C_{\text{low}}(C_{\text{int}},f_0)>0$ such that \begin{equation}
\int_M|f_0|d\mu_{g(t)}\ge C_{\text{low}}. \end{equation} \end{enumerate} \end{lemma}
\begin{proof} \begin{enumerate} \item Let $u$ be a solution of \eqref{PCFVolNew1}, \eqref{PCFVolNew2}. We then know that $u(t)\in \mathcal{C}_A$. So, with \eqref{APrioriBound} we have for all $t\ge0$ that \begin{equation}\label{EstimateNabla1} \begin{split}
\|\nabla_{\bar g}u(t)\|^2_{L^2(M,\bar g)}&=2E_{f}(u(t))-\int_M(2\bar Ku(t)-f\mathrm{e}^{2u(t)})d\mu_{\bar g}\\
&=2E_{f}(u(t))+\int_M(2|\bar K|u(t)+f\mathrm{e}^{2u(t)})d\mu_{\bar g}\\
&\le2E_{f}(u_0)+|\bar K|\log(A)+A\max_{x\in M}f(x), \end{split} \end{equation} where we used the fact that $\int_M 2u(t)d\mu_{\bar g}\le\log(A)$ by \eqref{Jensen-consequence} and since $\int_M\mathrm{e}^{2u(t)}d\mu_{\bar g}\equiv A$. With this and \autoref{Onofri-ungleichung} we can now estimate \begin{align*} A&=\int_M\mathrm{e}^{2u(t)}d\mu_{\bar g}=\mathrm{e}^{2\bar u(t)}\int_M\mathrm{e}^{2(u(t)-\bar u(t))}d\mu_{\bar g}\\
&\le\mathrm{e}^{2\bar u(t)}C_{\text{MT}}\exp(\eta_1\|\nabla_{\bar g}(2u(t))\|^2_{L^2(M,\bar g)})\\
&\le\mathrm{e}^{2\bar u(t)}C_{\text{MT}}\exp(4\eta_1(2E_{f}(u_0)+|\bar K|\log(A)+A\max_{x\in M}f(x)))\\ &=:C_{\text{up}}\mathrm{e}^{2\bar u(t)}, \end{align*} with $C_{\text{up}}=C_{\text{up}}(A,C_{\text{MT}},E_{f}(u_0),f,\bar K,\eta_1)>0$ and therefore \[\bar u(t)\ge\frac12\log\left(\frac{A}{C_{\text{up}}}\right)=:m_0(A,C_{\text{MT}},E_{f}(u_0),f,\bar K,\eta_1)\in \mathds R.\] So, for a solution $u(t)\in \mathcal{C}_A$ of \eqref{PCFVolNew1}, \eqref{PCFVolNew2} we get the uniform bound \[m_0\le\bar u(t)\le\frac12\log(A).\] \item Let $u$ be a solution of \eqref{PCFVolNew1}, \eqref{PCFVolNew2}. So, $u(t)\in \mathcal{C}_A$. With \autoref{Onofri-ungleichung}, \eqref{UniBarU}, and \eqref{EstimateNabla1} we directly get for any $p\in\mathds R$ that \begin{equation}\label{C_int} \begin{split} \int_M\mathrm{e}^{2pu(t)}d\mu_{\bar g}&=\mathrm{e}^{2p\bar u(t)}\int_M\mathrm{e}^{2p(u(t)-\bar u(t))}d\mu_{\bar g}\\
&\le\mathrm{e}^{2p\bar u(t)}C_{\text{MT}}\exp(4\eta_2p^2\|\nabla_{\bar g}u(t)\|^2_{L^2(M,\bar g)})\\ &\le C_{\text{int}}, \end{split} \end{equation} where $C_{\text{int}}=C_{\text{int}}(A, C_{\text{MT}},E_{f}(u_0),f, \bar K, \eta_1,\eta_2,p)>0$. \item Similar to \cite[Lemma 2.3]{Str20} we see by the choice of $f_0$, H\"older's inequality, and \eqref{C_int} that \begin{equation}\label{AbschInt}
0<\left|\int_M\sqrt{|f_0|}d\mu_{\bar g}\right|^2\le\int_M|f_0|\mathrm{e}^{2u(t)}d\mu_{\bar g}\int_M\mathrm{e}^{-2u(t)}d\mu_{\bar g}\le C_{\text{int}}\int_M|f_0|\mathrm{e}^{2u(t)}d\mu_{\bar g} \end{equation} which shows the claim. \end{enumerate} So, \autoref{Properties2} is proven. \end{proof}
Now we can turn to the proofs of the main results.
\subsection{Short-Time Existence}\label{SectionShortTimeExistence} Let $A>0$. We are looking for a short-time solution of \eqref{PCFVolNew1} with initial data \eqref{PCFVolNew2}. Using the Gauss equation \eqref{GaussEquation} we can rewrite \eqref{PCFVolNew1}, \eqref{PCFVolNew2} in the following way: \begin{align} \partial_tu(t)&=f-K_{g(t)}-\alpha_A(t)\nonumber\\ &=\mathrm{e}^{-2u(t)}\Delta_{\bar g}u(t)+\bar K\left(\frac1A-\mathrm{e}^{-2u(t)}\right)+f-\frac1A\int_Mf\mathrm{e}^{2u(t)} d\mu_{\bar g}; \label{HG1}\\ u(0)&=u_0\in \mathcal{C}_{p,A}:=\left\{u\in W^{2,p}(M,\bar g)\mid \int_M\mathrm{e}^{2u}=A\right\},\label{HG2} \end{align} with $p>2$, where \[\alpha_A(t)=\frac{1}{A}\left(\int_M fd\mu_{g(t)}-\bar K\right).\] To find a solution of \eqref{HG1}, \eqref{HG2}, we consider the linear equation \begin{align} \partial_tu(t)&=\mathrm{e}^{-2v(t)}\Delta_{\bar g}u(t)+\bar K\left(\frac1A-\mathrm{e}^{-2v(t)}\right)+f-\frac1A\int_Mf \mathrm{e}^{2v(t)}d\mu_{\bar g}; \label{LineareHG1}\\ u(0)&=u_0\in\mathcal{C}_{p,A},\label{LineareHG2} \end{align}
and use a fixed point argument in the space $(X,\|\cdot\|_X):=(C_tC_x,\|\cdot\|_{C_tC_x})$. First we observe that for $v\in C_tC_x$, equation \eqref{LineareHG1} is strongly parabolic. Furthermoren, with $p>2$ and the fact that $M$ is compact, we have $u_0\in \mathcal{C}_{p,A}\subset H^2(M,\bar g)$, and therefore $u_0\in L^\infty(M,\bar g)$.
For the fixed point argument we fix $R=R(u_0):=\|u_0\|_{L^\infty(M,\bar g)}+1$. For fixed $T>0$, let \[ X=C_tC_x=C([0,T], C(M,\bar g))\hookrightarrow L^\infty_tL^\infty_x\] with
\[\|u\|_X=\max_{t\in[0,T],\: x\in M}|u(x,t)|.\]
For $v\in X$, by \cite[Theorem 7.32]{Lie96} and the appendix, we get a unique solution $u_v\in W^{2,1}_p=W^{1,p}_tL^p_x\cap L^p_tW^{2,p}_x$ of \eqref{LineareHG1}, \eqref{LineareHG2} for $t\in [0,T]$, $x\in M$. On $X_R=\{ U\in X\mid \| U\|_X\le R\}$, we now define the function $\Phi$ as follows: for $v\in X_R$, let $\Phi(v)=:u_v$ be the unique solution of \eqref{LineareHG1}, \eqref{LineareHG2}. First, we want to show that $\Phi:X_R\to X_R$ if $T>0$ is chosen small enough.
\begin{lemma} If $T>0$ is fixed with
\begin{equation}
\label{eq:T-condition}
T \le \left( |\bar K|\mathrm{e}^{2(\|u_0\|_{L^\infty(M,\bar g)}+1)}+\|f\|_{L^\infty(M,\bar g)}\left(1+ \frac{\mathrm{e}^{2(\|u_0\|_{L^\infty(M,\bar g)}+1)}}{A}\right) \right)^{-1}
\end{equation} and $v\in X_R$, then $\Phi(v)\in X_R$. \end{lemma}
\begin{proof} With \autoref{max-principle} (ii) we directly get \begin{equation}\label{BoundUv}
\|\Phi(v)\|_{L^\infty_tL^\infty_x} = \|u_v\|_{L^\infty_tL^\infty_x} \le \|u_0^+\|_{L^\infty(M,\bar g)}+T d_T \end{equation} where \begin{align*}
d_T &\le |\bar K|\mathrm{e}^{2\|v\|_{L^\infty_tL^\infty_x}}+\|f\|_{L^\infty(M,\bar g)}+\frac{\|f\|_{L^\infty(M,\bar g)}\mathrm{e}^{2\|v\|_{L^\infty_tL^\infty_x}}}{A}\\
&\le |\bar K|\mathrm{e}^{2R}+\|f\|_{L^\infty(M,\bar g)}\left(1+ \frac{\mathrm{e}^{2R}}{A}\right),\\ \end{align*} hence \begin{align*}
\|\Phi(v)\|_{L^\infty_tL^\infty_x} & \le T\left(|\bar K|\mathrm{e}^{2R}+\|f\|_{L^\infty(M,\bar g)}\left(1+ \frac{\mathrm{e}^{2R}}{A}\right)\right)+\|u^+_0\|_{L^\infty(M,\bar g)}\\
&\le 1+ \|u_0\|_{L^\infty(M,\bar g)} = R, \end{align*}
by (\ref{eq:T-condition}) and since $R=\|u_0\|_{L^\infty(M,\bar g)}+1$, which shows the claim. \end{proof}
We now use Schauder's fixed point Theorem \cite{Schauder1930} to show the following proposition. \begin{proposition}\label{existence}
If $u_0 \in \mathcal{C}_{p,A}\subset W^{2,p}(M,\bar g)$ and $T>0$ is fixed with \eqref{eq:T-condition}, then there exists a short-time solution $u \in X \cap C^{\infty}(M \times (0,T))$ of \eqref{HG1}, \eqref{HG2}.\\
Moreover, any such solution satisfies $u \in C([0,T), H^1(M,\bar g))$. \end{proposition}
\begin{proof}
{\bf Step 1:} First we recall Schauder's Theorem: It asserts that if $H$ is a nonempty, convex, and closed subset of a Banach space $B$ and $F$ is a continuous mapping of $H$ into itself such that $F(H)$ is a relatively compact subset of $H$, then $F$ has a fixed point.
In our case, $B\hat{=}X=C([0,T];C(M,\bar g))$, $H\hat{=}X_R=\{u\in X\mid \|u\|_X=\|u\|_{C_tC_x}\le R\}$, and $F\hat{=}\Phi$. So to show the existence of a fixed point of $\Phi$ in $X_R$, it remains to show that \begin{enumerate} \item $\Phi: X_R\to X_R$ ist continuous and \item $\Phi(X_R)\subset X_R$ is relatively compact. \end{enumerate}
In a first step we show that $\Phi:X_R\to X_R$ ist continuous. For this, let $(v_n)_{n\in\mathds N}\subset X_R$ be a sequence with $\|v_n-v\|_X\to0$ for $n\to\infty$ with $v\in X_R$. With \autoref{sec:appendix} we know that for all $v_n$ there exists $u_n\in W^{2,1}_p$, $p>2$, which is the unique solution of \eqref{LineareHG1}, \eqref{LineareHG2} such that
\[\|u_n\|_{W^{2,1}_p}\le C(\|u_0\|_{W^{2,p}(M,\bar g)}+\|d_n\|_{L^p_tL^p_x})\] with \[d_n(t):=\bar K\left(\frac1A-\mathrm{e}^{-2v_n(t)}\right)+f-\frac1A\int_Mf\mathrm{e}^{2v_n(t)}d\mu_{\bar g}.\] Since $v_n\to v$ in $C_tC_x$ and therefore $v_n\to v$ in $L^\infty_tL^\infty_x$, we know that $v_n\to v$ in $L^p_tL^p_x$ for all $p$. Furthermore, since the exponential map is continuous, we have $\mathrm{e}^{\pm2v_n}\to\mathrm{e}^{\pm2v}$ in $L^p_tL^p_x$ for all $p$, and therefore $d_n\to d$ in $L^p_tL^p_x$ for all $p$.
Hence, for every $\varepsilon>0$ there exist $N_V, N_d\in\mathds N$ such that
\[\|v_n-v\|_{L^p_tL^p_x}<\varepsilon\quad\text{for all }n\ge N\quad\quad\text{and}\quad\quad \|d_n-d\|_{L^p_tL^p_x}<\varepsilon\quad\text{for all }n\ge N,\] with $N:=\max\{N_V, N_d\}$.
Furthermore we have the estimate \begin{align*}
\|\mathrm{e}^{2v_n}-\mathrm{e}^{2v}\|_{L^\infty_tL^\infty_x}&=\|(\mathrm{e}^{2v_n-2v}-1)\mathrm{e}^{2v}\|_{L^\infty_tL^\infty_x}\le \|\mathrm{e}^{2v_n-2v}-1\|_{L^\infty_tL^\infty_x}\|\mathrm{e}^{2v}\|_{L^\infty_tL^\infty_x}\\
&\le\|2v_n-2v\|\mathrm{e}^{\|2V_n-2V\|_{L^\infty_tL^\infty_x}}\|\mathrm{e}^{2v}\|_{L^\infty_tL^\infty_x}<2\varepsilon\mathrm{e}^{2\varepsilon} \mathrm{e}^{2R}, \end{align*}
and similarly $\|\mathrm{e}^{-2v_n}-\mathrm{e}^{-2v}\|_{L^\infty_tL^\infty_x}<2\varepsilon\mathrm{e}^{2\varepsilon} \mathrm{e}^{2R}$.
Considering now the difference $u_n-u$, where $u_n=\Phi(v_n)$ and $u=\Phi(v)$, we see that $u_n-u$ fulfils the equation \begin{align*} \partial_t(u_n-u)(t)&=\mathrm{e}^{-2v_n(t)}\Delta_{\bar g}u_n(t)+d_n(t)-\mathrm{e}^{-2v(t)}\Delta_{\bar g}u(t)- d(t)\\ &=\mathrm{e}^{-2v_n(t)}\Delta_{\bar g}(u_n-u)(t)+(\mathrm{e}^{-2v_n(t)}-\mathrm{e}^{-2v(t)})\Delta_{\bar g}u(t)+d_n(t)-d(t) \end{align*} with \begin{align*}
\|u_n-u\|_{W^{2,1}_p}&\le C\|(\mathrm{e}^{-2v_n}-\mathrm{e}^{-2v})\Delta_{\bar g}u+d_n-d\|_{L^p_tL^p_x}\\
&\le C\left(\|\mathrm{e}^{-2v_n}-\mathrm{e}^{-2v}\|_{L^\infty_tL^\infty_x}\|\Delta_{\bar g}u\|_{L^p_tL^p_x}+\|d_n-d\|_{L^p_tL^p_x}\right)\\
&\le C(2\varepsilon\mathrm{e}^{2\varepsilon}\mathrm{e}^{2R}\|\Delta_{\bar g}u\|_{L^p_tL^p_x}+\varepsilon)\quad\text{for } n\ge N. \end{align*}
Since $\|\Delta_{\bar g}u\|_{L^p_tL^p_x}$ is finite and $\varepsilon>0$ was arbitrary, we see that $\|\Phi(v_n)-\Phi(v)\|_{W^{2,1}_p}\to0$ for $n\to\infty$. So, we get
\[\|\Phi(v_n)-\Phi(v)\|_{X}\le C\|\Phi(v_n)-\Phi(v)\|_{C^\alpha}\le C\|\Phi(v_n)-\Phi(v)\|_{W^{2,1}_p}\to 0\quad\text{for }n\to\infty\] which shows the continuity of $\Phi:X_R\to X_R$.
In a second step we show that $\Phi(X_R)$ is relatively compact. For this let $(u_n)_{n\in\mathds N}\subset \Phi(X_R)$ be an arbitrary sequence in the image of $\Phi$. So, again with \autoref{sec:appendix}, we see that for every $u_n\in \Phi(X_R)$ there exists a $v_n\in X_R$ with $\Phi(v_n)=u_n$ such that \begin{align*}
\|u_n\|_{W^{2,1}_p}&\le C(\|u_0\|_{W^{2,p}(M,\bar g)}+\|d_n\|_{L^p_tL^p_x})\\ &\le C
\left(\|u_0\|_{W^{2,p}(M,\bar g)}+\frac{T|\bar K|}{A}+\|\bar K\mathrm{e}^{-2v_n}\|_{L^p_tL^p_x}+\|f\|_{L^p_tL^p_x}+\left\|\frac1A\int_Mf\mathrm{e}^{2v_n}d\mu_{\bar g}\right\|_{L^p_tL^p_x}\right)\\
&\le C\left(\|u_0\|_{W^{2,p}(M,\bar g)}+\frac{T|\bar K|}{A}+|\bar K|\mathrm{e}^{2R}+T\|f\|_{L^\infty(M,\bar g)}+\frac{T}A\|f\|_{L^\infty(M,\bar g)}\mathrm{e}^{2R}\right)\\ &\le C(A, f, \bar K, R, T,u_0)=:C_d. \end{align*} So, $(u_n)_{n\in\mathds N}$ is uniformly bounded in $W^{2,1}_p((0,T)\times M)$. Using now that $W^{2,1}_p((0,T)\times M)$ is continuously embedded in $C^\alpha([0,T]\times M)$ for some $0<\alpha<1$ and this on the other hand is compactly embedded in $C^\beta([0,T]\times M)$ for some $0<\beta<\alpha<1$ we can conclude the claim.\\ We have thus proved that $\Phi$ has a fixed point $u$ in $X_R$, which then is a (strong) solution $u \in W^{2,1}_p((0,T)\times M)$ of \eqref{HG1}, \eqref{HG2}.
{\bf Step 2:} We now show that $u \in C^\infty(M \times (0,T))$. To see this, we first note the trivial fact that $u \in W^{2,1}_p((0,T)\times M)$ is a strong solution of \eqref{LineareHG1}, \eqref{LineareHG2} with $v = u$. Since then $v \in W^{2,1}_p((0,T)\times M) \subset C^\alpha([0,T]\times M)$, \cite[Theorems 5.9 and 5.10]{Lie96} imply the existence of a classical solution $\tilde u \in X \cap C^{2+\alpha',1+\alpha'}_{loc}((0,T)\times M)$ of \eqref{LineareHG1}, \eqref{LineareHG2} with $v = u$ for some $\alpha'>0$. Here $C^{2+\alpha',1+\alpha'}_{loc}((0,T)\times M)$ denotes the space of functions $f \in C^{2,1}((0,T)\times M)$ with the property that $\partial_t f$ and all derivatives up to second order of $f$ with respect to $x \in M$ are locally $\alpha'$-H\"older continuous. In particular, $\tilde u \in W^{2,1}_p((\varepsilon,T-\varepsilon)\times M)$ for $\varepsilon \in (0,T)$. The function $w:= u- \tilde u \in W^{2,1}_p((\varepsilon,T-\varepsilon)\times M)$ is then a strong solution of the initial value problem $$ \partial_t w(t)=\mathrm{e}^{-2v(t)}\Delta_{\bar g}w(t) \quad \text{for $t \in (\varepsilon,T-\varepsilon)$}, \qquad w(\varepsilon)= u(\varepsilon,\cdot)-\tilde u(\varepsilon,\cdot). $$
By \autoref{max-principle} (ii) we then have $|w| \le \|u(\varepsilon,\cdot)-\tilde u(\varepsilon,\cdot)\|_{L^\infty(M,\bar g)}$ on $(\varepsilon,T-\varepsilon)\times M$, whereas $\|u(\varepsilon,\cdot)-\tilde u(\varepsilon,\cdot)\|_{L^\infty(M,\bar g)} \to 0$ as $\varepsilon \to 0$ by the continuity of $u$ and $\tilde u$. It thus follows that $u \equiv \tilde u$ on $(0,T)\times M)$, and therefore $u \in C^{2+\alpha',1+\alpha'}_{loc}((0,T)\times M)$. Since $u$ solves \eqref{LineareHG1}, \eqref{LineareHG2} with $v = u \in C^{2+\alpha',1+\alpha'}_{loc}((0,T)\times M)$, we can apply \cite[Theorems 5.9]{Lie96} and the above argument again to get $u \in C^{4+\alpha'',2+\alpha''}_{loc}((0,T)\times M)$ for some $\alpha''>0$. Repeating this argument inductively, we get $u \in C^{k,\frac{k}{2}}_{loc}((0,T)\times M)$ for every $k>0$, and hence $u \in C^\infty(M \times (0,T))$.\\
{\bf Step 3:} It remains to show that any solution $u \in X \cap C^{\infty}((0,T)\times M)$ of \eqref{HG1}, \eqref{HG2} also satisfies $u \in C([0,T), H^1(M,\bar g))$. Since $u \in C^{\infty}((0,T)\times M)$, only the continuity in $t=0$ needs to be proved. Setting $\phi(t)= \|u(t)\|_{H^1(M,\bar g)}^2$ for $t \in (0,T)$, we see that \begin{align*}
\frac{1}{2}(\phi(t_2)-\phi(t_1)) &= \frac{1}{2} \int_{t_1}^{t_2}\partial_t \|u(t)\|_{H^1(M,\bar g)}^2\,dt =\int_{t_1}^{t_2}\int_{M} \Bigl(u(t) \partial_t u(t) + \nabla u(t) \nabla \partial_t u(t)\Bigr)d\mu_{\bar g}dt \\ &=\int_{t_1}^{t_2}\int_{M} \Bigl(u(t) \partial_t u(t) - [\Delta u(t)] \partial_t u(t)\Bigr)d\mu_{\bar g}dt \end{align*} and therefore, by H\"older's inequality, \begin{align*}
\frac{1}{2}|\phi(t_2)-\phi(t_1)|&\le \int_{t_1}^{t_2} \int_{M}\bigl(|u||\partial_t u|+ |\Delta u||\partial_t u|\bigr)d\mu_{\bar g}dt\\
&\le C \|\partial_t u\|_{L^p((0,T) \times M)} \bigl(\|u\|_{L^p((0,T) \times M)} + \|\Delta u\|_{L^p((0,T) \times M)} \bigr) (t_2-t_1)^\beta\\
&\le C \|u\|_{W^{1,2}_p((0,T) \times M)}(t_2-t_1)^\beta, \end{align*} for $0<t_1<t_2<T$ with some $\beta >0$ depending on $p>2$, which implies that the function $\phi$ is uniformly continuous and therefore bounded on $(0,T)$.
We now assume by contradiction that $u$ is not continuous at $t=0$ with respect to the $H^1(M,\bar g)$-norm. Then there exists a sequence $(t_n)_{n\in\mathds N}$ in $(0,T)$ and $\varepsilon>0$ with $t_n \to 0^+$ as $n \to \infty$ and \begin{equation}
\label{eq:contradiction-inequality}
\|u(t_n)-u_0\|_{{H^1(M,\bar g)}} \ge \varepsilon \qquad \text{for all $n \in \mathds N$.} \end{equation}
Since $\|u(t_n)\|_{H^1(M,\bar g)}^2 = \phi(t_n)$ remains bounded as $n \to \infty$, we conclude that, passing to a subsequence, the sequence $u(t_n)$ converges weakly in $H^1(M,\bar g)$ and therefore strongly in $L^2(M, \bar g)$. Since the strong $L^2$-limit of $u(t_n)$ must be $u_0=u(0)$ as a consequence of the fact that $u \in X$, we deduce that $u(t_n) \rightharpoonup u_0$ weakly in ${H^1(M,\bar g)}$ as $n \to \infty$. Combining this information with \autoref{sec:appendix} from the appendix, we deduce that \begin{equation} \label{eq:limsup-liminf-ineq}
\limsup_{n \to \infty}\|u(t_n)\|_{H^1(M,\bar g)}^2 \le \|u_0\|_{H^1(M,\bar g)}^2 \le \liminf_{n \to \infty}\|u(t_n)\|_{H^1(M,\bar g)}^2 \end{equation}
and therefore $\|u(t_n)\|_{H^1(M,\bar g)} \to \|u_0\|_{H^1(M,\bar g)}$. Note here that this part of \autoref{sec:appendix} applies since $u$ solves \eqref{LineareHG1}, \eqref{LineareHG2} with $v = u \in W^{2,1}_p((0,T)\times M) \subset C^\alpha([0,T]\times M)$ for some $\alpha>0$. From (\ref{eq:limsup-liminf-ineq}) and the uniform convexity of the Hilbert space ${H^1(M,\bar g)}$, we conclude that $u(t_n) \to u_0$ strongly in $H^1(M,\bar g)$, contrary to (\ref{eq:contradiction-inequality}). \end{proof}
\subsection{Uniqueness}\label{Uniqueness} We now show that the solution from \autoref{existence} is unique. \begin{lemma} \label{lemma-uniqueness-nonlinear-short-time} Let $u_0 \in W^{2,p}(M,\bar g)$, $p>2$, and $T>0$ be fixed with \eqref{eq:T-condition}. Then the short-time solution of $u \in X \cap C^{\infty}(M \times (0,T))$ of \eqref{HG1}, \eqref{HG2} given by \autoref{existence} is unique. \end{lemma}
\begin{proof} Let $u_1, u_2 \in X \cap C^{\infty}(M \times (0,T))$ be two solutions of \eqref{HG1}, \eqref{HG2}. The difference $u:= u_1-u_2 \in X \cap C^{\infty}(M \times (0,T))$ then fulfils \begin{equation}\label{diff-equation-uniqueness} \begin{split} \partial_t u(t)&=\mathrm{e}^{-2u_1(t)}\Delta_{\bar g}u_1(t)-\mathrm{e}^{-2u_2(t)}\Delta_{\bar g}u_2(t)\\
&\phantom{aaaaa}-\bar K(\mathrm{e}^{-2u_1(t)}-\mathrm{e}^{-2u_2(t)})-\frac1A\int_Mf(\mathrm{e}^{2u_1(t)}-\mathrm{e}^{2u_2(t)})d\mu_{\bar g} \\ &= \mathrm{e}^{-2u_1(t)}\Delta_{\bar g}u(t) + \Delta_{\bar g}u_2(t)\bigl( \mathrm{e}^{-2u_1(t)} -\mathrm{e}^{-2u_2(t)}\bigr)\\
&\phantom{aaaaa}-\bar K(\mathrm{e}^{-2u_1(t)}-\mathrm{e}^{-2u_2(t)})-\frac1A\int_Mf(\mathrm{e}^{2u_1(t)}-\mathrm{e}^{2u_2(t)})d\mu_{\bar g}\quad \text{for $t \in (0,T)$} . \end{split} \end{equation}
In the following, the letter $C$ denotes different positive constants. Multiplying (\ref{diff-equation-uniqueness}) with $2u$ and integrating over $M$ gives \begin{align}
\frac{d}{dt}& \|u(t)\|_{L^2(M,\bar g)}^2 = 2 \int_{M} u(t) \partial_t u(t) d\mu_{\bar g} \nonumber\\
&=2\int_M\mathrm{e}^{-2u_1(t)}u(t)\Delta_{\bar g}u(t)d\mu_{\bar g} + 2\int_Mu(t)\Delta_{\bar g}u_2(t)\bigl( \mathrm{e}^{-2u_1(t)} -\mathrm{e}^{-2u_2(t)}\bigr)d\mu_{\bar g}\\
&\qquad-2\int_M\bar Ku(t)(\mathrm{e}^{-2u_1(t)}-\mathrm{e}^{-2u_2(t)})d\mu_{\bar g}-\frac2A\int_Mf(\mathrm{e}^{2u_1(t)}-\mathrm{e}^{2u_2(t)})d\mu_{\bar g}\int_Mu(t)d\mu_{\bar g}\nonumber\\
&\le 2 \int_M \mathrm{e}^{-2u_1(t)} u(t) \Delta_{\bar g}u(t) + 2\int_M V(t,x)u^2(t) + 2\rho(t) \|u(t)\|_{L^2(M,\bar g)} \int_M |u(t)|d\mu_{\bar g} \nonumber\\
&\le 2 \left(-\int_M \mathrm{e}^{-2u_1(t)} |\nabla_{\bar g} u(t)|_{\bar g}^2 + 2 \int_M \mathrm{e}^{-2u_1(t)} u(t) \langle\nabla_{\bar g} u_1(t),\nabla_{\bar g}u(t)\rangle_{\bar g}d\mu_{\bar g}\right)\nonumber\\
&\qquad + 2\|V(t,\cdot)\|_{L^p(M,\bar g)} \|u(t)\|_{L^{2p'}(M,\bar g)}^2 + C \|u(t)\|_{L^2(M,\bar g)}^2 \nonumber\\
&\le C \|\nabla_{\bar g}u_1(t)\|_{L^4(M,\bar g)} \|u(t)\|_{L^4(M,\bar g)} \|\nabla_{\bar g}u(t)\|_{L^2(M,\bar g)}\nonumber\\
&\qquad+ 2\|V(t,\cdot)\|_{L^p(M,\bar g)} \|u(t)\|_{L^{2p'}(M,\bar g)}^2 +C \|u(t)\|_{L^2(M,\bar g)}^2 \nonumber\\
&\le C \Bigl(\|u_1(t)\|_{H^2(M,\bar g)} \|u(t)\|_{H^1(M,\bar g)}^2 + 2\|V(t,\cdot)\|_{L^p(M,\bar g)} \|u(t)\|_{H^1(M,\bar g)}^2 + \|u(t)\|_{L^2(M,\bar g)}^2\Bigr) \nonumber\\
&\le C \Bigl(\|u_1(t)\|_{H^2(M,\bar g)} + 2\|V(t,\cdot)\|_{L^p(M,\bar g)}+1\Bigr) \|u\|_{H^1(M,\bar g)}^2,\label{gronwall-H1-1} \end{align} with functions $V \in L^p((0,T) \times M) \cap C^\infty((0,T) \times M)$ and $\rho \in L^\infty(0,T)$. Here we used the Sobolev embeddings $H^1(M,\bar g) \hookrightarrow L^\rho(M)$ for $\rho \in [1,\infty)$. Multiplying (\ref{diff-equation-uniqueness}) with $-2\Delta u$ and integrating over $M$ yields \begin{align}
\frac{d}{dt} &\|\nabla_g u(t)\|_{L^2(M,\bar g)}^2 = 2 \int_M \nabla u(t) \nabla \partial_t u(t)d\mu_{\bar g}= -2 \int_M \Delta_g u(t) \partial_t u(t)d\mu_{\bar g} \nonumber\\
&\le - 2 \int_M \mathrm{e}^{-2u_1(t)} |\Delta_{\bar g}u(t)|^2 d\mu_{\bar g} +2 \int_M V(x,t)|u(t)||\Delta u(t)|d\mu_{\bar g} \nonumber\\
&\le -\kappa \|\Delta_{\bar g}u(t)\|_{L^2(M,\bar g)}^2 +
2\|V(t,\cdot)\|_{L^p(M,\bar g)} \|u\|_{L^\alpha(M,\bar g)} \|\Delta_g u\|_{L^2(M,\bar g)} \nonumber\\
&\le -\kappa \|\Delta_{\bar g}u(t)\|_{L^2(M,\bar g)}^2 +
\frac{1}{\kappa} \|V(t,\cdot)\|_{L^p(M,\bar g)}^2 \|u\|_{L^\alpha(M,\bar g)}^2 + \kappa \|\Delta_g u\|_{L^2(M,\bar g)}^2 \nonumber\\
&= \frac{1}{\kappa} \|V(t,\cdot)\|_{L^p(M,\bar g)}^2 \|u\|_{L^\alpha(M,\bar g)}^2 \le C \|V(t,\cdot)\|_{L^p(M,\bar g)}^2 \|u\|_{H^1(M,\bar g)}^2, \label{gronwall-H1-2} \end{align} where we used first H\"older's inequality with $\alpha = \frac{2p}{p-2}$, then Young's inequality and finally Sobolev embeddings again. Here we note that, by making $C>0$ larger if necessary, we may assume that the constants are the same in (\ref{gronwall-H1-1}) and (\ref{gronwall-H1-2}). Combining these estimates gives \begin{equation}
\label{eq:gronwall-condition}
\frac{d}{dt} \|u(t)\|_{H^1(M,\bar g)}^2 \le g(t) \|u(t)\|_{H^1(M,\bar g)}^2 \qquad \text{for $t \in (0,T)$} \end{equation}
with the function $g \in L^1(0,T)$ given by $g_1(t)= C \Bigl(\|u_1(t)\|_{H^2(M,\bar g)} + 3\|V(t,\cdot)\|_{L^p(M,\bar g)}+1\Bigr)$. Integrating and using the fact that $u \in C([0,T), H^1(M,\bar g))$ by \autoref{existence} with $u(0)=u_1(0)-u_2(0)=0$, we see that $$
\|u(t)\|_{H^1(M,\bar g)}^2 \le \int_0^t g(s)\|u(s)\|_{H^1(M,\bar g)}^2\,ds \qquad \text{for $t \in [0,T)$.} $$
It then follows from Gronwall's inequality \cite{CazHar99} that $\|u(t)\|_{H^1(M,\bar g)}^2 \equiv 0$ on $[0,T)$, hence $u_1 \equiv u_2$. \end{proof}
\subsection{Global Existence} From Section \ref{SectionShortTimeExistence} and Section \ref{Uniqueness} we know that there exists a unique solution \[u\in C([0,T],C(M))\cap C([0,1],H^1(M,\bar g))\cap C^\infty((0,T)\times M),\] of the initial value problem \eqref{HG1}, \eqref{HG2}. In particular we know that $u\in L^\infty_tL^\infty_x$ for $t\in[0,T]$, where $T>0$ is given by \eqref{eq:T-condition}. In this section we want to show that $u$ posses an $L^\infty$-a-priori bound on any time interval $[0,T]$, $T<\infty$, and therefore, $u$ is the unique global solution of \eqref{HG1}, \eqref{HG2}. For this we partially follow the idea of \cite[Chapter 6]{BuzSchStr16}. \begin{lemma}\label{UpperULowerK} For every $T>0$, there exists $\mathcal{M}(T)>0$ such that we have
\[\sup_{t\in[0,T]}\|u(t)\|_{L^\infty(M,\bar g)}\le \mathcal{M}(T).\] \end{lemma}
\begin{proof} Let
\[I:=\left\{t\ge 0\:\Big|\:\begin{array}{l} \text{$u$ is a solution of \eqref{HG1} on $(0,t]\times M$}\\ \text{with initial data $u(0)\in\mathcal{C}_{p,A}$}\end{array}\right\},\] $T_{\text{max}}:=\sup I$, and $T_k\subset I$ a sequence in $I$ such that $T_k\to T_{\max}$ for $k\to\infty$.
For any $t\in[0,T_k]$ and any $x_{\text{max}}(t)\in M$ where \[u(t,x_{\text{max}}(t))=\max_{x\in M}u(t,x)\ge0\]
we have with $\partial_tu(t)=\Delta_{g(t)}u(t)-\mathrm{e}^{-2u(t)}\bar K+f-\alpha(t)$ and the upper bound for $|\alpha|$ which is given by \begin{equation}
\alpha_0:=\max\{|\alpha_1|,|\alpha_2|\}, \end{equation} that \begin{equation}\label{BoundPartialU} \begin{split}
\frac{d}{dt}\left[u(t,x_{\text{max}}(t))\right]=\partial_tu(t,x_{\text{max}}(t))&\le |\bar K|\mathrm{e}^{-2u(t,x_{\text{max}}(t))}+f(x_{\text{max}}(t))+\alpha_0\\
&\le|\bar K|+\|f\|_{L^\infty(M,\bar g)}+\alpha_0, \end{split} \end{equation} where we used that $\nabla_{\bar g}u(t,x_{\text{max}}(t))=0$ and therefore \[\frac{d}{dt}\left[u(t,x_{\text{max}}(t)\right]=\partial_tu(t,x_{\text{max}}(t))+\nabla_{\bar g}u(t,x_{\text{max}}(t))\dot x_{\text{max}}(t)=\partial_tu(t,x_{\text{max}}(t)).\]
Integrating \eqref{BoundPartialU} on both side with respect to $t$ and taking the supremum over $t$ yields (together with the fact that $u(0)=u_0\in \mathcal{C}_{p,A}$) \begin{equation}\label{BoundU} \begin{split}
\sup_{\substack{t\in[0,T_k]\\ x\in M}}u(t,x)&\le T_k(|\bar K|+\|f\|_{L^\infty(M,\bar g)}+\alpha_0)+\sup_{x\in M}u_0(x)\\
&\to T_\text{max}(|\bar K|+\|f\|_{L^\infty(M,\bar g)}+\alpha_0) +\sup_{x\in M}u_0(x)=:\mathcal{M}_1(T_{\max})<\infty \end{split} \end{equation} for $k\to\infty$ which shows the upper bound for $u$.
Analogously, at any point $x_{\text{min}}(t)\in M$ where \[u(t,x_{\text{min}}(t))=\min_{x\in M}u(t,x)\le0\]
we have with $\partial_tu(t)=\Delta_{g(t)}u(t)-\mathrm{e}^{-2u(t)}\bar K+f-\alpha(t)$, the fact that $\bar K<0$, and the upper bound for $|\alpha|$ given by $\alpha_0$
that \begin{equation}\label{BoundPartialU2}
\frac{d}{dt}\left[u(t,x_{\text{min}}(t))\right]=\partial_tu(t,x_{\text{min}}(t))\ge -\|f\|_{L^\infty(M,\bar g)}-\alpha_0. \end{equation} Integrating \eqref{BoundPartialU2} on both side with respect to $t$ and taking the infimum over $t$ yields (together with the fact that $u(0)=u_0\in\mathcal{C}_{p,A}$) \begin{equation}\label{BoundU2} \begin{split}
\inf_{\substack{t\in[0,T_k]\\ x\in M}}u(t,x)&\ge -T_k(\|f\|_{L^\infty(M,\bar g)}+\alpha_0)+\inf_{x\in M}u_0(x)\\
&\to -T_\text{max}(\|f\|_{L^\infty(M,\bar g)}+\alpha_0)+\inf_{x\in M}u_0(x) =:\mathcal{M}_2(T_{\max})>-\infty \end{split} \end{equation} for $k\to\infty$ which shows the lower bound for $u$.
So, we get \begin{equation}\label{UniformBoundU} \begin{split}
\sup_{\substack{t\in[0,T]\\ x\in M}}|u(t,x)|&\le \max\{|\mathcal{M}_1(T)|,|\mathcal{M}_2(T)|\}\\
&\le T(|\bar K|+\|f\|_{L^\infty(M,\bar g)}+\alpha_0)+\sup_{x\in M}|u_0(x)|=:\mathcal{M}(T) \end{split} \end{equation} which shows the claim. \end{proof}
In fact, with the help of \eqref{APrioriBound} we can turn \eqref{UniformBoundU} into a uniform estimate for all time.
\begin{lemma}\label{Linfty}
Let $u$ be the global, smooth solution of \eqref{HG1} with $u(0)=u_0\in \mathcal{C}_{p,A}$. Then we have that $\sup_{t>0}\|u(t)\|_{L^\infty(M,\bar g)}\le C_{\uni}<\infty$. \end{lemma}
\begin{proof} We follow the proof of \cite[Lemma 2.5]{Str20}.
By using the fact that $u(t)$ is a volume preserving solution of \eqref{HG1} with $u(0)=u_0\in \mathcal{C}_{p,A}$ and therefore $\int_M\mathrm{e}^{2u(t)}d\mu_{\bar g}\equiv A$, we get with \eqref{Jensen-consequence} and the fact that $\bar K<0$ that \begin{equation}\label{LowerBoundEnergy} \begin{split}
E_{f}(u(t))&=\frac12\|\nabla_{\bar g}u(t)\|^2_{L^2(M, \bar g)}+\int_M\bar Ku(t)d\mu_{\bar g}-\frac12\int_Mf\mathrm{e}^{2u(t)}d\mu_{\bar g}\\ &\ge\frac{\bar K}{2}\int_M2u(t)d\mu_{\bar g}-\frac12\int_Mf\mathrm{e}^{2u(t)}d\mu_{\bar g}\\
&\ge\frac{\bar K}{2}\log(A)-\frac{A}{2}\|f\|_{L^\infty(M,\bar g)}>-\infty. \end{split} \end{equation}
Defining
\[F(t):=\int_M|\partial_tu(t)|^2d\mu_{g(t)}=\int_M|\partial_tu(t)|^2\mathrm{e}^{2u(t)}d\mu_{\bar g}\] and using the uniform lower bound of $E_{f}$ given by \eqref{LowerBoundEnergy}, we then get from \eqref{EnergyEstimate} or \eqref{APrioriBound}, respectively, the estimate \begin{equation}\label{UniformEstimate}
\int_0^\infty F(t)dt=\int_0^\infty\int_M|\partial_tu(t)|^2d\mu_{g(t)}dt\le E_{f}(u_0)+\frac{|\bar K|}{2}|\log(A)|+\frac{A}{2}\|f\|_{L^\infty(M,\bar g)}. \end{equation}
Hence, for any $T>0$ we find $t_T\in[T,T+1]$ such that \begin{equation}\label{EstimateFtT}
F(t_T)=\inf_{t\in(T,T+1)}F(t)\le E_{f}(u_0)+\frac{|\bar K|}{2}|\log(A)|+\frac{A}{2}\|f\|_{L^\infty(M,\bar g)}. \end{equation} So, at time $t_T$ we get with \eqref{PCFVol1}, H\"olders inequality, \eqref{Uniformexp}, and \eqref{EstimateFtT} that \begin{equation}\label{DeltaUniform} \begin{split}
\|\Delta_{\bar g}&u(t_T)\|_{L^{\frac32}(M,\bar g)}\\
&\le\|\mathrm{e}^{2u(t_T)}\partial_tu(t_T)\|_{L^{\frac32}(M,\bar g)}+\|\bar K\|_{L^{\frac32}(M,\bar g)}+\|\mathrm{e}^{2u(t_T)}f\|_{L^{\frac32}(M,\bar g)}+\|\mathrm{e}^{2u(t_T)}\alpha(t_T)\|_{L^{\frac32}(M,\bar g)}\\
&\le\|\mathrm{e}^{u(t_T)}\|_{L^6(M,\bar g)}F(t_T)^{\frac12}+|\bar K|+\left(\int_M\mathrm{e}^{3u(t_T)}|f|^{\frac32}d\mu_{\bar g}\right)^{\frac23}+\left(\int_M\mathrm{e}^{3u(t_T)}|\alpha(t_T)|^{\frac32}d\mu_{\bar g}\right)^{\frac23}\\
&\le C_{\text{int}}^{\frac16}(A,E_{f}(u_0),f,\bar K,\eta_1,\eta_2, 3)\left(E_f(u_0)+\frac{|\bar K|}{2}|\log(A)|+\frac{A}{2}\|f\|_{L^\infty(M,\bar g)}\right)^{\frac12}+|\bar K|\\
&\phantom{aaaaa}+C_{\text{int}}^{\frac23}\left(A,E_{f}(u_0),f,\bar K,\eta_1,\eta_2, \frac{3}{2}\right)(\|f\|_{L^\infty(M,\bar g)}+\alpha_0)\\ &=:C_{10}\left(A,E_{f}(u_0),f,\bar K,\eta_1,\eta_2, \frac32,3\right). \end{split} \end{equation}
Furthermore, with Sobolev's embedding theorem we have $W^{2,\frac{3}{2}}\subset C^{0,\frac23}$. Therefore we get with Poincar\'e's inequality, the Calder\'on--Zygmund inequality for closed surfaces, and with \eqref{DeltaUniform} that \begin{equation} \begin{split}
\|u(t_T)-\bar u(t_T)\|^{\frac32}_{L^\infty(M,\bar g)}&\le C\|u(t_T)-\bar u(t_T)\|^{\frac32}_{W^{2,\frac32}(M,\bar g)}\le C\|\nabla_{\bar g}^2u(t_T)\|^{\frac32}_{L^{\frac32}(M,\bar g)}\\
&\le C\|\Delta_{\bar g}u(t_T)\|^{\frac32}_{L^{\frac32}(M,\bar g)}\le C C_{10}^{\frac32}, \end{split} \end{equation} and therefore with \eqref{UniBarU} we obtain the uniform bound \begin{equation}
\|u(t_T)\|_{L^\infty(M,\bar g)}\le CC_{10}+\max\left\{|m_0|,\frac12|\log(A)|\right\}. \end{equation} Upon shifting time by $t_T$, from \eqref{UniformBoundU} we now get \begin{equation} \begin{split}
\sup_{s\in[T+1,T+2]}\|u(s)\|_{L^\infty(M,\bar g)}&\le \sup_{s\in[t_T,T+2]}\|u(s)\|_{L^\infty(M,\bar g)}\\
&\le 2(|\bar K|+\|f\|_{L^\infty(M,\bar g)}+\alpha_0)+\sup_{x\in M}|u(t_T,x)|\\
&\le 2(|\bar K|+\|f\|_{L^\infty(M,\bar g)}+\alpha_0)+CC_{10}+\max\left\{|m_0|,\frac12|\log(A)|\right\}. \end{split} \end{equation} Since $T>0$ is arbitrary, the claim follows. \end{proof}
\subsection{Convergence of the Flow}\label{SectionConvergenceFlow} Let $f_0\le0$ be a smooth, nonconstant function with$\max_{x\in M}f_0(x)=0$. Following here the argumentation of \cite{Str20}, and using \eqref{UniformEstimate}, we know that for a suitable sequence $t_l\to\infty$, $l\to\infty$, with associated metrics $g_l=g(t_l)$ we obtain convergence \begin{equation}\label{Conv1}
\int_M|\partial_tu(t_l)|^2d\mu_{g(t_l)}=\int_M|f_0-K_{g_l}-\alpha(t_l)|^2d\mu_{g(t_l)}\to0\quad\text{for }l\to\infty. \end{equation} Provided that we can also show convergence of the associated sequence of metrics $g(t_l)$ to a limit metric $g_A^\infty=\mathrm{e}^{2u_A^\infty}\bar g$ with Gauss curvature $K_{g_A^\infty}$, it then follows that $K_{g_A^\infty}=f_0-\alpha_A^\infty$ for a constant $\alpha^\infty_A$. Later we will have a closer look at this constant $\alpha^\infty_A$.
\begin{lemma}\label{KonvergenzF}
For $F(t)=\int_M|\partial_tu(t)|^2d\mu_{g(t)}$ as above, we have $F(t)\to0$ for $t\to\infty$. \end{lemma}
\begin{proof} First we consider the evolution equation of the curvature $K_{g(t)}$ and of $\alpha(t)$. From the Gauss equation \eqref{GaussEquation} we get for the curvature that \begin{equation}\label{EvolutionCurvature} \begin{split} \partial_tK_{g(t)}&=\partial_t(-\mathrm{e}^{-2u(t)}\Delta_{\bar g}u(t)+\mathrm{e}^{-2u(t)}\bar K)\\ &=-2\partial_tu(t)K_{g(t)}-\Delta_{g(t)}\partial_tu(t)\\ &=2K_{g(t)}(K_{g(t)}-f_0+\alpha(t))+\Delta_{g(t)}(K_{g(t)}-f_0+\alpha(t))\\ &=2(K_{g(t)}-f_0+\alpha(t))^2+2(f_0-\alpha(t))(K_{g(t)}-f_0+\alpha(t))+\Delta_{g(t)}(K_{g(t)}-f_0+\alpha(t)). \end{split} \end{equation}
With \eqref{tildealpha} we get for the evolution equation for $\alpha(t)$: \begin{equation}\label{EvolutionBeta} \frac{d}{dt}\alpha(t)=\frac2A\int_Mf_0\mathrm{e}^{2u(t)}\partial_tu(t)d\mu_{\bar g}=\frac2A\int_Mf_0(f_0-K_{g(t)}-\alpha(t))d\mu_{g(t)}. \end{equation}
So, with \eqref{EvolutionCurvature} and \eqref{EvolutionBeta} we arrive at \begin{equation}\label{Combined} \begin{split} \partial_t(K_{g(t)}-f_0-\alpha(t))&-\Delta_{g(t)}(K_{g(t)}-f_0+\alpha(t))\\ &=2(K_{g(t)}-f_0+\alpha(t))^2+2(f_0-\alpha(t))(K_{g(t)}-f_0+\alpha(t))\\ &\phantom{aaaaa}+\frac2A\int_Mf_0(K_{g(t)}-f_0+\alpha(t))d\mu_{g(t)}. \end{split} \end{equation}
Following the proof of Lemma 3.1\ in \cite{Str20} we therefore get \begin{equation}\label{EvolutionF1} \begin{split}
\frac12&\frac{d}{dt}\int_M|f_0-K_{g(t)}-\alpha(t)|^2d\mu_{g(t)}\\ &=\int_M\left(\left(\partial_tK_{g(t)}+\left(\frac{d}{dt}\alpha(t)\right)\right)(K_{g(t)}-f_0+\alpha(t))-(K_{g(t)}-f_0-\alpha(t))^3\right)d\mu_{g(t)}\\
&=-\int_M|\nabla_{g(t)}(K_{g(t)}-f_0+\alpha(t))|_{g(t)}^2d\mu_{g(t)}+2\int_M(f_0-\alpha(t))(K_{g(t)}-f_0+\alpha(t))^2d\mu_{g(t)}\\ &\phantom{aaaaa}+\int_M(K_{g(t)}-f_0+\alpha(t))^3d\mu_{g(t)}, \end{split} \end{equation} where we used in the second step the fact that \[\left(\frac{d}{dt}\alpha(t)\right)\int_M(K_{g(t)}-f_0+\alpha(t))d\mu_{g(t)}=0\] by \eqref{VolPres}.
With H\"older's inequality we can estimate \begin{equation}\label{Estimate1}
\int_M(K_{g(t)}-f_0+\alpha(t))^3d\mu_{g(t)}\le\|\partial_tu(t)\|^3_{L^3(M,g(t))}\le \|\partial_tu(t)\|_{L^2(M,g(t))}\|\partial_tu(t)\|^2_{L^4(M,g(t))} \end{equation}
and by \autoref{eindSob} we further get with the uniform bound for $u\in C_tC_x$ that \begin{equation}\label{Estimate2} \begin{split}
\|&\partial_tu(t)\|^2_{L^4(M,g(t))}\\
&=\left(\int_M|\partial_tu(t)|^4\mathrm{e}^{2u(t)}d\mu_{\bar g}\right)^{\frac12}\le\mathrm{e}^{\|u\|_{L^\infty_tL^\infty_x}}\|\partial_tu(t)\|^2_{L^4(M,\bar g)}\\
&\le\mathrm{e}^{\|u\|_{L^\infty_tL^\infty_x}}\sqrt{C_{\text{GNL}}}\|\partial_tu(t)\|_{L^2(M,\bar g)}\|\partial_tu(t)\|_{H^1(M,\bar g)}\\
&=\mathrm{e}^{\|u\|_{L^\infty_tL^\infty_x}}\sqrt{C_{\text{GNL}}}\left(\int_M|\partial_tu(t)|^2\mathrm{e}^{2u(t)}\mathrm{e}^{-2u(t)}d\mu_{\bar g}\right)^{\frac12}\left(\int_M|\partial_tu(t)|^2\mathrm{e}^{2u(t)}\mathrm{e}^{-2u(t)}d\mu_{\bar g}+\int_M|\nabla_{\bar g}\partial_tu(t)|^2_{\bar g}d\mu_{\bar g}\right)^{\frac12}\\
&=\mathrm{e}^{\|u\|_{L^\infty_tL^\infty_x}}\sqrt{C_{\text{GNL}}}\left(\int_M|\partial_tu(t)|^2\mathrm{e}^{-2u(t)}d\mu_{g(t)}\right)^{\frac12}\left(\int_M|\partial_tu(t)|^2\mathrm{e}^{-2u(t)}d\mu_{g(t)}+\int_M|\nabla_{g(t)}\partial_tu(t)|^2_{g(t)}d\mu_{g(t)}\right)^{\frac12}\\
&\le \mathrm{e}^{\|u\|_{L^\infty_tL^\infty_x}}\max\{\mathrm{e}^{\|u\|_{L^\infty_tL^\infty_x}},\mathrm{e}^{2\|u\|_{L^\infty_tL^\infty_x}}\}\sqrt{C_{\text{GNL}}}\|\partial_tu(t)\|_{L^2(M,g(t))}\|\partial_tu(t)\|_{H^1(M,g(t))}\\
&\eqqcolon\tilde C_2\|\partial_tu(t)\|_{L^2(M, g(t))}\|\partial_tu(t)\|_{H^1(M,g(t))}, \end{split} \end{equation} where we used the fact that
\[\int_M|\nabla_{\bar g}\partial_tu(t)|^2_{\bar g}d\mu_{\bar g}=\int_M |\nabla_{g(t)}\partial_tu(t)|^2_{g(t)}d\mu_{g(t)}=:G(t).\]
Plugging in \eqref{Estimate2} into \eqref{Estimate1} we arrive at \begin{equation}\label{EstimateF} \begin{split}
\int_M(K_{g(t)}-f_0+\alpha(t))^3d\mu_{g(t)}&\le \tilde C_2\|\partial_tu(t)\|^2_{L^2(M,g(t))}\|\partial_tu(t)\|_{H^1(M,g(t))}\\
&\le \frac{\tilde C_2^2}{2}\|\partial_tu(t)\|^4_{L^2(M,g(t))}+\frac12\|\partial_tu(t)\|^2_{H^1(M,g(t))}\\ &\le\tilde C_2^2 F^2(t)+\frac12(F(t)+G(t)), \end{split} \end{equation} where we used Young's inequality in the second step.
With $\alpha_0=\max\{|\alpha_1|,|\alpha_2|\}>0$ we furthermore have that
\[2\int_M(f_0-\alpha(t))(K_{g(t)}-f_0+\alpha(t))^2d\mu_{g(t)}\le 2(\|f_0\|_{L^\infty(M,\bar g)}+\alpha_0)F(t)=:\tilde C_3(\alpha_0,f_0)F(t)\]
So, \eqref{EvolutionF1} yields \begin{equation}\label{EvolutionF2} \begin{split} \frac{d}{dt}F(t)+G(t)&\le 2\left(\tilde C_3F(t)+\tilde C^2_2F^2(t)+\frac12 F(t)\right)\\ &=(2\tilde C_3+1)F(t)+2\tilde C^2_2F^2(t)\\ &=:\tilde C_4F(t)+2\tilde C^2_2 F^2(t). \end{split} \end{equation}
We recall that with \eqref{UniformEstimate} we have $\liminf_{t\to\infty}F(t)=0$ and therefore we know that there exist $t_l\to\infty$ with $F(t_l)\to0$ as $l\to\infty$, see \eqref{Conv1}.
By integrating \eqref{EvolutionF2} over $(t_l,t)\subset(t_l,T)$ and taking the supremum over $(t_l,T)$ we get with $\int_{t_l}^TG(t)dt\ge0$ that \begin{align*} \sup_{t\in(t_l,T)}F(t)&\le F(t_l)+\tilde C_4\int_{t_l}^TF(t)dt+2\tilde C^2_2\int_{t_l}^TF^2(t)dt\\ &\le F(t_l)+\tilde C_4\int_{t_l}^TF(t)dt+2\tilde C^2_2\sup_{t\in(t_l,T)}F(t)\int_{t_l}^TF(t)dt\\ &\le F(t_l)+\tilde C_4\int_{t_l}^TF(t)dt+2\tilde C^2_2\sup_{t\in(t_l,T)}F(t)\int_{t_l}^\infty F(t)dt. \end{align*}
With \eqref{UniformEstimate} we also have $\int_{t_l}^\infty F(t)dt\to0$ for $l\to\infty$. So, for $T>0$ big enough such that for $t_l<T$ big enough we have that $2\tilde C^2_2\int_{t_l}^\infty F(t)dt$ is small enough to guarantee that $1-2\tilde C^2_2\int_{t_l}^\infty F(t) dt>0$ and therefore the term $2\tilde C^2_2\sup_{t\in(t_l,T)}F(t)\int_{t_l}^\infty F(t) dt$ can be absorbed on the left hand side. So, we get \[\sup_{t\in(t_l,T)}F(t)\le\frac{1}{\left(1-2\tilde C^2_2\int_{t_l}^\infty F(t) dt\right)} \left(F(t_l)+\tilde C_4\int_{t_l}^TF(t) dt\right).\]
Letting $T\to\infty$ yields \[\sup_{t\in(t_l,\infty)}F(t)\le\frac{1}{\left(1-\tilde C^2_2\int_{t_l}^\infty F(t) dt\right)} \left(F(t_l)+\tilde C_4\int_{t_l}^\infty F(t) dt\right)\to0\quad\text{as }l\to \infty\] which shows the claim. \end{proof}
To prove now the convergence of the flow, let $A>0$ and $u_0\in \mathcal{C}_{p,A}$, $p>2$. Furthermore let $f\in C^\infty(M)$ be a smooth, nonconstant function, and $(f_0,\lambda)\in C^\infty(M)\times\mathds R$ the unique pair such that \[f=f_0+\lambda\] with $f_0\le0$, $f_0$ nonconstant, and $\max_{x\in M}f_0(x)=0$. Since by \autoref{Properties1} the additive rescaled prescribed Gauss curvature flow \eqref{PCFVolNew1} is invariant under adding or subtracting a constant $C>0$ to the function $f$, for all functions \[f\in\{f_0+\lambda\mid\lambda\in\mathds R\}\] we consider the same flow given by \begin{equation}\label{HG1b} \partial_tu(t)=f_0-K_{g(t)}-\alpha_A(t)\quad\text{in }(0,T)\times M, \end{equation} which is \eqref{PCFVolNew1} with $f$ replaced by $f_0$.
With \eqref{APrioriBound} we know that
\[\frac12\int_M(|\nabla_{\bar g}u(T)|^2_{\bar g}+2\bar Ku(T)-f_0\mathrm{e}^{2u(T)})d\mu_{\bar g}=E_{f_0}(u(T))\le E_{f_0}(u(0)).\] So, we get with \eqref{Jensen-consequence} that \begin{align*}
\frac12\int_M|\nabla_{\bar g}u(T)|^2_{\bar g}d\mu_{\bar g}&=E_{f_0}(u(T))-\int_M\bar Ku(T)d\mu_{\bar g}+\frac12\int_Mf_0\mathrm{e}^{2u(T)}d\mu_{\bar g}\\
&\le E_{f_0}(u(T))+|\bar K|\int_M u(T)d\mu_{\bar g}\\
&\le E_{f_0}(u(0))+\frac{|\bar K|}{2}|\log(A)|. \end{align*}
So, $u$ is uniformly (in $T$) bounded in $H^1(M,\bar g)$, i.e., $\|u\|_{L^\infty_tH^1_x}\le C$.
We now consider $u_l:=u(t_l)$ for a suitable sequence $t_l\to\infty$. By the Theorem of Banach-Alao\u{g}lu we know that $(u_l)_l$ is weak$^*$ relatively compact in $H^1(M,\bar g)$ and therefore (since $H^1$ is reflexive) also weak relatively compact. This means that that there exists a subsequence $u_{l_k}$ which we again call $u_l$ such that $u_l\to u_A^\infty$ weakly in $H^1(M,\bar g)$ and therefore strongly in $L^2(M,\bar g)$ (by a direct consequence of the Rellich--Kondrachov embedding Theorem). Furthermore with \eqref{LowerBoundtildealpha} and \eqref{UpperBoundtildealpha} we know that $\alpha_l:=\alpha(t_l)\to \alpha_A^\infty$ as $l\to\infty$. Moreover we have $\mathrm{e}^{\pm u_l}\to \mathrm{e}^{\pm u_A^\infty}$ (as $l\to\infty$) in $L^p(M,\bar g)$ for any $2\le p<\infty$. Indeed, with \autoref{Linfty} and \eqref{EstimateExp} we have \begin{align*}
\|\mathrm{e}^{u_l}-\mathrm{e}^{u_A^\infty}\|^p_{L^p(M,\bar g)}&=\int_M\mathrm{e}^{pu_l}|1-\mathrm{e}^{u_A^\infty-u_l}|^pd\mu_{\bar g}\le \mathrm{e}^{pC_{\text{uni}}}\int_M|1-\mathrm{e}^{u_A^\infty-u_l}|^pd\mu_{\bar g}\\
&\le \mathrm{e}^{pC_{\text{uni}}}\int_M|u_A^\infty-u_l|^p\mathrm{e}^{p|u_A^\infty-u_l|}|d\mu_{\bar g}\\
&\le\mathrm{e}^{pC_{\text{uni}}}\mathrm{e}^{2pC_{\text{uni}}}\int_M|u_A^\infty-u_l|^{p-2}|u_A^\infty-u_l|^2d\mu_{\bar g}\\
&\le \mathrm{e}^{3pC_{\text{uni}}}(2C_{\text{uni}})^{p-2}\|u_A^\infty-u_l\|^2_{L^2(M,\bar g)}\to 0\quad\text{as }l\to \infty. \end{align*} Replacing $u_l$ by $-u_l$ we get also $\mathrm{e}^{-u_l}\to \mathrm{e}^{-u_A^\infty}$ in $L^p(M,\bar g)$ as $l\to\infty$ for any $p<\infty$. Moreover, with \autoref{Linfty} and \autoref{KonvergenzF} we also have $\mathrm{e}^{2u_l}\partial_tu_l\to0$ in $L^2(M,\bar g)$ as $l\to\infty$. Furthermore we have \begin{align*}
\|\mathrm{e}^{2u_l}\alpha_l-\mathrm{e}^{2u_A^\infty}\alpha_A^\infty\|_{L^2(M,\bar g)}&\le\|\mathrm{e}^{2u_l}(\alpha_l-\alpha_A^\infty)\|_{L^2(M,\bar g)}+\|\alpha_A^\infty(\mathrm{e}^{2u_l}-\mathrm{e}^{2u_A^\infty})\|_{L^2(M,\bar g)}\\
&\le\|\mathrm{e}^{2u_l}\|_{L^\infty(M,\bar g)}|\alpha_l-\alpha_A^\infty|A^{\frac12}+|\alpha_A^\infty|\|\mathrm{e}^{2u_l}-\mathrm{e}^{2u_A^\infty}\|_{L^2(M,\bar g)}\\ &\to 0\quad\text{for }l\to \infty. \end{align*}
So, considering our evolution equation \eqref{HG1}, we therefore get \begin{align*} \Delta_{\bar g}u_l&=\mathrm{e}^{2u_l}\partial_tu_l+\bar K-\mathrm{e}^{2u_l}f_0+\mathrm{e}^{2u_l}\alpha_l\\ &\to \bar K-\mathrm{e}^{2u_A^\infty}f_0+\mathrm{e}^{2u_A^\infty}\alpha^\infty_A=:(\Delta_{\bar g}u)_A^\infty \end{align*}
in $L^2(M,\bar g)$. Since the Laplace operator $\Delta_{\bar g}$ is closed we know that $(\Delta_{\bar g}u)_A^\infty=\Delta_{\bar g}u_A^\infty$. Hence $\|\Delta_{\bar g}(u_l-u_A^\infty)\|_{L^2(M,\bar g)}\to0$ as $l\to\infty$. So, we even have strong convergence $u_l\to u_A^\infty$ in $H^2(M,\bar g)$ and uniformly.
Thus, passing to the limit $l\to\infty$ in the equation \[\mathrm{e}^{2u_l}\partial_tu_l-\Delta_{\bar g}u_l=-\bar K+\mathrm{e}^{2u_l}f_0-\mathrm{e}^{2u_l}\alpha_l\] we get the identity \[-\Delta_{\bar g}u^\infty_A=-\bar K+\mathrm{e}^{2u^\infty_A}f_0-\mathrm{e}^{2u^\infty_A}\alpha^\infty_A\] and therefore
\[K_{g^\infty_A}=f_0-\alpha^\infty_A=f_0+\frac{1}{A}\left(\bar K+\int_M|f_0|d\mu_{g^\infty_A}\right)\] which shows the convergence of the flow.
\subsection{\texorpdfstring{The Sign of the Constant $\alpha^\infty_A$}{The Sign of the Constant}} In this subsection we prove \autoref{sign-changing} and \autoref{sec:stat-minim-probl-1-cor-2-theorem}, with other words, under certain assumptions we can now further estimate the expression
\[\frac{1}{A}\left(\bar K+\int_M|f_0|d\mu_{g^\infty_A}\right)\] to show that it is positive.
The proof of \autoref{sec:stat-minim-probl-1-cor-2-theorem} is already covered by the proof of \autoref{sec:stat-minim-probl-1-cor-2}. So we can turn to \autoref{sign-changing}.
\begin{proof}[Proof of \autoref{sign-changing}]
We have seen in \autoref{UpperULowerK} that in the case where $u_0 \equiv \frac{1}{2}\log(A) \in \mathcal{C}_{p,A}$, the uniform $L^\infty$-bound on the global solution of the initial value problem \eqref{HG1}, \eqref{HG2} only depends on $A$ and an upper bound on $\|f\|_{L^\infty(M,\bar g)}$. In other words, if $A>0$ and $c>0$ are fixed, then there exists $\tau>0$ with the property that \[
\sup_{t>0}\|u(t)\|_{L^\infty(M,\bar g)}\le \tau \]
for every $f \in C^\infty(M)$ with $\|f\|_{L^\infty(M,\bar g)} \le c$ and the corresponding solution $u$ of the initial value problem \eqref{HG1}, \eqref{HG2} with $u_0 \equiv \frac{1}{2}\log(A) \in \mathcal{C}_{p,A}$. Consequently,
we also have $\|u^\infty\|_{L^\infty(M,\bar g)} \le \tau$ under the current assumptions on $f$, which implies that \begin{align*}
\lambda&=\frac1A\left(\bar K- \int_M f \mathrm{e}^{2u^\infty}d\mu_{\bar g}\right)= \frac1A\left(\bar K +cA - \int_M (f+c) \mathrm{e}^{2u^\infty}d\mu_{\bar g}\right)\\
&\ge c + \frac{\bar K}{A} - \|f+c\|_{L^1(M,\bar g)}\|\mathrm{e}^{2u^\infty}\|_{L^\infty(M,\bar g)} \ge c + \frac{\bar K}{A} - \|f+c\|_{L^1(M,\bar g)}\mathrm{e}^{2\tau}. \end{align*}
Hence, if $\|f+c\|_{L^1(M,\bar g)} <\varepsilon:=\frac{c + \frac{\bar K}{A}}{\mathrm{e}^{2\tau}}$, we have $\lambda>0$. \end{proof}
\section{Appendix} As before, let $(M,\bar g)$ be a two-dimensional, smooth, closed, connected, oriented Riemann manifold endowed with a smooth background metric $\bar g$. For a domain $\Omega \subset M \times \mathds R$ and $p \ge 1$, we let $W^{2,1}_p(\Omega)$ denote the space of functions $u \in L^p(\Omega)$ which have weak derivatives $Du$, $D^2u$ and $\partial_tu$ in $L^p(\Omega)$. In the following, we fix $p>2$, which implies that \begin{equation}
\label{eq:embedding-hoelder} \text{$W^{2,1}_p(\Omega)$ is continuously embedded in $C^\alpha(\overline \Omega)$ for some $\alpha = \alpha(p)>0$,} \end{equation} see e.g.~\cite[Lemma 3.3]{LadSolUra68}. We consider the linear parabolic problem \begin{equation}
\label{eq:linear-parabolic} \partial_tu(x,t) = a(x,t)\Delta_{\bar g} u(x,t) + c(x,t)u(x,t) +d(x,t), \end{equation} with $a,c,d \in C(\overline \Omega)$ and $d \in L^p(\Omega)$. We say that a function $u \in W^{2,1}_p(\Omega)$ is a (strong) solution of (\ref{eq:linear-parabolic}) in $\Omega$ if (\ref{eq:linear-parabolic}) holds almost everywhere in $\Omega$. Specifically, we consider (\ref{eq:linear-parabolic}) on the cylindrical domains $\Omega_T= M \times (0,T) $ and $\widetilde \Omega_T= M \times (-\infty,T) $ in the following.
In particular, we consider strong solutions of (\ref{eq:linear-parabolic}) together with the initial condition \begin{equation}
\label{eq:linear-parabolic-initial} u(0,x)=u_0(x) \qquad \text{in $M$} \end{equation} with $u_0 \in W^{2,p}(M,\bar g)$, which is supposed to hold in the (initial) trace sense.
\begin{proposition} \label{sec:appendix} Let $T>0$, $a,c \in C(\overline\Omega_T)$ with $a_T:= \min \limits_{(x,t)\in\overline\Omega_T}a(x,t) >0$, let $d \in L^p(\Omega_T)$ for some $p>2$, and let $u_0 \in W^{2,p}(M,\bar g)$.
Then the initial value problem (\ref{eq:linear-parabolic}), (\ref{eq:linear-parabolic-initial}) has a unique strong solution $u \in W^{2,1}_p(\Omega_T)$. Moreover, $u$ satisfies the estimate \begin{equation}
\label{eq:W2-1-p-a-priori}
\|u\|_{W^{2,1}_p(\Omega_T)} \le C \Bigl( \|u_0\|_{W^{2,p}(M,\bar g)} + \|d\|_{L^p(\Omega_T)}\Bigr) \end{equation}
with a constant $C>0$ depending only on $\|a\|_{L^\infty(\Omega_T)}$, $\|c\|_{L^\infty(\Omega_T)}$ and $a_T$. Moreover, $C$ does not increase after making $T$ smaller.\\ If, moreover, $a,c,d \in C^\alpha(\Omega_T)$ for some $\alpha>0$, then $u \in C(\overline \Omega_T) \cap C^{2,1}(\Omega_T)$ is a classical solution of (\ref{eq:linear-parabolic}), (\ref{eq:linear-parabolic-initial}), and we have the inequality \begin{equation}
\label{eq:H-1-inequality-appendix}
\|u_0\|_{H^1(M,\bar g)} \ge \limsup_{t \to 0^+}\|u(t)\|_{H^1(M,\bar g)} \end{equation} \end{proposition}
\begin{proof} In the following, the letter $C$ stands for various positive constants depending only on $\|a\|_{L^\infty(\Omega_T)}$, $\|c\|_{L^\infty(\Omega_T)}$, and $a_T$, and which do not increase after making $T$ smaller.
{\bf Step 1:} We first assume that we are given a strong solution $u \in W^{2,1}_p(\Omega_T)$ of (\ref{eq:linear-parabolic}), (\ref{eq:linear-parabolic-initial}) with $u_0\equiv0 \in W^{2,p}(M,\bar g)$. We then define $v: \widetilde\Omega_T \to \mathds R$ by $$ v(x,t)= \left \{
\begin{aligned}
&u(x,t),&& \qquad \text{for }t>0;\\
&0,&& \qquad \text{for }t \le 0.
\end{aligned} \right. $$ Then $v \in W^{2,1}_p(\widetilde \Omega_T)$ solves (\ref{eq:linear-parabolic}) with $a,c,d$ replaced by suitable extensions $\tilde a,\tilde c, \in L^\infty(\widetilde \Omega_T)$, $\tilde d\in L^p(\widetilde\Omega_T)$ satisfying $\tilde a(x,t)=a(x,0)$, $\tilde c(x,t)=c(x,0)$ and $\tilde d(x,t)= 0$ for $t\le0$, $x \in M$.
Therefore, \cite[Theorem 7.22]{Lie96} gives rise to the uniform bound \begin{equation}
\label{eq:uniform-liebermann-v}
\|D^2 v\|_{L^p(\widetilde \Omega_T)} + \|\partial_tv\|_{L^p(\widetilde \Omega_T)} \le C \Bigl(\|\tilde d\|_{L^p(\widetilde \Omega_T)}+ \|v\|_{L^p(\widetilde \Omega_T)}\Bigr). \end{equation} This translates into the estimate \begin{equation}
\label{eq:uniform-liebermann-u}
\|D^2 u\|_{L^p(\Omega_T)} + \|\partial_tu\|_{L^p(\Omega_T)} \le C \Bigl(\|d\|_{L^p(\Omega_T)}+ \|u\|_{L^p(\Omega_T)}\Bigr). \end{equation}
Moreover, setting $V(t):=\|u(t)\|_{L^p(M,\bar g)}^p$ for $t \in \mathds R$, we have $V(0)=0$ and \begin{align*}
\dot V(t)&= p\int_{M} |u(t)|^{p-2}u(t) \partial_tu(t)d\mu_{\bar g} \le pV(t)^{\frac{1}{p'}} \|\partial_tu(t)\|_{L^p(M,\bar g)}\\
&\le
p\left(\frac{V(t)}{p'}+\frac{\|\partial_tu(t)\|^p_{L^p(M,\bar g)}}{p}\right)=\frac{p}{p'}V(t)+\|\partial_tu(t)\|^p_{L^p(M,\bar g)} \end{align*} for $t \in (0,T)$, therefore \begin{align*}
V(t) &= \int_0^t \dot V(s)\,ds \le \frac{p}{p'}\int_0^t V(s)\,ds + \|\partial_tu\|_{L^p(\Omega_t)}^p\\
&\le \frac{p}{p'}\int_0^t V(s)\,ds + C \Bigl(\|d\|_{L^p(\Omega_t)}^p+ \|u\|_{L^p(\Omega_t)}^p\Bigr)\le C \left(\int_0^t V(s)\,ds + \|d\|_{L^p(\Omega_t)}^p \right). \end{align*}
By Gronwall's inequality we get $V(t) \le C \|d\|_{L^p(\Omega_t)}^p$ and thus \begin{equation}
\label{eq:gronwall-consequence}
\|u(t)\|_{L^p(M,\bar g)} \le C \|d\|_{L^p(\Omega_t)}\qquad \text{for $t \in [0,T]$.} \end{equation} This already implies the uniqueness of strong solutions of (\ref{eq:linear-parabolic}), (\ref{eq:linear-parabolic-initial}), since the difference $u$ of two solutions $u_1,u_2 \in W^{2,1}_p(\Omega_T)$ of (\ref{eq:linear-parabolic}), (\ref{eq:linear-parabolic-initial}) satisfies (\ref{eq:linear-parabolic}), (\ref{eq:linear-parabolic-initial}) with $u_0=0$ and $d=0$. Moreover, if $u \in W^{2,1}_p(\Omega_T)$ is a strong solution of (\ref{eq:linear-parabolic}), (\ref{eq:linear-parabolic-initial}), then the function $\hat u \in W^{2,1}_p(\Omega_T)$ given by $\hat u(x,t):= u(x,t)- u_0(x)$ safisfies (\ref{eq:linear-parabolic}), (\ref{eq:linear-parabolic-initial}) with $u_0=0$ and $d$ replaced by $\hat d$ given by $$ \hat d(x,t)= d(x,t)+a(x,t) \Delta_{\bar g} u_0(x)+c(x,t) u_0(x). $$ Consequently, combining (\ref{eq:uniform-liebermann-u}) and (\ref{eq:gronwall-consequence}), and using an interpolation estimate for $Du$, we find that \begin{align*}
\|u\|_{W^{2,1}_p(\Omega_T)} &\le \|\hat u\|_{W^{2,1}_p(\Omega_T)}+\|u_0\|_{W^{2,p}(M,\bar g)}\le C\left(\|\hat d\|_{L^p(\Omega_T)}+\|\hat u\|_{L^p(\Omega_T)}\right)+\|u_0\|_{W^{2,p}(M,\bar g)}\\
&\le C\|\hat d\|_{L^p(\Omega_T)}+\|u_0\|_{W^{2,p}(M,\bar g)}\le C\left(\|d\|_{L^p(\Omega_T)}+\|u_0\|_{W^{2,p}(M,\bar g)}\right), \end{align*} as claimed in (\ref{eq:W2-1-p-a-priori}).
{\bf Step 2 (Existence):} In the case where $a,c,d \in C^\alpha(\Omega_T)$ and $u_0\in C^{2+\alpha}(M)$, the existence of a classical solution $u \in C(\overline \Omega_T) \cap C^{2,1}(\Omega_T)$ of (\ref{eq:linear-parabolic}), (\ref{eq:linear-parabolic-initial}) follows as in \cite[Theorem 5.14]{Lie96}.
In the general case we consider (\ref{eq:linear-parabolic}), (\ref{eq:linear-parabolic-initial}) with coefficients $a_n, c_n, d_n \in C^\alpha(\overline\Omega_T)$, $u_{0,n}\in C^{2+\alpha}(M)$, in place of $a,c,d, u_0$ with the property that $a_n \to a$, $c_n \to c$ in $L^\infty(\Omega_T)$, $d_n \to d \in L^p(\Omega_T)$ as well as $u_{0,n}\to u_0$ in $W^{2,p}$. The associated unique solutions $u_n \in C(\overline \Omega_T) \cap C^{2,1}(\Omega_T)$ are uniformly bounded in $W^{2,1}_p(\Omega_T)$ by (\ref{eq:W2-1-p-a-priori}), and therefore we have $u_n \rightharpoonup u$ in $W^{2,1}_p(\Omega_T)$ after passing to a subsequence. For every $\phi \in C^\infty_c(\Omega_T)$, we then have \begin{align*} &\int_{\Omega_T}\Bigl(\partial_tu(x,t) - a(x,t)\Delta_{\bar g} u(t.x) - c(x,t)u(x,t) - d(x,t)\Bigr)\phi(x,t) d\mu_{\bar g}(x)dt\\ &= \lim_{n \to \infty} \int_{\Omega_T}\Bigl(\partial_tu_n(x,t) - a_{n}(x,t)\Delta_{\bar g} u_n(x,t) - c_{n} (x,t)u_n(x,t) - d_{n}(x,t)\Bigr)\phi(x,t) d\mu_{\bar g}(x)dt = 0, \end{align*} and from this we deduce that $\partial_tu(x,t) - a(x,t)\Delta_{\bar g} u(x,t) - c(x,t)u(x,t) - d(x,t)= 0$ almost everywhere in $\Omega_T$, so $u$ is a strong solution of (\ref{eq:linear-parabolic}).\\ {\bf Step 3:} It remains to show the inequality~(\ref{eq:H-1-inequality-appendix}) in the case where $a,c,d \in C^\alpha(\Omega_T)$ for some $\alpha>0$. Since $u \in C(\overline{\Omega_T}) \cap C^{2,1}(\Omega_T)$ in this case and therefore $$
\|u_0\|_{L^2(M,\bar g)}= \lim_{t \to 0^+}\|u(t)\|_{L^2(M,\bar g)}, $$ it suffices to show that \begin{equation}
\label{eq:H-1-inequality-appendix-sufficient}
\|\nabla u_0\|_{L^2(M,\bar g)} \ge \limsup_{t \to 0^+}\|\nabla u(t)\|_{L^2(M,\bar g)}. \end{equation} If $u_0 \in C^{2+\alpha}(M)$ for some $\alpha>0$, this follows by \cite[Theorem 5.14]{Lie96} with $\lim$ in place of $\limsup$, since the function $t \mapsto u(t)$ is continuous from $[0,T) \to C^{2+\alpha}(M)$ in this case. Moreover, in this case we have, by H\"older's and Young's inequality, \begin{align*}
\frac{d}{dt} \|\nabla u(t)\|_{L^2(M,\bar g)}^2 &= -\int_M \partial_tu(t) \Delta u(t)d\mu_{\bar g} \\
&= - \int_M \Bigl(a(t)|\Delta u(t)|^2 + c(t) u(t)\Delta u(t) +d(t) \Delta u(t) \Bigr)d\mu_{\bar g}\\
&\le - a_T \|\Delta_{\bar g} u(t)\|_{L^2(M,\bar g)}^2 + \|c(t) u(t) + d(t)\|_{L^2(M,\bar g)} \|\Delta_{\bar g} u(t)\|_{L^2(M,\bar g)}\\
&\le - a_T \|\Delta_{\bar g} u(t)\|_{L^2(M,\bar g)}^2 + a_T\|\Delta_{\bar g} u(t)\|_{L^2(M,\bar g)}^2 + \frac{1}{4a_T} \|c(t) u(t) + d(t)\|_{L^2(M,\bar g)}^2\\
&= \frac{1}{4a_T} \|c(t) u(t) + d(t)\|_{L^2(M,\bar g)}^2, \end{align*} and therefore \begin{equation}
\label{eq:intermediate-inequality-appendix}
\|\nabla u(t)\|_{L^2(M,\bar g)}^2 \le \|\nabla u(0)\|_{L^2(M,\bar g)}^2 + \frac{1}{4a_T} \int_0^t \|c(s) u(s) + d(s)\|_{L^2(M,\bar g)}^2 \,ds \qquad \text{for $t>0$.} \end{equation} In the general case, we consider (\ref{eq:linear-parabolic}), (\ref{eq:linear-parabolic-initial}) with a sequence of initial conditions $u_{n,0}$ in place of $u_0$, where $u_{n,0} \to u_0$ in $H^2(M)$. The associated unique solutions $u_n \in C(\overline \Omega_T) \cap C^{2,1}(\Omega_T)$ are uniformly bounded in $W^{2,1}_p(\Omega_T)$ by (\ref{eq:W2-1-p-a-priori}), and they are also uniformly bounded in $C^{2,1}([\varepsilon,T]\times M)$ by \cite[Theorem 5.15]{Lie96} for every $\varepsilon \in (0,T)$. Fix $t \in (0,T)$. Passing to a subsequence, we may assume that $u_n \rightharpoonup u$ in $W^{2,1}_p(\Omega_T)$, $u_n \to u$ strongly in $C^{0}(\overline{\Omega_T})$ and $u_n(t) \to u(t)$ strongly in $C^1(M)$. As in Step 2, we see, by testing with $\phi \in C^\infty_c(\Omega_T)$, that $\partial_tu(x,t) -a(x,t)\Delta_{\bar g} u(x,t) - c(x,t)u(x,t) - d(x,t)= 0$ almost everywhere in $\Omega_T$, so $u$ is the unique strong solution of (\ref{eq:linear-parabolic}), (\ref{eq:linear-parabolic-initial}). Moreover, by (\ref{eq:intermediate-inequality-appendix}) we have \begin{align*}
\|\nabla u(t)\|_{L^2(M,\bar g)}^2 &= \lim_{n \to \infty} \|\nabla u_n(t)\|_{L^2(M,\bar g)}^2 \\
&\le \lim_{n \to \infty}\left( \|\nabla u_n(0)\|_{L^2(M)}^2 + \frac{1}{4a_T} \int_0^t \|c(s) u_n(s) + d(s)\|_{L^2(M,\bar g)}^2 \,ds \right)\\
&= \|\nabla u(0)\|_{L^2(M,\bar g)}^2 + \frac{1}{4a_T} \int_0^t \|c(s) u(s) + d(s)\|_{L^2(M,\bar g)}^2 \,ds. \end{align*} It thus follows that $$
\|\nabla u(t)\|_{L^2(M,\bar g)}^2- \|\nabla u(0)\|_{L^2(M,\bar g)}^2 \le \frac{1}{4a_T} \int_0^t \|c(s) u(s) + d(s)\|_{L^2(M,\bar g)}^2 \,ds $$ and therefore $$
\limsup_{t \to 0} \Bigl(\|\nabla u(t)\|_{L^2(M,\bar g)}^2- \|\nabla u(0)\|_{L^2(M,\bar g)}^2\Bigr) \le \frac{1}{4a_T} \lim_{t \to 0^+} \int_0^t \|c(s) u(s) + d(s)\|_{L^2(M,\bar g)}^2 \,ds = 0, $$ as claimed in (\ref{eq:H-1-inequality-appendix-sufficient}). \end{proof}
Next we prove a maximum principle for solutions of (\ref{eq:linear-parabolic}),~(\ref{eq:linear-parabolic-initial}). We need the following preliminary lemma.
\begin{lemma}
\label{prelim-1-appendix}
Let $T>0$.
\begin{itemize}
\item[(i)] For any function $u \in C^2(M)$ we have
$$
\int_{\{x\in M\mid u(x)>0\}}\Delta_{\bar g} u d\mu_{\bar g}\le 0.
$$
\item[(ii)] Let $u,\rho \in C^{1}([0,T])$ be functions with $u(0) \le 0$ and $\rho(T) \ge 0$. Then
\begin{equation}
\label{eq:lemma-claim-prelim}
\int_{\{t\in[0,T]\mid u(t)>0\}}\bigl(\rho(t) \partial_tu(t) + \kappa u(t)\bigr) \,dt \ge 0 \qquad \text{with}\quad \kappa:= \sup_{s \in (0,T)}\partial_t\rho(s).
\end{equation}
\item[(iii)] Let $u \in C^{2,1}(\Omega_T) \cap C^{0,1}(\overline\Omega_T)$, $\rho \in C^{0,1}(\overline\Omega_T)$ be functions with $u \le 0$ on $M\times \{0\}$ and $\rho \ge 0$ on $M\times \{T\}$. Then we have
\begin{equation}
\label{eq:lemma-claim}
\begin{split}
&\int_{\{(x,t)\in M\times[0,T]\mid u(x,t)>0\}}(\rho(x,t)\partial_t u(x,t) + \kappa u(x,t) - \Delta_{\bar g} u(x,t))d\mu_{\bar g}(x)dt \ge 0 \\
&\text{with}\quad\kappa:= \sup_{(s,x)\in M\times (0,T)}\partial_t\rho(s,x).
\end{split}
\end{equation}
\end{itemize}
\end{lemma}
\begin{proof}
(i) By Lebesgue's theorem, it suffices to prove
\begin{equation}
\label{eq:lemma-claim-eps}
\int_{\{x\in M\mid u(x)>\varepsilon_n\}}\Delta_{\bar g} ud\mu_{\bar g}\le 0
\end{equation}
for a sequence $\varepsilon_n \to 0^+$. By Sard's Lemma, we may choose this sequence such that $\Omega_\varepsilon:= \{x\in M\mid u(x)>\varepsilon_n\}$ is an open set of class $C^1$, whereas the outer unit vector field of $\Omega_\varepsilon$ is given by $(x,t)\mapsto- \frac{\nabla_{\bar g} u(x,t)}{|\nabla_{\bar g} u(x,t)|_{\bar g}}$. Hence (\ref{eq:lemma-claim-eps}) follows from the divergence theorem.\\
(ii) The set $\{t\in[0,T]\mid u(t)>0\}$ is a union of at most countably many open intervals $I_j$, $j \in \mathds N$. For any such interval, partial integration gives
$$
\int_{I_j }\Bigl(\rho(t) \partial_tu(t)+ \partial_t\rho(t) u(t)\Bigr) \,dt = \left\{
\begin{aligned}
&0, && \qquad \text{if $T \not \in \overline I_j$;}\\
&\rho(T) u(T) \ge 0, && \qquad \text{if $T \in \overline I_j$.}
\end{aligned}
\right.
$$
Consequently, $$ \int_{\{t\in[0,T]\mid u(t)>0\}}\rho(t) \partial_tu(t)\,dt \ge - \int_{\{t\in[0,T]\mid u(t)>0\}}\partial_t\rho(t) u(t) \,dt \ge - \int_{\{t\in[0,T]\mid u(t)>0\}}\kappa u(t) \,dt $$ with $\kappa$ given in (\ref{eq:lemma-claim-prelim}). This shows the claim.\\ (iii) This is a direct consequence of (i), (ii) and Fubini's theorem.
\end{proof}
\begin{proposition}
\label{max-principle} (Maximum principle)\\
Let $T>0$, $a, c \in C(\overline\Omega_T)$ with $a_T:= \min \limits_{(x,t)\in\overline\Omega_T}a(x,t) >0$, let $d \in L^p(\Omega_T)$ for some $p>2$ with $d_T := \sup_{(x,t)\in\Omega_T}d(x,t)< \infty$, and let $u_0 \in W^{2,p}(M,\bar g)$. Moreover, let $u \in W^{2,1}_p(\Omega_T)$ be the unique solution of (\ref{eq:linear-parabolic}), (\ref{eq:linear-parabolic-initial}).
\begin{itemize}
\item[(i)] If $u_0 \le 0$ on $M$ and $d_T \le 0$, then $u \le 0$ on $\Omega_T$.
\item[(ii)] If $c \equiv 0$ on $\Omega_T$, then \begin{equation}
\label{max-principle-est}
u(x,t) \le \|u_0^+\|_{L^\infty(M,\bar g)} + t d_T \qquad \text{for $t \in [0,T],\: x \in M$.} \end{equation} \end{itemize} \end{proposition}
\begin{proof} (i) {\bf Step 1:} We consider the special case $a \in C^{0,1}(\overline\Omega_T)$, $u_0\le 0$ and $d_T\le -\varepsilon$ for some $\varepsilon>0$. We put $\rho:= \frac{1}{a} \in C^{0,1}(\overline \Omega_T)$ and $\kappa:= \sup \limits_{(s,x)\in M \times (0,T)}\partial_t\rho(s,x)$ as in~(\ref{eq:lemma-claim}). Moreover, we consider the function $$ \breve{u}\in W^{2,1}_p(\Omega_T), \qquad \breve{u}(x,t)= \mathrm{e}^{- \breve{\kappa} t} u(x,t) $$
with $\breve{\kappa} = \frac{|\kappa|}{\min_{(x,t)\in \overline\Omega_T}\rho(x,t)} + \|c\|_{L^\infty(\Omega_T)}$, noting that $\breve{u}$ satisfies \begin{equation}
\label{eq:max-princ-mod-eq}
\begin{split} \rho(x,t) \partial_t\breve{u}(x,t)& -\Delta_{\bar g} \breve{u}(x,t) + \kappa \breve{u}(x,t)\\ &=\mathrm{e}^{-\breve{\kappa}t}\Bigl(u(x,t)(\rho(x,t)c(x,t)-\rho(x,t)\breve{\kappa}+\kappa)+\rho(x,t)d(x,t)\Bigr)\\ &\le - \rho(x,t) \varepsilon \mathrm{e}^{- \breve{\kappa} t} \qquad \text{almost everywhere in $\{(x,t)\in \Omega_T\mid\breve{u}(x,t)>0\}$.}
\end{split} \end{equation}
We now let $(u_n)_{n\in\mathds N}$ be a sequence in $C^{2,1}(\Omega_T) \cap C^{0,1}(\overline\Omega_T)$ with $u_n(x,0)\le0$ and $u_n \to \breve{u}$ in $W^{2,1}_p(\Omega_T)$. Since the functions $g_n:= 1_{\{(x,t)\in M\times[0,T]\mid u_n(x,t)>0\}}$ are bounded in $L^{p'}(\Omega_T)$, we may pass to a subsequence such that $g_n \rightharpoonup g$ in $L^{p'}(\Omega_T)$, where $g \ge 0$ and $g \equiv 1$ in $\{(x,t)\in M\times[0,T]\mid \breve{u}(x,t) >0\}$, since $u_n \to \breve{u}$ uniformly as a consequence of (\ref{eq:embedding-hoelder}) and therefore $g_n \to 1$ pointwisely on $\{(x,t)\in M\times[0,T]\mid \breve{u}(x,t)>0\}$. Applying Lemma~\ref{prelim-1-appendix} (iii) to $u_n$, we find that \begin{align*}
0&\le \int_{\{(x,t)\in M\times[0,T]\mid u_n(x,t)>0\}} \Bigl( \rho(x,t) \partial_tu_n(t) -\Delta_{\bar g} u_n(x,t) + \kappa u_n(x,t) \Bigr) d\mu_{\bar g}(x)dt\\
&= \int_{M \times (0,T)} g_n(x,t) \Bigl( \rho(x,t) \partial_tu_n(x,t) -\Delta_{\bar g} u_n(x,t) + \kappa u_n(x,t) \Bigr) d\mu_{\bar g}(x)dt \end{align*} for all $n \in \mathds N$ and therefore \begin{align*}
0 &\le \lim_{n \to \infty} \int_{M \times (0,T)} g_n(x,t) \Bigl( \rho(x,t) \partial_tu_n(x,t) -\Delta_{\bar g} u_n(x,t) + \kappa u_n(x,t) \Bigr) d\mu_{\bar g}(x)dt\\
&= \int_{M \times (0,T)} g(x,t) \Bigl( \rho(x,t) \partial_t\breve{u}(x,t) -\Delta_{\bar g} \breve{u}(x,t) + \kappa \breve{u}(x,t)\Bigr) d\mu_{\bar g}dt\\
&\le - \int_{M\times (0,T)} g(x,t) \rho(x,t) \varepsilon \mathrm{e}^{- \breve{\kappa} t} d\mu_{\bar g}(x)dt \le - \int_{\{(x,t)\in M\times(0,T)\mid \breve{u}(x,t)>0\}}\rho(x,t) \varepsilon \mathrm{e}^{- \breve{\kappa} t} d\mu_{\bar g}(x)dt . \end{align*} We thus conclude that $\{(x,t)\in M\times(0,T)\mid \breve{u}(x,t)>0\} = \{(x,t)\in M\times(0,T)\mid u(x,t) >0\} = \varnothing$ and therefore $u \le 0$ in $M\times (0,T)$.\\ {\bf Step 2:} In the special case where $a \in C^{0,1}(\overline\Omega_T)$, $u_0\le 0$ and $d_T\le 0$, we may apply Step 1 to the functions $u_\varepsilon \in W^{2,1}_p(\Omega_T)$ defined by $u_\varepsilon(x,t)= u(x,t)- \varepsilon t$, which yields that $u_\varepsilon \le 0$ for every $\varepsilon>0$ and therefore $u \le 0$ in $\Omega_T$.\\ {\bf Step 3:} In the general case, we consider a sequence $a_n \in C^{0,1}(\overline\Omega_T)$ with $a_n \to a$ in $C(\overline \Omega_T)$, and we let $u_n$ denote the associated solutions of (\ref{eq:linear-parabolic}),~(\ref{eq:linear-parabolic-initial}) with $a$ replaced by $a_n$. As in the end of the proof of Proposition~\ref{sec:appendix}, we then find that, after passing to a subsequence, $u_n \rightharpoonup \tilde u$ in $W^{2,1}_p(\Omega_T)$, where $\tilde u$ is a solution of (\ref{eq:linear-parabolic}),~(\ref{eq:linear-parabolic-initial}). By uniqueness, we have $u= \tilde u$. Moreover, since $u_n \le 0$ for all $n$ by Step 3, we have $u = \tilde u \le 0$, as required.\\
(ii) We consider the function $v \in W^{2,1}_p(\Omega_T)$ given by $v(x,t)= u(x,t)- \|u_0^+\|_{L^\infty(M)}- t d_T$, which, by assumption, satisfies (\ref{eq:linear-parabolic}), (\ref{eq:linear-parabolic-initial}) with $c \equiv 0$, $d- d_T$ in place of $d$ and $u_0 - \|u_0^+\|_{L^\infty(M)}$ in place of $u_0$. Then (i) yields $v \le 0$ in $\Omega_T$, and therefore $u$ satisfies (\ref{max-principle-est}). \end{proof}
\printbibliography
\end{document} | arXiv |
Thomas J. Osler
Thomas Joseph Osler (April 26, 1940 – March 26, 2023) was an American mathematician, national champion distance runner, and author.
Thomas J. Osler
Osler at whiteboard in 2020
Born(1940-04-26)April 26, 1940
Camden, New Jersey, U.S.
DiedMarch 26, 2023(2023-03-26) (aged 82)
Alma mater
• Drexel University (BS)
New York University (PhD)
Scientific career
FieldsMathematics
InstitutionsRowan University
Early life and education
Born in 1940 in Camden, New Jersey,[1] Osler was a graduate of Camden High School in 1957 and then studied physics at Drexel University, graduating in 1962.[2][3] He completed his PhD at the Courant Institute of Mathematical Sciences of New York University,[4] in 1970. His dissertation, Leibniz Rule, the Chain Rule, and Taylor's Theorem for Fractional Derivatives, was supervised by Samuel Karp.[5]
Career
Osler taught at Saint Joseph's University and the Rensselaer Polytechnic Institute[6] before joining the mathematics department at Rowan University in New Jersey in 1972;[7] he was a full professor at Rowan University until his death.[4]
In mathematics, Osler is best known for his work on fractional calculus.[8][9][10] He also gave a series of product formulas for $\pi $ that interpolate between the formula of Viète and that of Wallis.[11]
In 2009, the New Jersey Section of the Mathematical Association of America gave him their Distinguished Teaching Award.[12][13] A mathematics conference was held at Rowan University in honor of his 70th birthday in 2010.[6]
Running
Osler won three national Amateur Athletic Union championships at 25 km (1965), 30 km and 50 mi (1967).[14][15] Osler won the 1965 Philadelphia Marathon, finishing the race in freezing-cold weather in a time of 2:34:07.[16] In the course of his career he has won races of nearly every length from one mile to 100 miles.
Osler was involved in the creation of the Road Runners Club of America with Olympian Browning Ross; together they were elected as co-secretaries in 1959[17] and were among the four first official elected officers of the newly formed club.[18] He served on the Amateur Athletic Union Standards Committee in 1979.[19] He has been credited with helping to popularize the idea of walk breaks among US marathon runners.[1][3]
In 1980, Osler was inducted into the Road Runners Club of America Hall of fame.[17][20]
Running publications
Osler was the author of several books and booklets on running:
• Guide to Long Distance Running (a 20-page booklet coauthored with Edward Dodd) was published in 1965 by the South Jersey Track Club.[21]
• The Conditioning of Distance Runners (a 29-page booklet) was published in 1967 by the Long Distance Log.[1][3][21] It was reprinted in 1984–1985 in Runner's World magazine[22][23] and reprinted with a new foreword by Amby Burfoot in 2019.[24]
• Serious Runner's Handbook: Answers to Hundreds of your Running Questions (187 pages) was published by World Publications in 1978.[25]
• Ultramarathoning: The Next Challenge (299 pages, coauthored with Edward Dodd) was also published by World Publications, in 1979.[26]
Personal life and death
Osler was a resident of Glassboro, New Jersey.[12]
Osler died on March 26, 2023, at the age of 82.[27]
References
1. Benyo, Richard; Henderson, Joe (2002). ""Tom Osler"". Running Encyclopedia. Human Kinetics. ISBN 0736037349.
2. "It All Adds Up: Running, teaching and math". Rowan Today. Rowan University. September 16, 2009.
3. Englehart, Richard (September 2008). "Like a Cat Chases Mice". Marathon & Beyond.
4. "Tom Osler, PhD". Faculty and Staff. Rowan University Mathematics Department. Retrieved March 27, 2023.
5. Thomas J. Osler at the Mathematics Genealogy Project
6. "Oslerfest: Prominent mathematicians to pay tribute to legendary Rowan prof". Rowan Today. Rowan University. April 12, 2010.
7. "Osler honored for distinguished teaching by Mathematical Association of America". Rowan Today. Rowan University. April 17, 2009.
8. Yang, Xiao-Jun; Gao, Feng; Ju, Yang (2020). "Section 2.3: Osler fractional calculus". General Fractional Derivatives with Applications in Viscoelasticity. Academic Press. pp. 107–111. ISBN 9780128172094.
9. Almeida, Ricardo (2019). "Further properties of Osler's generalized fractional integrals and derivatives with respect to another function". The Rocky Mountain Journal of Mathematics. 49 (8): 2459–2493. doi:10.1216/RMJ-2019-49-8-2459. hdl:10773/27488. MR 4058333. S2CID 214139065.
10. Nishimoto, Katsuyuki (1977). "Osler's cut and Nishimoto's cut". Journal of the College of Engineering of Nihon University, Series B. 18: 9–13. MR 0486359.
11. Arndt, Jörg; Haenel, Christoph (2001). "12.8 Viète ✕ Wallis = Osler". π Unleashed. Berlin Heidelberg New York: Springer-Verlag. pp. 160–162. ISBN 3-540-66572-2.
12. Shryock, Bob (May 7, 2009). "Running Man". South Jersey Times.
13. "New Jersey Section Archives". Mathematical Association of America. Retrieved November 23, 2020.
14. "United States Champions (with Local Connections)". Retrieved November 23, 2020.
15. United States Championships (Men). GBR Athletics. Retrieved November 25, 2020.
16. "Osler Captures Phila. Marathon", Asbury Park Press, December 27, 1965. Accessed November 24, 2020. "Philadelphia – Tom Osler of the South Jersey Track Club, 25-year-old New York University graduate student from Camden, N.J., scored an easy victory in the Ruthrauff Marathon race yesterday through Fairmount Park. Osier braved sub-freezing temperatures and stiff winds to cover the 26 miles, 385 yards in two hours, 34 minutes and seven seconds."
17. "History of Road Runners Club of America" (PDF). Road Runners Club of America. Retrieved November 24, 2020.
18. "50th Anniversary Report". Road Runners Club of America. Retrieved November 24, 2020.
19. "Pertinent Trivia" (PDF). Measurement News (88): 14. March 1988.
20. "Distance Running History". Road Runner's Club of America. Retrieved November 24, 2020.
21. Morison, Ray Leon (June 1975). An Annotated Bibliography of Track and Field Books Published in the United States Between 1960–1974 (PDF) (Master's thesis). San Jose State University. pp. 23, 33 – via Education Resources Information Center.
22. Osler, Tom (December 1984). "The Conditioning of Distance Runners (part 1)". Runner's World: 52–57, 87.
23. Osler, Tom (January 1985). "The Conditioning of Distance Runners (part 2)". Runner's World: 44–47, 80.
24. Osler, Thomas J. (1967). The Conditioning of Distance Runners (2019 ed.). Y42K Publishing. ISBN 9781710036725.
25. Osler, Tom (1978). Serious Runner's Handbook: Answers to Hundreds of Your Running Questions. Mountain View, California, USA: World Publications, Inc. ISBN 0-89037-126-1. Briefly reviewed in "Books". The Marine Corps Gazette. 1978. pp. 57–60; see in particular p. 59.
26. Osler, Tom; Dodd, Ed (1979). Ultramarathoning: The Next Challenge. Mountain View, California, USA: World Publications, Inc. ISBN 0-89037-169-5. See also Edwards, Sally (September 1983). "Ultramarathoning—A Dying Sport". UltraRunning Magazine. The book Ultramarathoning by Tom Osler and Ed Dodd had a shelf life of about 2 years, with 6,000 copies printed before the publisher (World Publications) discontinued it.
27. "Thomas J. Osler". South Jersey Times. March 27, 2023. Retrieved March 27, 2023 – via Legacy.com.
External links
• Thomas J. Osler publications indexed by Google Scholar
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Distance was 20 miles from 1930 to 1932
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| Wikipedia |
\begin{document}
\begin{abstract} We describe a family $\{\Psi_{\alpha, \beta}\}$ of polygon exchange transformations parameterized by points $(\alpha, \beta)$ in the square $[0, \half] \times [0, \half]$. Whenever $\alpha$ and $\beta$ are irrational, $\Psi_{\alpha, \beta}$ has periodic orbits of arbitrarily large period. We show that for almost all parameters, the polygon exchange map has the property that almost every point is periodic. However, there is a dense set of irrational parameters for which this fails. By choosing parameters carefully, the measure of non-periodic points can be made arbitrarily close to full measure. These results are powered by a notion of renormalization which holds in a more general setting. Namely, we consider a renormalization of tilings arising from the Corner Percolation Model. \end{abstract} \maketitle
\section{Introduction} \name{sect:introduction} Let $X$ be a finite disjoint union of polygons in the plane. A {\em polygon exchange map} of $X$, $T:X \to X$, cuts $X$ into finitely many polygonal pieces, and applies a translation to each piece so that the image $T(X)$ has full area in $X$. There is some ambiguity of definition on the boundaries of the pieces.
Polygon exchange maps are natural generalizations of interval exchange maps, and yet comparatively little is understood about the dynamics of a generic polygon exchange map. However, some polygon exchange maps are well understood using the idea of renormalization. As a simple example of renormalization, a first return map of $T$ to a union of polygonal subsets might be affinely conjugate to the original map $T$. Once a renormalization procedure is found, we can hope to exploit it to deduce detailed information about the dynamical system. Papers on polygon exchange maps following this philosophy include \cite{AKT01}, \cite{LKV04}, \cite{L07} and \cite{Schwartz10}.
In this paper, we give the first example of a two dimensional parameter space of polygon exchange maps which is invariant under a renormalization operation. In our case, this means that each map in the family admits a return map which is affinely conjugate to a map in the family. (This family will be called $\{\til \Psi_{\alpha, \beta}\}$.) We then exploit this renormalization operation to understand the dynamical behavior of these maps.
The polygon exchange maps we describe are in fact {\em rectangle exchange maps}. That is, all the polygons used define the map are rectangles with horizontal and vertical sides. To define these maps, consider the planar lattice $\Lambda \subset \R^2$ generated by the vectors $(\half, \half)$ and $(-\frac{1}{2}, \half)$. This lattice contains $\Z^2$ as an index two subgroup. Let $Y$ be the torus $\R^2/\Lambda$. A fundamental domain for the action of $\Lambda$ by translation on $\R^2$ is given by the union of the two squares $$A_1=[0,\thalf) \times [0, \thalf) \and A_{-1}=[0,\thalf) \times [\thalf,1).$$ Let $N$ be the finite set of four elements, \begin{equation} \name{eq:Nset} N=\{(1,0), (-1,0), (0,1), (0,-1)\} \subset \R^2. \end{equation} We think of $Y \times N$ as a disjoint union of four copies of the torus $Y$. Fix two parameters $\alpha, \beta \in [0, \half]$. We define the rectangle exchange map $\Psi_{\alpha, \beta}:Y \times N \to Y \times N$ according to the following rule. If $(x,y) \in A_s \pmod{\Lambda}$ with $s \in \{\pm 1\}$ and $\v=(a,b) \in N$, then \begin{equation} \name{eq:psi intro} \Psi_{\alpha, \beta}\big((x,y),\v\big)=\big((x+b s\alpha,y+a s \beta) \pmod{\Lambda},(bs,as)\big). \end{equation} Note that $(bs,as) \in N$. So fixing this data, only one coordinate changes in moving from $(x,y)$ to $(x+b s\alpha,y+a s \beta)$. Figure \ref{fig:polygon exchange} illustrates a map in this family.
\begin{figure}
\caption{This illustrates the map $\Psi=\Psi_{\alpha, \beta}$ defined in equation \ref{eq:psi intro}. Above the line indicates the sets $A_s^{(a,b)}=A_s \times \{(a,b)\}$, and below illustrates their images under $\Psi$. In both cases, the tori are drawn $Y \times \{(1,0)\}$, $Y \times \{(-1,0)\}$, $Y \times \{(0,1)\}$ and $Y \times \{(0,-1)\}$, from left to right.}
\end{figure}
These maps have many periodic trajectories. In fact, \begin{theorem} \name{thm:large periods} Whenever $\alpha$ and $\beta$ are irrational, there are points in $Y \times N$ which are periodic under $\Psi_{\alpha, \beta}$ of arbitrary large period. \end{theorem}
\begin{remark} It follows that $\Psi_{\alpha, \beta}$ is not conjugate to a product of interval exchange maps. \end{remark}
Every periodic point has an open neighborhood of points which are periodic and have the same period. It is natural to ask ``what is the total area of periodic points?'' Let $\lambda$ be Lebesgue measure on $Y \times N$, rescaled so that $\lambda(Y \times N)=1$. Let $M(\alpha, \beta)$ denote the $\lambda$-measure of the periodic points, i.e., $$M(\alpha, \beta)=\lambda \{p \in Y \times N~:~\Psi_{\alpha, \beta}^n(p)=p\quad \textrm{for some $n \geq 1$.}\}.$$ Our renormalization operation allows us to prove the following theorems.
\begin{theorem}[Periodicity almost everywhere] \name{thm:periodicity} $M(\alpha, \beta)=1$ for Lebesgue-almost every parameter $(\alpha, \beta) \in [0, \half] \times [0, \half]$. \end{theorem}
However, this result does not hold for all irrational pairs $(\alpha, \beta)$.
\begin{theorem} \name{thm:4} For any $\epsilon>0$, there are irrationals $\alpha$ and $\beta$ so that $M(\alpha, \beta)<\epsilon$. \end{theorem}
From this together with basic observations about the action of renormalization on the parameter space, we obtain: \begin{corollary} \name{cor:5} There is a dense set of irrational parameters $(\alpha, \beta)$ so that $M(\alpha, \beta) \neq 1$. \end{corollary}
Questions involving the measure-theoretic prevalence of periodic orbits for piecewise isometries are common in the literature. Probably the first questions of this form appear in \cite{Ashwin97} and \cite[\S6]{Goetz00}. The above theorems highlight the subtlety of this question. For the family $\{\Psi_{\alpha, \beta}\}$, we utilize a renormalization procedure to analyze $M(\alpha,\beta)$. We were able to understand the value of this function for almost every every pair $(\alpha, \beta)$ in Theorem \ref{thm:periodicity}, and for very specific pairs in Theorem \ref{thm:4} and Corollary \ref{cor:5}. But, it is reasonable to ask if there is a nice characterization of the set $$\{(\alpha, \beta)~:~M(\alpha, \beta) \neq 1\}.$$ Or for instance, what is this set's Hausdorff dimension? These finer questions remain unanswered and appear difficult.
\subsection{Renormalizing the polygon exchange maps} \name{sect:intro renormalization} A {\em renormalization} of a polygon exchange map $T:X \to X$, is the choice of a finite union $Y$ of polygonal subsets of $X$ with disjoint interiors such that the first return map $T_Y:Y \to Y$ is also a polygon exchange map.
For the maps $\Psi_{\alpha, \beta}$, we actually renormalize on a double cover. Let $\til Y=\R^2/\Z^2$, and note that the natural projection $\pi:\til Y \to Y$ is a double cover. We define $\til A_s=\pi^{-1}(A_s)$ for $s \in \{\pm 1\}$. Then we define the lift of the map $\Psi_{\alpha, \beta}$ to be the map $\til \Psi_{\alpha, \beta}:\til Y \times N \to \til Y \times N$ given by \begin{equation} \name{eq:psi til} \til \Psi_{\alpha, \beta}\big((x,y),\v\big)=\big((x+b s\alpha,y+a s \beta) \pmod{\Z^2},(bs,as)\big), \end{equation} where $s \in \{\pm 1\}$ is chosen so that $(x,y) \in \til A_s$.
The maps $\til \Psi_{\alpha, \beta}$ are parameterized by a choice of $(\alpha, \beta)$ from the square $[0,\half] \times [0, \half]$. We will show when $(\alpha, \beta)$ is taken from the open square $(0,\half) \times (0, \half)$, a certain return map of $\til \Psi_{\alpha, \beta}$ is affinely conjugate to a map of the form $\til \Psi_{f(\alpha), f(\beta)}$. Here $f$ is the map \begin{equation} \name{eq:f} f:[0, \thalf) \to [0, \thalf]\quad \textrm{is given by} \quad f(t)=\frac{t}{1-2 t} \pmod{G}, \end{equation} where $G$ is the group of isometries of $\R$ preserving $\Z$. This group is generated by $t \mapsto -t$ and $t \mapsto 1-t$, so the interval $[0, \half]$ represents a fundamental domain for the group action. We use $t \pmod{G}$ to denote the unique $g(t) \in [0, \half]$ with $g \in G$. To define the return map under consideration we define the rectangle \begin{equation} \name{eq:Z} Z=[\alpha, 1-\alpha) \times [\beta,1-\beta) \subset \til Y. \end{equation} We define $\wh \Psi$ be the first return map of $\til \Psi_{\alpha,\beta}$ to $Z \times N$. This map is affinely conjugate to the map $\til \Psi_{f(\alpha), f(\beta)}$ via a conjugating map of the form \begin{equation} \name{eq:phi} \phi:Z \times N \to \til Y \times N; \quad \phi(x,y,\v)=\big(\psi_\alpha(x), \psi_\beta(y), \v\big). \end{equation} Here, we have used $\psi_t$ with $t \in \{\alpha, \beta\}$ to denote the maps \begin{equation} \name{eq:psi} \psi_t:[t, 1-t) \to \R /\Z; \quad \psi_t(x)=\begin{cases} \frac{x-\frac{1}{2}}{1-2 t}+\frac{1}{2} & \textit{if $\exists n \in \Z$ s.t. $n \leq \frac{t}{1-2 t}<n+\half$,}\\ \frac{\frac{1}{2}-x}{1-2 t} & \textit{otherwise.} \end{cases} \end{equation} The two cases correspond to the possibility that we use an orientation preserving or reversing element of $G$ to move $\frac{t}{1-2 t}$ into $[0, \half]$.
We now formally state our renormalization theorem. \begin{theorem}\name{thm:ren} Fix parameters $\alpha, \beta \in (0, \half)$. Define $f$, $Z$, and $\phi$ as above. The first return map $\wh \Psi$ of $\til \Psi_{\alpha,\beta}$ to $Z \times N$ satisfies $$\phi \circ \wh \Psi=\til \Psi_{f(\alpha),f(\beta)} \circ \phi.$$ \end{theorem}
The primary case of interest is when $\alpha$ and $\beta$ are irrational. Then, $f(\alpha)$ and $f(\beta)$ are also irrational. Therefore we can apply the above renormalization infinitely many times.
\subsection{Corner Percolation and Truchet tilings} \name{sect:intro corner} We will understand the family of rectangle exchange maps $\{\Psi_{\alpha, \beta}\}$ using a combinatorial tool we call the {\em arithmetic graph}, following Schwartz. (See \cite{S07}, for instance). In our case, this fundamental tool is connected to the corner percolation model introduced by B\'alint T\'oth, and studied in depth by G\'abor Pete \cite{Pete08}. (We give a different treatment of the topic in this paper.)
The {\em corner percolation tiles} are the four $1 \times 1$ square tiles decorated by arcs as below. \begin{center} \includegraphics[height=0.5in]{corners} \end{center} Consider tilings of the plane by corner percolation tiles centered at the points in $\Z^2$. Any two adjacent tiles meet along a common edge. We will say that such a tiling is a {\em corner percolation tiling} if for each pair of adjacent tiles meeting along a common edge $e$, either both the arcs of the tiles touch $e$, or neither of the arcs touch $e$. So, in a corner percolation tiling, the arcs of the tiles join to form a family of simple curves in the plane. These are the {\em curves} of the tiling.
The {\em Truchet tiles} are the two $1 \times 1$ squares decorated by arcs as below. \begin{center} \includegraphics[height=0.5in]{tiles} \end{center} We call the left tile $T_{-1}$ and the right tile $T_{1}$. The subscripts were chosen to indicate the slope of segments formed by straightening the arcs to segments.
Given a function $\tau:\Z^2 \to \{\pm 1\}$, the {\em Truchet tiling determined by $\tau$} is the tiling of the plane formed by placing a copy of the tile $T_{\tau(m,n)}$ centered at the point $(m,n)$ for each $(m,n) \in \Z^2$. We denote this tiling by $\tiling{\tau}$. Variations of these tilings were first studied for aesthetic reasons by S\'ebastien Truchet in the early 1700s \cite{Truchet}, and this version of tiles were first described by Smith and Boucher
\cite{Smith87}. An example of a Truchet tiling relevant to this paper is given in figure \ref{fig:arithmetic graph}.
There is a two-to-one map from the corner percolation tiles to the Truchet tiles given by taking the union of the decorations of a corner percolation tile and its rotation by 180 degrees. By applying this map to each tile in a corner percolation tiling, we obtain a {\em corner percolation induced Truchet tiling}. \begin{proposition} \name{prop:corner perc induced} Let $\tau:\Z^2 \to \{\pm 1\}$. The following statements are equivalent. \begin{enumerate} \item The Truchet tiling $[\tau]$ is induced by a corner percolation tiling. \item There are maps $\Z \to \{\pm 1\}$ given by $m \mapsto \omega_m$ and $n \mapsto \eta_n$ so that $\tau(m,n)=\omega_m \eta_n$. \item For each $m,n \in \Z$, we have the following identity involving a product of values of $\tau$: $$\tau(m,n) \tau(m+1,n) \tau(m,n+1) \tau(m+1,n+1)=1.$$ \end{enumerate} \end{proposition} The easiest way to prove this statement is to prove that the first and second statements are equivalent to the third. We leave the proof to the reader.
\subsection{Dynamics on Truchet tilings} \name{sect:intro dynamics on truchet} We will explain how to think of the space of Truchet tilings as a dynamical system.
Consider the unit square with horizontal and vertical sides centered at the origin. We let $N$ be the collection of four inward pointed unit normal vectors based at the midpoints of the edges of this square, as in equation \ref{eq:Nset}. \begin{center} \includegraphics[height=1in]{directions3} \end{center}
Let $\sT$ denote the collection of all maps $\Z^2 \to \{\pm 1\}$. The collection of maps $\sT$ should be given the product topology (or equivalently, the topology of pointwise convergence on compact sets).
We will define a dynamical system on $\sT \times N$. First we give an informal definition. Choose $(\tau, \v) \in \sT \times N$. The inward normal $\v \in N$ is a vector based at a midpoint of an edge of the square centered at the origin. The Truchet tiling determined by $\tau$ places the tile $T_{\tau(0,0)}$ at the origin. We drag the vector inward along an arc of this tile keeping the vector tangent to the arc. After a quarter turn, we end up as a vector pointed out of the square centered at the origin. So, the vector points into a square adjacent to the square at the origin. We translate the tiling and this vector so that this adjacent square becomes centered at the origin.
Formally, this is the dynamical system $\Phi_0:\sT \times N \to \sT \times N$ given by \begin{equation} \name{eq:Phi} \Phi_0 \big(\tau, (a,b)\big)=\big(\tau \circ S_{s(b,a)}, s(b,a)\big), \end{equation} where $s=\tau(0,0) \in \{\pm 1\}$ and $S_{s(b,a)}$ is the translation of $\Z^2$ by the vector $s(b,a)$.
It is important to note that because the corner percolation induced Truchet tilings are translation invariant, they are also $\Phi_0$ invariant. So, $\Phi_0$ restricts to an action on corner percolation induced Truchet tilings.
Let $\Omega_{\pm}$ to denote the collection of all maps $\Z \to \{\pm 1\}$. The set $\Omega_\pm$ is a shift space. We define the {\em shift map} $\sigma: \Omega_\pm \to \Omega_\pm$ by \begin{equation} \name{eq:shift} [\sigma(\omega)]_n=\omega_{n+1}. \end{equation} When $\Omega_\pm$ is equipped with its natural topology, $\sigma$ is a homeomorphism of $\Omega_\pm$.
Consider the map $\Omega_{\pm} \times \Omega_{\pm} \to \sT$ given by $(\omega, \eta) \mapsto \tau_{\omega, \eta}$, where $\tau_{\omega,\eta}$ denotes the map \begin{equation} \name{eq:tau omega} \tau_{\omega, \eta}: \Z^2 \to \{\pm 1\}; \quad \tau_{\omega, \eta}(m,n)=\omega_m \eta_n \end{equation} as in statement 2 of Proposition \ref{prop:corner perc induced}. This map is two-to-one, and the image is the collection of corner percolation induced Truchet tilings. There is a natural lift of the action of $\Phi_0$ on the image to the space $X=\Omega_\pm \times \Omega_\pm \times N$. This lift is the map
$\Phi:X \to X$ given by \begin{equation} \name{eq:Phi2} \Phi\big(\omega, \eta, (a,b)\big)=\big(\sigma^{sb} (\omega), \sigma^{sa} (\eta), s(b,a)\big), \quad \textrm{with $s=\omega_0 \eta_0 \in \{\pm 1\}$.} \end{equation}
\subsection{Overview} \name{sect:overview} We have now introduced enough of the mathematical objects appearing in the paper, so we can give an overview of the ideas of this paper.
The rectangle exchange maps $\til \Psi_{\alpha, \beta}$ are factors of the map $\Phi$ in the sense that for all irrational $\alpha$ and $\beta$, there is an embedding \begin{equation} \name{eq:factor} \pi:\til Y \times N \to X=\Omega_\pm \times \Omega_\pm \times N \quad \text{so that} \quad \pi \circ \til \Psi_{\alpha, \beta}=\Phi \circ \pi. \end{equation} The map $\pi$ can be extended to a continuous embedding from a coding space as is often done for interval exchange maps. See the coding construction for interval exchange maps in \cite{KH95}, for instance.
We will describe a renormalization operation for the map $\Phi$. Using the map $\pi$, we are able to restrict this renormalization operation to a renormalization of the rectangle exchange maps $\til \Psi_{\alpha, \beta}$. This enables us to prove Theorem \ref{thm:ren}.
We are able to prove our measure theoretic results using a detailed analysis of the renormalization of these rectangle exchange maps. Of particular importance is a finite dimensional cocycle defined over the renormalization dynamics of the parameter space. We call this the {\em return time cocycle}, because the cocycle conveys information about return times of the rectangle exchange maps to the subsets we use to define the return maps for our renormalization.
We will now outline the proof of Periodicity Almost Everywhere (Theorem \ref{thm:periodicity}): \begin{enumerate} \item For each $\alpha, \beta \in [0,\half]$, we define the measure $\nu_{\alpha, \beta}$ on $X$ to be $\lambda \circ \pi^{-1}$, where $\lambda$ is the Lebesgue probability measure on $\til Y \times N$ and $\pi$ is the embedding which was mentioned in equation \ref{eq:factor} and depends on $\alpha$ and $\beta$ . These measures are $\Phi$-invariant. \name{item:factor} \item We define the notion of a stable periodic orbit of $\Phi$ and let ${\mathit NS} \subset X$ be the collection of points without a stable periodic orbit. If $\alpha$ and $\beta$ are irrational, then any point $z$ which is periodic under ${\widetilde \Psi}_{\alpha, \beta}$ satisfies $\pi(z) \not \in {\mathit NS}$. \name{item:stable} \item We define a nested sequence of Borel subsets $\sO_k \subset X$ so that ${\mathit NS} = \bigcap_{k=0}^\infty \sO_k$ up to sets of $\nu_{\alpha, \beta}$-measure zero. Then, $\nu_{\alpha, \beta}({\mathit NS})$ is the limit of a decreasing sequence, $\lim_{k \to \infty} \nu_{\alpha, \beta}(\sO_k)$. \name{item:limit} \item \name{item:inequality} Using the return time cocycle, we are able to find an expression for $\nu_{\alpha, \beta}(\sO_k)$. We then show there is a continuous function $g:(0,\half] \times (0, \half] \to \R$ which is strictly less than one on its domain so that for all $k \geq 1$, $$\nu_{\alpha, \beta}(\sO_{k}) \leq g\big(f^{k-1}(\alpha),f^{k-1}(\beta)\big) \nu_{\alpha, \beta}(\sO_{k-1}),$$ where $f$ is defined as in equation \ref{eq:f}. It follows that if the orbit $\{(f \times f)^k(\alpha, \beta)~:~k \geq 0\}$ has an accumulation point in $(0,\half] \times (0, \half]$, then $\nu_{\alpha, \beta}({\mathit NS})=0$ as desired. (We remark that $g(\alpha, \beta)$ tends to one if either $\alpha$ or $\beta$ tends to zero.) \item We show that Lebesgue-a.e. pair $(\alpha, \beta)$ recurs under $f \times f$. \name{item:zero} \end{enumerate}
We will now say a few words about the proof of Theorem \ref{thm:4}, which says that irrational parameters $(\alpha, \beta)$ exist so that the total measure of periodic points of $\Psi_{\alpha, \beta}$ is as close to zero as we like. By the above argument, if the measure of the non-periodic points of $\Psi_{\alpha, \beta}$ is to be positive, then the orbit of $(\alpha, \beta)$ under $f \times f$ must diverge in the sense that $$\limsup_{k \to \infty} \min \big(f^k(\alpha), f^k(\beta_k)\big)=0.$$ To find such an $(\alpha, \beta)$, we observe that $f$ is semi-conjugate to the shift map on the full one-sided shift space defined over a countable alphabet. So, we can describe a pair $(\alpha, \beta)$ in terms of a symbolic coding of its $f \times f$-orbit. We understand the cocycle mentioned above in terms of this symbolic coding, and show that for appropriate choices $\nu_{\alpha,\beta}({\mathit NS})$ can be made as close to one as we like.
\subsection{Background on polygon exchange transformations} \name{sect:background} Few general results about rectangle and polygon exchange transformations are known. Our lack of understanding is highlighted by a question of Gowers \cite{G00}: are all rectangle exchanges recurrent? It is known that (vast generalizations of) polygon exchange transformations have zero entropy \cite{GH97}. And in \cite{H81}, a criterion is provided for a rectangle exchange to be minimal.
A {\em piecewise rotation} is a collection of polygons $X$ in $\R^2$ together with a map $T:X \to X$. The map $T$ cuts $X$ into finitely many polygonal pieces and applies an orientation preserving Euclidean isometry to each piece. The image $T(X)$ must have full area in $X$.
If on each polygonal piece, $T$ performs either a translation or a rotation by a rational multiple of $\pi$, then there is a natural construction of a polygon exchange map $S:Y \to Y$ together with a covering map $c:Y \to X$ so that $c \circ S=T \circ c$. Thus, studying such a {\em rational piecewise rotation} is closely related to studying a polygon exchange map.
There are several examples of renormalizable piecewise rotations. In \cite{AKT01}, a family of piecewise rotations is studied. Renormalization is used to understand a few of the maps in this family whose pieces are rotated by rational multiplies of $\pi$. Another example of a renormalizable piecewise rotation is provided in \cite{LKV04}. And in \cite{L07}, a general theory of renormalization of piecewise rotations is developed. In all these cases, periodic points are shown to be of full measure in the dynamical system.
Another topic of the papers \cite{AKT01}, \cite{LKV04} and \cite{L07} is to understand the dynamics on the set of points whose orbits are not periodic. (E.g., we would like to know if the dynamics are minimal or uniquely ergodic on this set.) These questions could be asked about maps in the family $\{\Psi_{\alpha, \beta}\}$, but we postpone investigating these questions until a subsequent paper.
In \cite{GP04}, renormalization arguments are used to explain that natural return maps of piecewise rotations may be piecewise rotations with a countable collection of polygons of continuity. By an observation of Hubert, this holds even for rectangle exchange maps and products of interval exchange maps \cite[\S 6.1]{GP04}.
Outer billiards also gives rise to polygon exchange maps. Fix a convex polygon $P$ in $\R^2$. There are two continuous choices of maps $\phi$ from $\R^2 \sm P$ to the space of tangent lines of $P$ so that each point $Q \in \R^2 \sm P$ is sent to a tangent line containing $Q$. Choose such a $\phi$. For a typical point $Q$, $\phi(Q)$ intersects $P$ in exactly one point $Z$, which is a vertex of $P$. We define $T(Q)$ to be the point obtained applying the central reflection through $Z$ to the point $Q$. This defines the {\em outer (or dual) billiards map}, a map $T:\R^2 \sm P \to \R^2 \sm P$. ($T$ is well defined and invertible off a finite number of rays.) We refer the reader to \cite{Tab} for an introduction to the subject.
The square of the outer billiards map is a piecewise translation of $\R^2 \sm P$. Return maps of $T^2$ to polygonal sets give possible sources of polygon exchange maps. Maps of these forms are studied \cite{Tab95}, \cite{BC09}, and \cite{Schwartz10}.
A {\em polytope exchange transformation} is the 3-dimensional analog of a polygon exchange. Recently, Schwartz described a renormalization scheme for a polytope exchange map arising from a compactification of a first return map of outer billiards map in the Penrose kite \cite{Schwartz11}. Due to the complexity of this polytope exchange, Schwartz's renormalization result is proved with the aid of a computer. It is believed that other outer billiards systems should exhibit similar phenomena.
The polygon exchange transformations that arise in this paper were concocted to share properties with the polytope and polygon exchange maps studied in \cite{Schwartz11}. In particular, the Truchet tilings we study share many properties with the arithmetic graph studied in \cite{S07}, \cite{S09} and \cite{Schwartz11}. Namely, both give decorations of the plane by simple curves which may be closed or bi-infinite.
\subsection{Outline} In section \ref{sect:curve following}, we formally define a curve following map for a Truchet tiling. Roughly, this map is the same as the definition of $\Phi_0$ only we do not translate the tiling.
In section \ref{sect:arithmetic graph}, we describe the construction of the arithmetic graph, which connects the curve following map to the dynamics of our polygon exchange maps. We use this construction to define the map $\pi$ as in equation \ref{eq:factor}.
In section \ref{sect:renormalizing tilings}, we explain how to renormalize the curve following map for Truchet tilings arising from corner percolation. For most such tilings, we find a subset of tiles such that the return map of the curve following map to this subset is conjugate to the curve following map of a different tiling. This is the most important observation of the paper.
Subsection \ref{sect:renormalization2} takes the renormalization of the curve following map and promotes it to a renormalization of the map $\Phi$ defined in equation \ref{eq:Phi2}. Earlier subsections of section \ref{sect:corner} explain necessary background and definitions necessary to describe this version of renormalization. We define the set ${\mathit NS} \subset X$ of non-stable periodic orbits in section \ref{sect:periodic orbits}. Proposition \ref{prop:stable} implies that periodic points $z$ of $\til \Psi_{\alpha, \beta}$ satisfy $\pi(z) \not \in {\mathit NS}$ when $\alpha$ and $\beta$ are irrational. These observations were part of statement (\ref{item:stable}) of the outline of the proof of Theorem \ref{thm:periodicity} (Periodicity almost everywhere).
In section \ref{sect:rectangle exchanges}, we explain how the renormalization of $\Phi$ induces the renormalization of the polygon exchange maps $\til \Psi_{\alpha, \beta}$ described by Theorem \ref{thm:ren}.
The next two sections of the paper deal with our measure theoretic results. We define the return time cocycle and state a theorem which describes the cocycle's relevance in subsection \ref{sect:limit formula}. This relevance includes a connection to the decreasing sequence of sets $\sO_k$ mentioned in statement (\ref{item:limit}) of the outline of the proof of Theorem \ref{thm:periodicity}. The later subsections are concerned with explaining the construction of the cocycle and proving the main theorem of the section.
In section \ref{sect:calc}, we utilize the cocycle to prove our measure theoretic results. Subsection \ref{sect:recurrent case}, proves statement (\ref{item:inequality}) of the outline of the Periodicity Almost Everywhere theorem. Subsection \ref{sect:non-recurrent case} proves Theorem \ref{thm:4} and Corollary \ref{cor:5} of the introduction, which guarantee the existence of parameters for which the map $\Psi_{\alpha,\beta}$ is not periodic almost everywhere.
Finally, section \ref{sect:param space} concerns the dynamical behavior of the maps $f$ and $f \times f$. The map $f$ was defined in equation \ref{eq:f} and $f \times f$ is the action of renormalization on the parameter space. Many of our results are predicated on the understanding of these maps, e.g. statement (\ref{item:zero}) of the outline of the proof of Theorem \ref{thm:periodicity}. Our analysis of these maps is fairly standard, so we have postponed this discussion to the end of the paper.
\section{Following curves in Truchet tilings} \name{sect:curve following}
We now present a useful concept for understanding Truchet tilings and the dynamics of the map $\Phi_0:\sT \times N \to \sT \times N$ defined in equation \ref{eq:Phi}. Recall that $\Phi_0(\tau, \v)$ moved the vector $\v \in N$ along the curve of the tile of the tiling $[\tau]$ centered at the origin, and then translated to keep the vector pointed into the square at the origin.
We will now consider what happens if we forget the translation. In this case, the tiling remains fixed while the vector has moved away from the origin. Formally, we fix a Truchet tiling $[\tau]$ determined by a map $\tau:\Z^2 \to \{\pm 1\}$ and define the {\em curve following map} to be \begin{equation} \name{eq:C} \sC:\Z^2 \times N \to \Z^2 \times N; \quad \big((m,n),(a,b)\big) \mapsto \big((m+sb,n+sa),s(b,a)\big), \end{equation} where $s=\tau(m,n)$. This map considers the inward pointing unit normal in direction $(a,b)$ based at a midpoint of an edge of the unit square centered at $(m,n)$. It moves the vector forward along the curve of the tile centered at $(m,n)$, keeping the vector tangent to the curve, and stops as soon as the vector leaves the tile. The new resulting vector points into the square centered at $(m+sb,n+sa)$ and points in direction $s(b,a)$.
We can recover the behavior of powers of the the map $\Phi_0$ applied to pairs of the form $(\tau, \v)$ from the curve following map for $\tau$. To do this, define the map \begin{equation} \name{eq:S0} \sS_0:\Z^2 \times N \to \sT \times N; \quad (m,n, \v) \mapsto (\tau \circ S_{m,n}, \v), \end{equation} where $S_{m,n}:\Z^2 \to \Z^2$ is the translation $(x,y) \mapsto (x,y)+(m,n)$. Either by inspection or induction, it can be shown that for all $\tau \in \sT$ and all $k \in \Z$, \begin{equation} \name{eq:curve following relation1} \Phi_0^k(\tau,\v)=\sS_0 \circ \sC^k\big(0,0,\v\big). \end{equation} Informally, the right hand side just waits to translate until we have moved $k$ steps forward, but we translate by the composition of the translations used when evaluating $\Phi_0^k(\tau,\v)$.
Similarly, we can recover the behavior of the map $\Phi$. Fix $\omega$ and $\eta$ and define $\tau$ by $\tau(m,n)=\omega_m \eta_n$ as in \ref{eq:tau omega}. Then we define an analog of $\sS_0$ and see that it satisfies a similar identity involving the curve following map of $\tau$. \begin{equation} \name{eq:S} \sS:\Z^2 \times N \to X; \quad (m,n,\v) \mapsto \big(\sigma^m(\omega),\sigma^n(\eta),\v). \end{equation} \begin{equation} \name{eq:curve following relation} \Phi^k(\omega, \eta,\v)=\sS \circ \sC^k\big(0,0,\v\big). \end{equation} Here $X=\Omega_\pm \times \Omega_\pm \times N$ is the domain of $\Phi$ as in the introduction.
\section{Construction of the arithmetic graph } \name{sect:arithmetic graph} \begin{comment} \boldred{This section needs a little work:} \begin{enumerate} \item I'd like it to be shorter. Maybe add a picture explaining. \item Referee's suggestion: (Assuming this is correct) maybe it would be worth also explaining the main construction here in a different way: The set
$$S(x, y) = \{(x, y) + m \alpha + n \beta| m, n \in \Z^2\}$$ is invariant under $\Psi_{\alpha, \beta}$, and the arithmetic graph construction produces a planar curve (with 4 possible local decorations) for each orbit. The union of the curves attached to $S(x, y)$ coincides with the union of curves in the Truchet tiling $\tau_{\alpha, \beta, x, y}$. \end{enumerate} \end{comment} In this section, we fully explain the connection between the family of polygon exchange maps $\{\Psi_{\alpha, \beta}\}$ and Truchet tilings which arise from corner percolation.
Consider the polygon exchange map $\til \Psi_{\alpha, \beta}:\til Y \times N \to \til Y \times N$ defined in Equation \ref{eq:psi intro}. Let $(x_0, y_0) \in \til Y=\R^2 / \Z^2$ and choose a $\v \in N$. Then let $\big((x_1, y_1), \bw\big)=\til \Psi_{\alpha, \beta} \big((x_0,y_0),\v\big)$. Observe that (modulo $\Z^2$) we have $$(x_1,y_1)-(x_0,y_0) \in \{(\pm \alpha, 0), (0, \pm \beta)\}.$$ Fixing $(x,y) \in \til Y$, we define the map \begin{equation} \name{eq:M} M:\Z^2 \times N \to \til Y \times N; \quad M(m,n, \v)=(x+m \alpha, y+n \beta, \v). \end{equation} The argument above shows that $M(\Z^2 \times N)$ is $\til \Psi_{\alpha, \beta}$-invariant. Note also that so long as $\alpha$ and $\beta$ are irrational, the map $M$ is injective.
\begin{definition}[Arithmetic Graph] The {\em arithmetic graph} associated to the irrational parameters $(\alpha, \beta) \in (0,\half) \times (0, \half)$ and the point $(x,y) \in \til Y$ is the directed graph whose vertices are points in $\Z^2 \times N$ with an edge running from $(m_0, n_0, \v)$ to $(m_1, n_1, \bw)$ if and only if $$\til \Psi_{\alpha, \beta} \circ M(m_0, n_0, \v)=M(m_1, n_1, \bw).$$ \end{definition}
We will show that the arithmetic graph associated to $(\alpha, \beta)$ and $(x,y) \in \til Y$ is closely related to a Truchet tiling. Define \begin{equation} \name{eq:omega from x} \omega_m=\begin{cases} 1 & \textrm{if $x+m\alpha \in [0, \half)$}\\ -1 & \textrm{if $x+m\alpha \in [\half,1)$} \end{cases} \and \eta_n=\begin{cases} 1 & \textrm{if $y+n \beta \in [0, \half)$}\\ -1 & \textrm{if $y+n \beta \in [\half,1)$.} \end{cases} \end{equation} In these definitions, $x+m \alpha$ and $y+n \beta$ are taken to lie in $\R/\Z$. We then define $\tau$ according to the rule $\tau(m,n)=\omega_m \eta_n$.
\begin{proposition} \name{prop:graph iso} Fix irrationals $\alpha, \beta \in (0, \half)$ and fix any $(x,y) \in \til Y$. Let $\omega$, $\eta$ and $\tau$ be as above. Then there is an edge joining $(m_0, n_0, \v)$ to $(m_1, n_1, \bw)$ in the arithmetic graph if and only if the curve following map of $\tau$ satisfies $$\sC \big((m_0, n_0), \v\big)=\big((m_1, n_1), \bw\big).$$ \end{proposition} \begin{proof} We must show that for each $(m,n,\v) \in \Z^2 \times N$, we have $$\til \Psi_{\alpha, \beta} \circ M(m,n,\v)=M \circ \sC(m,n,\v).$$ Let $\v=(a,b)$ and $s=\tau(m, n)$. Then by the definitions of $\sC$ and $M$, we have $$\begin{narrowarray}{0pt}{rcl} M \circ \sC\big(m,n, \v) & = & M\big((m+sb,n+sa),s(b,a)\big) \\ & = & \Big(\big(x+(m+sb)\alpha,y+(n+sa)\beta\big),s(b,a)\Big). \end{narrowarray}$$ Observe that by definition of $\tau$ and $s$, we have $(x+m\alpha,y+n\beta)\in \til A_s$. It follows that $$\begin{narrowarray}{0pt}{rcl} \til \Psi_{\alpha, \beta} \circ M(m,n,\v) & = & \til \Psi_{\alpha, \beta}(x+m\alpha, y+n\beta, \v) \\ & = & \big(x+m\alpha+bs\alpha, y+n\beta+as\beta, (bs,as)\big). \end{narrowarray}$$ \end{proof}
We define the embedding map $\pi$ which appeared in section \ref{sect:overview} of the introduction by \begin{equation} \name{eq:pi} \pi:\til Y \times N \to X; \quad (x,y,\v) \mapsto (\omega, \eta, \v) \end{equation} with $\omega$ and $\eta$ defined in terms of $\alpha$, $\beta$, $x$ and $y$ as in equation \ref{eq:omega from x}. We show this map satisfies equation \ref{eq:factor}:
\begin{proposition} \name{prop:ag conj} If $\alpha$ and $\beta$ are irrational, then $\pi \circ {\widetilde \Psi}_{\alpha, \beta}=\Phi \circ \pi$. \end{proposition} \begin{proof} Fix $x$, $y$ and $\v$. Define $\omega$ and $\eta$ so that $\pi(x,y,\v)=(\omega, \eta, \v)$. Then, we have $$\Phi \circ \pi(x,y,\v)=\Phi(\omega, \eta,\v)=\sS \circ \sC(0,0,\v),$$ by equation \ref{eq:curve following relation}. By Proposition \ref{prop:graph iso}, we continue: $$\Phi \circ \pi(x,y,\v)=\sS \circ M^{-1} \circ \til \Psi_{\alpha, \beta} \circ M(0,0,\v).$$ Here we can invert $M$ because irrationality of $\alpha$ and $\beta$ implies the map $M$ is injective. We claim that the map $$\sS \circ M^{-1}:M(\Z^2 \times N) \to X \quad \text{is given by} \quad \sS \circ M^{-1}=\pi.$$ This will conclude the proof since $M(0,0,\v)=(x,y,\v)$. To prove this claim, we demonstrate that $\sS=\pi \circ M$. Fix any $(i,j,\bw) \in \Z^2 \times N$. Then $$\pi \circ M(i,j,\bw)=\pi(x+i \alpha, y+j\beta, \bw)=(\omega', \eta', \bw).$$ Here, $\omega'$ and $\eta'$ are defined as in equation \ref{eq:omega from x}, but with $x$ replaced by $x+i\alpha$ and $y$ by $y+j\beta$. By definition of $\omega$, $\eta$, $\omega'$ and $\eta'$, we have the desired identity $$(\omega', \eta' ,\bw)=\big( \sigma^{i}(\omega), \sigma^{j}(\eta), \bw\big)=\sS(i,j,\bw).$$ \end{proof}
\section{Renormalization of the Truchet Tilings} \name{sect:renormalizing tilings} In this section, we will explain how the Truchet tilings induced by corner percolation tilings exhibit a ``renormalization operation.'' We call this operation a renormalization, because when interpreted dynamically the operation corresponds to a renormalization of the map $\Phi:X \to X$ defined in equation \ref{eq:Phi2} of the introduction.
\subsection{Renormalization}
For any $\omega \in \Omega_{\pm}$, we define the subset $K(\omega) \subset \Z$ to be \begin{equation} \name{eq:K1} K(\omega)=\set{n \in \Z}{$\omega_n \neq -1$ or $\omega_{n+1}\neq 1$} \cap \set{n \in \Z}{$\omega_{n-1}\neq-1$ or $\omega_{n} \neq 1$}.
\end{equation} That is, $K(\omega)$ is the collection $n$ so that $\omega_n$ is not part of a subword of the form $-+$.
Throughout this section, we will fix $\omega, \eta \in \Omega_\pm$, and define $\tau=\tau_{\omega, \eta}$ as in equation \ref{eq:tau omega} (i.e., $\tau(m,n)=\omega_m \eta_n$). By Proposition \ref{prop:corner perc induced}, all Truchet tilings induced by corner percolation are of this form. To describe the renormalization of $[\tau]$, we construct the two sets $K(\omega)$ and $K(\eta)$. We make the following assumption about these sets: \begin{equation} \name{eq:assumption} \text{The sets $K(\omega)$ and $K(\eta)$ have neither upper nor lower bounds.} \end{equation} We will assume that $\omega$ and $\eta$ satisfy this assumption throughout this section. This condition guarantees that there exist increasing bijections $$\kappa_1:\Z \to K(\omega) \and \kappa_2:\Z \to K(\eta).$$ Each of these bijections is unique up to precomposition with a translation of $\Z$.
We are now ready to define the renormalization of the tiling $[\tau]$. The renormalized tiling is defined by removing rows and columns of tiles from $[\tau]$ and then sliding the remaining tiles together. We define the {\em set of centers of the kept squares} to be $${\overline{K}}=K(\omega) \times K(\eta).$$ We also define the bijection $$\kappa=\kappa_1 \times \kappa_2:\Z^2 \to {\overline{K}}.$$ This enables us to define the {\em renormalization of $\tau$} to be the map $\tau'=\tau \circ \kappa.$ We call $[\tau']$ the renormalization of $[\tau]$. It is uniquely defined up to translation.
The tiles whose centers lie in the set $\Z^2 \sm {\overline{K}}$ are a union of rows and columns. The tiling $[\tau']$ can be obtained from the tiling $[\tau]$ by collapsing all columns of tiles with centers in $\Z^2 \sm {\overline{K}}$ to vertical lines, and collapsing all rows of tiles with centers in $\Z^2 \sm {\overline{K}}$ to horizontal lines. See figure \ref{fig:even rectangles}.
This paper exploits the relationship between the tiling $[\tau]$ and the renormalized tiling $[\tau']$, which we will informally state now and state formally in the theorem below. First, whenever four tiles form a loop these four tiles are removed by the renormalization operation. Second, the renormalization operation preserves the identities of any curve in the tiling which is not a loop of length four. That is, some tiles making the curve may be removed, but once the remaining tiles are slid together again, there is a new curve which visits the remaining tiles of the curve in the same order. Third, this process shortens all closed loops visiting more than four tiles. We then hope to apply this process repeatedly, shrinking long loops until they eventually become loops of length four and disappear. This gives a mechanism to detect closed loops in the tiling.
In stating a theorem which makes this relationship between $[\tau]$ and $[\tau']$ rigorous, we will utilize the curve following map defined in section \ref{sect:curve following}. We define $\sC$ and $\sC'$ to be the curve following maps defined in equation \ref{eq:C} with respect to the tilings $[\tau]$ and $[\tau']$, respectively. We also define ${\wh \sC}:{\overline{K}} \times N \to {\overline{K}} \times N$ to be the first return map of $\sC$ to ${\overline{K}} \times N$. That is, when $(m,n,\v) \in {\overline{K}} \times N$, we define \begin{equation} \name{eq:sR} {\wh \sC}(m,n,\v)=\sC^k(m,n,\v) \quad \text{where} \quad k=\min~\{j>0~:~\sC^j(m,n,\v)\in {\overline{K}} \times N\}. \end{equation} Informally, the map ${\wh \sC}$ takes a inward unit normal to a square whose center lies in ${\overline{K}}$, then moves the vector along the curve of the tiling $[\tau]$ until the vector returns to a square whose center lies in ${\overline{K}}$.
\begin{theorem}[Tiling Renormalization] \name{thm:tiling renormalization} Assume $\omega, \eta \in \Omega_\pm$ satisfy the assumption given in equation \ref{eq:assumption}. In this case: \begin{enumerate} \item The first return map ${\wh \sC}$ of $\sC$ to ${\overline{K}} \times N$ is well defined on all of ${\overline{K}} \times N$. \item Define $\til \kappa:\Z^2 \times N \to {\overline{K}} \times N$ by $\til \kappa\big((m,n),\v\big)=\big(\kappa(m,n),\v)$. Then, $${\wh \sC} \circ \til \kappa=\til \kappa \circ \sC'.$$ \item The following statements are equivalent for any $(m,n, \v) \in \Z^2 \times N$. \begin{enumerate} \item There is no $k>0$ so that $\sC^k(m,n,\v) \in {\overline{K}} \times N$. \item There is no $k<0$ so that $\sC^k(m,n,\v) \in {\overline{K}} \times N$. \item $\sC^4(m,n,\v)=(m,n,\v)$. \end{enumerate} \item If there is a $p$ so that $\sC^p(m,n,\v)=(m,n,\v)$, then there is a $k>0$ so that $\sC^k(m,n,\v) \not \in {\overline{K}}$. \end{enumerate} \end{theorem}
We will also need to understand the return times of $\sC$ to ${\overline{K}} \times N$ in terms of the tiling $[\tau]$. This is relevant to our measure theoretic results. For $(m,n,\v) \in {\overline{K}} \times N$, we define the return time function $$R(m,n,\v)=\min~\{j>0~:~\sC^j(m,n,\v)\in {\overline{K}} \times N\}.$$ For $k=R(m,n,\v)$, we have ${\wh \sC}(m,n,\v)=\sC^k(m,n,\v)$. See equation \ref{eq:sR}.
We can describe the return time in terms of the number of nearby rows and columns excised to produce $[\tau']$. To explain this we define a new {\em excision} function $$E:{\overline{K}} \times N \to \Z_+; \quad E(m,n,\v)=\min~\{j>0~:~(m,n)+j\v \in {\overline{K}}\}.$$ This represents one more than the number of adjacent rows or columns that will be removed, starting with the square opposite the edge in direction $\v$. This is always well defined so long as $\omega$ and $\eta$ satisfy \ref{eq:assumption}.
\begin{theorem}[Tiling Return Time] \name{thm:return time} Suppose $\omega, \eta \in \Omega_\pm$ satisfy the assumption given in equation \ref{eq:assumption}. Choose any $\v=(a,b) \in N$. Define $s=\omega_m \eta_n$ and $\bw=(sb,sa) \in N$. Then, $$R(m,n,\v)=2E(m,n, \bw)-1.$$ \end{theorem}
\begin{remark} By equation \ref{eq:C}, $\bw=(sb,sa)$ is the directional component of $\sC(\omega, \eta, \v)$. \end{remark}
\subsection{Proofs}\name{sect:ren proofs} In this section we prove the renormalization theorems of the previous subsection. As above, we fix $\omega$ and $\eta$.
First we investigate loops visiting four squares in the tiling. \begin{proposition} If $\sC^4(m,n,\v)=(m,n,\v)$, then for all $k$ we have $\sC^k(m,n,\v) \not \in {\overline{K}} \times N$. \end{proposition} \begin{proof} Suppose $(m,n,\v)$ is tangent to a loop of length four. Four Truchet tiles coming together to make a loop of length four come in exactly one configuration. Since the map $(\omega, \eta) \to \tau_{\omega, \eta}$ is two-to-one, there are exactly two local choices of $\omega$ and $\eta$ which give rise to a loop of length four. These choices are shown below: \begin{center} \includegraphics[scale=0.5]{p4a} \end{center} All of the squares in either of these pictures lie in $\Z^2 \sm {\overline{K}}$. \end{proof}
We will now explain another possibility for what the curve through $(m,n,\v)$ looks like assuming $(m,n) \not \in {\overline{K}}$.
\begin{definition} A {\em horizontal box} is a subset of $\Z^2$ of the form $$H=\big\{(i,j) \in \Z^2~:~\text{$i \in \{m+1, \ldots, m+2\ell\}$ and $j \in \{n, n+1\}$}\big\},$$ where $\ell, m, n \in \Z$ are constants with $\ell \geq 1$ so that $$\omega(m+i)=(-1)^i \quad \text{for $i=1, \ldots, 2\ell$}, \and \eta(n)=\eta(n+1).$$ A {\em vertical box} is a subset of $\Z^2$ of the form $$V=\big\{(i,j) \in \Z^2~:~\text{$i \in \{m, m+1\}$ and $j \in \{n+1, \ldots, n+2\ell\}$}\big\},$$ where $\ell, m, n \in \Z$ are constants with $\ell \geq 1$ so that $$\omega(m)=\omega(m+1), \and \eta(n+i)=(-1)^i \quad \text{for $i=1, \ldots, 2\ell$}.$$ In both cases, we call $\ell$ the {\em length parameter} of the box. \end{definition}
The tiles whose centers belong to a horizontal box must look like one of the following cases when $\ell=3$: \begin{center} \includegraphics[scale=0.5]{horizontal_box} \end{center} Each horizontal box has a {\em central curve}, which visits all squares with centers in the horizontal box. This curve is depicted in black above. The collection of tiles whose centers lie in a vertical box looks the same as the above pictures after applying a reflection in the line $x=y$.
\begin{lemma} \name{lem:xor} Suppose $(m,n) \in \Z^2 \sm {\overline{K}}$ and $\v \in N$. Then exactly one of the following statements holds. \begin{enumerate} \item $\sC^4(m,n,\v)=(m,n,\v).$ \item $(m,n,\v)$ is tangent to the central curve of a horizontal box. \item $(m,n,\v)$ is tangent to the central curve of a vertical box. \end{enumerate} \end{lemma} \begin{proof} Suppose $(m,n) \in \Z^2 \sm {\overline{K}}$. This means that either $m \not \in K(\omega)$ or $n \not \in K(\eta)$. By reflection in the line $y=x$, we may assume without loss of generality that $m \not \in K(\omega)$. This means that there is a choice of $m' \in \{m-1,m\}$ so that \begin{equation} \name{eq:omp} \omega_{m'}=-1 \and \omega_{m'+1}=1. \end{equation} Assuming this, we can draw all tiles with centers in the set $\{m', m'+1\} \times \{n-1,n,n+1\}$. There are eight possibilities: \begin{center} \includegraphics[scale=0.5]{hbox0} \end{center} We have colored the tilings by the following rules. All curves through $(m',n)$ and $(m'+1,n)$ have been colored black or gray. The black curves are either closed loops of length four, or they are central curves of a horizontal box (with $\ell=1$). The gray curves are not yet part of a horizontal or vertical box and we need to do further analysis. The curves drawn in white and outlined are irrelevant to us because they do not (locally) pass through the tiles with centers $(m',n)$ or $(m'+1,n)$.
We further analyze the gray curves which come in pairs as above. Each gray curve visits two tiles of six in the above picture. For each gray curve, there is a choice of $n' \in \{n-1,n\}$ so that the curve visits only tiles with centers in the set $\{m',m'+1\} \times \{n',n'+1\}$. Furthermore we have $$\eta_{n'}=-1 \and \eta_{n'+1}=1.$$ Now we consider extending the tiling left and right. There are a total of four ways to extend depending on the choices of $\omega_{m'-1}$ and $\omega_{m'+2}$. The four possible collections of tiles with centers in the set $\{m'-1,m',m'+1, m'+2\} \times \{n',n'+1\}$ are show below: \begin{center} \includegraphics[scale=0.5]{vbox0} \end{center} Observe that in all cases, the gray curve is either a closed loop of length four, or is a central curve in a vertical box (with $\ell=1$).
The above argument shows that each $(m,n,\v)$ satisfies one of the three statements in the lemma. We need to show the statements are mutually exclusive. Clearly when $\sC^4(m,n,\v)=(m,n,\v)$, we can not have that $(m,n,\v)$ is tangent to a curve in a horizontal or vertical box. Now suppose that $(m,n,\v)$ was tangent to central curves of both horizontal and vertical boxes. Because of the $(m,n)$ lies in the horizontal box, there is an $m' \in \{m-1,m\}$ so that equation \ref{eq:omp} holds. Because $(m,n)$ lies in a vertical box, there is an $m'' \in \{m-1,m\}$ so that $\omega_{m''}=\omega_{m''+1}$. This leaves two possibilities: $$\omega_{m-1} \omega_m \omega_{m+1}=-++ \quad \text{or} \quad \omega_{m-1} \omega_m \omega_{m+1}=--+.$$ A similar argument shows that $$\eta_{n-1} \eta_n \eta_{n+1}=-++ \quad \text{or} \quad \eta_{n-1} \eta_n \eta_{n+1}=--+.$$ Therefore, the tiles with centers in $\{m-1,m,m+1\} \times \{n-1,n,n+1\}$ have the following four possible configurations: \begin{center} \includegraphics[scale=0.5]{hvbox0} \end{center} In the above pictures, the central curve of the horizontal box containing $(m,n)$ is colored black, and the central curve of the vertical box containing $(m,n)$ is colored gray. Observe that these curves are disjoint. This implies statements (2) and (3) are mutually exclusive. \end{proof}
We call a horizontal (resp. vertical) box {\em maximal} if it is not contained in a larger horizontal (resp. vertical) box.
\begin{proposition} Assume $\omega, \eta \in \Omega_\pm$ satisfy the assumption given in equation \ref{eq:assumption}. Then, every horizontal (resp. vertical) box is contained in a maximal horizontal (resp. vertical) box. \end{proposition} \begin{proof} Suppose a horizontal box was not contained in a largest maximal box. Then it would be contained in arbitrary large horizontal box. Let $(m,n)$ be the point in the box with smallest coordinates. Then, we see that there are arbitrary long intervals $I$ containing $m$ so that $\omega$ alternates on $I$. But this is ruled out by the assumption given in equation \ref{eq:assumption}. A similar statement holds for vertical boxes. \end{proof}
\begin{proposition} \name{prop:leaving boxes} Suppose $(m,n,\v)$ is tangent to the central curve in a maximal horizontal or vertical box $B$. Then, the smallest $k>0$ so that $\sC^k(m,n,\v)$ is no longer tangent to the central curve of $B$ satisfies $\sC^k(m,n,\v) \in {\overline{K}} \times N$. Similarly, the largest $k<0$ so that $\sC^k(m,n,\v)$ is no longer tangent to the central curve of $B$ satisfies $\sC^k(m,n,\v) \in {\overline{K}} \times N$. \end{proposition} \begin{proof} We prove the statement for $k>0$; the other statement has a similar proof. Let $(m',n',\v')=\sC^k(m,n,\v)$. Then $\v'$ is horizontal if $B$ is horizontal, and $\v'$ is vertical if $B$ is vertical. Suppose without loss of generality that $B$ and $\v'$ are horizontal. If $(m',n') \not \in {\overline{K}}$, then Lemma \ref{lem:xor} implies that $(m',n')$ is tangent to the central curve of a new horizontal or vertical box $B'$ and that the central curves of $B$ and $B'$ are disjoint. Since $(m',n',\v')$ is the initial entrance to the horizontal box $B'$ and $v'$ is horizontal, we know that $B'$ is horizontal. Observe that horizontal boxes can be joined so that their central curves connect only if $B \cup B'$ is a larger horizontal box. This contradicts maximality of $B$. \end{proof}
We now can prove our renormalization theorems.
\begin{proof}[Proof of Theorems \ref{thm:tiling renormalization} and \ref{thm:return time}.] Statement (1) of the theorem follows from statement (2). We will now simultaneously prove statement (2) of the Tiling Renormalization Theorem and the Return Time Theorem. Choose any $(m',n',\v) \in \Z^2 \times N$ and write $\v=(a,b) \in N$. Define $$(m,n)=\kappa(m',n'), \quad s=\tau'(m',n')=\tau(m,n) \and \bw=(sb,sa).$$ Then we have $$\sC'(m',n',\v)=\big((m',n')+\bw,\bw\big) \and \sC\big(m,n,\v)=\big((m,n)+\bw,\bw\big).$$ The two statements we wish to prove follow respectively from $${\wh \sC}(m, n, \v)=\til \kappa\big((m', n')+\bw, \bw) \and R(m,n, \v)=2E\big((m,n)+\bw,\bw\big)-1.$$ For these equations, we may assume without loss of generality that $\v$ is vertical. This means that $\bw$ is horizontal and we can write $\bw=(c,0)$ taking $c=sb \in \{\pm 1\}$ and $a=0$.
First the consider the case that $(m,n)+\bw \in {\overline{K}}$. This is the center of the square containing $\sC(m,n,\v)$, which means $$R(m, n,\v)=1 \and E(m,n,\bw)=1,$$ proving this case of the Return Time Theorem. In addition, we have $${\wh \sC}(m, n, \v)=\sC(m, n, \v)=(m+c,n,\bw).$$ Because both $m \in K(\omega)$ and $m+c\in K(\omega)$ with $c \in \{\pm 1\}$, we have $$\kappa_1^{-1}(m+c)=\kappa_1^{-1}(m)+c=m'+c,$$ because $\kappa_1:\Z \to K(\omega)$ is an order preserving bijection. We have therefore shown a special case of statement (2) of the Tiling Renormalization theorem, $${\wh \sC}(m, n, \v)=(m+c,n,\bw)=\til \kappa\big((m', n')+\bw, \bw).$$
Otherwise we have $(m,n)+\bw \not \in {\overline{K}}$. Here, $\sC(m,n,\v)$ is tangent to the central curve of a maximal horizontal or vertical box $B$. Observe that $(m,n) \in {\overline{K}}$, so $\sC(m,n,\v)=(m+c,n,\bw)$ is the first time the curve enters this box. Since $\bw$ is horizontal, the box $B$ must be a horizontal box. Let $\ell$ denote the length parameter of the maximal horizontal box $B$. If $c=1$, this means that $$\omega_{m+k}=(-1)^k \quad \text{for $k=1, \ldots, 2\ell$}.$$ If $c=-1$, this means that $$\omega_{m+k-2\ell-1}=(-1)^k \quad \text{for $k=1, \ldots, 2 \ell$}.$$ Note that $\ell$ is the maximal number with this property. Therefore, $\kappa_1(m'+c)=m+c(2\ell+1)$. That is, $\kappa_1$ must skip over $2 \ell$ numbers to reach $\kappa_1(m'+c)$. The orbit $\sC^i(m,n,\v)$ follows the central curve of $B$ and then returns to ${\overline{K}}$ by Proposition \ref{prop:leaving boxes}. By inspection of horizontal boxes, we can then observe \begin{enumerate} \item[(a)] $R(m,n,\v)=1+4 \ell.$ \item[(b)] ${\wh \sC}(m,n,\v)=\sC^{1+4\ell}(m,n,\v)=\big(m+c(2\ell+1),n,\bw\big)$. \item[(c)] $E(m,n,\bw)=2 \ell+1.$ \end{enumerate} Statements (a) and (c) imply $R(m,n, \v)=2E(m,n,\bw)-1$. By (b) and observations above, $${\wh \sC}(m, n, \v)=\big(m+c(2\ell+1),n,\bw\big)=\til \kappa(m+c,n,\bw).$$ This finishes the proof of statement (2) of the Tiling Renormalization Theorem and proof of the Return Time Theorem.
Statement (3) of Theorems \ref{thm:tiling renormalization} follows from Lemma \ref{lem:xor} and Proposition \ref{prop:leaving boxes}. If $(m,n) \not \in {\overline{K}}$ and $\sC^4(m,n,\v) \neq (m,n,\v)$ then $(m,n,\v)$ is tangent to a central curve of a maximal horizontal or vertical box. Under positive or negative iteration by $\sC$ it must leave the box, and when it does it enters the set ${\overline{K}} \times N$.
We now consider statement (4). Suppose $(m,n,\v)$ is periodic under $\sC$ and never visits the set $(\Z^2 \sm {\overline{K}}) \times N$. Then this periodic orbit is confined to a region of the tiling consisting of tiles with centers in the set $$X=\{a, a+1, \ldots, b\} \times \{a', a'+1, \ldots, b'\},$$ where there are $c$ and $c'$ so that for all $(m,n) \in X$ $$\omega(m)=\begin{cases} 1 & \textrm{if $m < c$} \\ -1 & \textrm{if $m \geq c$} \end{cases} \and \eta(n)=\begin{cases} 1 & \textrm{if $n < c'$} \\ -1 & \textrm{if $n \geq c'$.} \end{cases}$$ But such a portion of a tiling can have no closed curves. See the example below. \begin{center} \includegraphics[scale=0.5]{region} \end{center} \end{proof}
\section{Dynamical Renormalization} \name{sect:corner} This section culminates in a description of a renormalization of the dynamical system $\Phi:X \to X$ defined in Equation \ref{eq:Phi2}.
\subsection{Background on shift spaces} \name{sect:topology} Recall that $\Omega_\pm$ denotes the space of all bi-infinite sequences in the alphabet $\{\pm 1\}$. We will now describe some of the general structure associated with shift spaces in this context. For further background on shift spaces see \cite{LindMarcus}, for instance.
A {\em word} in the alphabet $\{\pm 1\}$ is an element $w$ of a set $\{\pm 1\}^{\{1,\ldots,n\}}$ for some $n$, called the {\em length} of $w$. We write $w=w_1 \ldots w_n$ with $w_i \in \{\pm 1\}$ to denote a word. To simplify notation of the elements in $\{\pm 1\}$, we use $+$ to denote $1$ and $-$ to denote $-1$. So the word $w$ where $w_1=1$ and $w_2=-1$ can be written $w=+-$. Adjacency indicates the {\em concatenation} of words; if $w$ and $w'$ are words of length $n$ and $n'$ respectively, then $$ww'=w_1 \ldots w_{n} w'_1 \ldots w'_{n'}.$$
The choice of a word $w=w_1 \ldots w_n$ and an integer $b$ determines a {\em cylinder set}, $${\mathit{cyl}}(w,b)=\set{\omega \in \Omega_{\pm}}{$\omega_{i-b}=w_{i}$ for all $i=1, \ldots, n$}.$$ Whenever $b \in \{1, \ldots, n\}$, we can also denote the cylinder set ${\mathit{cyl}}(w,b)$ by $${\mathit{cyl}}(w_1 \ldots \widehat w_b \ldots w_n),$$ with the hat indicating that $w_b$ represents the zeroth entry of the those $\omega$ in the cylinder set. We equip $\Omega_\pm$ with the topology generated by the cylinder sets. The topological space $\Omega_\pm$ is homeomorphic to a Cantor set.
Recall that the shift map $\sigma:\Omega_\pm \to \Omega_\pm$ is defined by $\sigma(\omega)_n=\omega_{n+1}$ as in Equation \ref{eq:shift}.
A {\em shift-invariant measure} on $\Omega_\pm$ is a Borel measure $\mu$ satisfying $$\mu \circ \sigma^{-1}(A)=\mu(A) \quad \textrm{for all Borel subsets $A \subset \Omega_\pm$}.$$ Full shift spaces admit a plethora of shift-invariant probability measures.
\subsection{Invariant measures} \name{sect:tilings from shift spaces} Recall the definition of $\Phi:X \to X$ where $X=\Omega_\pm \times \Omega_\pm \times N$ as in equation \ref{eq:Phi2}, \begin{equation} \name{eq:Phi2b} \Phi\big(\omega, \omega', (a,b)\big)=\big(\sigma^{sb} (\omega), \sigma^{sa} (\omega'), s(b,a)\big) \quad \textrm{with $s=\omega_0 \omega'_0 \in \{\pm 1\}$.} \end{equation}
The following gives a natural construction of $\Phi$-invariant measures.
\begin{proposition} \name{prop:product measures} Suppose $\mu$ and $\mu'$ are shift invariant probability measures on $\Omega_\pm$. Let $\mu_N$ be the discrete probability measure on $N$ so that $\mu_N(\{\v\})=\frac{1}{4}$ for each $\v \in N$. Then $\mu \times \mu' \times \mu_N$ is a $\Phi$-invariant probability measure on $X$. \end{proposition}
The proof is just to observe that each Borel set $A \subset X$ can be decomposed into pieces on which the action of $\Phi$ is a power of a shift on each $\Omega_\pm$-coordinate and a permutation on $N$. The power and permutation are taken to be constant on each piece.
\begin{comment} \begin{proof} Let $A \subset X$ be Borel. We can decompose $A$ into eight disjoint subsets $A_1, \ldots, A_8$ so that for each $i \in \{1,\ldots,8\}$ there is an $s \in \{\pm 1\}$ and an $(a,b) \in N$ so that each $(\omega,\omega',\v) \in A_i$ satisfies $\omega_0 \omega'_0=s$ and $\v=(a,b)$. Then the restriction of $\Phi$ to $A_i$ satisfies
$$\Phi|_{A_i}=\sigma^{sb} \times \sigma^{sa} \times \pi,$$ where $\pi$ is any permutation sending $(a,b)$ to $(sb,sa)$. Observe that any choice of $\mu \times \mu' \times \mu_N$ preserves the measure of each $A_i$. Since $\Phi$ is invertible, it also preserves the measure of each $A$. \end{proof} \end{comment}
\subsection{Periodic orbits} \name{sect:periodic orbits} Suppose $(\omega, \eta,\v) \in X$ is periodic under $\Phi$. We say $(\omega, \eta,\v)$ has a {\em stable periodic orbit of period $n$} if $n$ is the smallest positive integer for which there are open neighborhoods $U$ and $V$ of $\omega$ and $\eta$ respectively for which $$\omega' \in U \and \eta'\in V \quad \text{implies} \quad \Phi^n(\omega', \eta',\v)=(\omega', \eta',\v).$$
\begin{remark} Not all periodic orbits are stable. When $\omega_n=1$ and $\eta_n=1$ for all $n \in \Z$, we have $\Phi^2(\omega, \eta,\v)=(\omega, \eta,\v)$ for all $\v$, but $(\omega, \eta,\v)$ is not a stable periodic orbit of any period. \end{remark}
The following proposition characterizes the points with stable periodic orbits.
\begin{proposition}[Stability Proposition] \name{prop:stable}
The following statements hold. \begin{enumerate}
\item $(\omega, \eta,\v) \in X$ has a stable periodic orbit if and only if the curve of the tiling $[\tau_{\omega, \eta}]$ passing through the normal $\v$ to the square centered at the origin is closed.
\item If $(\omega, \eta,\v) \in X$ has a periodic orbit but not a stable periodic orbit, then either $\omega$ or $\eta$ is periodic under the shift map $\sigma$.
\end{enumerate} \end{proposition}
\begin{proof}[Proof of Proposition \ref{prop:stable}.]
First suppose the curve of the tiling $[\tau_{\omega,\eta}]$ through the normal $\v$ to the square centered at the origin is closed. There are integers $m$ and $n$ so that all tiles visited by this closed curve have centers in the set $[-m,m] \times [-n,n]$. We define $$U={\mathit{cyl}}(\omega_{-m} \omega_{-m+1} \ldots \wh \omega_0 \ldots \omega_m) \and V={\mathit{cyl}}(\eta_{-n} \eta_{-n+1} \ldots \wh \eta_0 \ldots \eta_n).$$ Observe that every tiling determined by $\omega' \in U$ and $\eta' \in V$ looks the same for the set of tiles with centers in $[-m,m] \times [-n,n]$. In particular, every such tiling has the same closed curve through the normal $\v$ to the square centered at the origin. This always gives a periodic orbit of the same period as $(\omega, \eta, \v)$.
Now suppose $(\omega, \eta, \v)$ has period $k$ but the associated curve of the tiling $[\tau_{\omega,\eta}]$ is not closed. Recall the definition of the curve following map given in Section \ref{sect:curve following}. Define $m$ and $n$ so that the curve following map for $[\tau_{\omega, \eta}]$ satisfies $\sC^k(0,0,\v)=(m,n,\v)$. Because the loop has not closed, $m \neq 0$ or $n \neq 0$. But because $(\omega, \eta,\v)$ has period $k$, we have $\sigma^m(\omega)=\omega$ and $\sigma^n(\eta)=\eta$. See equation \ref{eq:curve following relation}. So, $\omega$ is periodic or $\eta$ is periodic. We can see that $(\omega, \eta, \v)$ does not have a stable periodic orbit, since we can always perturb $\omega$ and $\eta$ within any $U$ and $V$ to destroy periodicity but to ensure that the curve following map $\sC_0$ of the perturbed tiling satisfies $\sC^k_0(0,0,\v)=(m,n,\v)$. \end{proof}
\begin{remark}[Closed curves in the arithmetric graph] In polygonal billiards and polygonal outer billiards, a periodic orbit is called {\em stable} if periodic paths with the same combinatorial type do not disappear when sufficiently small changes are made to the polygon. The fact that closed curves in the arithmetic graph correspond to stable periodic orbits also holds true in the study of outer billiards in polygons. See \cite{S09} for the case when the polygon is a kite. A periodic billiard path in a triangle gives rise to a so-called hexpath in the hexagonal tiling of the plane. This hexpath is always periodic up to a translation, and the periodic billiard path is stable if and only if this translation is trivial, i.e. the hexpath closes up. See \cite{HooSch09}. Both these statements have generalizations to all polygons which can be obtained by appropriately interpreting known combinatorial criteria for stability. See \cite{Tab} for these combinatorial criteria. \end{remark}
\begin{comment} \begin{remark}[Drift and aperiodicity] There is one trivial mechanism for a positive measure set of aperiodic orbits to appear. Let $\mu$ and $\mu'$ be shift invariant probability measures on $\Omega_\pm$, and set $\nu=\mu \times \mu' \times \mu_N$ as in Proposition \ref{prop:product measures}. If the symbols $1$ and $-1$ appear with unequal probability according to either the measure $\mu$ or $\mu'$, then there will be a subset of $A \subset X$ with positive $\nu(A)>0$ so that no point $(\omega, \eta, \v) \in A$ has a stable periodic orbit. To see this, it can be observed that the function $$M:X \to \{\pm 1\}; \quad (\omega, \eta, (a,b)\big) \mapsto b \omega_0+a \omega_0$$ is $\Phi$-invariant. With such an asymmetric choice of $\mu$ or $\mu'$, there is $\Phi$ has non-trivial drift on each of the two sets $M^{-1}(\pm 1)$, meaning that $$\int_{M^{-1}(\pm 1)} a ~d\nu\big(\omega, \eta, (a,b)\big) \neq 0 \quad \textrm{or} \quad \int_{M^{-1}(\pm 1)} b ~d\nu\big(\omega, \eta, (a,b)\big) \neq 0.$$ This erases the possibility of statement (4) of Proposition \ref{prop:stable} holding $\nu$-a.e.. Measures discussed in this paper weight $1$ and $-1$ equally, so this observation plays no roll in this paper. \end{remark} \end{comment}
\begin{comment} \begin{corollary} \name{cor:restatement} Let $\mu$ and $\mu'$ be shift-invariant measures on $\Omega_\pm$. Let $P_n \subset X$ be the set of all $(\omega, \omega',\v)$ with stable periodic orbits of period $n$. Fix an edge $e$ of the tiling of the plane by squares centered at the integer points as in the theorem of the introduction. Then, $\mu \times \mu' \times \mu_N(P_n)$ is equal to the $\mu \times \mu'$ measure of those $(\omega, \omega')$ so that the curve of the tiling $\tiling{\tau_{\omega, \omega'}}$ through $e$ is closed and visits $n$ squares (counting multiplicities). \end{corollary}
The proof follows from \hyperref[prop:stable]{the Stability Proposition} together with the observation that both quantities are translation invariant. The fact that the horizontal or vertical orientation of $e$ is irrelevant follows from the fact that curves of the tiling alternate intersecting horizontal and vertical edges. We omit a detailed proof of this corollary. \end{comment}
\begin{comment} \subsection{An invariant function and drift} \name{sect:drift} In this section, we prove a {\em drift} theorem, which implies that for some of the $\Phi$-invariant measures of the form $\mu \times \mu' \times \mu_N$, a positive measure of points have non-periodic trajectories.
We now give a vague description of the idea of drift in this context. Here, {\em drift} refers to some ``average'' motion of points in $\R^2$ according to the dynamics of following curves. Integrating over a stable periodic orbit yields zero drift. So, if according to some invariant measure the drift is non-zero, then there must be a positive measure of points whose orbits are not stably periodic.
The first observation of this section is that there is a simple $\Phi$ invariant function on $X=\Omega \times \Omega' \times N$.
\begin{lemma}[Invariant function] \name{lem:M} Let $M\big(\omega, \omega', (a,b)\big)=b \omega_0+a \omega_0'$. This is a $\Phi$-invariant function from $X$ to $\{\pm 1\}$. \end{lemma} \begin{proof}[Sketch of proof] We partition the space $\Omega_\pm \times \Omega_\pm \times N$ into $16$ subsets $G\big(s,s',(a,b)\big)$ according to choices of $s,s' \in \{\pm 1\}$ and $(a,b) \in N$. These groups are defined $$G\big(s,s',(a,b)\big)=\big\{\big(\omega, \omega', (a,b)\big) \in \Omega_\pm \times \Omega_\pm \times N~:~ \textrm{$\omega_0=s$ and $\omega'_0=s'$}\}.$$ Write $\sG$ for the set of these 16 subsets. Let $\sim$ be the strongest equivalence relation on $\sG$ for which $G_1 \sim G_2$ whenever $\Phi(G_1)$ intersects $G_2$. The equivalence classes can be computed by drawing the graph where the nodes are elements of $\sG$ and the arrows are drawn from $G_1$ to $G_2$ whenever $\Phi(G_1)$ intersects $G_2$; the equivalence classes are then the connected components of this graph. One of the two maximal equivalence classes is shown below. \begin{center} \begin{xy} (10,20)*+{G\big(1,1,(1,0)\big)}="a"; (50,20)*+{G\big(1,-1,(0,1)\big)}="b"; (90,20)*+{G\big(-1,-1,(-1,0)\big)}="c"; (130,20)*+{G\big(-1,1,(0,-1)\big)}="d"; (10,0)*+{G\big(1,1,(0,1)\big)}="ap"; (50,0)*+{G\big(1,-1,(-1,0)\big)}="bp"; (90,0)*+{G\big(-1,-1,(0,-1)\big)}="cp"; (130,0)*+{G\big(-1,1,(1,0)\big)}="dp"; {\ar@{->} "a";"b"}; {\ar@{->} "b";"c"}; {\ar@{->} "c";"d"}; {\ar@{<-} "ap";"bp"}; {\ar@{<-} "bp";"cp"}; {\ar@{<-} "cp";"dp"}; {\ar@{<->} "a";"ap"}; {\ar@{<->} "b";"bp"}; {\ar@{<->} "c";"cp"}; {\ar@{<->} "d";"dp"}; {\ar@{->}@/_{1.5pc}/ "d";"a"}; {\ar@{->}@/_{1.5pc}/ "ap";"dp"}; \end{xy} \end{center} Note that $M\equiv1$ on this equivalence class. The function $M$ takes the value $-1$ on the eight remaining subsets. \end{proof}
We now consider the drift of a measure restricted to one of the two invariant sets provided by the above lemma. This enables us to prove the following theorem.
\begin{theorem}[Drift Theorem] Suppose $\mu$ and $\mu'$ are shift-invariant probability measure on $\Omega_\pm$ satisfying $$p=\int_{\Omega_\pm} \omega_0~d\mu(\omega) \and q=\int_{\Omega_\pm} \omega'_0~d\mu'(\omega').$$ Then the $\mu \times \mu' \times \mu_N$ measure of the set of all $(\omega, \omega',\v)$
without stable periodic orbits is at least $\max \{|p|, |q|\}$. \end{theorem} \begin{proof} \red{{\bf Referee's comment:} If you must have the Drift Theorem in the paper, maybe you can organize it differently. You could say at the start of the proof that $$\int_{P_s} (a,b)~d(\ldots)=0$$ by statement (4) of Proposition 13. On the other hand, you’ll show below that $$\int X_s (a,b)~d(\ldots)=\frac{1}{4}(q,p).$$ }
If $f$ and $g$ are two functions on a space $Y$ and $\nu$ is a measure on $X$, denote the pair of values consisting of the integrals of $f$ and $g$ by $$\int_Y (f,g)~d\nu=\big( \int_Y f~d\nu, \int_Y g~d\nu\big).$$
Let $X_s=M^{-1}(\{s\})$ for $s \in \{\pm 1\}$. Let $\nu=\mu \times \mu' \times \mu_N$. We would like to compute the integral $$I=\int_{X_s} (a,b)~d \nu\big(\omega, \omega',(a,b)\big).$$ Here $(a,b)$ varies over the four values of the vectors in $N$. Define $$X_s(a,b)=\{(\omega, \omega') \in \Omega_\pm \times \Omega_\pm~:~M\big(\omega, \omega',(a,b)\big)=s\}.$$ We have $X_s=\bigcup_{\v \in N} X_s(\v)$ and \begin{equation} \name{eq:I} I=\frac{1}{4} \sum_{(a,b) \in N} \int_{X_s(a,b)} (a,b)~d(\mu \times \mu')(\omega, \omega'), \end{equation} with the integral taken over all pairs $(\omega,\omega')$ with $(a,b)$ fixed by the sum. The $\frac{1}{4}$ appears because of the removal of $\mu_N$. Consider the case $(a,b)=(1,0)$. Note that $M\big(\omega,\omega',(1,0)\big)=\omega'_0$, so that whenever $(\omega, \omega', \v) \in X_s(1,0)$ we have $s \omega'_0=1$. Therefore, $$\int_{X_s(1,0)} (1,0)~d(\mu \times \mu')(\omega, \omega')=\int_{X_s(1,0)} (s \omega'_0,0) ~d(\mu \times \mu')(\omega, \omega').$$ Similarly, in the case $(a,b)=(-1,0)$, we have $M\big(\omega,\omega',(-1,0)\big)=-\omega'_0$, so that whenever $(\omega, \omega', \v) \in X_s(-1,0)$ we have $s \omega'_0=-1$. Thus, $$\int_{X_s(-1,0)} (-1,0)~d(\mu \times \mu')(\omega, \omega')=\int_{X_s(-1,0)} (s \omega'_0,0) ~d(\mu \times \mu')(\omega, \omega').$$ Observe that $X_s(1,0)=\Omega_\pm \times \{\omega'~:~\omega'_0=s\}$ and $X_s(-1,0)=\Omega_\pm \times \{\omega'~:~\omega'_0=-s\}$ are disjoint and their union is $\Omega_\pm \times \Omega_\pm$. Therefore, for two terms of the sum in equation \ref{eq:I} we have $$\int_{X_s(1,0)} (1,0)~d(\mu \times \mu')+\int_{X_s(-1,0)} (-1,0)~d(\mu \times \mu')= s \int_{\Omega_\pm \times \Omega_\pm} (\omega'_0,0) d(\mu \times \mu')=(sq,0).$$ Similar analysis holds for the cases $(a,b)=(0,\pm 1)$ and show that the total integrals is given by $I=\frac{s}{4}(q,p)$.
Let $P_s$ denote the set of all $\big(\omega, \omega',(a,b)\big) \in X_s$ which have stable periodic orbits. This set is $\Phi$-invariant, and statement (4) of Proposition \ref{prop:stable} guarantees that $$\int_{P_s} (a,b)~d(\mu \times \mu' \times \mu_N)=0.$$
Consider $\big(\omega, \omega',(a,b)\big) \in X$ and $\Phi(\big(\omega, \omega',(a,b)\big)=\big(\eta, \eta', s(b,a)\big)$. Observe that $(a,b)+s(b,a)$ lies the set of four elements $\{(\pm 1, \pm 1\}$. Therefore, for any $\Phi$-invariant set $A \subset \Omega_\pm \times \Omega_\pm \times N$ with $$\int_{A} (a,b)~d(\mu \times \mu' \times \mu_N)=(x,y)$$
we have the naive bound $|x| \leq \frac{1}{2} \mu \times \mu' \times \mu_N(A)$ and $|y| \leq \frac{1}{2} \mu \times \mu' \times \mu_N(A)$.
We apply this naive bound to the invariant set $X_s \smallsetminus P_s$, we see $$\frac{s}{4}(q,p)=I=\int_{X_s \smallsetminus P_s} (a,b)~d(\mu \times \mu' \times \mu_N)$$
so that $\mu \times \mu' \times \mu_N(X_s \smallsetminus P_s) \geq \half \max \{|p|, |q|\}$. Since this is true for each $s \in \{\pm 1\}$, we have that the total measure of points in $X$ without stable periodic orbits is at least $\max \{|p|, |q|\}$ as desired. \end{proof} \end{comment}
\subsection{The collapsing map} \name{sect:collapsing} In the tiling renormalization procedure described in section \ref{sect:renormalizing tilings}, we took any $\omega$ and $\eta$ in $\Omega_\pm$ and removed all subwords of the form $-+$ to build new elements $\omega'$ and $\eta'$ in $\Omega_\pm$. The tiling $[\tau_{\omega',\eta'}]$ was shown to have a similar structure to the tiling $[\tau_{\omega,\eta}]$. The choice of $\omega'$ and $\eta'$ was only canonical up to a power of the shift map. In order to use this tiling renormalization procedure to understand the map $\Phi$ will will need to make the choice canonical. We do this via a map we call the {\em collapsing map}.
The idea of the collapsing function $c$ mentioned at the beginning of this section is to remove any substrings of the form $-+$ and then slide the remaining entries together toward the zeroth entry. For example, $$c(\ldots \underline{-+}+--\underline{-+}\underline{-+}\widehat+-\underline{-+}++\ldots)=\ldots +--\widehat+-++\ldots,$$ where underlined entries have been removed. There are two potential reasons why $c(\omega)$ may not be well defined. First, the zeroth entry might be removed by this process, so we lose track of the indexing. Second, the remaining list may not be bi-infinite.
We will now build up to a formal definition of the collapsing map. We define the set $S \subset \Omega_\pm$ to be the union of two cylinder sets, $$S={\mathit{cyl}}(\widehat{-}+) \cup {\mathit{cyl}}(-\widehat{+}).$$ We can restate the definition of the set $K(\omega)$ given in equation \ref{eq:K1} as \begin{equation} \name{eq:K2} K(\omega)=\{k \in \Z~:~\sigma^k(\omega) \not \in S\}. \end{equation} We call $\omega$ {\em unbounded-collapsible} if $K(\omega)$ has no upper nor lower bound. Our definition of $\omega'$ depended on an order preserving bijection $\Z \to K(\omega)$. Such a bijection is guaranteed to exist if $\omega$ is unbounded-collapsible, but there are many possible choices. If $0 \in K(\omega)$, we call $\omega$ {\em zero-collapsible} and define $i \mapsto k_i$ to be the unique order preserving bijection $\Z \to K(\omega)$ so that $k_0=0$. We call $\omega$ {\em collapsible} if it is both unbounded- and zero-collapsible. We use $C \in \Omega_\pm$ to denote the set of collapsible $\omega$, and define the collapsing map to be $$c:C \to \Omega_\pm; \quad [c(\omega)]_i=\omega_{k_i}.$$
We briefly record some properties of the collapsing map. \begin{theorem}[Properties of the collapsing map]\quad \name{thm:collapsing} \begin{enumerate} \item The map $c:C \to \Omega_\pm$ is a continuous surjection. \item \name{item:crenormalization} If $\wh \sigma:C \to C$ is the first return map of $\sigma$ to $C$, then $$c \circ \wh \sigma(\omega)= \sigma \circ c(\omega) \quad \text{for all $\omega \in C$.}$$ \item \name{item:collapsing measures} If $\mu$ is a shift-invariant measure on $\Omega_\pm$ then so is $\mu \circ c^{-1}$. \item \name{item:omegaalt2} Define $\omega^{\textrm{alt}} \in \Omega_\pm$ by $\omega^{\textrm{alt}}_n=(-1)^n$. If $\mu$ is a finite shift-invariant measure on $\Omega_\pm$, then $$\mu(\set{\omega \in \Omega_\pm}{$\omega$ is not unbounded-collapsible})=2 \mu(\{\omega^{\textrm{alt}}\}).$$ \end{enumerate} \end{theorem} \begin{proof}[Sketch of proof.] Suppose $\eta \in \Omega_\pm$. Then the collection of preimages, $c^{-1}(\eta)$, is contained in the collection of all $\omega \in \Omega_\pm$ obtained by inserting a non-negative power of the word $-+$ between each of the symbols in $\eta$. The only restriction is that a positive power must be inserted between every pair of symbols of the form $-+$. In particular, $c$ is a surjection. This discussion can also be used to prove that the preimage of a cylinder set is a union of cylinder sets intersected with $C$. So, $c$ is continuous.
To see statement (\ref{item:crenormalization}), observe that $\wh \sigma(\omega)=\sigma^n(\omega)$ where $n$ is the smallest positive entry in $K(\omega)$. The proof then follows from the definition of the collapsing map.
Statement (\ref{item:collapsing measures}) follows from two observations. The restriction of a $\sigma$-invariant measure to $C$ is $\widehat \sigma$-invariant. The pullback of a $\widehat \sigma$-invariant measure under $c$ is $\sigma$-invariant by (\ref{item:crenormalization}).
Statement (\ref{item:omegaalt2}) follows from the Poincar\'e Recurrence Theorem. If $\omega$ is not unbounded-collapsible, then $\sigma^n(\omega)$ converges to the periodic orbit $\{\omega^{\textrm{alt}}, \sigma(\omega^{\textrm{alt}})\}$ either as $n \to +\infty$ or $n \to -\infty$. The Poincar\'e Recurrence Theorem implies that the set of $\omega$ which are not unbounded-collapsible and do not belong to $\{\omega^{\textrm{alt}}, \sigma(\omega^{\textrm{alt}})\}$ has $\mu$-measure zero. \end{proof}
We close with the definition of two functions which will be important in the next subsection. These are the forward and backward return times of $\sigma$ to $C$. \begin{equation} \name{eq:r+} \begin{array}{c} \displaystyle r_+:C \to \Z_+; \quad r_+(\omega)=\min \{n>0~:~\sigma^n(\omega) \in C\}.\\ \displaystyle r_-:C \to \Z_+; \quad r_-(\omega)=\min \{n>0~:~\sigma^{-n}(\omega) \in C\}. \end{array} \end{equation} Observe that these functions are well-defined for every $\omega \in C$.
\subsection{Renormalization Theorems} \name{sect:renormalization2} In this section, we describe general renormalization results for the map $\Phi:X \to X$, where $X=\Omega_\pm \times \Omega_\pm \times N$.
Define $\sR_1 \subset X$ to be the set of ``once renormalizable'' elements of $X$, \begin{equation} \name{eq:R1} \sR_1=C \times C \times N. \end{equation} That is, $\sR_1$ is the collection of all $(\omega, \eta, \v)$ where $\omega$ and $\eta$ are both collapsible. The renormalization mentioned is the map \begin{equation} \name{eq:rho} \rho:\sR_1 \to X; \quad (\omega, \eta, \v) \mapsto \big(c(\omega), c(\eta), \v\big). \end{equation} The manner in which $\rho$ renormalizes the map $\Phi$ is described by the theorem below.
Before stating the theorem, we define some important subsets of $X$: $$P_4=\{\textrm{$x\in X$~:~$x$ has a stable periodic orbit of period $4$}\}.$$ $${\mathit NUC}=\set{(\omega, \eta, \v)\in X}{either $\omega$ or $\eta$ is not unbounded-collapsible}.$$ The points in $P_4$ correspond to loops in a tiling of smallest possible size. The points in ${\mathit NUC}$ consist of all $(\omega, \eta,\v)$ so that $\omega$ and $\eta$ fail to satisfy the assumption \ref{eq:assumption} necessary for the Tiling Renormalization Theorem of Section \ref{sect:renormalizing tilings} to hold. With this in mind, we restate that theorem in this context.
\begin{theorem}[Dynamical Renormalization] \name{thm:renormalization} \hspace{1em}
\begin{enumerate} \item The first return map $\wh \Phi:\sR_1 \to \sR_1$ of $\Phi$ to $\sR_1$ is well defined and invertible. \item If $x \in \sR_1$, we have $\rho \circ \wh \Phi(x)=\Phi \circ \rho(x).$ \item The following statements are equivalent for any $x \in X \sm {\mathit NUC}$. \begin{enumerate} \item There is no $k>0$ so that $\Phi^k(x) \in \sR_1$. \item There is no $k<0$ so that $\Phi^k(x) \in \sR_1$. \item $x \in P_4$. \end{enumerate} \item A point $x \in \sR_1$ has a stable periodic orbit if and only if $\rho(x)$ has a stable periodic orbit. Moreover, $\rho(x)$ has strictly smaller period than $x$. \end{enumerate} \end{theorem}
We omit the proof of this theorem. It follows from Theorem \ref{thm:tiling renormalization} using the connection between curve following and the map $\Phi$ described in Section \ref{sect:curve following}. See equation \ref{eq:curve following relation}.
Statements (3) and (4) of the Renormalization Theorem are useful for detecting stable periodic orbits. A periodic orbit is shortened when applying $\rho$. If we can apply $\rho$ infinitely many times, then eventually the orbit becomes period four, and then the orbit vanishes under one more application of $\rho$. This is the basic observation enabling us to compute the total measures of periodic points for some measures.
For applications, we will need to compute the return time function $R_1:\sR_1 \to \Z_+$ of $\Phi$ to $\sR_1$. We do this in terms of the functions $r_+$ and $r_-$ defined in equation \ref{eq:r+} below.
\begin{lemma}[Dynamical Return Time] \name{lem:returns} Fix $(\omega, \eta, \v) \in \sR_1$. Let $(a,b)=\v$. Define $s=\omega_0 \eta_0$ and $\bw=(sb,sa)$. Then, $$R_1(\omega, \eta, \v)=\begin{cases} 2 r_+(\omega)-1 & \text{if $\bw=(1,0)$,} \\ 2 r_-(\omega)-1 & \text{if $\bw=(-1,0)$,} \\ 2 r_+(\eta)-1 & \text{if $\bw=(0,1)$,} \\ 2 r_-(\eta)-1 & \text{if $\bw=(0,-1)$.} \end{cases}$$ \end{lemma} This lemma follows directly from Theorem \ref{thm:return time}, so we omit the proof.
\section{Renormalization of the Rectangle Exchange Maps} \name{sect:rectangle exchanges}
In this section, we explain how the renormalization of the map $\Phi$ described in section \ref{sect:renormalization2} induces a renormalization of the polygon exchange maps $\til \Psi_{\alpha, \beta}$ defined in the introduction.
The first subsection provides necessary prerequisite details involving coding of rotations.
\subsection{Coding Rotations} \name{sect:rot} Let $\alpha \in \R$. The rotation by $\alpha$ is the map $$T_\alpha:\R/\Z \to \R/\Z; \quad x \mapsto x+\alpha.$$ Given any $x \in \R/\Z$ we construct an element of $\Omega_\pm$ via coding, $$\varsigma:\R/\Z \to \Omega_\pm; \quad \varsigma(x)_n= \begin{cases} 1 & \textrm{if $T_\alpha^n(x) \in [0, \half)$}\\ -1 & \textrm{otherwise.} \end{cases}$$ Observe that $\varsigma$ semiconjugates the rotation to the shift map on $\Omega_\pm$. That is, \begin{equation} \name{eq:semiconj} \sigma \circ \varsigma(x)=\varsigma \circ T_\alpha(x) \quad \text{for all $x \in \R/\Z$.} \end{equation} The map $\varsigma$ is an embedding so long as $\alpha$ is irrational. Since Lebesgue measure $\lambda$ is invariant under $T_\alpha$, we can pull back Lebesgue measure to obtain a $\sigma$-invariant measure on $\Omega_\pm$, namely \begin{equation} \name{eq:mu alpha} \mu_\alpha=\lambda \circ \varsigma^{-1}. \end{equation}
Recall that a rotation is conjugate to its inverse via an orientation reversing isometry of the circle. Moreover, if we choose the particular orientation reversing isometry $$\iota:t \mapsto \half-t \pmod{1},$$ we see that $\varsigma \circ \iota$ is the coding map of $T_{-\alpha}$ (modulo a set of Lebesgue measure zero consisting of the orbits of $0$ and $\half$). In particular, the two measures $\mu_\alpha$ and $\mu_{-\alpha}$ are equal. Because we will be primarily interested in the measures which arise from this construction, it is natural for us to only consider rotations $T_\alpha$ with $\alpha \in [0,\half]$.
This observation explains the connection between rotations and the group $G$ of isometries of $\R$ preserving $\Z$. Explicitly, $G$ is the group of maps of the form $$g:\R \to \R; \quad t \mapsto rt+n \quad \text{with $n \in \Z$ and $r \in \{\pm 1\}$.}$$ For the following theorem, we will need to make more observations and definitions involving this group. The interval $[0,\half]$ is a fundamental domain for the $G$-action on $\R$. We define the map $$o:\R \to \{\pm 1\}; \quad t \mapsto \begin{cases} 1 & \text{if $\exists n \in \Z$ so that $t+n \in [0, \half]$} \\ -1 & \text{otherwise.} \end{cases} $$ This map records the orientation of the element $g \in G$ which caries $t$ into $[0,\half]$. If there is ambiguity, the map chooses positive sign.
Recall the definition of the collapsible elements $C \subset \Omega_\pm$ and the collapsing map $c:C \to \Omega_\pm$. The shift map $\sigma$ on $\Omega_\pm$ was renormalized in a sense by the collapsing map, because the collapsing map semiconjugates the first return $\wh \sigma$ of the shift map to $C$ to the shift map. See statement (\ref{item:crenormalization}) of Theorem \ref{thm:collapsing}.
The following theorem explains how the collapsing map interacts with the rotation via coding. The theorem observes the existence of a renormalization in the sense used in the theory of interval exchange maps. In this setting a {\em renormalization} is simply a return map to an interval which is conjugate up to a dilation to an interval exchange map on the same number of intervals. (A rotation is an interval exchange defined using two intervals.)
\begin{theorem}[Rotation Renormalization] \name{thm:rotren} Assume $\alpha \in [0, \half)$. \begin{enumerate} \item \name{item:collapsing interval} The preimage of the collapsible sequences, $\varsigma^{-1}(C)$, is the interval $C_\alpha=[\alpha, 1-\alpha)$. \item \name{item:ret id} In particular, the first return map $\wh T_\alpha$ of the rotation $T_\alpha$ to $C_\alpha$ satisfies $$\varsigma \circ \wh T_\alpha(x)=\wh \sigma \circ \varsigma(x) \quad \text{for all $x \in C_\alpha$.}$$ \item \name{item:iet ret} The first return map $\wh T_\alpha:C_\alpha \to C_\alpha$ is the rotation by $\alpha$ modulo $1-2\alpha$. \item \name{item:psi} Let $\gamma=f(\alpha)$, where $f(\alpha)$ denotes the element of $[0, \half]$ which is $G$-equivalent to $\frac{\alpha}{1-2\alpha}$ as in equation \ref{eq:f} of the introduction. As in equation \ref{eq:psi}, define the dilation $$\psi=\psi_\alpha:[\alpha, 1-\alpha) \to \R/\Z; \quad \psi(x)= \begin{cases} \frac{x-\frac{1}{2}}{1-2\alpha}+\frac{1}{2} & \textrm{if $o(\frac{\alpha}{1-2\alpha})=1$,} \\ \frac{\frac{1}{2}-x}{1-2\alpha} & \textrm{if $o(\frac{\alpha}{1-2\alpha})=-1$.} \end{cases}$$ This dilation has the following properties: \begin{enumerate} \item $\psi \circ \wh T_\alpha(x)=T_\gamma \circ \psi(x)$ for all $x \in C_\alpha$. \item If $\varsigma'$ is the coding map for $T_\gamma$, then $c \circ \varsigma(x)=\varsigma' \circ \psi(x)$ for $\lambda$-almost every $x \in C_\alpha$. \end{enumerate} \end{enumerate} \end{theorem}
We make several comments about this theorem. First, it should be observed that the return map $\wh T_\alpha$ defines a renormalization in the interval exchange sense. Once we know that $\wh T_\alpha$ is a rotation by $\alpha$ modulo $1-2\alpha$, we know that any surjective dilation $C_\alpha \to \R/\Z$ will conjugate $\wh T_\alpha$ to either $T_\gamma$ or $T_{-\gamma}$, depending on orientation. However, when $\alpha$ is irrational, there is a unique choice of a dilation that respects codings as in statement (4b). Second, when $\alpha$ is irrational, we can really think of this as a pullback of the renormalization happening on $\Omega_\pm$. This is because $\varsigma$ is injective, and $\varsigma \circ T_\alpha=\sigma \circ \varsigma$. In this case, we could alternately define $$\wh T_\alpha=\varsigma^{-1} \circ \wh \sigma \circ \varsigma \and \psi=(\varsigma')^{-1} \circ c \circ \varsigma.$$
The following describes the action of the renormalizing map $c$ on measures of the form $\mu_\alpha$ as defined in equation \ref{eq:mu alpha}.
\begin{corollary}[Action on Measures] \name{cor:collapsing rotations} Suppose $0 \leq \alpha < \frac{1}{2}$ and let $\gamma$ be as in statement \ref{item:psi} of Theorem \ref{thm:rotren}. Then, $$\mu_\alpha \circ c^{-1}=(1-2\alpha) \mu_\gamma.$$ \end{corollary} \begin{proof} This follows from statement (\ref{item:psi}b) and the fact that the length of $C_\alpha$ is $1-2\alpha$. \end{proof}
It will also be useful to record the values of the function $r_+$ and $r_-$ defined in equation \ref{eq:r+}. For $x \in C_\alpha$, the quantities $r_+\circ\varsigma(x)$ and $r_-\circ\varsigma(x)$ record the first return times of $T_{\alpha}$ and $T^{-1}_{\alpha}$ to $C_\alpha$, respectively. \begin{lemma}[Rotation Return Times] \name{lem:rrt} For Lebesgue-almost every $x \in C_\alpha$, we have $$r_+\circ\varsigma(x)=2 \left\lfloor \frac{x}{1-2\alpha} \right\rfloor+1 \and r_-\circ\varsigma(x)=2 \left\lfloor \frac{1-x}{1-2\alpha} \right\rfloor+1,$$ where $\floor{t}$ denotes the greatest integer less than or equal to $t$. \end{lemma}
We have an alternate formula for the return times, which will be useful later.
\begin{corollary} \name{cor:rot ret} Suppose $f(\alpha)=r(\frac{\alpha}{1-2\alpha}-n)$ for $n \in \Z$ and $r \in \{\pm 1\}.$ If $r=1$, then for $\mu_\alpha$-a.e. collapsible $\omega$, we have: $$r_+(\omega)=\begin{cases} 2n+3 & \text{if $c(\omega) \in {\mathit{cyl}}(\wh -+)$} \\ 2n+1 & \text{otherwise,} \end{cases} \and r_-(\omega)=\begin{cases} 2n+3 & \text{if $c(\omega) \in {\mathit{cyl}}(-\wh +)$} \\ 2n+1 & \text{otherwise.} \end{cases}$$ If $r=-1$, then for $\mu_\alpha$-a.e. collapsible $\omega$, we have: $$r_+(\omega)=\begin{cases} 2n-1 & \text{if $c(\omega) \in {\mathit{cyl}}(\wh +-)$} \\ 2n+1 & \text{otherwise,} \end{cases} \and r_-(\omega)=\begin{cases} 2n-1 & \text{if $c(\omega) \in {\mathit{cyl}}(+\wh -)$} \\ 2n+1 & \text{otherwise.} \end{cases}$$ \end{corollary}
We now give proofs of the Rotation Renormalization Theorem and Rotation Return Time Lemma. We will conclude this subsection with a proof of the Corollary.
\begin{proof}[Proof of Theorem \ref{thm:rotren} and Lemma \ref{lem:rrt}] We begin by proving statement (\ref{item:collapsing interval}) of the Theorem. If $x \in [1-\alpha,1)$, then $\varsigma(x) \in {\mathit{cyl}}(\wh - +)$. And, if $x \in [0,\alpha)$, then $\varsigma(x) \in {\mathit{cyl}}(- \wh +)$. In either case $\varsigma(x)$ is not zero-collapsible. If $x \in [\alpha, \half)$, then $\varsigma(x) \in {\mathit{cyl}}(+ \wh +)$, and if $x \in [\half, 1-\alpha)$ then $\varsigma(x) \in {\mathit{cyl}}(\wh - -)$. In these cases, $x$ is zero-collapsible. We also observe that $x \in [\alpha, 1-\alpha)$ is always unbounded-collapsible, because the only way an infinite sequence of alternating signs can appear from coding a $T_\alpha$ is when $\alpha=\half$.
Statement (\ref{item:ret id}) follows from statement (\ref{item:collapsing interval}) and equation \ref{eq:semiconj}.
We now prove statement (\ref{item:iet ret}) of the Theorem and the formula for $r_+$ given in the Lemma. To do this, we provide an pseudo-code algorithm to produce the first return $\wh T_\alpha(x) \in [\alpha, 1-\alpha)$: \begin{enumerate} \item[(0)] Set $i=0$ and $x_0=x$. \item[(1)] If $x_i+\alpha \in [\alpha, 1-\alpha)$ then $\wh T_\alpha(x)=x_i+\alpha$. Stop, because we have found $\wh T_\alpha(x)$. \item[(2)] Set $x_{i+1}=x_i-1+2 \alpha$. \item[(3)] Iterate $i$. (Set $i$ to be $i+1$.) Return to step $1$. \end{enumerate} For the moment assume this procedure terminates with $\wh T_\alpha(x)=x_n+\alpha$. (We prove this occurs for some $n$ below.) Observe that $x_i=x_0+i(2 \alpha-1)$ for all $i$, so that $x_n+\alpha$ is indeed equivalent to $x+\alpha$ modulo $1-2\alpha$. Therefore, $\wh T_\alpha$ is indeed a rotation by $\alpha$ modulo $1-2\alpha$.
We now explain why the algorithm terminates. If it fails to terminate with $i=0$, then $x_0+\alpha \geq 1-\alpha$. Observe that $x_i+\alpha=x+i(2\alpha-1)+\alpha$ is a decreasing sequence and that $x_i-x_{i+1}=1-2\alpha$. Since the sequence $\{x_i+\alpha\}$ iteratively decreases by an amount equal to the length of $[\alpha, 1-\alpha)$, there is precisely one integer $n$ for which $x_n+\alpha \in [\alpha, 1-\alpha)$. This integer is given by $n=\lfloor \frac{x}{1-2\alpha}\rfloor$.
Now suppose $0 \leq i<n$. Then $T(x_i)=x_i+\alpha \in [1-\alpha, 1)$, and $T^2(x_i)=x_i+2\alpha-1=x_{i+1}$. So by induction, $T^{2i}(x)=x_i$ and $T^{2i+1}(x)=x_i+\alpha$ for $0 \leq i \leq n$. Moreover, we have $T^{2n+1}(x)=x_n+\alpha$ is the first return of $T$ to $[\alpha, 1-\alpha)$. Thus, $r_+\circ\varsigma(x)=2n+1$ as desired.
Now we verify the formula for $r_-\circ\varsigma(x)$ given in the Lemma. Observe that the map $x \mapsto 1-x$ conjugates $T_\alpha$ to $T^{-1}_\alpha$ and sends $C_\alpha$ to $C_\alpha$ almost-everywhere. Therefore, we have $$r_-\circ\varsigma(x)=r_+\big(\varsigma(1-x)\big)$$ almost everywhere. (This only fails at the point $\alpha \in C_\alpha$.)
Finally, we prove statement (\ref{item:psi}) of the Theorem. Observe that $\psi$ is a bijective dilation $C_\alpha \to \R/\Z$. By the remarks below the theorem, the dilation conjugates $\wh T_\alpha$ to a rotation by $\pm \gamma$. Since the orientation preserving nature of the element of $g \in G$ carrying $\frac{\alpha}{1-2\alpha}$ matches the orientation preserving nature of $\psi$, we know the dilation conjugates $\wh T_\alpha$ to $T_\gamma$. This proves statement (a). Statement (b) follows from the fact that the map $\psi$ respects the labeling of intervals by $\pm 1$ almost-everywhere. As in the definition of $\varsigma$, we have labeled the interval $[0, \half)$ by $+1$ and $[\half, 1)$ by $-1$. Observe $$\psi\big([0, {\textstyle \half}) \cap C_\alpha\big)=[0, {\textstyle \half}) \and \psi\big([{\textstyle \half},1) \cap C_\alpha\big)=[{\textstyle \half},1)$$ almost-everywhere (with ambiguities at endpoints). Since this is true almost-everywhere, statement (b) follows from statement (a). \end{proof}
\begin{proof}[Proof of Corollary \ref{cor:rot ret}] Since we only need this statement $\mu_\alpha$-a.e., we can assume that $\omega=\varsigma(x)$. By the lemma, the formulas in the corollary for $r_+$ are equivalent to formulas for the function $$m(x)=\left\lfloor \frac{x}{1-2\alpha} \right\rfloor.$$ Note that $m(x)$ takes only two values on $[\alpha, 1-\alpha)$. Since $m$ is an increasing function, these two values are $m(\alpha)$ and $m(\alpha)+1$.
Consider the case when $r=1$. Then, $n \leq \frac{\alpha}{1-2\alpha} \leq n+\half$. The function $m(x)$ takes the two values $n$ and $n+1$, with the discontinuity happening at the point $y=(1-2\alpha)(n+1)$. We compute $$\psi(y)=n+1-\frac{1}{2(1-2\alpha)}+\half=1+n-\frac{\alpha}{1-2\alpha}=1-\gamma,$$ where $\gamma=f(\alpha)$. So $z=\psi(y)$ is also the first point (from left to right) for which $z \in[\half,1)$ but $T_\gamma(z) \in [0, \half)$. The points $x \in [\alpha,1-\alpha)$ to the right of $y$ are characterized by the fact that $\psi(x) \geq z$. This is equivalent to the condition that $$\varsigma \circ \psi(x)=c \circ \varsigma(x) \in {\mathit{cyl}}(\wh -+).$$ In other words, the larger value is taken if and only if $c(\omega) \in {\mathit{cyl}}(\wh -+)$.
The proof in the case of $r=-1$ is similar. We have $n-\half < \frac{\alpha}{1-2\alpha} < n.$ This time the discontinuity occurs when $y=n(1-2\alpha)$. Then we have $$\psi(y)=\frac{1}{2(1-2\alpha)}-n=\half+\frac{\alpha}{1-2\alpha}-n=\half-\gamma.$$ Let $z=\psi(y)$. Recall that $\psi$ is orientation reversing and sends the endpoints of $[\alpha, 1-\alpha)$ to $\half$. So, the points $x \in [\alpha, 1-\alpha)$ to the left of $y$ are characterized by the fact that $\psi(x) \in [\half-\gamma,\half)$. Equivalently, we have $$\varsigma \circ \psi(x)=c \circ \varsigma(x) \in {\mathit{cyl}}(\wh +-).$$
To see the equations for $r_-$, it suffices to use the orientation reversing involution $t \mapsto 1-t$, which conjugates $T_\alpha$ to its inverse, nearly preserves $[\alpha, 1-\alpha)$, and switches the labeling of subintervals by $\pm 1$. \end{proof}
\subsection{Rectangle exchange transformations} \name{sect:ret rot} Fix $\alpha$ and $\beta$ in $[0,\half)$. Recall we defined an embedding $\pi:\til Y \times N \to X$ from the discussion of the arithmetic graph in equation \ref{eq:pi}. An alternate definition for this embedding can be given using the coding maps $\varsigma$ and $\varsigma'$ of the rotations $T_\alpha$ and $T_\beta$, respectively. Namely, we have \begin{equation} \name{eq:pi2} \pi:\til Y \times N \to X; \quad \pi(x,y,\v)=\big(\varsigma(x), \varsigma'(y), \v\big). \end{equation} By Proposition \ref{prop:ag conj}, we have $\pi \circ \til \Psi_{\alpha, \beta}=\Phi \circ \pi,$ where $\til \Psi_{\alpha, \beta}$ is the rectangle exchange map defined in equation \ref{eq:psi til} of the introduction.
So long as $\alpha$ and $\beta$ are irrational, the map $\pi$ is an embedding. We can use this embedding to pullback the renormalization of the map $\Phi$ defined in section \ref{sect:renormalization2} to a renormalization of these rectangle exchange maps. This yields the following theorem.
\begin{theorem}[Rectangle Exchange Renormalization] \name{thm:rectren} Assume $\alpha, \beta \in [0, \half)$. \begin{enumerate} \item \name{item:once ren} Let $Z$ be the rectangle $Z=[\alpha,1-\alpha) \times [\beta, 1-\beta)$. The union of four rectangles $Z \times N$ is the preimage $\pi^{-1}(\sR_1)$ of the once renormalizable elements of $X$. \item \name{item:ret id2} The first return map $\wh \Psi$ of the rectangle exchange $\til \Psi_{\alpha, \beta}$ to $Z \times N$ satisfies $$\pi \circ \wh \Psi(x)=\wh \Phi \circ \pi(x) \quad \text{for all $x \in Z \times N$.}$$ \item \name{item:phi} The map $\phi:Z \times N \to \til Y \times N$ defined by $\phi=\psi_\alpha \times \psi_\beta \times \textit{id}$ as in equation \ref{eq:phi} satisfies the following statements: \begin{enumerate} \item $\phi \circ \wh \Psi(z)=\til \Psi_{f(\alpha),f(\beta)} \circ \phi(z)$ for all $z \in Z \times N$. \item Let $\pi'$ be the embedding $\til Y \times N \to X$ defined as in equation \ref{eq:pi2}, but using the coding maps for $T_{f(\alpha)}$ and $T_{f(\beta)}$. Then, $$\pi' \circ \phi(z)=\rho \circ \pi(z) \quad \text{for Lebesgue-almost every $z \in Z \times N$.}$$ \end{enumerate} \end{enumerate} \end{theorem} These statements indicate that the first return map $\til \Psi$ of $\til \Psi_{\alpha, \beta}$ to the union $Z \times N$ of rectangles is affinely conjugate to $\til \Psi_{f(\alpha), f(\beta)}$. So, this describes a renormalization in the rectangle exchange sense. The theorem also indicates compatibility with the renormalization of the map $\Phi:X \to X$. In fact, so long as $\alpha$ and $\beta$ are irrational, we have the alternate almost everywhere equivalent definitions, $$\wh \Psi=\pi^{-1} \circ \wh \Phi \circ \pi \and \phi=(\pi')^{-1} \circ \rho \circ \pi.$$
We also record the action on measures. The pushforward of Lebesgue measure under the embedding $\pi$ is the measure $\mu_\alpha \times \mu_\beta \times \mu_N$ on $X$. Here $\mu_\alpha$ and $\mu_\beta$ are defined as in the previous section and $\mu_N$ is the uniform measure on $N$. \begin{corollary}[Action of $\rho$ on Measures] \name{cor:rho act} Let $\nu=\mu_\alpha \times \mu_\beta \times \mu_N$ and $\nu'=\mu_{f(\alpha)} \times \mu_{f(\alpha)} \times \mu_N$. Then, $$\nu \circ \rho^{-1}=(1-2\alpha)(1-2\beta) \nu'.$$ \end{corollary} The proof follows from the above renormalization theorem and Corollary \ref{cor:collapsing rotations}.
\section{The Return Time Cocycle} \name{sect:cocycle} In this section, we state our main formula for computing the total measure of the set $$NS=\{x\in X~:~ \text{$x$ does not have a stable periodic orbit under $\Phi$}\}$$ with respect to the measures coming from rectangle exchange maps.
\subsection{The Cocycle Limit Formula} \name{sect:limit formula} Assume $\alpha$ and $\beta$ are irrationals in $(0,\half)$. Set $\nu=\mu_\alpha \times \mu_\beta \times \mu_N$. Our formula is given using the following data: \begin{enumerate} \item \name{item:O} We find a nested sequence of Borel sets, $$X=\sO_0 \supset \sO_1 \supset \sO_2\ldots \quad \text{so that} \quad {\mathit NS}=\bigcap_{i=0}^\infty \sO_n$$ up to a set of $\nu$-measure zero. Thus we have $\nu({\mathit NS})=\lim_{n \to \infty} \nu(\sO_n).$ \item We now define the {\em return time cocycle} $N(\alpha,\beta,k):\R^4 \to \R^4$ over the dynamics of $f \times f$. (The transformation $f$ acting on the irrationals in $(0,\half)$ was defined in equation \ref{eq:f}.) Using $\alpha$ and $\beta$, we define $m,n \in \Z$ and $r,n \in \{\pm 1\}$ according to the formula: $$f(\alpha)=r(\frac{\alpha}{1-2 \alpha}-m) \and f(\beta)=s(\frac{\beta}{1-2\beta}-n),$$ We define $N(\alpha, \beta,0)$ to be the identity matrix and define \begin{equation}\name{eq:N} N(\alpha,\beta,1)=\left[\begin{array}{rrrr} 2m+r & 1 & 0 & 2m+r \\ 2m & 1 & 0 & 2m \\ 0 & 2 n+s & 2 n+s & 1 \\ 0 & 2 n & 2 n & 1 \\ \end{array}\right]. \end{equation} This matrix has determinant $rs \in \{\pm 1\}$. We extend inductively by defining $$N(\alpha, \beta,k+1)=N\big(f^k(\alpha),f^k(\beta), 1\big) N(\alpha, \beta,k) \quad \text{for $k \geq 1$.}$$ \item We define a one-dimensional cocycle $D$ over the dynamics of $f \times f$. This cocycle is defined by setting $D(\alpha, \beta,0)=1$ and \begin{equation}\name{eq:D} D(\alpha, \beta,k)=\prod_{j=0}^{k-1} \big(1-2f^j(\alpha)\big)\big(1-2f^j(\beta)\big) \quad \text{for $k \geq 1$.} \end{equation} \item We define the vector $$\bn_{\alpha, \beta}=\left(\alpha(1-2\beta), \frac{1-2\alpha}{2}, \beta(1-2\alpha), \frac{1-2\beta}{2}\right).$$ \end{enumerate} \begin{theorem}[Cocyle formula]\name{thm:cocycle formula} Let $\alpha, \beta \in (0,\half)$ be irrational and let $k>0$. Define $$\nu=\mu_\alpha \times \mu_\beta \times \mu_N, \quad d_k=D(\alpha,\beta,k) \and \bn_k=\bn_{f^k(\alpha),f^k(\beta)}.$$ Letting $\1 \in \R^4$ denote the vector all of whose entries are one, we have $$\nu(\sO_{k+1})=d_k \bn_k \cdot N(\alpha,\beta,k) \1.$$ \end{theorem}
We have the following consequence by statement (1) above.
\begin{corollary}[Limit formula] \name{cor:limit formula} For irrationals $\alpha, \beta \in (0,\half)$ we have $$\nu({\mathit NS})=\lim_{k \to \infty} d_k \bn_k \cdot N(\alpha,\beta,k) \1.$$ \end{corollary}
\subsection{The return time cocycle} Our renormalization of $\Phi:X \to X$ described in section \ref{sect:renormalization2} is useful for measuring the prevalence of stable periodic trajectories on $X=\Omega_\pm \times \Omega_\pm \times N$. To begin to understand this, we recall some of the structure of the renormalization. We defined $\wh \Phi: \sR_1 \to \sR_1$ to be the first return map to a Borel subset $\sR_1 \subset X$. We found a Borel measurable map $\rho:\sR_1 \to X$ so that $$\rho \circ \wh \Phi(x)=\Phi \circ \rho(x) \quad \text{for each $x \in \sR_1$}.$$
We showed that the $\Phi$-orbit of an $x \in X$ always visits $\sR_1$ unless it belongs to the set $P_4$ of stable periodic orbits of period four, or if it belongs to the set ${\mathit NUC}$ of points $x=(\omega, \eta, \v)$ with $\omega$ or $\eta$ not unbounded collapsible. We view the case of $x \in {\mathit NUC}$ as rare, and justify this because ${\mathit NUC}$ has zero measure with respect to many product measures $\mu \times \mu' \times \mu_N$. (A criterion for this can be found in statement (\ref{item:omegaalt2}) of Theorem \ref{thm:collapsing}.) We make the following definition: \begin{definition} Let $\nu$ be a Borel measure on $X$. We say $\nu$ is {\em robustly renormalizable} if for all integers $n \geq 0$ we have $\nu \circ \rho^{-n}({\mathit NUC})=0$. \end{definition} \begin{remark} So long as $\alpha$ and $\beta$ are irrational, the measures $\nu=\mu_\alpha \times \mu_\beta \times \mu_N$ are robustly renormalizable. Corollary \ref{cor:rho act} describes $\nu \circ \rho^{-n}$ in this case and statement (\ref{item:omegaalt2}) of Theorem \ref{thm:collapsing} implies $\nu \circ \rho^{-n}({\mathit NUC})=0$. \end{remark}
To understand iterations of $\rho$, for each $n \geq 1$ define the subsets $$\sR_n=\rho^{-n}(X) \and \sO_n=\bigcup_{m \in \Z} \Phi^m(\sR_n).$$ We say that $x \in \sR_n$ is {\em $n$-times renormalizable.} The set $\sO_n$ is the smallest $\Phi$-invariant subset of $X$ containing $\sR_n$. When $x \in \sO_n$, we say that {\em the orbit of $x$ is $n$-times renormalizable}.
Recall that the renormalization $\rho$ has the property that $x \in \sR_1$ has a stable periodic orbit if and only if $\rho(x)$ has a stable periodic orbit, and that $\rho(x)$ has a strictly smaller period. By the discussion above the definition, if $\nu$ is robustly renormalizable, then $$\rho^n(\sR_n \sm \sO_{n+1})=P_4, \qquad \text{$\nu \circ \rho^{-n}$-a.e..}$$ (If we can't apply $\rho$ once more at some point in the orbit of $x \in \rho^n(\sR_n \sm \sO_{n+1})$, it must be that either $x \in P_4$ or $x \in {\mathit NUC}$.) In particular, almost every point in $\sO_n \sm \sO_{n+1}$ has a stable period orbit. Conversely, suppose $x$ has a stable periodic orbit of period larger than four. The fact that $\rho$ decreases periods guarantees that $x \in \sO_n \sm \sO_{n+1}$ for some $n$.
We can use the above argument to compute the measure of all points with a stable periodic orbit. The complement of this set is $$NS=\{\textrm{$(\omega, \eta, \v)\in X$ without a stable periodic orbit}\}.$$
\begin{corollary} \name{cor:limit} If $\nu$ is robustly renormalizable, then $$\nu({\mathit NS})=\lim_{n \to \infty} \nu(\sO_n).$$ \end{corollary} \begin{proof} The above argument shows that the following holds $\nu$-a.e., taking $\sO_0=X$. $$X \sm {\mathit NS}=\bigcup_{n=0}^\infty (\sO_n \sm \sO_{n+1}) \and {\mathit NS}=\bigcap_{n=0}^\infty \sO_n.$$ This is a nested intersection, so the conclusion follows. \end{proof}
Because of this Corollary, we wish to iteratively compute the measures of the sets $\sO_n$. For this, we need some understanding of the return times to $\sR_n$. For integers $n>0$, we define $$R_n:\sR_n \to \Z_+; \quad R_n(x)=\min \{m>0~:~\Phi^m(x) \in \sR_n\}.$$ The existence of this number is provided by statement (1) of the Theorem \ref{thm:renormalization}. Observe that if $\nu$ is $\Phi$-invariant then we have \begin{equation} \name{eq:ri} \nu(\sO_n)=\int_{\sR_n} \ret{n}{x}~d\nu(x). \end{equation} This demonstrates the importance of knowing the return times.
Let $\nu$ be a $\Phi$-invariant measure on $X$. We interpret $\rho$ as a measure preserving map from the measure space
$(\sR_1, \sB, \nu|_{\sR_1})$ to the space $(X,\sB, \nu \circ \rho^{-1})$ with $\sB$ denoting the Borel $\sigma$-algebra. Recall that $\rho$ is a {\em measurable isomorphism $(\textit{mod}~0)$} if there are subsets $Z_1 \subset \sR_1$ with $\nu(Z_1)=0$ and $Z_2 \subset X$ with $\nu \circ \rho^{-1}(Z_2)=0$ so that the restriction of $\rho$ to $\sR_1 \sm Z_1$ is a bijection onto $X \sm Z_2$ with measurable inverse. In this case, there is an inverse map $$\rho_{\nu}^{-1}:X \sm Z_2 \to \sR_1 \sm Z_1.$$ We call this map the {\em measurable inverse} of $\rho$ with respect to $\nu$. We abuse notation by considering $\rho_{\nu}^{-1}$ to be a map from $X$ to $\sR_1$, but note that it is defined only $\nu$-almost everywhere.
\begin{remark} \name{rem:invertibility} So long as $\alpha$ and $\beta$ are irrational, $\rho$ has a measurable inverse with respect to $\nu=\mu_\alpha \times \mu_\beta \times \mu_N$. This is because the coding map $\pi:\til Y \times N \to X$ given in equation \ref{eq:pi2} is a measurable isomorphism from $\til Y \times N$ equipped with Lebesgue measure to $X$ equipped with the measure $\nu$. This follows from the facts that $\pi$ is injective and $\nu$ is the pushforward of Lebesgue measure under $\pi$. Utilizing statement (\ref{item:phi}b) of Theorem \ref{thm:rectren}, we can explicitly describe the measurable inverse as $$\rho_{\nu}^{-1}=\pi\circ \phi^{-1} \circ (\pi')^{-1}$$ \end{remark}
\begin{remark} Measures for which $\rho$ is not measurably invertible can be analyzed as below utilizing conditional expectations. See \cite{Htruchet1}. \end{remark}
We now generalize the return time definition to a linear operator on the space of all Borel measurable functions on $X$. Suppose $f$ is a Borel measurable function on $X$. We define the {\em retraction of $f$ to $\sR_1$} to be the function $r_f:\sR_1 \to \R$ given by $$r_f(x)=\sum_{i=0}^{\ret{1}{x}-1} f \circ \Phi^i(x).$$ We think of this as a generalization of the return time, since for the constant function ${\mathbbm{1}}$ we have $\ret{1}{x}=r_{\mathbbm{1}}(x)$.
Now assume that $\rho_\nu^{-1}$ is a measurable inverse of $\rho$ with respect to $\nu$ as above. Then for any $\Phi$-invariant set $A \subset \sO_1$ and any $\nu$-integrable $f:X \to \R$, we have $$\int_{A} f~d \nu=\int_{A \cap \sR_1} r_f(x)~d\nu=\int_{\rho(A \cap \sR_1)} r_f \circ \rho^{-1}_\nu(y)~d(\nu \circ \rho^{-1})(y).$$
This motivates the definition of a linear operator on functions $X \to \R$: \begin{equation} \name{eq:cocycle1} C(\nu, 1): L^1(\nu) \to L^1(\nu \circ \rho^{-1}); \quad f \mapsto r_f \circ \rho^{-1}_\nu. \end{equation} From the above remarks, it has the property that \begin{equation} \name{eq:cocycle i} \int_{A} f~d\nu=\int_{\rho(A \cap \sR_1)} C(\nu,1)(f)~d(\nu \circ \rho^{-1}). \end{equation} We would like to apply this operation repeatedly, so we make the following definition.
\begin{definition} Let $\nu$ be a robustly renormalizable measure, and define $\nu_n=\nu \circ \rho^{-n}$ for integers $n \geq 0$. We say $\nu$ is {\em robustly invertible} if for each $n \geq 1$, the renormalization $\rho$ thought of as a measurable map from $(X,\sB,\nu_{n-1})$ to $(X,\sB,\nu_{n})$ has a measurable inverse $\rho^{-1}_n:X \to \sR_1$. This means for $\nu_{n-1}$-a.e. $x \in X$ and $\nu_{n}$-a.e. $y \in X$ we have $$\rho_n^{-1} \circ \rho(x)=x \and \rho \circ \rho_n^{-1}(y)=y.$$ \end{definition}
Suppose that $\nu$ is robustly invertible, and define $\nu_n=\nu \circ \rho^{-n}$ and $\rho^{-1}_n$ as in the definition above so that $\nu_0=\nu$. Observe that we can compose the operators $C(\nu_n,1)$ constructed as in equation \ref{eq:cocycle1}. Each operator $C(\nu_n,1)$ sends $L^1(\nu_n)$ to $L^1(\nu_{n+1})$, so for integers $n \geq 0$ and $m \geq 1$ define $$C(\nu_n,m):L^1(\nu_n) \to L^1(\nu_{n+m}); \quad C(\nu_n,m)= C(\nu_{n+m-1},1)\circ \ldots \circ C(\nu_{n+1},1) \circ C(\nu_n,1).$$ Taking $C(\nu,0)$ to be the identity operator on $L^1(\nu)$, these operators form cocycle over the renormalization dynamics of $\rho$ acting on the space of robustly invertible $\Phi$-invariant measures. That is they satisfy the identity $$C(\nu,m+k)=C(\nu \circ \rho^{-m},k) \circ C(\nu,m) \quad \text{for all $m,k \geq 0$}.$$ We prove that this cocycle satisfies a generalization of equation \ref{eq:cocycle i}. \begin{lemma}[Integral Formula] \name{lem:cocycle} Suppose $\nu$ is robustly invertible. Then for all integers $n \geq 1$, all Borel measurable $\Phi$-invariant sets $A \subset \sO_n$, and all $\nu$-integrable $g:A \to \R$, we have $$\int_{A} f~d\nu=\int_{\rho^n(A\cap \sR_n)} C(\nu, n)(f)(x)~d\nu \circ \rho^{-n}(x).$$ \end{lemma} \begin{proof} This formula follows by inductively applying equation \ref{eq:cocycle i}. We demonstrate how it works for the first iteration. Let $B=\rho(A \cap \sR_1)$. The set $B$ is $\Phi$-invariant by statement (2) of Theorem \ref{thm:renormalization}. In addition, $B \subset \sO_{n-1}$. We let $g=C(\nu,1)(f)$. Then by equation \ref{eq:cocycle i}, we have $$\int_{A} f~d\nu=\int_B g~d\nu \circ \rho^{-1}.$$ Assuming $n-1 \geq 1$, we can apply equation \ref{eq:cocycle i} again. \end{proof}
\subsection{Step functions} We are interested in the behavior of cocycle $C(\nu,n)$, when $\nu$ is taken from the space of measures coming from our rectangle exchange maps. This space of measures is $\rho$ invariant up to scaling. The scaling constant is given by the one-dimensional cocycle $D$ defined in equation \ref{eq:D}.
\begin{proposition}\name{prop:measure scaling} Let $\nu=\mu_\alpha \times \mu_\beta \times \mu_N$. Then, for all $k \geq 0$ we have $$\nu \circ \rho^{-k}=D(\alpha, \beta, k)~\mu_{f^k(\alpha)} \times \mu_{f^k(\beta)} \times \mu_N.$$ \end{proposition} \begin{proof} This follows from an inductive application of Corollary \ref{cor:rho act}. \end{proof}
We will see that $C(\nu,k)$ preserves a finite dimensional subspace of step functions containing the constant function ${\mathbbm{1}}$, so long as $\nu$ has the form above. This reduces the equation for integrating such a step function over $\sO_n$ given in Lemma \ref{lem:cocycle} to working with a finite dimensional cocycle. In this subsection, we find a $6$-dimensional invariant subspace. In the following subsection, we observe that ${\mathbbm{1}}$ belongs to a four dimensional invariant subspace. This allows us to drop the dimension of the cocycle to four.
We partition the space $X$ into six non-empty pieces ${\mathcal S}_1, {\mathcal S}_2, \ldots, {\mathcal S}_6$ and define the linear embedding into the space of of Borel measurable functions on $X$, \begin{equation} \name{eq:epsilon} \epsilon:\R^6 \to \sM(X); \quad \text{$\epsilon(\bp)(x)=\bp_i$ if $x \in {\mathcal S}_i$.} \end{equation} We say $x \in {\mathcal S}_i$ has {\em step class $i$}. These sets have combinatorial definitions given below.
First we define two sets. The {\em set of directions} consists of the terms {\em horizontal} and {\em vertical}. We define the {\em set of sign pairs} to be $\{--,-+,+-,++\}$. This is the set of words of length $2$ in the alphabet $\{\pm 1\}$. To each element $x=(\omega, \eta, \v) \in X$ with $\v=(a,b)$, we assign a unique direction and sign pair. Recall the definition of $\Phi$, $$\Phi(x)=\big(\sigma^{sb}(\omega), \sigma^{sb}(\eta),(sb,sa)\big) \quad \text{with $s=\omega_0 \eta_0$}.$$ This assignment of direction and sign pair to $x$ is given by the following chart.
\begin{center}
\begin{tabular}{|c|c|c|} \hline Value of $(sb,sa)$ & Direction & Sign pair \\ \hline $(1,0)$ & horizontal & $\omega_0 \omega_1$ \\ \hline $(-1,0)$ & horizontal & $\omega_{-1} \omega_0$ \\ \hline $(0,1)$ & vertical & $\eta_0 \eta_1$ \\ \hline $(0,-1)$ & vertical & $\eta_{-1} \eta_0$ \\ \hline \end{tabular} \end{center} If $x \in X$ has horizontal direction and sign pair $-+$, we call $x$ a {\em $-+$-horizontal step}. We use similar language to describe all combinations of directions with sign pairs.
We use these terms to define the six step classes. Each $x \in X$ belongs to exactly one class. \begin{citemize} \item We say $x$ has {\em step class $1$} if $x$ is a $(-+)$-horizontal step. \item We say $x$ has {\em step class $2$} if $x$ is a $(+-)$-horizontal step. \item We say $x$ has {\em step class $3$} if $x$ is a $(++)$- or $(--)$-horizontal step. \item We say $x$ has {\em step class $4$} if $x$ is a $(-+)$-vertical step. \item We say $x$ has {\em step class $5$} if $x$ is a $(+-)$-vertical step. \item We say $x$ has {\em step class $6$} if $x$ is a $(++)$- or $(--)$-vertical step. \end{citemize} This defines a partition of $X$ into the six sets ${\mathcal S}_1, \ldots, {\mathcal S}_6 \subset X$, and defines the function $\epsilon$ as in equation \ref{eq:epsilon}.
\begin{comment}
A {\em step} will indicate information which depends only on information at the origin, and the next square visited by the curve through the normal $\v$ leaving the square centered at the origin. Define $(\eta, \eta', \v')=\Phi(\omega, \omega',\v)$. Letting $s=\omega_0 \omega_0'$, we have $\v'=(sb,sa)$. See equation \ref{eq:Phi2}. If $\v'=(1,0)$ or $\v'=(-1,0)$, we call $x$ a {\em horizontal step}, otherwise we call $x$ a {\em vertical step}.
Suppose
Suppose $x=(\omega, \omega', \v)$ is a horizontal step as above
, and let $(\eta, \eta', \v')=\Phi(\omega, \omega',\v)$ as above. We say the sign pair of $x$ is
In this case $\omega'=\eta'$, but $\eta=\sigma^{sb}(\omega)$. We call $x$ a {\em $\omega_0 \omega_1$-horizontal step} if $sb=1$ and a {\em $\omega_{-1} \omega_0$-horizontal step} if $sb=-1$. So, we have defined the term {\em $w$-horizontal step} for each word $w$ of length $2$, i.e. $w \in \{--,-+,+-,++\}$. If $x$ is either a $(++)$-horizontal step or a $(--)$-horizontal step, then we call $x$ a {\em matching horizontal step}.
Similarly, if $x$ is a vertical step, then $\omega=\eta$ and $\eta'=\sigma^{sa} (\omega')$. If $sa=1$ we define $x$ to be a {\em $\omega_0' \omega_1'$-vertical step}, and if $sa=-1$ we define $x$ to be a {\em $\omega_{-1}' \omega_0'$-vertical step}. If $x$ is either a $(++)$-vertical step or a $(--)$-vertical step, we call $x$ a {\em matching vertical step}. \end{comment}
We will need to integrate a step function $\epsilon(\bp)$ over $X$ with respect to the measure $$\nu=\mu_\alpha \times \mu_\beta \times \mu_N.$$ To do this, we define the vector $\m_{\alpha,\beta} \in \R^6$ according to the rule $$\m_{\alpha,\beta}=\big(\nu({\mathcal S}_1), \ldots, \nu({\mathcal S}_6)\big).$$ This choice guarantees that we have the formula \begin{equation} \name{eq:meas vect} \int_X \epsilon(\bp)~d\mu=\m_{\alpha,\beta} \cdot \bp. \end{equation} We have the following explicit formula for $\m_{\alpha,\beta}$:
\begin{proposition}\name{prop:meas vect} We have $\m_{\alpha, \beta}=\thalf(\alpha, \alpha, 1-2\alpha, \beta, \beta, 1-2\beta).$ \end{proposition} \begin{proof} Let $x=(\omega,\eta,\v)$ be taken at random from $X$ according to the measure $\nu$. Let $\v=(a,b)$ and $s=\omega_0 \eta_0$ so that the directional component of $\Phi(x)$ is $\bw=(sb,sa)$. The probability that $\bw=(1,0)$ is $1/4$. Given this, $x$ is a $(-+)$ step if $\omega \in {\mathit{cyl}}(\wh - +)$. The $\mu_\alpha$ measure of ${\mathit{cyl}}(\wh - +)$ is $\alpha$. Similarly, we see that the probability of $x \in {\mathcal S}_i$ given that $\v=(1,0)$ is given by the $i$-th entry of the vector $$(\alpha, \alpha, 1-2\alpha,0,0,0).$$ The same holds vector holds for the case $\v=(-1,0)$. Given that $\v=(0,1)$ or $\v=(0,-1)$, the probability of $x \in {\mathcal S}_i$ is given by the entries of $$(0,0,0,\beta,\beta,1-2\beta).$$ We get $\m_{\alpha, \beta}$ by averaging the two vectors above. \end{proof}
For the following theorem, we define $\chi_i$ to be the characteristic function of ${\mathcal S}_i$, and define $\be_i \in \R^6$ to be the standard basis vector with $1$ in position $i$.
\begin{theorem}[Collapsed Steps] \name{thm:returns2} Suppose $x \in \sR_1$ has return time $\ret{1}{x}=4k+1$ and $\rho(x) \in {\mathcal S}_j$. Then, for all $i \in \{1, \ldots, 6\}$ we have $r_1(\chi_i;x)=\be_j \cdot K \be_i$ with $$K=\left[\begin{array}{rrrrrr} k & k-1 & 2 & 0 & 0 & 2k \\ k & k+1 & 0 & 0 & 0 & 2k \\ k & k & 1 & 0 & 0 & 2k \\ 0 & 0 & 2k & k & k-1 & 2 \\ 0 & 0 & 2k & k & k+1 & 0 \\ 0 & 0 & 2k & k & k & 1 \\ \end{array}\right].$$ \end{theorem}
Observe that the $j$-th row of $K$ gives the number of each step type which appears in the set $\{x, \Phi(x), \ldots, \Phi^{4k}\}$ provided $\rho(x) \in {\mathcal S}_j$ and $\ret{1}{x}=4k$. We prove this theorem at the end of this subsection.
The utility of the Lemma is the following. So long as the return time function is $\nu$-a.e. constant on each $\rho^{-1}({\mathcal S}_i)$, the cocycle $C(\nu,1)$ will preserve the subspace $\epsilon(\R^6)$. Indeed, if $\ret{1}{x}=4k+1$ for $\nu$-a.e. $x \in \rho^{-1}({\mathcal S}_j)$, then \begin{equation} \name{eq:cocycle obs} C(\nu,1)\big(\epsilon(\be_i)\big)(y)=\epsilon(K \be_i)(y) \quad \text{for $\nu \circ \rho^{-1}$-a.e. $y \in {\mathcal S}_j$,} \end{equation} with $K$ as in the above Theorem. In the case of measures of the form $\nu=\mu_\alpha \times \mu_\beta \times \mu_N$ the condition of being almost everywhere constant on $\rho^{-1}({\mathcal S}_j)$ is guaranteed (indirectly) by Corollary \ref{cor:rot ret}. In this case, we can extend linearly to understand the action of $C(\nu,1)$ on the subspace $L$. Details are in the proof of the following Lemma.
\begin{lemma}[Finite Dimensional Cocycle] \name{lem:matrices rectangle case} Let $\nu=\mu_\alpha \times \mu_\beta \times \mu_N$ with $\alpha, \beta \in [0,\half)$. The operator $C(\nu,1)$ preserves the space of step functions $\epsilon(\R^6)$. Determine $r,s \in \{\pm 1\}$ and $m,n \in \Z$ according to the rule $$f(\alpha)=r(\frac{\alpha}{1-2 \alpha}-m) \and f(\beta)=s(\frac{\beta}{1-2\beta}-n).$$ Then, the action of $C(\nu,1)$ on $\epsilon(\R^6)$ satisfies $$C(\nu,1)\big(\epsilon(\bp)\big)=\epsilon (M \bp) \qquad \text{$\nu \circ \rho^{-1}$-a.e.,}$$ with the matrix $M=M(\alpha,\beta,1)$ given by $$M=\left[\begin{array}{rrrrrr} m+\frac{r+1}{2} & m+\frac{r-1}{2} & 2 & 0 & 0 & 2m+1+r \\ m+\frac{r-1}{2} & m+\frac{r+1}{2} & 0 & 0 & 0 & 2m-1+r \\ m & m & 1 & 0 & 0 & 2m \\ 0 & 0 & 2 n+1+s & n+\frac{s+1}{2} & n+\frac{s-1}{2} & 2 \\ 0 & 0 & 2n-1+s & n+\frac{s-1}{2} & n+\frac{s+1}{2} & 0 \\ 0 & 0 & 2n & n & n & 1 \\ \end{array}\right]. $$ \end{lemma} The matrix $M$ above has entries which are all non-negative integers, and has determinant $rs \in \{\pm 1\}$.
We extend the definition of $M(\alpha, \beta, 1)$ to a cocycle. We define $M(\alpha, \beta,0)$ to be the identity matrix. We inductively define $$M(\alpha, \beta,k+1)=M\big(f^k(\alpha),f^k(\beta),1\big) M(\alpha, \beta, k) \quad \text{for $k \geq 1$.}$$ We have the following.
\begin{corollary}\name{cor:cocycle integral 1} Let $\alpha, \beta \in (0,\half)$ be irrational and set $\nu=\mu_\alpha \times \mu_\beta \times \mu_N$. Let $\bp \in \R^6$ and set $g=\epsilon(\bp)$. For $k \geq 0$, define $$\m_k=\m_{f^k(\alpha),f^k(\beta)}\in \R^6, \quad d_k=D(\alpha, \beta,k) \in \R, \and M_k=M(\alpha,\beta,k).$$ Then, $$\int_{\sO_k} g~d\nu=d_k (\m_k \cdot M_k \bp).$$ \end{corollary} \begin{proof} It follows by inductively applying Lemma \ref{lem:matrices rectangle case} that $$C(\nu,k)(g)=\epsilon(M_k \bp) \quad \text{$\nu \circ \rho^{-k}$-a.e..}$$ And therefore by Proposition \ref{prop:measure scaling} and equation \ref{eq:meas vect}, we have $$\int_X C(\nu,k)(g)(x)~d\nu \circ \rho^{-k}(x)=d_k (\m_k \cdot M_k \bp).$$ Then the conclusion follows from Lemma \ref{lem:cocycle} with $A=\sO_k$ so that $\rho^k(A \cap \sR_k)=X$. \end{proof}
The remainder of this section is devoted to proofs of Theorem \ref{thm:returns2} and Lemma \ref{lem:matrices rectangle case}. \begin{proof}[Proof of Lemma \ref{lem:matrices rectangle case} assuming Theorem \ref{thm:returns2}] Fix $\alpha$ and $\beta$ as in the Lemma. This determines the constants $m$, $n$, $r$ and $s$ as well as the matrix $M$. By linearity, it is sufficient to prove that for each $i, j \in \{1, \ldots, 6\}$, we have $$ C(\nu,1)\big(\epsilon(\be_i)\big)(y)=\epsilon(M \be_i)(y) \quad \text{for $\nu \circ \rho^{-1}$-a.e. $y \in {\mathcal S}_j$,} $$ By definition of $\epsilon$, for all $y \in {\mathcal S}_j$ we have $$\epsilon(M \be_i)(y)=\be_j \cdot (M \be_i).$$ By equation \ref{eq:cocycle obs}, to prove the Lemma, it is sufficient to check the following statements: \begin{enumerate} \item There is a constant $k$ so that $\ret{1}{x}=4k+1$ for $\nu$-a.e. $x \in \rho^{-1}({\mathcal S}_j)$. \item Defining $K$ using $k$ as in the Theorem, we have $\be_j \cdot (M \be_i)=\be_j \cdot (K \be_i)$. \end{enumerate}
We will carry this argument out for one $j$, and leave the remaining cases to the reader. Suppose $j=1$. Then we are interested in the case when $x \in \rho^{-1}({\mathcal S}_1)$. This means that $y=\rho(x)$ is a $(-+)$-horizontal step. Let $x=(\omega, \eta, \v)$. Since $y=\big(c(\omega), c(\eta), \v\big)$ is a horizontal step, we know $\v=(0,b)$ for $b \in \{\pm 1\}$. Let $s=\omega_0 \eta_0$ and $\bw=(sb,0)$. By Theorem \ref{thm:return time}, we know that the return time of $x$ to $\sR_1$ is given by $$R_1(x)=2 E(0,0, \bw)-1,$$ with this quantity $e=E(0,0,\bw)$ indicating one more than the number of columns removed to the right by the operation $\rho$ if $sb=1$ and one more than the number of columns to the left removed if $sb=-1$. See above Theorem \ref{thm:return time}. We break into cases depending on $r$ and $sb$. If $sb=1$, since $y$ is a $(-+)$-horizontal step, we have $y \in {\mathit{cyl}}(\wh - +)$ and therefore $$E(0,0,\bw)=r_+(\omega)=\begin{cases} 2m+3 & \text{if $r=1$}\\ 2m+1 & \text{if $r=-1$} \end{cases}$$ by Corollary \ref{cor:rot ret}. Similarly, if $sb=-1$, we have $y \in {\mathit{cyl}}(- \wh +)$ and $$E(0,0,\bw)=r_-(\omega)=\begin{cases} 2m+3 & \text{if $r=1$}\\ 2m+1 & \text{if $r=-1$} \end{cases}$$ Either way, the following choice of $k$ satisfies statement (1) above: $$k=m+\frac{r+1}{2}$$ Statement (2) claims $\be_1 \cdot (M \be_i)=\be_1 \cdot (K \be_i)$ for all $i$. This is just an observation. \end{proof}
\begin{proof}[Proof of Theorem \ref{thm:returns2}] We will prove the theorem in the case that $\rho(x)$ is a horizontal step, so $j=1,2,3$. The vertical case follows from symmetry.
Suppose $\rho(x)$ is a $(r s)$-horizontal step with $r,s \in \{\pm 1\}$. Let $x=(\omega, \eta, \v)$. First observe that if the return time $R_1(x)=1$, then we know $(rs) \neq (-+)$. Otherwise, $\omega$ would fail to be zero-collapsible. In addition, if $R_1(x)=1$ then $\rho(x)$ is also an $(rs)$-horizontal step. When $R_1(x)=1$, we have $k=0$. In this case we can check that for $j=2,3$ and $i = 1, \ldots, 6$ we have $\be_j \cdot K \be_i$ equals one if $i=j$ and zero otherwise. (This is the observation that the second and third rows of $K$ are the corresponding rows of the identity matrix when $k=0$.)
Now suppose that $R_1(x)=4k+1$ and $k \geq 1$. Then, $\Phi(x) \not \in \sR_1$. Recall the definition of horizontal box given in section \ref{sect:ren proofs}. The corresponding curve in the tiling associated to $(\omega, \eta)$ enters a maximal horizontal box $B$. Observe that the length parameter of $\ell$ can be computed using Theorem \ref{thm:return time}. (It is half of $E(0,0,\bw)-1$ if $\bw$ is the directional component of $\Phi(x)$.) Therefore, $\ell=k$. So, the curve of the tiling follows the central curve of the horizontal box $B$. When it leaves the horizontal box it returns to $\sR_1$ Possible pictures of this curve are shown below in the case that $\rho(x)$ is a $(-+)$-horizontal step, and the length parameter of the box is $\ell=2$: \begin{center} \includegraphics[scale=0.5]{horizontal_box_aug} \end{center} The sequence of step classes associated to $\Phi^i(x)$ for $i=0, \ldots 4k$ can be determined by examining the adjacent pairs of tiles passed through by the central curve extended into the two neighboring squares. The cases of $x$ and $\Phi^{4k}(x)$ correspond to the pairs of tiles at the two ends. The left one is an $(r-)$-horizontal step and the right end is a $(+s)$-horizontal step. Along the central curve of the horizontal box, we pass through $\ell=2k$ $(++)$- and $(--)$-vertical steps, $k$ $(-+)$-horizontal steps, and $k-1$ $(+-)$-horizontal steps. The total count of each type of step ${\mathcal S}_i$ gives the values of $e_j \cdot K \be_i$ for $j\in \{1,2,3\}$ and $i\in \{1, \ldots, 6\}$ as desired. \end{proof}
\subsection{Simplifications} \name{sect:simp} In this subsection, we perform some minor optimizations to the formula given in Corollary \ref{cor:cocycle integral 1}.
The first observation is that we can integrate $\epsilon(\bp)$ over $\sO_1$ without applying the cocycle. We define a new vector $\bq_{\alpha, \beta}$ to be the vector whose $i$-th entry is $\nu(\sO_1 \cap {\mathcal S}_i)$. Then by definition we have \begin{equation}\name{eq:q int} \int_{\sO_1} \epsilon(\bp)~d\nu=\bq_{\alpha,\beta} \cdot \bp. \end{equation}
\begin{proposition} \name{prop:q} For $\alpha, \beta \in [0,\half)$ and $\nu=\mu_\alpha \times \mu_\beta \times \mu_N$, we have $$\bq_{\alpha,\beta}=\half \Big(\alpha(1-2\beta), \alpha(1-2\beta), 1-2\alpha, \beta(1-2\alpha), \beta(1-2\alpha), 1-2\beta\Big).$$ \end{proposition} \begin{proof} Corollary \ref{cor:cocycle integral 1} gives an alternate version of the integral in equation \ref{eq:q int}. Namely, $$\int_{\sO_1} \epsilon(\bp)~d\nu=(1-2\alpha)(1-2\beta) \m_{f(\alpha),f(\beta)} \cdot M(\alpha,\beta,1) \bp.$$ Let $M=M(\alpha,\beta,1)$. We must have $$\bq_{\alpha,\beta}=(1-2\alpha)(1-2\beta) M^T \m_{f(\alpha),f(\beta)}.$$ Thus, we have reduced the problem to a calculation. The matrix $M$ is defined as in Lemma \ref{lem:matrices rectangle case} using the constants $m, n\in \Z$ and $r,s \in \{\pm 1\}$ which satisfy $$f(\alpha)=r(\frac{\alpha}{1-2\alpha}-m) \and f(\alpha)=s(\frac{\alpha}{1-2\alpha}-n).$$ Using these equations, we can show by direct computation that $$M^T \m_{f(\alpha),f(\beta)}=\half\left(\frac{\alpha}{1-2\alpha},\frac{\alpha}{1-2\alpha},\frac{1}{1-2\beta},\frac{\beta}{1-2\beta}, \frac{\beta}{1-2\beta},\frac{1}{1-2\alpha}\right).$$ The conclusion follows by multiplying through by $(1-2\alpha)(1-2\beta)$. \end{proof}
It follows that we have the following slightly simpler formula: \begin{corollary}\name{cor:cocycle integral 2}
Let $g=\epsilon(\bp)$. For $k \geq 0$, set $$\bq_k=\bq_{f^k(\alpha),f^k(\beta)}\in \R^6, \quad d_k=D(\alpha, \beta,k) \in \R, \and M_k=M(\alpha,\beta,k).$$ Then, $$\int_{\sO_{k+1}} g~d\nu=d_k (\bq_k \cdot M_k \bp).$$ \end{corollary} \begin{proof} This proof mirrors the proof of Corollary \ref{cor:cocycle integral 1}. By Lemma \ref{lem:matrices rectangle case}, $$C(\nu,k)(g)=\epsilon(M_k \bp) \quad \text{$\nu \circ \rho^{-k}$-a.e..}$$ By Proposition \ref{prop:measure scaling} and equation \ref{eq:q int}, $$\int_{\sO_1} C(\nu,k)(g)(x)~d\nu \circ \rho^{-k}(x)=d_k (\bq_k \cdot M_k \bp).$$ We apply Lemma \ref{lem:cocycle} to the case of $A=\sO_{k+1}$ so that $\sO_1=\rho^k(A \cap \sR_k).$ \end{proof}
For our final trick, we reduce the dimension of the cocycle to four. We observe that the right multiplication by the cocycle $M(\alpha, \beta,n)$ leaves invariant a four-dimensional subspace. To explain this, we introduce the following linear projection $\bpi:\R^6 \to \R^4$ and section $\s:\R^4 \to \R^6$ satisfying $\bpi \circ \s={\textrm{id}}$. \begin{equation} \name{eq:pi1} \bpi(a,b,c,d,e,f)=(a+b,c,d+e,f) \and \s(a,c,d,f)=(\frac{a}{2}, \frac{a}{2}, c, \frac{d}{2},\frac{d}{2},f). \end{equation}
\begin{proposition} Multiplication by $M^T=M(\alpha,\beta,k)^T$ leaves invariant the subspace $\s(\R^4)$. Moreover, for all $\v \in \R^4$ we have $$M(\alpha,\beta,k)^T \circ \s(\v)=\s \circ N(\alpha,\beta,k)^T(\v),$$ where $N$ is the cocycle defined in equation \ref{eq:N}. \end{proposition} The proof is just a calculation to verify the equation in the Proposition in the case $n=1$. The general case follows from the cocycle identity.
We define the projection of the vector $\bq_{\alpha,\beta}$ defined in Proposition \ref{prop:q} to be the row vector \begin{equation} \name{eq:n} \bn_{\alpha,\beta}=\bpi(\bq_{\alpha,\beta})=\left(\alpha(1-2\beta), \frac{1-2\alpha}{2}, \beta(1-2\alpha), \frac{1-2\beta}{2}\right). \end{equation} Also note that $\bq_{\alpha,\beta}=\s(\bn_{\alpha,\beta})$.
We apply Corollary \ref{cor:cocycle integral 2} to the special case when $\bp=\1$. To ease notation, for $k \geq 0$ make the following definitions: $$M_k=M(\alpha, \beta,k). \qquad N_k=N(\alpha, \beta,k).$$ $$\bq_k=\bq_{f^k(\alpha),f^k(\beta)}. \qquad \bn_k=\bn_{f^k(\alpha),f^k(\beta)}.$$ We use $\1_6 \in \R^6$ and $\1_4 \in \R^4$ to denote vectors all of whose entries are one. We have $$\nu(\sO_{k+1})=d_k \bq_k \cdot M_k \1_6=d_k \pi(M_k^T \bq_k)\cdot \1_4=d_k (N_k^T \bn_k)\cdot \1_4 .$$ This is our Cocycle Formula, proving Theorem \ref{thm:cocycle formula}.
\section{Cocycle Calculations} \name{sect:calc}
\subsection{The recurrent case} \name{sect:recurrent case} The goal of this subsection is to prove the following theorem concerning measures of the set ${\mathit NS} \subset X$ of points without a stable periodic $\Phi$-orbit. Recall that our renormalization action on parameters for rectangle exchange maps was closely related to the map $f:[0,\half) \to [0,\half]$ defined in equation \ref{eq:f}.
\begin{theorem}[Recurrent Case] \name{thm:recurrent case} Let $\alpha, \beta \in (0,\half)$ be irrational and let $\nu=\mu_\alpha \times \mu_{\beta} \times \mu_N$. If the sequence of points $$\big\{\big(f^k(\alpha),f^k(\beta)\big)\big\}$$ has an accumulation point $(x,y)$ with $x \geq 0$ and $y \geq 0$, then $\nu({\mathit NS})=0$. \end{theorem}
We make use of statement (\ref{item:O}) of section \ref{sect:limit formula} which provides the limit formula $$\nu({\mathit NS})=\lim_{k \to \infty} \nu(\sO_k).$$ Recall that the sequence of sets $\sO_k$ were nested, so this sequence is decreasing. Therefore to show $\nu({\mathit NS})=0$, it is sufficient to show that there is an $\epsilon>0$ so that for infinitely many $k$ we have $\nu(\sO_{k+1})<(1-\epsilon) \nu(\sO_{k})$. Thus the theorem is implied by the following lemma.
\begin{lemma}[Scaling Lemma] \name{lem:scaling} For all $\alpha, \beta \in (0,\half)$ and all $k \geq 1$, we have $$\nu(\sO_{k+1}) \leq g\big(f^{k+1}(\alpha), f^{k+1}(\beta)\big) \nu(\sO_{k})$$ where $g(x,y)=1-\frac{4}{3} xy$. \end{lemma}
\begin{remark}[Slow divergence] Observe that the scaling lemma actually implies that if the $f \times f$ orbit of $(\alpha, \beta)$ satisfies $$\prod_{k=1}^\infty g\big(f^k(\alpha),f^k(\beta)\big)=0,$$ then almost every orbit is periodic, $\nu({\mathit NS})=0$. \end{remark}
For two vectors $\v$ and $\bw$ in $\R^n$, we say $\v \leq \bw$ {\em entrywise} if $\v_i \leq \bw_i$ for $i=1, \ldots n$. We will see that it is sufficient to show the following:
\begin{lemma}[Scaling Lemma II] \name{lem:scaling2} For all $\gamma, \delta \in (0,\half)$, we have the entrywise inequality $$D(\gamma,\delta,1) N(\gamma,\delta,1)^T \bn_{f(\gamma),f(\delta)} \leq g\big(f(\gamma),f(\delta)\big) \bn_{\gamma, \delta},$$ where $g(x,y)$ is as defined in the previous lemma. \end{lemma} \begin{proof}[Proof of Lemma \ref{lem:scaling} given Lemma \ref{lem:scaling2}] We utilize the limit formula in Theorem \ref{thm:cocycle formula} to relate $\nu(\sO_k)$ to $\nu(\sO_{k+1})$. Define $$\v=D(\alpha,\beta,k-1) N(\alpha,\beta,k-1) \1.$$ Set $\gamma=f^{k-1}(\alpha)$ and $\delta=f^{k-1}(\beta)$. Then by Theorem \ref{thm:cocycle formula}, we have $$\nu(\sO_k)=\bn_{\gamma,\delta} \cdot \v \and \nu(\sO_{k+1})=\Big(D(\gamma,\delta,1) N(\gamma,\delta,1)^T \bn_{f(\gamma),f(\delta)}\big) \cdot \v.$$ So the entrywise inequality implies $$\nu(\sO_{k+1}) \leq g\big(f(\gamma), f(\delta)\big) \nu(\sO_{k}).$$ \end{proof}
\begin{proof}[Proof of Lemma \ref{lem:scaling2}] We use notation similar to that of section \ref{sect:limit formula} for defining $m$, $n$, $r$ and $s$. We define these constants so that $$r f(\gamma)+m=\frac{\gamma}{1-2\gamma} \and s f(\delta)+n=\frac{\delta}{1-2\delta}.$$ We break the vector $\bn_{f(\gamma),f(\delta)}$ into two pieces, writing $\bn_{f(\gamma),f(\delta)}=\ba-\bb$ with $$\ba=\big(f(\gamma),\frac{1-2f(\gamma)}{2},f(\delta),\frac{1-2f(\delta)}{2}\big) \and \bb=2f(\gamma)f(\delta)\big(1,0,1,0\big).$$ Define $d=D(\gamma,\delta,1)=(1-2\gamma)(1-2\delta)$ and $N=N(\gamma,\delta,1)$. The matrix $N$ is given exactly as in equation \ref{eq:N}. We have $$\bn_{\gamma,\delta}= d N^T \ba.$$ This is not an accident; it comes from the fact that $\ba=\pi(\m_{\gamma,\delta})$ and the meaning of these quantities (which were defined in section \ref{sect:cocycle}). But the statement can also be verified by calculation. For instance, the first entry of $d N^T \ba$ is given by $$\begin{narrowarray}{3pt}{rcl} (d N^T \ba)_1 & = & (1-2\gamma)(1-2\delta)\Big[(2m+r)f(\gamma)+(2m)(\frac{1-2f(\gamma)}{2})\Big] \\ & = & (1-2\gamma)(1-2\delta)[r f(\gamma)+m]=(1-2\delta)\gamma=(\bn_{\gamma,\delta})_1. \end{narrowarray}$$
We have shown that $$dN^T \bn_{f(\gamma),f(\delta)}=\bn_{\gamma,\delta}-dN^T \bb.$$ To simplify expressions below let $\bz=dN^T \bb$ and $\bn=\bn_{\gamma,\delta}.$ We will show that $$\bz_i/\bn_i \geq \frac{4}{3} f(\gamma) f(\delta) \quad \text{for $i \in \{1,2,3,4\}$.}$$ This will conclude the proof. We have $$\bn=\left(\gamma(1-2\delta),\frac{1-2\gamma}{2},\delta(1-\gamma),\frac{1-2\delta}{2}\right).$$ $$\bz=(1-2\gamma)(1-2\delta) 2f(\gamma)f(\delta)\big(2m+r,2n+s+1,2n+s,2m+r+1\big).$$ To prove the theorem, we will provide lower bounds for the quantities $\bz_i/(f(\gamma)f(\delta)\bn_i).$ In the cases below, we use the observation \begin{equation} \name{eq:useful ineq} \frac{\gamma}{1-2\gamma} \leq m+\half. \end{equation} We begin with $i=1$, and break into two cases. In case $m=0$, we have $r=1$ and therefore, $$\frac{\bz_1}{f(\gamma)f(\delta) \bn_1} =\frac{2(1-2\gamma)}{\gamma} \geq 4 > \frac{4}{3}.$$ In the remaining cases, we have $m \geq 1$ and we use the fact that $2m+r \geq 2m-1$. $$\frac{\bz_1}{f(\gamma)f(\delta) \bn_1} = \frac{(4m+2r)(1-2\gamma)}{\gamma} \geq \frac{2(2m-1)(1-2\gamma)}{\gamma} \geq \frac{4m-2}{m+\half} \geq \frac{4}{3}.$$ The case of $i=4$ is given by: $$\frac{\bz_4}{f(\gamma)f(\delta) \bn_4}=4(1-2\gamma)(2m+r+1)$$ In case $m=0$, we have $r=1$ and $\gamma<1/4$. Therefore, when $m=0$, we have $$\frac{\bz_4}{f(\gamma)f(\delta) \bn_4}=8(1-2\gamma) > 2 > \frac{4}{3}.$$ Otherwise, we have $m \geq 1$ and $\gamma>1/4$. Using equation \ref{eq:useful ineq}, we see $$\frac{\bz_4}{f(\gamma)f(\delta) \bn_4} \geq \frac{4\gamma(2m+r+1)}{m+\half}> \frac{2m+r+1}{m+\half} \geq \frac{2m}{m+\half} \geq \frac{4}{3}.$$ The remaining two indices follow by symmetry. (Observe that the action of switching $\gamma$ with $\delta$ has the effect of swapping the first and third and second and fourth entries of all vectors involved.) \end{proof}
\subsection{The non-recurrent case} \name{sect:non-recurrent case} Consider any four sequences of integers $m_i, n_i \geq 0$ and $r_i, s_i \in \{\pm 1\}$ defined for $i \geq 0$ so that $$(m_i,r_i) \neq (0,-1) \and (n_i,s_i) \neq (0,-1) \quad \text{for all $i \geq 0$.}$$ We call a collection of these four sequences an {\em itinerary}. Theorem \ref{thm:coding} implies that for any itinerary, there is a unique pair $(\alpha, \beta)$ so that \begin{equation} \name{eq:forward} f^{i+1}(\alpha)=r_i\left(\frac{f^i(\alpha)}{1-2f^i(\alpha)}-m_i\right) \and f^{i+1}(\beta)=s_i\left(\frac{f^i(\beta)}{1-2f^i(\beta)}-n_i\right). \end{equation} We call $(\alpha, \beta)$ the pair {\em determined} by the itinerary. Theorem \ref{thm:coding} also gives a mild restriction on the itinerary which guarantees the irrationality of $\alpha$ and $\beta$.
The itinerary is relevant for computing the cocycle $N(\alpha, \beta,k)$, which is the main ingredient in the formula $$\mu_\alpha \times \mu_\beta \times \mu_N(\sO_{k+1})=D(\alpha, \beta,k) \bn_{f^k(\alpha),f^k(\beta)} \cdot N(\alpha, \beta, k) \1.$$ The limit of these quantities as $n \to \infty$ gives the measure of all points without stable periodic orbits. See Section \ref{sect:limit formula}.
We will investigate itineraries of a particular form, and show that we can make choices which guarantee that the measure of $\sO_n$ decays as slow as we wish.
\begin{definition}[Upward and Downward Itineraries] Consider an itinerary $I$ consisting of sequences $\{m_i\}$, $\{n_i\}$, $\{r_i\}$ and $\{s_i\}$ as above. Let $k \geq 1$. \begin{itemize} \item We say $I$ is a $k$-upward itinerary if $$(m_i,r_i,n_i,s_i)=\begin{cases} (0,1,1,1) & \text{if $i=0$,} \\ (0,1,0,1) & \text{if $1 \leq i \leq k-1$,} \\ (1,1,0,1) & \text{if $i=k$.} \end{cases}$$ \item We say $I$ has a $k$-rightward itinerary if $$(m_i,r_i,n_i,s_i)=\begin{cases} (1,1,0,1) & \text{if $i=0$,} \\ (0,1,0,1) & \text{if $1 \leq i \leq k-1$,} \\ (0,1,1,1) & \text{if $i=k$.} \end{cases}$$ \end{itemize} \end{definition} If $(\alpha, \beta)$ has a $k$-upward itinerary then $\beta \geq \frac{1}{3}$. And, if $(\alpha, \beta)$ has a $k$-rightward itinerary then $\alpha \geq \frac{1}{3}$. (See Proposition \ref{prop:special itineraries}.) This explains our choice of terminology.
We make use of a shift map on itineraries. If $I$ is an itinerary and $k \geq 1$ is an integer, we define $\sigma^k(I)$ to be the collection of sequences formed by dropping the first $k$ values of each of the four sequences making up $I$, and re-indexing so that each sequence begins at zero.
\begin{definition}[Understandable Itineraries] Let $\{k_j \geq 1\}$ be a sequence of integers defined for $j \geq 0$. Using $\{k_j\}$, we define the auxiliary sequence $\{{a}_j\}$ inductively by the rule $${a}_{0}=0, \and {a}_{j+1}={a}_j+k_j+1 \quad \text{for $j \geq 0$}.$$ We say the {\em $\{k_j\}$-understandable itinerary} $I$ is the itinerary determined by the following rules. \begin{enumerate} \item For any even $j \geq 0$, the itinerary $\sigma^{{a}_j}(I)$ is a $k_j$-upward itinerary. \item For any odd $j \geq 1$, the itinerary $\sigma^{{a}_j}(I)$ is a $k_j$-rightward itinerary. \end{enumerate} \end{definition}
Because of Theorem \ref{thm:recurrent case}, we are interested in pairs $(\alpha, \beta)$ such that the collection of all limit points of the $f \times f$-orbit of $(\alpha, \beta)$ is contained in the set $$\set{(x,y) \in [0, \half] \times [0, \half]}{$x=0$ or $y=0$}.$$ The holds for the pair $(\alpha, \beta)$ determined by a $\{k_j\}$-understandable itinerary precisely when $\liminf k_j=\infty$. The following results will imply that we get a (large) positive measure set of non-periodic points if $\{k_j\}$ grows sufficiently quickly.
\begin{proposition}[Decay Control]\name{prop:decay control} There is a function $K_0:(0,1) \to \Z$ satisfying the following statement. For each $\epsilon_0>0$, whenever $(\alpha, \beta)$ has a $k_0$-upward itinerary with $k_0>K_0(\epsilon_0)$ then $\nu(\sO_1)>1-\epsilon_0,$ where $\nu=\mu_\alpha \times \mu_\beta \times \mu_N.$ \end{proposition}
\begin{theorem}[Decay Control] \name{thm:decay control} There is a function $K:\Z \times (0,1) \to \Z$ satisfying the following statement. For any sequence $\{\epsilon_j\}_{j \geq 1}$ with $0<\epsilon_j<1$, if $\{k_j\}_{j \geq 0}$ is a sequence satisfying $$k_{j} \geq K(k_{j-1},\epsilon_{j}) \quad \text{for all} \quad j \geq 1,$$ then the pair $(\alpha, \beta)$ determined by the $\{k_j\}$-understandable itinerary with auxiliary sequence $\{a_j\}$ satisfies $$\nu(\sO_{{a}_{j}+1})>(1-\epsilon_{j}) \nu(\sO_{{a}_{j-1}+1}) \quad \text{for all $j \geq 1$,}$$ where $\nu=\mu_\alpha \times \mu_\beta \times \mu_N$. \end{theorem}
The Decay Control Proposition and Theorem together imply Theorem \ref{thm:4} of the introduction, which can be restated as saying that for any $\eta>0$, there exists a pair of irrationals $(\alpha, \beta)$ so that $\nu({\mathit NS})>1-\eta.$ \begin{proof}[Proof of Theorem \ref{thm:4} given the Decay Control results] Fix any $\eta>0$, and fix any sequence $\{\epsilon_j\}_{j \geq 0}$ of numbers in $(0,1)$ so that $$\prod_{j=0}^\infty (1-\epsilon_j)>1-\eta.$$ Choose a sequence $\{k_j\}_{j \geq 0}$ so that $k_0>K_0(\epsilon_0)$ and $k_j \geq K(k_{j-1},\epsilon_j)$ for all $j \geq 1$. Then the Decay Control Proposition implies $\nu(\sO_1)>1-\epsilon_0$. The theorem implies that $$\nu(\sO_{{a}_{j}+1})>(1-\epsilon_{j}) \nu(\sO_{{a}_{j-1}+1})$$ for all $j \geq 1$. By statement (\ref{item:O}) of section \ref{sect:limit formula}, we have $$\nu({\mathit NS})=\lim_{j \to \infty} \nu(\sO_{{a}_{j}+1}) \geq \lim_{j \to \infty} \prod_{i=0}^j (1-\epsilon_i) > 1-\eta$$ as desired. Finally, we observe that $\alpha$ and $\beta$ are irrational, by Theorem \ref{thm:coding}. This is true for any pair determined by an understandable itinerary. \end{proof}
We will now give a proof of Corollary \ref{cor:5}, which states that the set $$P=\{(\alpha,\beta)~:~\mu_\alpha \times \mu_\beta \times \mu_N({\mathit NS})>0\}$$ is dense.
\begin{proof}[Proof of Corollary \ref{cor:5} given the Decay Control results] Theorem \ref{thm:4} gives a pair $(\alpha, \beta) \in P$. Whenever $(\alpha', \beta')$ satisfies $f^n(\alpha')=\alpha$ and $f^n(\beta')=\beta$, we also have $(\alpha',\beta') \in Y$ because Theorem \ref{thm:ren} gives a return map of ${\widetilde \Psi}_{\alpha', \beta'}$ which is affinely conjugate to ${\widetilde \Psi}_{\alpha, \beta}$. Theorem \ref{thm:coding} implies that the itinerary map gives a semiconjugacy from the shift map on a shift space to the action of $f \times f$. Since the collection of preimages of a point in a shift space is dense, the collection of preimages of $(\alpha,\beta)$ under $f \times f$ must be dense. \end{proof}
We will build up to a proof of the Decay Control Proposition and Theorem Theorem. The following is a necessary calculation, which follows from an inductive argument using equation \ref{eq:forward}. (We carry out a similar calculation in the proof of Lemma \ref{lemma:decay ineq} below.)
\begin{proposition}[Starting Points of Itineraries] \name{prop:special itineraries} If $(\alpha, \beta)$ has a $k$-upward itinerary then $$\frac{1}{3+2k} \leq \alpha \leq \frac{3}{8+6k} \and \frac{1}{3} \leq \beta \leq \frac{3+2k}{8+6k}.$$ If $(\alpha, \beta)$ has a $k$-rightward itinerary then $$\frac{1}{3} \leq \alpha \leq \frac{3+2k}{8+6k} \and \frac{1}{3+2k} \leq \beta \leq \frac{3}{8+6k}.$$ \end{proposition} \begin{comment} \begin{proof} \boldred{Is this really needed?} We will only consider the case when $(\alpha, \beta)$ has a $k$-upward itinerary. Note that $\alpha_{k+1}, \beta_{k+1} \in [0,\half]$. By equation \ref{eq:backward}, we see \newcommand{\iv}[2]{{\left[#1,#2\right]}} $$\textstyle \alpha_k \in \iv{\frac{1}{3}}{\frac{3}{8}} \and \beta_k\in \iv{0}{\frac{1}{4}}.$$ Then since $(m_i,r_i,n_i,s_i)=(0,1,0,1)$ for $i=1,\ldots, k-1$ we see $$\textstyle \alpha_1=\frac{\alpha_k}{1+2(k-1)\alpha_k} \and \beta_1=\frac{\beta_k}{1+2(k-1)\beta_k}.$$ Therefore we have $$\textstyle \alpha_1 \in \iv{\frac{1}{2k+1}}{\frac{3}{6k+2}} \and \beta_1 \in \iv{0}{\frac{1}{2k+2}}.$$ Finally by one more application of equation \ref{eq:backward}, we see $$\textstyle \alpha_0 \in \iv{\frac{1}{2k+3}}{\frac{3}{6k+8}} \and \beta_0 \in \iv{\frac{1}{3}}{\frac{2k+3}{6k+8}}.$$ \end{proof} \end{comment}
We will now prove the Decay Control Proposition. This proof reveals some of the ideas appearing in the proof of the Decay Control Theorem. \begin{proof}[Proof of the Decay Control Proposition] It suffices to show that for any $\epsilon_0>0$ and for sufficiently large $k_0$ we have $\nu(\sO_{1})>1-\epsilon_{0}.$ By Theorem \ref{thm:cocycle formula}, we have $$\mu_\alpha \times \mu_\beta \times \mu_N(\sO_{1})=1-4\alpha \beta.$$ We know that $(\alpha, \beta)$ will have a $k_0$-upward itinerary. Proposition \ref{prop:special itineraries} then confines $(\alpha, \beta)$ to a rectangle where $$1-4\alpha\beta \geq 1-\frac{3(3+2k_0)}{(8+6k_0)^2}.$$ The quantity on the right tends to $1$ as $k_0 \to \infty$. So choosing a sufficiently large $k_0$ makes $1-4\alpha \beta$ larger than $1-\epsilon_0$. \end{proof}
Now consider the Decay Control Theorem. We begin by interpreting the quantities under consideration using the cocycle. Fix $j \geq 1$ and set $$\alpha'=f^{{a}_{j-1}}(\alpha), \quad \beta'=f^{{a}_{j-1}}(\beta), \quad \gamma=f^{{a}_{j}}(\alpha), \and \delta=f^{{a}_{j}}(\beta).$$ Define the vector \begin{equation} \name{eq:z2} \bz=D(\alpha, \beta,{a}_{j-1}) N(\alpha, \beta, {a}_{j-1}) \1. \end{equation} By our cocycle formula (Theorem \ref{thm:cocycle formula}), we have the following two identities: \begin{equation} \name{eq:decay1} \nu(\sO_{{a}_{j-1}+1})=\bn_{\alpha',\beta'} \cdot \bz. \end{equation} \begin{equation} \name{eq:decay2} \nu(\sO_{{a}_{j}+1})=\big(D(\alpha',\beta',k_{j-1}+1) N(\alpha',\beta',k_{j-1}+1)^T \bn_{\gamma,\delta} \big) \cdot \bz. \end{equation} To show that the value of the second equation is nearly as large as the first equation, it suffices to prove an entrywise inequality involving the vectors that show up in these equations. Specifically, we will show that there is a function $K$ as in the theorem so that whenever $k_{j}>K(k_{j-1},\epsilon_j)$ we have the entrywise inequality \begin{equation} \name{eq:entrywise ineq} D(\alpha',\beta',k_{j-1}+1) N(\alpha',\beta',k_{j-1}+1)^T \bn_{\gamma,\delta} > (1-\epsilon_j) \bn_{\alpha',\beta'} \end{equation} This statement is proved in the following lemma.
\begin{lemma}[Decay Control Inequality] \name{lemma:decay ineq} Fix any $\epsilon>0$. Assume that $(\alpha, \beta)$ has $k$-upward itinerary. Define $$\gamma=f^{k+1}(\alpha) \and \delta=f^{k+1}(\beta).$$ There is a $K=K(k,\epsilon)$ so that for any $k'>K$, if $(\gamma,\delta)$ has a $k'$-rightward itinerary, then we have the entrywise inequality $$D(\alpha,\beta,k+1) N(\alpha,\beta,k+1)^T \bn_{\gamma,\delta} > (1-\epsilon) \bn_{\alpha,\beta}.$$ The same statement holds with the same function $K(k, \epsilon)$ when the notion of `upward' is swapped with `rightward.' \end{lemma}
\begin{remark}[On the proof of the lemma] We prove Lemma \ref{lemma:decay ineq} by calculation, which seems unfortunate. However, we can go back to the argument above the lemma to see why this is necessary. The argument we use only utilizes knowledge of the portion of the itinerary with indices $j$ satisfying $a_{j-1} \leq j < a_{j+1}$. Since the vector $\bz$ defined in equation \ref{eq:z2} depends on an earlier part of the itinerary, we have no a priori control over the value of $\bz$.
So, to control the decay in moving from the value of equation \ref{eq:decay1} to equation \ref{eq:decay2}, we are forced to prove the entrywise inequality in equation \ref{eq:entrywise ineq}.
For further commentary, let $\bx=\bn_{\alpha',\beta'}$ and $$\by=D(\alpha,\beta,k+1) N(\alpha,\beta,k+1)^T \bn_{\gamma,\delta}.$$ These vectors have some geometric meaning. The vector $\bx$ represents the $\nu'=\mu_{\alpha'} \times \mu_{\beta'} \times \mu_N$ measures of the intersections of $\sO_1$ with four subsets of $X$. (These four sets are $\sS_1 \cup \sS_2$, $\sS_3$, $\sS_4 \cup \sS_5$ and $\sS_6$. See equation \ref{eq:n} and the definition of $\bq$ above equation \ref{eq:q int}.) By definition of the cocycle, the entries of the vector $\by$ represent the $\nu'$ measures of $\sO_{k+2}$ intersected with the same four sets. Since $\sO_{k+2} \subset \sO_1$, we see $\by < \bx$ entrywise. Moreover, we could prove directly that as $k_j \to \infty$ that we have $\nu'(\sO_{k+2}) \to 1$ and $\nu'(\sO_1) \to 1.$ (This explains the situation when $\bz=\1$.) However, as $k_j \to \infty$ some of the entries of $\bx$ tend to zero. If $\by$ is obtained from $\bx$ by disproportionately decreasing the values of small entries of $\bx$, then $\by$ could be arranged not to satisfy $\by > (1-\epsilon) \bx$ while still satisfying the conditions $$\by \cdot \1 \approx \bx \cdot \1 \and \by<\bx.$$ We view this as forcing us to do a calculation to guarantee $\by>(1-\epsilon) \bx$ as $k_j \to \infty$. \end{remark}
\begin{proof}We will only do the version of the lemma without swapping terms. This second case follows from the first by symmetry.
We will do all our calculations in terms of $\gamma$ and $\delta$. Let $\alpha_i=f^i(\alpha)$ and $\beta_i=f^i(\beta)$. Knowing that $(\alpha, \beta)$ has a $k$-downward itinerary allows us to give formulas for some of these values via equation \ref{eq:forward}: $$\alpha_i=\begin{cases} \frac{1+\gamma}{1+2(1+\gamma)(1+k-i)} & \text{if $0 \leq i \leq k$,}\\ \gamma & \text{if $i=k+1$.} \end{cases} \qquad \beta_i=\begin{cases} \frac{1+\delta(2k+1)}{3+2 \delta(3k+1)} & \text{if $i=0$,}\\ \frac{\delta}{1+2 \delta(k+1-i)} & \text{if $1 \leq i \leq k+1$,} \end{cases} $$ It is relevant to compute $1-2\alpha_i$ and $1-2\beta_i$. We have $$1-2\alpha_i=\begin{cases} \frac{1+2(\gamma+1)(k-i)}{1+2(\gamma+1)(k+1-i)} & \text{if $0 \leq i \leq k$,}\\ 1-2\gamma & \text{if $i=k+1$.} \end{cases} \qquad 1-2\beta_i=\begin{cases} \frac{1+2\delta k}{3+2\delta(1+3k)} & \text{if $i=0$,}\\ \frac{1+2 \delta(k-i)}{1+2 \delta(k+1-i)} & \text{if $1 \leq i \leq k+1$,} \end{cases} $$ To simplify notation, we define the following quantities: $$\begin{array}{ccc} d=D(\alpha,\beta,k+1). & \quad & N=N(\alpha, \beta, k+1). \\ \v=\frac{1}{d} \bn_{\alpha, \beta}. & \quad & \bw=N^T \bn_{\gamma, \delta}. \end{array}$$ Fix $\epsilon>0$ as in the lemma. The goal of this proof is to show that for $k'$ sufficiently large, we have $\bw>(1-\epsilon)\v$ entrywise. We do this by calculation.
There is a lot of cancellation in the product for $d$: $$d=\prod_{j=0}^k (1-2\alpha_j)(1-2\beta_j)=\frac{1}{\big(1+2(1+\gamma)(k+1)\big)\big(3+2\delta(3k+1)\big)}.$$ By a calculation, we observe that $\v$ has a relatively simple expression: $$\begin{array}{cc} \displaystyle \v_1=(1+\gamma)(1+2\delta k) & \displaystyle \v_2= \thalf\big(1+2(1+\gamma)k\big)\big(3+\delta(2+6k)\big) \\ \displaystyle \v_3=\big(1+2(1+\gamma)k\big)(1+\delta(1+2k)) & \displaystyle \v_4=\thalf\big(1+2(1+\gamma)(1+k)\big)(1+2\delta k) \\ \end{array}$$ The value of $N$ can be computed from the knowledge that $(\alpha,\beta)$ has a $k$-upward itinerary: $$\begin{narrowarray}{3pt}{rcl} N & = & \left[\begin{array}{rrrr} 3 & 1 & 0 & 3 \\ 2 & 1 & 0 & 2 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{array}\right] \left(\left[\begin{array}{rrrr} 1 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{array}\right]^{k-1}\right) \left[\begin{array}{rrrr} 1 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 3 & 3 & 1 \\ 0 & 2 & 2 & 1 \end{array}\right] \\ & = & \left[\begin{array}{rrrr} 3 & 1+9k & 6k & 3(1+k) \\ 2 & 1+6k & 4k & 2(1+k) \\ 0 & 3(1+k) & 3+2k & 1+k \\ 0 & 2 & 2 & 1 \end{array}\right] \end{narrowarray} $$ By definition of $\bn_{\gamma,\delta}$ we have $$\textstyle \bn_{\gamma,\delta}=\big(\gamma(1-2\delta), \frac{1-2\gamma}{2}, (1-2\gamma)\delta,\frac{1-2\delta}{2}\big).$$ This allows us to compute $\bw=N^T \bn_{\gamma,\delta}$: $$\bw=\left[\begin{array}{r} 1+\gamma-6 \gamma \delta \\ \thalf\Big(3\big(1+2(1+\gamma)k\big) -2\delta(8 \gamma -1)(1+3k)\Big) \\ 1+2(1+\gamma)k-2 \delta(1-6 \gamma+2k-16 \gamma k) \\ \thalf\Big(1+2(1+\gamma)(1+k) -2\delta(8 \gamma(k+1)-k\Big) \\ \end{array}\right]$$
Now observe that the limits of $\v$ and $\bw$ as $\delta \to 0$ are equal and positive: $$\lim_{\delta \to 0} \v=\lim_{\delta \to 0} \bw= \left[\begin{array}{r} 1+\gamma \\ {\textstyle \frac{3}{2}} \big(1+2(1+\gamma)k\big) \\ 1+2(1+\gamma)k \\ \thalf\Big(1+2(1+\gamma)(1+k)\big) \\ \end{array}\right]$$ By continuity of $\v$ and $\bw$ as functions of $\gamma$ and $\delta$, we can take this convergence to be uniform in $\gamma$. Therefore, there is a constant $C>0$ so that $\bw_i/\v_i>1-\epsilon$ for all $i$ whenever $\delta<C$. By Proposition \ref{prop:special itineraries}, we can force $\delta<C$ by assuming that $(\gamma,\delta)$ has a $k'$-rightward itinerary with $$\frac{3}{8+6k'}<C.$$ \end{proof}
By the remarks made above the statement of the lemma, this lemma also proves the Decay Control Theorem.
\begin{comment} \subsection{\boldred{Old stuff to be eliminated?}}
It will be more useful for us to write this as \begin{equation} \name{eq:backward} \alpha_i=\frac{r_i \alpha_{i+1}+m_i}{1+2(r_i \alpha_{i+1}+m_i)} \and \beta_i=\frac{s_i \beta_{i+1}+n_i}{1+2(s_i \beta_{i+1}+n_i)}. \end{equation} It should be observed that for any $k \geq 0$, that $\alpha$ is uniquely determined by $\alpha_{k+1}$ and the values of $m_i$ and $r_i$ for $i=1,\ldots,k$. Moreover so long as $\alpha_{k+1} \in [0, \half]$, each $m_i \geq 0$ and $(m_i,r_i) \neq (0,-1)$, we have $\alpha \in (0,\half]$. Symmetric statements hold for $\beta$.
The basic idea of this section is to specify the itinerary of a point. There is a unique pair $(\alpha, \beta)$ with this itinerary. This depends on \boldred{something in the next section}.
\begin{proposition}[Starting Points of Itineraries] \name{prop:special itineraries} Let $k \geq 2$. If $(\alpha, \beta)$ has a $k$-upward itinerary then $$(\alpha,\beta)\in \left[\frac{1}{3+2k},\frac{3}{8+6k}\right] \times \left[\frac{1}{3}, \frac{3+2k}{8+6k}\right].$$ If $(\alpha, \beta)$ have a $k$-rightward itinerary then $$(\alpha,\beta)\in \left[\frac{1}{3}, \frac{3+2k}{8+6k}\right] \times \left[\frac{1}{3+2k},\frac{3}{8+6k}\right].$$ \end{proposition} \begin{proof} We will only consider the case when $(\alpha, \beta)$ has a $k$-upward itinerary. Note that $\alpha_{k+1}, \beta_{k+1} \in [0,\half]$. By equation \ref{eq:backward}, we see \newcommand{\iv}[2]{{\left[#1,#2\right]}} $$\alpha_k \in \iv{\dfrac{1}{3}}{\frac{3}{8}} \and \beta_k\in \iv{0}{\frac{1}{4}}.$$ Then since $(m_i,r_i,n_i,s_i)=(0,1,0,1)$ for $i=1,\ldots, k-1$ we see $$\alpha_1=\frac{\alpha_k}{1+2(k-1)\alpha_k} \and \beta_1=\frac{\beta_k}{1+2(k-1)\beta_k}.$$ Therefore we have $$\alpha_1 \in \iv{\frac{1}{2k+1}}{\frac{3}{6k+2}} \and \beta_1 \in \iv{0}{\frac{1}{2k+2}}.$$ Finally by one more application of equation \ref{eq:backward}, we see $$\alpha_0 \in \iv{\frac{1}{2k+3}}{\frac{3}{6k+8}} \and \beta_0 \in \iv{\frac{1}{3}}{\frac{2k+3}{6k+8}}.$$ \end{proof}
The point of these itineraries is that the cocycle decays an arbitrarily small amount along these special itineraries. This is quantified by the following lemma. In the following lemma, for any fixed $(\alpha, \beta)$ we use the following notation for the quantities appearing in our cocycle limit formula, Theorem \ref{thm:limit formula}: $$d_k=D(\alpha, \beta,k), \quad N_k=N(\alpha, \beta,k), \and \bn_k=\bn_{f^k(\alpha),f^k(\beta)}.$$
\begin{lemma}[Decay control] \name{lem:decay control} For each integer $k \geq 2$ and all $\epsilon>0$ there is a constant $Y>0$ only depending on $k$ and $\epsilon$ so that whenever $$(\alpha_{k+1},\beta_{k+1}) \in (\frac{1}{4},\frac{1}{2}) \times (0,Y),$$ the $k$-upward preimage $(\alpha,\beta)$ of $(\alpha_{k+1},\alpha_{k+1}')$ satisfies the entrywise inequality $$d_{k+1} N_{k+1}^T \bn_{k+1}>(1-\epsilon) \bn_0.$$ Similarly, for each integer $k \geq 2$ and all $\epsilon>0$ there is a $X>0$ so that whenever $$(\alpha_{k+1},\alpha_{k+1}') \in (0,X) \times (\frac{1}{4},\frac{1}{2}),$$ the $k$-rightward preimage of $(\alpha_{k+1},\alpha_{k+1}')$ satisfies the same inequality. \end{lemma} \begin{proof} We only consider the first statement. The second follows by symmetry. The proof is a calculation. We set $(x,y)=(\alpha_{k+1},\alpha_{k+1}')$. Since $(n_{k+1},r_{k+1},n'_{k+1},r'_{k+1})$ is $(1,1,0,1)$, we have $(\alpha_{k},\alpha_{k}')=(\frac{1 + x}{3 + 2 x}, \frac{y}{1 + 2 y})$ by Equation \ref{eq:alpha i2}. We compute the first stage of the product in the statement of the lemma as $$\begin{narrowarray}{0pt}{rcl} \a & = & (1-2\alpha_k)(1-2\alpha_k') \bn_{x,y} N_{k} \\ & = & \frac{1}{2 (3+2 x) (1 + 2 y)}(2 (1 + x - 6 x y), 1 + 2 y - 8 x y, 2 (1 - 2 x) y,
3 + 2 x - 16 x y).\end{narrowarray}$$ In contrast, we have $$\b=\bn_{\alpha_k,\alpha'_k} = \frac{1}{2 (3+2 x) (1 + 2 y)}(2 (1 + x), 1 + 2 y, 2 y, 3 + 2 x).$$ Note that $x>\frac{1}{4}$. So for any $X_1<1$ there exists a $Y_1>0$ so that $y<Y_1$ guarantees that $\a_i>X_1 \b_i$ for $i \in \{1,2,4\}$. This is because $\lim_{y \to 0} \a_i=\lim_{y \to 0} \b_i>0$ whenever $x \geq \frac{1}{4}$. However, we have no control over the ratios of the third coordinates. We set $\a'=X_1(\b_1, \b_2, 0, \b_4)$, so that $\a' < \a < \b$ entrywise whenever $y<Y_1$.
Because $(n_i,r_i,n'_i,r'_i)=(0,1,0,1)$ for $1 \leq i \leq k-1$, for these $i$ we have $$(\alpha_i, \alpha'_i)=(\frac{\alpha_k}{1+2(k-i) \alpha_k}, \frac{\alpha'_k}{1+2(k-i) \alpha'_k})= (\frac{1 + x}{1 + 2 (k-i+1) (1 + x)}, \frac{y}{1 + 2 (k-i+1) y}).$$ Also $N_i=N_{0,1,0,1}$ for these $i$. Let $$M=\prod_{i=k-1}^1 N_i= \left[\begin{array}{rrrr} 1 & k-1 & 0 & k-1 \\ 0 & 1 & 0 & 0 \\ 0 & k-1 & 1 & k-1 \\ 0 & 0 & 0 & 1 \end{array}\right] \and $$ $$p=\prod_{i=1}^{k-1} (1-2\alpha_i)(1-2\alpha'_i)=\frac{(3 + 2 x) (1 + 2 y)}{(1 + 2 k (1 + x)) (1 + 2 k y)}.$$ Set $\bc'=p (\a' M)$, $\bc=p (\a M)$, $\bd=p (\b M)$. Now we have $\bc'<\bc<\bd$. We compute $$\bd=\frac{1}{2\big(1 + 2 k (1 + x)\big) (1 + 2 k y)}\big(2 (1 + x), 2k-1+2(k-1)x+2ky, 2 y, 1+2k+2kx+2(k-1)y\big).$$ Also observe that $\displaystyle \bc'=X_1 \big(\bd - p \b_3 (0,k-1,1,k-1)\big).$ As $b_3 \to 0$ uniformly in $x$ as $y \to 0$, and the second and fourth entries of $\bd$ are bounded uniformly away from zero, for any $X_2<X_1$, there is a $Y_2>0$ with $Y_2< Y_1$ so that $y<Y_2$ implies that $\bc_i>X_2 \bd_i$ for $i \in \{1,2,4\}$. We also have $\bc'_3=0$. Now set $$\be=\bn_{\alpha_1,\alpha'_1} = \frac{1}{2(1 + 2 k + 2 k x) (1 + 2 k y)} \left[\begin{array}{r} 2 (1 + x) \big(1 +2(k-1) y\big) \\
\big(2 k-1+2 (k-1) x\big) (1 + 2 k y) \\
2 \big(2k-1+2 (k-1) x\big) y \\ (1 + 2 k + 2 k x) \big(1 +2 (k-1) y\big) \end{array}\right].$$ Observe that for any $X_3<1$, there is a $Y_3>0$ so that $y<Y_3$ implies $\bd_i>X_3 \be_i$ for $i \in \{1,2,4\}$. For these $i$ we have $$X_2 X_3 \be_i<X_2 \bd_i<\bc_i < \be_i.$$ We set $\bc''=X_2 X_3(\be_1,\be_2,0,\be_4)$ so that $\bc''<\bc < \be$ entrywise whenever $y<\min \{Y_2,Y_3\}$.
Finally, we consider that $(n_0,r_0,n'_0,r'_0)=(0,1,1,1)$. To simplify notation set $$(w,z)=(\alpha_1, \alpha'_1)=\Big(\frac{1 + x}{1 + 2 k (1 + x)}, \frac{y}{1 + 2 k y}\Big).$$ Note that as $y \to 0$ we have $z \to 0$, while $w$ only depends on $x$. Moreover, since $x>\frac{1}{4}$, $w>\frac{5}{10k+4}>0$. We have $$(\alpha, \alpha')=(\alpha_0, \alpha'_0)= \left(\frac{w}{1 + 2 w}, \frac{1 + z}{3 + 2 z}\right)= \left(\frac{1 + x}{3 + 2k+2(k+1) x}, \frac{1 + (2 k+1) y}{3 + (2+6k) y}\right).$$ Set $\f'=(1-2\alpha_0)(1-2\alpha_0')\bc'' N_0$, $\f=(1-2\alpha_0)(1-2\alpha_0')\bc N_0$, and $\g=(1-2\alpha_0)(1-2\alpha_0')\be N_0$. Then $\f'<\f<\g$ whenever $y<\min \{Y_2,Y_3\}$. Since $\be=\bn_{\alpha_1,\alpha'_1}$, we have $$\g=\frac{1}{2(1 + 2 w) (3 + 2 z)}\big(2 w - 4 w z, 3 + 2z - 16 w z, 2 + 2z - 16 w z, 1 + 2w - 8 w z\big).$$ As before observe that $X_2 X_3 \g-\f'>\0$ and in fact $$\begin{narrowarray}{0pt}{rcl} X_2 X_3 \g-\f' & = & X_2 X_3 (1-2\alpha_0) (1-2\alpha_0') \be_3 (0,3,3,1) \\ & = &\frac{ X_2 X_3}{(1 + 2 w) (3 + 2 z)} (0,3(1-2w)z,3(1-2w)z,(1-2w)z). \end{narrowarray}$$ This vector tends to zero uniformly as $z \to 0$ and hence also as $y \to 0$. Therefore, for any $X_4<X_2 X_3$, there is a $Y_4>0$ with $Y_4<\min \{Y_2, Y_3\}$ so that $y<Y_4$ implies $X_4 \g < \f'$. In particular, we have $X_4 \g< \f<\g$. Finally, set $$\h=\bn_{\alpha, \alpha'}=\frac{1}{2(1 + 2 w) (3 + 2 z)}(2w,3+2z,2+2z,1+2w).$$ Observe that for any $X_5$ there is a $Y_5>0$ so that $y<Y_5$ guarantees that $\g>X_5 \h$. So therefore, we have $X_4 X_5 \h< X_4 \g<\f$ whenever $y<\min \{Y_4, Y_5\}$. Observe that $\f$ is the vector on the left side of the inequality in the statement of the lemma, and $\h$ is the vector on the right (without the constant of $1-\epsilon$). From the above argument, for any $\epsilon>0$, we can find a choose the $X_i$ so as to guarantee $X_4 X_5>1-\epsilon$, then there is a $Y=\min \{Y_4, Y_5\}$ so that $y<Y$ guarantees the inequality given in the lemma holds. \end{proof}
Given the lemma, we have the following algorithm, which proves that for any constant $\eta<1$ there are irrational $\alpha$ and $\alpha'$ so that the $\nu=\mu_\alpha \times \mu_{\alpha'} \times \mu_N$ measure of the set of points without stable periodic orbits is larger than $\eta$. We call this algorithm the {\em slow decay algorithm}.
Fix a sequence of positive constants $\epsilon_i<1$ for $i \geq 0$ so that the product $\prod_{i=0}^\infty (1-\epsilon_i) \geq \eta$. The following algorithm determines the constants $n_i$, $r_i$, $n_i'$ and $r_i'$ which will determine a pair $(\alpha, \alpha')$ with the desired properties. (We state this rigorously in the theorem below the algorithm.) These constants are described by prescribing the itinerary of $(\alpha, \alpha')$ as a sequence of special itineraries. We refer to the sequence $(\alpha_i, \alpha'_i)$ only for clarity. At any point in the algorithm, each $(\alpha_i, \alpha'_i)$ is only determined to live some box depending on what is known of the itinerary. \begin{enumerate}
\item[0.] Set $i=0$ and set $\ell_0=0$ and set $X_0= \half \epsilon_0$. \item[1.] Let $k_i=\min \{k \geq 2:\frac{3}{8+6k}<X_i\}.$ \item[2.] The first statement in the Decay Control Lemma gives us a $Y_{i+1}(k_i, \epsilon_{i+1})>0$. \item[3.] Set $\ell_{i+1}=\ell_{i}+k_i+1.$ We insist that $(\alpha_{\ell_{i+1}},\alpha_{\ell_{i+1}}') \in (\frac{1}{4},\frac{1}{2}) \times (0,Y_{i+1})$, and that $(\alpha_{\ell_i}, \alpha'_{\ell_i})=U_{k_i}(\alpha_{\ell_{i+1}},\alpha_{\ell_{i+1}}')$ so that the Decay Control Lemma applies. This determines the constants $n_j$, $r_j$, $n'_j$ and $r'_j$ for $j=\ell_i, \ldots, \ell_{i+1}-1$ and forces $(\alpha_{\ell_i}, \alpha'_{\ell_i})$ to lie in the box $[\frac{1}{3+2k_i},\frac{3}{8+6k_i}] \times [\frac{1}{3}, \frac{3+2k_i}{8+6k_i}]$ by Proposition \ref{prop:special itineraries}. \item[4.] Increment $i$. ($i$ becomes $i+1$.) \item[5.] Let $k_i=\min \{k \geq 2:\frac{3}{8+6k}<Y_i\}.$ \item[6.] The second statement in the Decay Control Lemma gives us a $X_{i+1}(k_i, \epsilon_{i+1})>0$. \item[7.] Set $\ell_{i+1}=\ell_{i}+k_i+1.$ We insist that $\alpha_{\ell_{i+1}} \in (0,X_{i+1}) \times (\frac{1}{4},\frac{1}{2})$, and that
$(\alpha_{\ell_i}, \alpha'_{\ell_i})=D_{k_i}(\alpha_{\ell_{i+1}},\alpha_{\ell_{i+1}}')$ so that the Decay Control Lemma applies. This determines the constants $n_j$, $r_j$, $n'_j$ and $r'_j$ for $j=\ell_i, \ldots, \ell_{i+1}-1$ and forces $(\alpha_{\ell_i}, \alpha'_{\ell_i})$ to lie in the box $[\frac{1}{3}, \frac{3+2k_i}{8+6k_i}] \times [\frac{1}{3+2k_i},\frac{3}{8+6k_i}]$. \item[8.] Increment $i$ and return to step $1$. \end{enumerate}
As $i$ tends to infinity, the restrictions on $(\alpha_0, \alpha'_0)$ become stricter. In the limit, we determine a pair $(\alpha_0, \alpha'_0)$. See Proposition \ref{prop:conjugate to shift}. This pair satisfies the following theorem, which implies Theorem \ref{thm:4} of the introduction.
\begin{theorem} Given the choices made above the algorithm, there is a unique pair of irrationals $(\alpha, \alpha')=(\alpha_0, \alpha'_0)$ so that the following holds for the sequence $(\alpha_k, \alpha'_k)$. Let $i \geq 0$. \begin{itemize} \item If $i$ is even, then $(\alpha_{\ell_{i+1}},\alpha_{\ell_{i+1}}') \in (0,Y_{i+1}) \times (\frac{1}{4},\frac{1}{2})$ and $(\alpha_{\ell_i}, \alpha'_{\ell_i})=U_{k_i}(\alpha_{\ell_{i+1}},\alpha_{\ell_{i+1}}')$. \item If $i$ is odd, then $(\alpha_{\ell_{i+1}},\alpha_{\ell_{i+1}}') \in (0,X_{i+1}) \times (\frac{1}{4},\frac{1}{2})$ and
$(\alpha_{\ell_i}, \alpha'_{\ell_i})=D_{k_i}(\alpha_{\ell_{i+1}},\alpha_{\ell_{i+1}}')$. \end{itemize} Moreover, we have $\mu_\alpha \times \mu_{\alpha'} \times \mu_N({\mathit NS})>\eta$. \end{theorem} \begin{proof} The values of $n_j$ and $r_j$ determine $\alpha$ uniquely. Proposition \ref{prop:conjugate to shift} implies that $\alpha$ is irrational. The same holds for $\alpha'$.
We need to show that the Decay Control Lemma applies at each of the times the algorithm visits steps 3 and 7. At step 3, we have $(\alpha_{\ell_i}, \alpha'_{\ell_i})=U_{k_i}(\alpha_{\ell_{i+1}},\alpha_{\ell_{i+1}}')$. We would the Decay Control Lemma to apply to the pair $(\alpha_{\ell_i}, \alpha'_{\ell_i})$. For this, we need $(\alpha_{\ell_{i+1}},\alpha_{\ell_{i+1}}') \in (\frac{1}{4},\frac{1}{2}) \times (0,Y_{i+1})$. This is realized in the next few steps. Since $(\alpha_{\ell_{i+1}},\alpha_{\ell_{i+1}}')$ is a $k_{i+1}$-rightward preimage. Therefore by Proposition \ref{prop:special itineraries}, $$(\alpha_{\ell_{i+1}},\alpha_{\ell_{i+1}}') \in (\frac{1}{3}, \frac{3+2k_{i+1}}{8+6k_{i+1}}) \times (\frac{1}{3+2k_{i+1}},\frac{3}{8+6k_{i+1}}) \subset (\frac{1}{4},\frac{1}{2}) \times (0,Y_{i+1}).$$ This verifies that the Decay Control Lemma applies at each step $3$. It applies at each step $7$ by a symmetric argument.
Let $\nu=\mu_\alpha \times \mu_{\alpha'} \times \mu_N$. It remains to show that $\nu({\mathit NS})>\eta$. We recall that $\nu({\mathit NS})$ is the limit of a descending sequence, $\nu({\mathit NS})=\lim_{j \to \infty} \nu(\sO_j)$. We will show that for all $i \geq 0$, $$\nu(\sO_{\ell_i+1}) > \prod_{k=0}^i (1-\epsilon_k).$$ We prove this inductively. For the base case of $i=0$, we need to show that $\nu(\sO_1)>1-\epsilon_0$. Recall that $\sO_1=X \sm P_4$. We have $\nu(P_4)=4 \alpha \alpha'$. This can be shown by Equations \ref{eq:n} and \ref{eq:rewritten limit2}, for instance. Since we chose $X_0=\half \epsilon_0$ and forced $\alpha<\frac{3}{8+6k}<X_0$ and $\alpha'<\half$, we have $\nu(P_4)<2 X_0=\epsilon_0$. Therefore, $\nu(\sO_1)=1-\nu(P_4)>1-\epsilon_0$ as desired.
To prove the inductive step, we will show that $\nu(\sO_{\ell_{i+1}+1}) > (1-\epsilon_{i+1}) \nu(\sO_{\ell_{i}+1})$ for all $i \geq 0$. Observe that by equation \ref{eq:rewritten limit2}, we have $$\nu(\sO_{\ell_{i}+1})= \Big(\prod_{j=0}^{\ell_{i}-1} (1-2\alpha_j)(1-2\alpha'_j)\Big) \Big(\bn_{\alpha_{\ell_{i}},\alpha'_{\ell_{i}}} \big(\prod_{j=\ell_{i}-1}^0 N_j\big) \1\Big).$$ Let $\z$ denote the column vector $$\z= \Big(\prod_{i=0}^{\ell_{i}-1} (1-2\alpha_i)(1-2\alpha'_i)\Big) \Big(\big(\prod_{i=\ell_{i}-1}^0 N_i\big) \1\Big), $$ so that $\nu(\sO_{\ell_{i}+1})=\bn_{\alpha_{\ell_{i}},\alpha'_{\ell_{i}}} \z$. By another application of equation \ref{eq:rewritten limit2}, $$\begin{narrowarray}{0pt}{rcl} \nu(\sO_{\ell_{i+1}+1}) & = & \Big(\prod_{j=0}^{\ell_{i+1}-1} (1-2\alpha_j)(1-2\alpha'_j)\Big) \Big(\bn_{\alpha_{\ell_{i+1}},\alpha'_{\ell_{i+1}}} \big(\prod_{j=\ell_{i+1}-1}^0 N_j\big) \1\Big) \\ & = &\Big(\prod_{j=\ell_i}^{\ell_{i+1}-1} (1-2\alpha_j)(1-2\alpha'_j)\Big) \Big(\bn_{\alpha_{\ell_{i+1}},\alpha'_{\ell_{i+1}}} \big(\prod_{j=\ell_{i+1}-1}^{\ell_i} N_j\big) \z\Big). \end{narrowarray}$$ Now we apply the Decay Control Lemma (with the choice of $(\alpha, \alpha')$ used in the lemma set to be $(\alpha_{\ell_i}, \alpha'_{\ell_i})$ here). Since $(\alpha_{\ell_i}, \alpha'_{\ell_i})=U_{k_i}(\alpha_{\ell_{i+1}},\alpha_{\ell_{i+1}}')$ or $(\alpha_{\ell_i}, \alpha'_{\ell_i})=D_{k_i}(\alpha_{\ell_{i+1}},\alpha_{\ell_{i+1}}')$, with the hypothesis of the lemma satisfied, we have the entrywise inequality $$ \Big(\prod_{j=\ell_i}^{\ell_{i+1}-1} (1-2\alpha_j)(1-2\alpha'_j)\Big) \Big(\bn_{\alpha_{\ell_{i+1}},\alpha'_{\ell_{i+1}}} \big(\prod_{j=\ell_{i+1}-1}^{\ell_i} N_j\big)\Big) >(1-\epsilon_{i+1}) \bn_{\alpha_{\ell_{i}},\alpha'_{\ell_{i}}}.$$ It follows that $\nu(\sO_{\ell_{i+1}+1})>(1-\epsilon_{i+1}) \bn_{\alpha_{\ell_{i}},\alpha'_{\ell_{i}}} \z=(1-\epsilon_{i+1}) \nu(\sO_{\ell_{i}+1})$. \end{proof} \begin{comment}
Now we will explain how we will explain how to make use of the fact that $(\alpha, \alpha')$ does not fall into the framework of Theorem \ref{thm:recurrent case}. We will see that without loss of generality, we may assume that for all $k$, we have $$(\alpha_k, \alpha_k') \in Y=\big[(0, \frac{1}{6}) \times (0, \frac{1}{2})\big] \cup \big[(0, \frac{1}{2}) \times (0, \frac{1}{6})\big].$$ Here, $Y$ denotes an $L$-shaped neighborhood of the possible limit points not handled by Theorem \ref{thm:recurrent case}. We define the following subsets of $Y$. $$H=(\frac{1}{4},\frac{1}{2}) \times (0, \frac{1}{6}), \quad V=(0, \frac{1}{6}) \times (\frac{1}{4},\frac{1}{2}) \and S=Y \sm (H \cup V).$$ It is useful to observe the following properties of the sequence $(\alpha_k, \alpha_k')$.
\begin{proposition}[Itinerary restriction] If $(\alpha_k, \alpha_k') \in H$ then $(\alpha_{k+1}, \alpha_{k+1}') \not \in V$, and if $(\alpha_k, \alpha_k') \in V$ then $(\alpha_{k+1}, \alpha_{k+1}') \not \in H$. \end{proposition} \begin{proof} To prove the first statement, observe that $(\alpha_k, \alpha_k') \in H$ implies $\alpha_k'<\frac{1}{6}$. Then $\alpha_{k+1}'=\frac{\alpha_k'}{1-2 \alpha_k'}<\frac{1}{4}.$ \end{proof}
\begin{proposition}[Infinite visits] Suppose $(\alpha_k, \alpha_k') \in Y$ for all $k$. Then, there are infinitely many $k$ so that $(\alpha_k, \alpha_k') \in H$ and infinitely many $k$ so that $(\alpha_k, \alpha_k') \in V$. \end{proposition} \begin{proof} We prove that $(\alpha_k, \alpha_k')$ visits $H$ infinitely often. Suppose not. Then $\alpha_k<\frac{1}{4}$ for all $k$. But then, $\alpha_{k+1}$ is determined from $\alpha_k$ from by applying the map $x \mapsto \frac{x}{1-2x}$. This map fixes zero, and strictly expands distances on $(0,\frac{1}{4})$. \end{proof}
\begin{lemma} Suppose that $(\alpha,\alpha') \in H$. Let $k$ be the smallest integer $k>0$ for which $(\alpha_k, \alpha'_k) \in H \cup V$. Assume that $(\alpha_k, \alpha'_k) \in V$. Let $\ell$ be the smallest integer $\ell>k$ so that $(\alpha_\ell, \alpha'_\ell) \in H$. Then there is a universal constant $C<1$ so that entrywise we have $$ \Big(\prod_{i=0}^{\ell-1} (1-2\alpha_i)(1-2\alpha'_i)\Big) \Big(\bn_{\alpha_{\ell},\alpha'_{\ell}} \big(\prod_{i=k-2}^0 N_i\big)\Big)<C\bn_{\alpha,\alpha'}.$$ \end{lemma} \begin{proof} Our plan is to analyze the product on the left hand side. By the itinerary restriction and the fact that $\ell$ was minimal, we know $(\alpha_{\ell-1}, \alpha'_{\ell-1}) \in S$. Therefore, $n_{\ell-1}=0$, $r_{\ell-1}=1$, $n_{\ell-1}'=0$ and $r_{\ell-1}'=1$. So $N_{\ell-1}=N_{0,1,0,1}$. Let $x=\alpha_{\ell-1}$ and $y=\alpha_{\ell-1}$. We compute the first part of the product as $$\begin{narrowarray}{0pt}{rcl} \a & = & (1-2 \alpha_{\ell-1})(1-2 \alpha_{\ell-1}') \bn_{\alpha_{\ell},\alpha_{\ell}'} N_{\ell-1} \\ & = & \frac{1}{2 (1 + 2 x) (1 + 2 y)}\big(2 x (1 - 2 y), 1 + 2 y - 8 x y, 2 (1 - 2 x) y, 1 + 2 x - 8 x y\big). \end{narrowarray}$$ We will compare this to what we would obtain had we started expanding the product at time $\ell-1$ instead of time $\ell$. Noting that $\alpha_{\ell-1}=\frac{x}{1+2x}$ and $\alpha'_{\ell-1}=\frac{y}{1+2y}$, we see $$\b=\bn_{\alpha_{\ell-1},\alpha_{\ell-1}'}= \frac{1}{2(1 + 2 x) (1 + 2 y)} \big(2 x, 1 + 2 y, 2 y, 1 + 2 x\big).$$ While there is no measurable change in the other entries, we see that the third entry changes: $$\a_3/\b_3=1-2x<\frac{1}{2} \quad \textrm{for all $(x,y) \in H$.}$$
Now observe that for all $i$ with $k \leq i<\ell-1$ we have $(\alpha_i, \alpha'_i) \in S \cup V$. In particular, this means that $\alpha_i< \frac{1}{4}$. Therefore, $n_i=0$ and $r_i=1$. We have no control over $n_i'$ and $r_i'$. However, observe that matrices of the form $N_{0,1,n_i',r_i'}$ acting on row vectors have $(0,0,1,0)$ as an eigenvector with eigenvalue $1$. We define the vectors $$\bc=\Big(\prod_{i=k}^{\ell-2} (1-2\alpha_i)(1-2\alpha'_i)\Big) \Big(\a \big(\prod_{i=\ell-2}^{k} N_i\big)\Big), \and$$ $$\bd=\Big(\prod_{i=k}^{\ell-2} (1-2\alpha_i)(1-2\alpha'_i)\Big) \Big(\b \big(\prod_{i=\ell-2}^{k} N_i\big)\Big).$$ It follows from this observation that $\bc_3<\bd_3/2$. Now define $\be=\bn_{\alpha_{k},\alpha_{k}'}$. We have that $\bd<\be$ entrywise. (This comes from the meaning of these vectors. E.g., $\be_1=\nu \circ \rho^{-k}\big(\sO_1 \cap ({\mathcal S}_1 \cup {\mathcal S}_2)\big)$ while $\bd_1=\nu \circ \rho^{-k}\big(\sO_{\ell-1-k} \cap ({\mathcal S}_1 \cup {\mathcal S}_2)\big)$. Note $\sO_{\ell-1-k} \subset \sO_1$.) In particular, we have $\bc_3<\be_3/2$.
By the minimality assumption on $k$ we have that for all $i$ with $1 \leq i < k$ we have $(\alpha_i, \alpha'_i) \in S$. Therefore, $n_{i}=0$, $r_{i}=1$, $n_{i}'=0$ and $r_{i}'=1$ for all such $i$. Consider the case of $i=k-1$. Define $$\f = (1-2 \alpha_{k-1})(1-2 \alpha_{k-1}') \bc N_{k-1} \and \g = (1-2 \alpha_{k-1})(1-2 \alpha_{k-1}') \be N_{k-1}$$ and set $\h=\bn_{\alpha_{k-1},\alpha_{k-1}'}$. By the same argument that proved $\a_3<\b_3/2$ and symmetry in the line $y=x$, we have $\g_1<\h_1/2$. Therefore, as $\f_1<\g_1$ we have $\f_1<\h_1/2$. We also now have $\f_3<\g_3/2<\h_3/2$ as $(0,0,1,0)$ is still an eigenvector of $N_{k-1}=N_{0,1,0,1}$.
Observe that by the itinerary restriction, we have $k-1 \geq 1$. Let $(z,w)=(\alpha_{k-1}, \alpha_{k-1}')$. This means that we have $$\g=\frac{1}{2(1 + 2 z) (1 + 2 w)} \big(2 z, 1 + 2 w, 2 z, 1 + 2 w\big).$$
Since $(\alpha_i, \alpha'_i) \in S$ when $1 \leq i \leq k-1$, we have $\alpha_{i-1}=\frac{\alpha_i}{1+2 \alpha_i}$ and $\alpha'_{i-1}=\frac{\alpha_i}{1+2 \alpha_i}$. Therefore by induction we have $\alpha=\frac{z}{1+2(k-1)z}$ and $\alpha=\frac{z}{1+2(k-1)z}$ \end{proof} \end{comment}
\section{Dynamics on the Parameter Space} \name{sect:param space} In this section, we investigate the dynamical behavior of the map: $$f:[0,\half) \to [0, \half]; \quad f(x)=\frac{x}{1-2x} \pmod{G},$$ where $G$ is the group of isometries of $\R$ preserving $\Z$. This map was first mentioned in equation \ref{eq:f} of the introduction. We also study the product map $f \times f$.
The map $f$ is somewhat similar an analog Gauss map which appears when studying continued fractions. In the first subsection, we develop this point of view with an emphasis on coding and detecting irrationality.
In the second subsection, we show that $f \times f$ is recurrent with respect to Lebesgue measure. \subsection{Coding and rationality}\name{sect:rationality f} We define $A$ to be the infinite alphabet $$A=\{ (n,r) \in \Z \times \{\pm 1\}~:~\text{$n \geq 0$ and $(n,r) \neq (0,-1)$}\}.$$ For each $(n,r) \in A$, we define the interval $$I_{n,r}=\left\{x \in [0,\half)~:~r(\frac{x}{1-2x}-n) \in [0,\half].\right\}.$$ Observe that the union of these intervals covers $[0,\half)$.
Let $\{(n_k,r_k) \in A\}$ be a sequence defined for $k \geq 0$. We say $\{(n_k,r_k)\}$ is a {\em coding sequence} for $x \in [0,\half)$ if $f^k(x)$ is well defined for all $k \geq 0$ and $$f^k(x) \in I_{n_k,r_k} \quad \text{for all $k \geq 0$}.$$ (The value $f^k(x)$ is always well defined unless there is a $k$ so that $f^k(x)=\half$.)
The main goal of this subsection is to prove the following: \begin{theorem}[Coding] \name{thm:coding} For each sequence $\{(n_k,r_k) \in A\}_{k \geq 0}$, there is a unique $x \in [0,\half)$ so that $\{(n_k,r_k)\}$ is a coding sequence for $x$. This $x$ depends continuously on the choice of $\{(n_k,r_k)\}$, when the collection of all sequences is given the shift space topology. Moreover $x$ is irrational unless there is an $K$ so that $(n_k,r_k)=(0,1)$ for all $k \geq K$. \end{theorem}
\begin{remark}[Continued Fractions] We may think of $x$ as determined by $\{(n_k,r_k)\}$ via: $$x=\cfrac{1}{2+\cfrac{1}{n_0+\cfrac{r_0}{2+\cfrac{1}{n_1+\cfrac{r_1}{2+\ldots}}}}}.$$ \end{remark}
The proof follows from understanding the action of $f$ on the intervals $I_{n,r}$. We compute $$I_{n,1}=\left[\frac{n}{1+2n}, \frac{2n+1}{4n+4}\right] \and I_{n,-1}=\left[\frac{2n-1}{4n}, \frac{n}{1+2n}\right].$$ Observe that $f$ restricts to a bijection $I_{n,r} \to [0,\half]$. The inverse of this restriction is \begin{equation} \name{eq:g} g_{n,r}:[0,\half] \to I_{n,r}; \quad g_{n,r}(x)=\frac{r x + n}{1 + 2 (r x + n)}. \end{equation}
We first prove the existence, uniqueness and continuity comments of the theorem. Further below, we give the proof of the irrationality condition.
\begin{proof}[Proof of existence, uniqueness and continuity] This follows from standard dynamics arguments involving Markov partitions for maps of the interval. The collection of all $I_{n,r}$ form a Markov Partition. The image of each interval under $f$ is $[0,\half]$.
Consider a finite sequence $\{(n_k,r_k)\}$ defined for $0 \leq k \leq K$. The set of points $x$ for which $f^k(x) \in I_{n_k,r_k}$ is given by \begin{equation} \name{eq:g seq} g_{n_0,r_0} \circ g_{n_1,r_1} \circ \ldots \circ g_{n_K,r_K}([0,\thalf]). \end{equation} Each such set is a closed interval. This applies continuity of the dependence of $x$ on the sequence (assuming $x$ is uniquely determined).
Now let $\{(n_k, r_k)\}$ be an infinite sequence. Let $J$ be the set of points $x$ so that $\{(n_k, r_k)\}$ is a coding sequence for $x$. The set $J$ is a nested intersection of sets of the form given in equation \ref{eq:g seq}. Therefore, $J$ is a non-empty closed interval. We must prove that $J$ has no more than one point. Suppose $J$ is not just a single point. Then, it contains an irrational $x$. Observe that under iteration, an irrational must visit the set $(\frac{1}{4}, \half)$ infinitely often. This is because if $f^i(x)<\frac{1}{4}$ for $i=1, \ldots k-1$, then $$f^k(x)=\frac{x}{1-2kx}.$$ So, eventually $f^k(x)>\frac{1}{4}$. If $x>\frac{1}{4}$ and is irrational, then $f$ is locally strictly expanding by a factor larger than one. Therefore, the length of $f^k(J)$ would have to tend to infinity as $k \to \infty$. This contradicts the assumption that $J$ was not just a single point. \end{proof}
\begin{proof}[Proof of the Irrationality Condition] Observe that there is a unique coding sequence for zero, consisting of the infinite sequence with $(n_k,r_k)=(0,1)$ for all $k$. So, it suffices to show that for all rational $p/q \in [0,\half)$ there is a $k$ so that $f^k({\textstyle \frac{p}{q}}) \in \{0,\thalf\}.$
Observe that the action of $f$ on reduced fractions in $\Q \cap [0,\half)$ is given by the formula $$f({\textstyle\frac{p}{q}})=\frac{p}{q-2p} \pmod{G}.$$ We define the ``complexity function'' $$\chi:\Q \to \N; \quad \frac{p}{q} \mapsto q,$$ where $\frac{p}{q}$ is assumed to be a reduced fraction with $q>0$. For all such $\frac{p}{q} \in (0,\half)$, we have $$\chi \circ f(\textstyle{\frac{p}{q}})=q-2p<q=\chi(\textstyle{\frac{p}{q}})$$ so the complexity drops by at least two when applying $f$. So, eventually the denominator must drop to a value of one or two. \end{proof}
\subsection{Measurable dynamics}\name{sect:measurable f} In this section we treat $f$ as a map on the set $I$ of irrationals in the interval $(0,\half)$. This set has full Lebesgue measure, and is invariant under $f$. Our main result is the following:
\begin{theorem}[Recurrence] \name{thm:recurrence} Let $\lambda$ denote Lebesgue measure on $I$. The action of $f \times f$ on $I^2$ is recurrent in the sense that for any Borel subset $A \subset I^2$, for $\lambda^2$-a.e. $(x,y) \in A$ there is an $n \geq 1$ so that $(f \times f)^n(x,y) \in A$. \end{theorem}
We actually prove the above statement by replacing $\lambda$ with an equivalent measure $m$. (Two measures are {\em equivalent} if they have the same null sets.)
\begin{lemma}[Invariant measure] The $\lambda$ equivalent measure $m$ on $I$ defined by $${m}(A)=\int_A \frac{1}{x}+\frac{1}{1-x}~dx.$$ is $f$-invariant (i.e., $m \circ f^{-1} = m$). \end{lemma} The measure $m$ should be thought of as analogous to the Gauss measure for continued fractions. \begin{proof} Let $\lambda$ denote Lebesgue measure on $I$. Consider the Radon-Nikodym derivative $$h(x)=\frac{d {m}}{d\lambda}(x)=\frac{1}{x}+\frac{1}{1-x}.$$ Observe that
$$\frac{d ({m} \circ f^{-1})}{d \lambda}(x)=\sum_{y \in f^{-1}(\{x\})} \frac{h(y)}{|f'(y)|}.$$
So, it suffices to show that this sum yields $h(x)$. We compute that $$\frac{h(y)}{|f'(y)|}=\frac{(1-2y)^2}{y(1-y)}.$$ We can write each $y \in f^{-1}(\{x\})$ as $y=g_{n,r}(x)$ as in equation \ref{eq:g}. We evaluate the sum in two portions. In the cases $r=1$ and $r=-1$, we respectively have
$$\sum_{n=0}^\infty \frac{h \circ g_{n,1}(x)}{|f' \circ g_{n,1}(x)|}=\sum_{n=0}^\infty \big(\frac{1}{n+x}-\frac{1}{1+n+x}\big)=\frac{1}{x}, \and$$
$$\sum_{n=1}^\infty \frac{h \circ g_{n,-1}(x)}{|f' \circ g_{n,-1}(x)|}=\sum_{n=1}^\infty \big(\frac{1}{n-x}-\frac{1}{1+n-x}\big)=\frac{1}{1-x}.$$ Combining these two sums, we see $\frac{d ({m} \circ f^{-1})}{d \lambda}(x)=h(x)$, as desired. \end{proof}
Note that the measure $m$ is infinite. If this were not the case, we would have Recurrence by the Poincar\'e recurrence theorem. However, the measure of sets of the form $(\epsilon,\half) \cap I$ is finite. Our proof of recurrence depends on controlling the possibility of the backward iterates of a set tending toward the set of points where one coordinate is zero. This control is given by the following.
\begin{lemma}[Plug Lemma] For $\epsilon>0$, define the following subsets of $I^2$: $$N_\epsilon=[0, \epsilon] \times I \cup I \times [0, \epsilon] \and P_\epsilon=(f \times f)(N_\epsilon) \sm N_\epsilon.$$ We have $\lim_{\epsilon \to 0} m \times m (P_\epsilon)=0.$ \end{lemma}
We call this the plug lemma, because any orbit starting in $N_\epsilon$ must pass through $P_\epsilon$ in order to reach the complement of $N_\epsilon$. The observation of the lemma is that not much can pass through the plug. Using this Lemma, we can prove recurrence:
\begin{proof}[Proof of the Recurrence Theorem] The proof is a general principal following from the Hopf Decomposition. (See \S 1.3 of \cite{Krengel85}, for instance.) If $f \times f$ were not recurrent, then there would be a wandering set $W \subset I \times I$ of positive $m \times m$ measure. That is, $W$ is a set so that the preimages $(f \times f)^{-k}(W)$ are pairwise disjoint for $k \geq 0$.
To simplify notation let $\mu=m \times m$ and $\phi=f \times f$.
We use the facts that $\cap_{\epsilon>0} N_\epsilon=\emptyset$ and $\lim_{\epsilon \to 0} \mu (P_\epsilon)=0$. By possibly making $W$ smaller, we can assume that there is an $\epsilon>0$ so that \begin{enumerate} \item $W \cap N_\epsilon= \emptyset$. \item $\mu(W)>\mu(P_\epsilon)$. \end{enumerate}
Now consider the sequence of sets $P_k$ and $W_k$ defined inductively according to the following rules. We define $P_0=W \cap P_\epsilon$ and $W_0=W \sm P_0$. For $k \geq 0$ define $$P_{k+1}=\phi^{-1}(W_k) \cap P_\epsilon \and W_{k+1}=\phi^{-1}(W_k) \sm P_{k+1}.$$ By invariance of $\mu$, the sequence $\mu(W_k)$ is decreasing. Because each $W_k$ is disjoint and lies in the complement of $N_\epsilon$ (which has finite measure with respect to $\mu$), we have $$\lim_{k \to \infty} \mu(W_k)=0.$$ Again by invariance of $\mu$, we have $\mu(W_k)=\mu(W_{k+1})+\mu(P_{k+1})$. It follows that for all $k \geq 0$, $$\mu(W)=\mu(W_k)+ \sum_{i=0}^k \mu(P_k) \quad \text{and so} \quad \mu(W)=\sum_{i =0}^\infty \mu(P_i).$$ Note that the sets $P_i$ are pairwise disjoint and lie in $P_\epsilon$. This contradicts the statement that $\mu(W)>\mu(P_\epsilon)$. \end{proof}
\begin{proof}[Proof of the Plug Lemma] We will assume $\epsilon<\frac{1}{4}$. We can write $P_\epsilon$ as a union of rectangles, $$P_{\epsilon}=\textstyle (\epsilon,\frac{\epsilon}{1-2\epsilon}] \times (\epsilon,\frac{1}{2}) \cup (\epsilon,\frac{1}{2}) \times (\epsilon,\frac{\epsilon}{1-2\epsilon}].$$ Therefore, $m \times m(P_\epsilon)$ is less than twice the product of the measures of these two intervals with respect to $m$. We have $$\textstyle m\big([\epsilon,\frac{\epsilon}{1-2\epsilon}]\big)=\log(\frac{1-\epsilon}{1-3 \epsilon}) \and m\big([\epsilon,\frac{1}{2}] = \log(\frac{1-\epsilon}{\epsilon}).$$ A calculation shows that as $\epsilon$ tends to zero, the product of these quantities tends to zero. \end{proof}
\begin{comment}
\subsection{Old treatment}
Note that the measure $m$ is infinite. If this were not the case, we would have Recurrence by the Poincar\'e recurrence theorem. However, the measure of sets of the form $(\epsilon,\half) \cap I$ is finite. Our proof of recurrence depends on controlling the possibility of an orbit tending toward the set of points where one coordinate is zero. This control is given by the following.
\begin{lemma}[Plug Lemma] For $\epsilon>0$, define the following subsets of $I^2$: $$N_\epsilon=[0, \epsilon] \times I \cup I \times [0, \epsilon] \and P_\epsilon=(f \times f)^{-1}(N_\epsilon) \sm N_\epsilon.$$ We have $\lim_{\epsilon \to 0} m \times m (P_\epsilon)=0.$ \end{lemma}
We call this the plug lemma, because any orbit of $f \times f$ beginning at a point in the compliment of $N_\epsilon$ must pass through the ``plug'' $P_\epsilon$ in order to reach $N_\epsilon$. The observation of the lemma is that not much can pass through the plug. Using this Lemma, we can prove recurrence:
\begin{proof}[Proof of the Recurrence Theorem] Because $\bigcap_{\epsilon>0} (P_\epsilon \cup N_\epsilon)=\emptyset$, it suffices to prove the theorem for a set $A$ so that $A \cap (P_{\epsilon'} \cup N_{\epsilon'})=\emptyset$ for some $\epsilon'>0$.
For each $\epsilon$ with $0<\epsilon<\epsilon'$ define the map $\phi_{\epsilon}:A \to A \cup P_{\epsilon}$ to be $$\phi_\epsilon(\bp)=(f \times f)^{n(\bp)}(\bp), \quad \textrm{where $n(\bp)=\min \{m\geq 1~:~(f \times f)^{m}(\bp) \in A \cup P_{\epsilon}\}$}.$$ We claim that $\phi_\epsilon$ is defined for ${m}^2$-almost every $\bp \in A$. Suppose not. Then, there is a $B \subset A$ of positive measure so that $$(f \times f)^k(B) \cap (A \cup P_\epsilon)=\emptyset \quad \text{for all $k \geq 1$}.$$ The union of all images of $B$ forms the set $$O=\bigcup_{k \geq 1} (f \times f)^k(B).$$ We have $O \subset I^2 \sm N_\epsilon$, so that ${m}^2(O)<\infty$. Also observe that both $B$ and $O$ lie inside $(f \times f)^{-1}(O)$. These sets are disjoint. So, $$m^2 \circ (f \times f)^{-1}(O) \geq m^2(O)+m^2(B)>m^2(O).$$ This contradicts the $f \times f$ invariance of $m^2$, proving $\phi_\epsilon$ is defined for ${m}^2$-a.e. $\bp \in A$.
Now define $\phi:A \to A$ so that whenever $\phi_\epsilon(\bp) \in A$ we have $\phi(\bp)=\phi_\epsilon(\bp)$. This choice is well defined since whenever $\phi_\epsilon(\bp) \in A$, we have $\phi_{\delta}(\bp)=\phi_\epsilon(\bp)$ for all $\delta<\epsilon$. (The first time $(f \times f)^{m}(\bp)$ enters $N_{\delta}$ is preceded by an entry into $N_{\epsilon}$.) Also $\phi$ is defined on $\phi_{\epsilon}^{-1}(A)$ for every $\epsilon$.
Observe that we can write $A$ as a disjoint union (modulo null sets), $$A=\phi_\epsilon^{-1}(P_\epsilon) \cup \phi_\epsilon^{-1}(A).$$ By definition of $\phi_\epsilon$, we have $m^2 \circ \phi_\epsilon^{-1}(P_\epsilon) \leq m^2(P_\epsilon).$ Thus, $$m^2 \circ \phi_\epsilon^{-1}(A) \geq m^2(A)-m^2(P_\epsilon).$$ Since ${m}^2 \circ \phi_{\epsilon}^{-1}(A)$ tends to ${m}^2(A)$ as $\epsilon \to 0$, we know that $\phi(\bp)$ is well defined for ${m}^2$-a.e. $\bp \in A$. Any such $\bp \in A$ returns to $A$. \end{proof}
\begin{proof}[Proof of the Plug Lemma] We will assume $\epsilon<\frac{1}{4}$. We will use the local inverses $g_{n,r}$ to describe $P_\epsilon=(f \times f)^{-1}(N_\epsilon) \sm N_\epsilon$. Observe that for $x \in [0, \epsilon]$, we also have $g_{0,1}(x) \in [0,\epsilon]$. For $n \geq 1$ we define $$J_n=g_{n,-1}([0,\epsilon]) \cup g_{n,1}([0,\epsilon])=\left[\frac{n - \epsilon}{2n-2\epsilon+1},\frac{n+\epsilon}{2n+2\epsilon+1}\right].$$ Observe that we can write $$P_\epsilon=\bigcup_{n=1}^\infty \big((J_n \times [\epsilon,\half]) \cup ([\epsilon,\half] \times J_n)\big).$$ We make the following computations: $${m}(J_n)=\log(n+\epsilon)+\log(n-\epsilon+1)-\log(n-\epsilon)-\log(n+\epsilon+1).$$ $$\sum_{n=1}^\infty {m}(J_n)=\log(1+\epsilon)-\log(1-\epsilon).$$ We also can compute ${m}( [\epsilon,\half])=\log(1-\epsilon)-\log(\epsilon).$ Therefore, we have $${m}^2(P_\epsilon)<2\big(\log(1+\epsilon)-\log(1-\epsilon)\big)\big(\log(1-\epsilon)-\log(\epsilon)\big).$$ A calculation then shows $\lim_{\epsilon \to 0} {m}^2(P_\epsilon)=0$, as desired. \end{proof} \end{comment}
\end{document} | arXiv |
Lower and upper bounds to the change of vorticity by transition from slip- to no-slip fluid flow
October 2014, 7(5): 1111-1132. doi: 10.3934/dcdss.2014.7.1111
On one multidimensional compressible nonlocal model of the dissipative QG equations
Shu Wang 1, , Zhonglin Wu 1, , Linrui Li 2, and Shengtao Chen 1,
College of Applied Sciences, Beijing University of Technology, PingLeYuan100, Chaoyang District, Beijing 100124, China, China, China
Basic Courses Department, Institute of Disaster Prevention, Yanjiao, Sanhe City, Hebei Province, 065201, China
Received January 2013 Revised June 2013 Published May 2014
In this paper we study the Cauchy problem for one multidimensional compressible nonlocal model of the dissipative quasi-geostrophic equations and discuss the effect of the sign of initial data on the wellposedness of this model. First, we prove the existence and uniqueness of local smooth solutions for the Cauchy problem for the model with the nonnegative initial data, which seems to imply that whether the well-posedness of this model holds or not depends heavily upon the sign of the initial data even for the subcritical case. Secondly, for the sub-critical case $1<\alpha\leq 2$, we obtain the global existence and uniqueness results of the nonnegative smooth solution. Next, we prove the global existence of the weak solution for $0<\alpha\le 2$ and $\nu>0$. Finally, for the sub-critical case $1<\alpha\leq 2$, we establish $H^\beta(\beta\geq 0)$ and $L^p(p\geq 2)$ decay rates of the smooth solution as $t\to\infty$. A inequality for the Riesz transformation is also established.
Keywords: dissipative quasi-geostrophic equations, Riesz transformation, Cauchy problem., sub-critical case, Multidimensional compressible nonlocal model.
Mathematics Subject Classification: 35A01, 35L45, 35L60, 35Q3.
Citation: Shu Wang, Zhonglin Wu, Linrui Li, Shengtao Chen. On one multidimensional compressible nonlocal model of the dissipative QG equations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 1111-1132. doi: 10.3934/dcdss.2014.7.1111
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Shu Wang Zhonglin Wu Linrui Li Shengtao Chen | CommonCrawl |
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Algebraic isomorphisms and $\mathcal {J}$-subspace lattices
by Jiankui Li and Oreste Panaia PDF
Proc. Amer. Math. Soc. 133 (2005), 2577-2587 Request permission
The class of $\mathcal {J}$-lattices was originally defined in the second author's thesis and subsequently by Longstaff, Nation, and Panaia. A subspace lattice $\mathcal {L}$ on a Banach space $X$ which is also a $\mathcal {J}$-lattice is called a $\mathcal {J}$-subspace lattice, abbreviated JSL. It is demonstrated that every single element of $Alg\mathcal {L}$ has rank at most one. It is also shown that $Alg\mathcal {L}$ has the strong finite rank decomposability property. Let $\mathcal {L}_1$ and $\mathcal {L}_2$ be subspace lattices that are also JSL's on the Banach spaces $X_1$ and $X_2$, respectively. The two properties just referred to, when combined, show that every algebraic isomorphism between $Alg\mathcal {L}_1$ and $Alg\mathcal {L}_2$ preserves rank. Finally we prove that every algebraic isomorphism between $Alg\mathcal {L}_1$ and $Alg\mathcal {L}_2$ is quasi-spatial.
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Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47L10
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Jiankui Li
Affiliation: Department of Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: [email protected]
Oreste Panaia
Affiliation: School of Mathematics, The University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia
Email: [email protected]
Received by editor(s): February 4, 2002
Received by editor(s) in revised form: April 17, 2003
Published electronically: April 15, 2005
Communicated by: David R. Larson
© Copyright 2005 American Mathematical Society
Journal: Proc. Amer. Math. Soc. 133 (2005), 2577-2587
MSC (2000): Primary 47L10
DOI: https://doi.org/10.1090/S0002-9939-05-07581-7 | CommonCrawl |
Full paper | Open | Published: 18 February 2019
Estimating global geodetic parameters using SLR observations to Galileo, GLONASS, BeiDou, GPS, and QZSS
K. Sośnica ORCID: orcid.org/0000-0001-6181-13071,
G. Bury1,
R. Zajdel1,
D. Strugarek1,
M. Drożdżewski1 &
K. Kazmierski1
All Galileo, GLONASS, QZSS, and BeiDou satellites are equipped with laser retroreflector arrays dedicated to satellite laser ranging (SLR). Using SLR data to new GNSS systems allows for estimating global geodetic parameters, such as Earth rotation parameters, global scale, and geocenter coordinates. In this study, we evaluate the quality of global geodetic parameters estimated on a basis of SLR tracking of new GNSS satellites and the combined solution based on SLR observations to GNSS and LAGEOS. We show that along with a progressive populating of Galileo orbital planes, the quality of geocenter components based on SLR–GNSS data has been improved to the level of 6 and 15 mm for equatorial and polar geocenter components, respectively. The scale of the reference frame and the geocenter coordinates in the combined LAGEOS + GNSS solutions are dominated by the LAGEOS data. Some noncore SLR stations provide by far more observations to GNSS than to LAGEOS, e.g., Russian and Chinese stations dedicated to supporting GLONASS and BeiDou constellations. The number of solutions for these stations can be increased by up to 40%, whereas the station coordinate repeatability can be improved from about 20–30 mm to the level of 15–20 mm when considering both SLR to LAGEOS and SLR to GNSS.
Satellite laser ranging (SLR) is the space geodetic technique used for many applications (Pearlman et al. 2019), such as the realization of the origin and scale of the International Terrestrial Reference Frames (ITRF, Altamimi et al. 2016), determination of global geodetic parameters, such as polar motion and length-of-day (LOD) excess (Pavlis 1994; Sośnica et al. 2014; Glaser et al. 2015; Bloßfeld et al. 2018), determination of low-degree harmonics of the Earth's gravity potential (Cox and Chao 2002; Bloßfeld et al. 2015; Sośnica et al. 2015a; Cheng and Ries 2017), orbit determination and validation for active satellites and space debris (Arnold et al. 2018; Strugarek et al. 2019; Kucharski et al. 2017), time transfer (Exertier et al. 2018), and verification of various general relativity effects (Ciufolini and Pavlis 2004; Pardini et al. 2017).
The SLR contribution to ITRF is based on observations to two LAGEOS satellites and two Etalons; however, the number of SLR observations to LAGEOS is more than ten times larger than that to Etalons (Appleby 1998). The contribution of Etalon to the ITRF realization is thus almost marginal. All new global navigation satellite systems (GNSS), such as GLONASS, Galileo, BeiDou, and Regional Navigation Satellite Systems, such as QZSS and NavIC, are equipped with laser retroreflector arrays dedicated to SLR tracking. Today, there are about 60 active GNSS satellites tracked by the International Laser Ranging Service (ILRS, Pearlman et al. 2002) stations. No active satellites, such as GNSS, are currently used for the ITRF realization, e.g., for the estimation of SLR station coordinates, geocenter coordinates, or Earth rotation parameters (ERPs). Some SLR stations, e.g., from the Russian SLR network, provide much more SLR observations to GNSS than to LAGEOS, because these stations were dedicated to support the GLONASS system.
Between 2014 and 2017, the ILRS conducted a series of intensive campaigns tracking GNSS, which resulted in a substantial increase in the number of tracked GNSS spacecraft and the enlargement of the number of collected data. In this study, we use SLR observations to 1 GPS, 31 GLONASS, 18 Galileo, 1 BeiDou in medium Earth orbits (MEO), 3 BeiDou in inclined geosynchronous orbits (IGSO), and 1 QZSS satellite in inclined eccentric geosynchronous orbit (see Fig. 1). Not all spacecraft were active at the same time due to the constellation modernization, e.g., only up to 24 GLONASS were simultaneously active. In 2014, only 3–4 Galileo in-orbit validation (IOV) satellites were active. In 2017, the number of active Galileo increased to 17 due to multiple launches of Galileo fully operational capability (FOC) spacecraft (see Figs. 1, 2). A large number of SLR data allow, e.g., for the evaluation of geophysical effects using solely SLR observations to GNSS (Bury et al. 2019).
Number of daily SLR observations to particular GNSS satellites between 2014.0 and 2017.4
Number of weekly SLR observations to LAGEOS and GNSS satellites between 2014.0 and 2017.4
The performance of the combined solution based on LAGEOS and GNSS was assessed by Sośnica et al. (2018a). However, the results from the SLR-to-GNSS-only solutions have not been evaluated so far. In this study, we present results from the determination of global geodetic parameters associated with the ITRF realization, such as ERPs, geocenter, the global scale, and SLR station coordinates, based on both SLR observations to GNSS and based on combined SLR-to-LAGEOS + GNSS data with different variants of handling satellite orbits and range biases.
We estimate parameters with 7-day intervals: SLR station coordinates, geocenter coordinates, range biases, whereas ERPs are estimated with 1-day intervals and parameterized as piece-wise linear (see Table 1). In the case of LAGEOS solutions, range biases are estimated only for selected 1–2 sites following the recommendations of the ILRS Data Handling File.Footnote 1 We use the International Earth Rotation and Reference Systems Service series IERS-14-C04 (Bizouard et al. 2018) as the a priori ERPs and the ILRS realization of the ITRF2014, i.e., SLRF2014, for the a priori station coordinates. One UT1–UTC parameter is fixed to IERS-14-C04 series to provide absolute orientation of the network. For LAGEOS, we estimate six Keplerian orbit parameters and five empirical parameters, i.e., a constant acceleration, periodic once-per-revolution accelerations in along-track S and periodic accelerations in cross-track W with the 7-day intervals:
$$\begin{aligned} \left\{ \begin{array}{lll} &{}R= - \\ &{}S= S_0 + S_{S1} \sin u + S_{C1} \cos u \\ &{}W= W_{S1} \sin u + W_{C1} \cos u \end{array}\right. \end{aligned}$$
where u is the satellite argument of latitude. No empirical parameters are estimated in the radial direction R.
Table 1 List of estimated parameters
For GNSS, six Keplerian and seven empirical parameters are estimated of the new Empirical CODE Orbit Model (ECOM2, Arnold et al. 2015) with the expansion up to twice-per-revolution parameters in the satellite-Sun direction. In the empirical orbit model for GNSS, the estimated parameters are decomposed into three orthogonal directions: the D axis pointing from a satellite toward Sun, the Y axis lying along the solar panels, and the B axis completing the right-handed coordinate orthogonal frame. The set of estimated empirical orbit parameters for GNSS includes (Arnold et al. 2015):
$$\begin{aligned} \left\{ \begin{array}{lll} &D= D_0 + D_{S2} \sin (2 \Delta u) + D_{C2} \cos (2 \Delta u) \\ &Y= Y_0 \\ &B= B_0 +B_{S1} \sin \Delta u + B_{C1} \cos \Delta u \end{array}\right. \end{aligned}$$
where \(\Delta u\) is the satellite argument of latitude with respect to the argument of latitude of the Sun. All orbit parameters are estimated without any constraints.
As a result, most of the parameters—SLR station coordinates, geocenter coordinates, ERPs—are common in the GNSS and LAGEOS solutions and can be stacked in the combination process. The only individually estimated parameters are range biases and satellite orbits (see Table 1).
We use the list of SLR core stations as recommended by the ILRS data handling file. The list of core stations is verified in every 7-day solution using the Helmert transformation. When the residuals after the 7-parameter Helmert transformation exceed the threshold of 25 mm for the north, east, or up component, the station is excluded from the list of core stations, and thus, the network constraints are not imposed thereon. Finally, we impose the no-net-rotation (NNR) and the no-net-translation (NNT) network minimum constraints using the verified list of core stations, whereas other station coordinates are estimated as free parameters.
We employ the same models for data reduction in the case of SLR observations to GNSS and to LAGEOS satellites. The Mendes and Pavlis (2004) model is used for the zenith path delay with a corresponding mapping function based on site-specific meteorological data. As the gravity potential model, we use EGM2008 (Pavlis et al. 2013) with a maximum expansion up to degree and order 30 and the ocean tide model FES2004 (Lyard et al. 2006). Solid Earth tides, pole tides, ocean pole tides, mean pole, ocean tidal loading displacements, and general relativistic corrections are modeled according to the IERS Conventions 2010 (Petit and Luzum 2010). The nontidal loading is here neglected, which may cause a systematic blue-sky effect up to 2 mm when fixing GNSS orbits (Bury et al. 2019). The center-of-mass corrections for LAGEOS satellitesFootnote 2 are applied as station and time dependent following the detector and detection procedure changes at individual SLR stations (Otsubo and Appleby 2003). For GNSS laser retroreflector offsets, we apply standard values provided by mission operators and distributed by the ILRS. We apply the time-variable retroreflector offsets with respect to the satellite center-of-mass for Galileo satellites,Footnote 3 which are caused by the fuel consumption. The time-variable offsets are, however, available only for Galileo.
We use the a priori multi-GNSS orbits provided by the Center for Orbit Determination in Europe (CODE, Prange et al. 2017). First, we generate 1-day normal equations (NEQs, see Fig. 3) based solely on SLR observations to GNSS in the Bernese GNSS Software (Dach et al. 2015). We stack five daily NEQs with the orbit transformations to continuous 5-day arcs because the 5-day arcs represent the optimum solution when using SLR data for the GNSS orbit determination (Bury et al. 2018). To allow a combination with the standard ILRS 7-day LAGEOS solutions, all solutions were transformed to the 7-day NEQs with the pre-elimination of all parameters exceeding the 7-day window and the pre-elimination of orbital parameters before stacking. The pre-elimination guarantees that the parameters are estimated as implicit parameters, which means that their values cannot explicitly be estimated, but that all other parameters assume the values as in the case when all parameters were explicitly estimated (Dach et al. 2015). This procedure employing the 5-day orbit pre-elimination provides more stable parameters when compared to direct 1-day solutions based on sparse SLR observations to multiple satellites from the GNSS constellations. We generate also a second solution, in which the GNSS orbital parameters are not estimated; thus, they are directly fixed to the a priori CODE orbits.
Processing scheme of the GNSS + LAGEOS combination
Range biases constitute one of the major error sources when processing SLR data to GNSS (Thaller et al. 2014, 2015). Estimation of range biases substantially increases the number of estimated parameters when estimated as station-satellite specific. Therefore, we estimate first the mean annual station-satellite range biases and reintroduce them as a priori values in the final solution, in order to stabilize the solution and to reduce the number of estimated parameters. The estimated range biases absorb not only the station-specific biases and circuit delays, but also the satellite signature effects and the differences between detector types employed at the SLR stations (Otsubo et al. 2001; Sośnica et al. 2015b). Finally, we generate solutions with the contribution of LAGEOS satellites. We use the relative weighting of 8 mm for LAGEOS and 40 mm for GNSS, which is dictated by the quality of the GNSS data from the SLR validation (Zajdel et al. 2017; Bruni et al. 2018).
Figure 4 shows how many 7-day solutions in % are possible to obtain for particular SLR stations when using LAGEOS-only, GNSS-only, and LAGEOS + GNSS observations. In total, 177 weekly SLR solutions were generated in 2014.0–2017.4. The mean number of stations present in 7-day solutions is 22, 20, and 24 in LAGEOS-only, GNSS-only, and LAGEOS \(+\) GNSS, respectively. Some stations, such as Haleakala (Hawaii, station 7119) and Arequipa (Peru, 7403), do not observe GNSS satellites at all. However, most of the SLR stations observe LAGEOS and GNSS on the regular basis, e.g., Yarragadee (Australia, 7090), Herstmonceux (UK, 7840), Changchun (China, 7237), and Mt. Stromlo (Australia, 7825).
The occurrence of SLR stations in 7-day solutions in the period 2014.0–2017.5 expressed in % (percentage of possible to obtain 7-day station positions)
Some stations provide by far more observations to GNSS than to LAGEOS, e.g., for Altay (Russia, 1879) 132 weekly solutions were possible when using LAGEOS data and 161 solutions using LAGEOS + GNSS data (22% more solutions), Komsomolsk (Russia, 1868)—41% more solutions, Arkhyz (Russia, 1886)—39% more solutions, Brasilia (Brazil, 7407)—30% more solutions. Other stations track both LAGEOS and GNSS, but provide more observations to GNSS, e.g., Beijing (China, 7249), Shanghai (China, 7821), Mendeleevo (Russia, 1874), and Wettzell (Germany, 7827). Most of these stations were build to support the GLONASS or BeiDou constellations by SLR tracking, time transfer, and GNSS clock synchronization using laser pulses (Meng et al. 2013); therefore, the GNSS targets have much higher priorities than the geodetic spherical satellites at those sites.
SLR-to-GNSS-only solutions
We evaluate the quality of derived ERPs by comparing with the IERS-14-C04 series which is based on the combination of four space geodetic techniques: GNSS, SLR, very long baseline interferometry (VLBI), and Doppler Orbitography Radiopositioning Integrated by Satellite (DORIS). Four different GNSS-only solutions are generated (Solution 1–4, see Table 2): one solution with the estimation of GNSS orbits based solely on SLR data (Solution 1), and three solutions with fixing the orbits to a priori CODE microwave-based solutions (Solutions 2–4). In Solutions 1 and 2, the range biases are fixed to the mean annual values estimated for every station-satellite pair. In Solution 3 and 4, the range biases are estimated as free parameters for every 7-day solution for every station-satellite pair (Solution 3) and for every system-satellite pair, i.e., one common value for all GLONASS satellites, all Galileo, etc. (Solution 4). We emphasize here that 'range biases' at the ranging stations refer both to 'true' instrumentation-based systematic effects plus effects induced by the large retroreflector arrays on the GNSS satellites.
Table 2 Comparison of estimated ERPs to the IERS-14-C04 series
Table 2 and Fig. 5 show that the largest RMS of pole coordinates of about 500–700 \(\upmu\)as is obtained for the solution in which the GNSS orbits are estimated on the basis of SLR data (Solution 1). The value of 600 \(\upmu\)as corresponds to 20 mm on the Earth's surface which is consistent with the quality of GNSS orbits derived from SLR data (Bury et al. 2018). The solution can be stabilized by fixing the GNSS orbits to microwave-based values (Solutions 2–4); however, when the biases are estimated every week as satellite-station-specific values (Solution 3), the RMSs of pole coordinates are similar to those from Solution 1. The ERP estimates can be stabilized through the re-substitution of mean annual range biases (Solution 2) or by reducing the number of estimated range biases by estimating system-specific values (Solution 4). However, some satellites show different performance, such as misbehaving GLONASS satellites in the plane #2 when compared to other GLONASS (Prange et al. 2017) or BeiDou IGSO and BeiDou MEO. Therefore, estimating one bias for the entire system may not properly account for all satellite-specific errors.
ERPs estimated in Solution 1 (GNSS est), 2 (GNSS fixed), and 5 (LAGEOS) compared to the IERS-14-C04 series
Figure 6 shows the geocenter coordinates derived from two GNSS-only solutions 1 and 2 as well as the LAGEOS-only solution (Solution 5). When fixing the GNSS orbits to the a priori values, the solution is stable, because the origin of the network is provided by the GNSS reference frame IGS14 with the accuracy of observation sensitivity to the reference frame origin and the orbit modeling accuracy. The satellite orbits are integrated around the IGS14 origin, which is fixed; thus, Solution 1 cannot realize a fully independent reference frame origin.
Geocenter coordinates from LAGEOS-1/2 (Solution 5, blue), GNSS-only with estimated parameters (Solution 1, green), and GNSS-only with fixed orbits (Solution 2, red)
Solutions 2–4 realize the reference origin through the determined GNSS orbits and the ground network. Until 2016, the GNSS-only solution is strongly dominated by GLONASS, because all 24 GLONASS satellites were active at that time and provided about 90% of all observations (see Fig. 2). The geocenter coordinates derived by GLONASS-only are known to be affected by substantial orbit errors (Fritsche et al. 2014; Arnold et al. 2015; Lutz et al. 2016). The spectral analysis of the geocenter coordinates shows harmonics corresponding to the draconitic year of GLONASS (and other GNSS satellites, see Fig. 6, right): first harmonic (353 days), second harmonic (177 days), third harmonic (118 days), fourth harmonic (88 days), etc., which indicate serious GLONASS orbit modeling deficiencies when based on SLR data only. The draconitic signals have the amplitudes up to 26 mm and 4 mm for the Z geocenter component in Solution 1 and Solution 2, respectively, and up to 9 mm and 3 mm for the equatorial components. Recently, serious orbit modeling issues for GLONASS satellites especially from the second orbital plane have been discovered (Dach et al. 2017; Prange et al. 2017) which are caused by malfunctioning of transmitter antenna panels or problems with proper maintenance of the yaw satellite orientation.
In the beginning of 2015, only three Galileo IOV satellites were active. In 2015, two Galileo satellites launched into incorrect eccentric orbits were activated (Sośnica et al. 2018b) and six additional FOC satellites were launched into three additional Galileo orbital planes, which resulted in a population of all Galileo planes by FOC satellites by the end of 2015.
The number of active Galileo satellites reached 17 in 2017, and the number of SLR observations to Galileo increased from about 240 observations per week in 2014 to 1150 observations in 2017. In 2014, Galileo constituted only 10% of all SLR-to-GNSS observations, whereas in mid-2017 the percentage of Galileo observations increased to 40%. Figure 6 shows the increasing contribution of Galileo satellites to the stabilization of GNSS-derived geocenter provided by SLR observations. Similar effect of the increasing contribution of Galileo on the quality of estimated pole coordinates and LOD is visible in Fig. 5, specially for the X-pole coordinate. The spectral analysis shows that in the solution with estimating GNSS orbits, the geocenter coordinates are strongly contaminated by orbit modeling issues, especially related to fewer observation opportunities in early years. However, starting from 2016, the geocenter coordinates can be derived from GNSS with the RMS of about 6 mm for the equatorial components and 15 mm for the Z component. In 2014, the stable geocenter and ERP solution with estimation of GNSS orbits (Solution 1) could be obtained only during the first ILRS intensive tracking, which took place in August–September 2015 and substantially increased the number of collected data (cf. Figs. 2, 6).
SLR-to-LAGEOS + GNSS solutions
LAGEOS-only solutions deliver the pole coordinates with the RMS of about 140–150 \(\upmu\)as, corresponding to 5 mm on the Earth's surface and the RMS of 120 \(\upmu\)s for LOD corresponding to 60 mm on the equator. When adding the SLR observations to GNSS (see Table 2), there is no significant improvement for the X- and Y-pole coordinates, because in both solutions the NNR constraint is imposed on the same set of core SLR stations, whose station coordinates improve only slightly by adding GNSS data.
The combined GNSS + LAGEOS Solution 6 with the estimation of GNSS orbits is strongly dominated by the LAGEOS observations because of the much lower number of estimated parameters in the case of LAGEOS and larger weights imposed on LAGEOS observations. For LAGEOS-only, 11 orbit parameters have to be estimated per 7 days, whereas for GNSS, 13 parameters are estimated for each satellite and each 5-day arc, which substantially increases the number of parameters. The a priori sigmas between LAGEOS observations and GNSS are \(\sigma _L{:}\sigma _G=8{:}40\) mm, which corresponds to the ratio of weights \(1/\sigma ^2_L{:}1/\sigma ^2_G=25{:}1\). We tested other sigma ratios between the LAGEOS and GNSS observations; however, increasing the GNSS weights always deteriorated the combination. Similar issues with inferior observation quality when compared to LAGEOS and issues with precise orbit determination are known for Etalon-1/2 which orbit at similar altitudes as GNSS satellites do. In this study, we use 55 satellites at the Etalon heights instead of two passive cannonballs used for operational ILRS products. Similar processing issues remain; however, GNSS satellites are active satellites; thus, the microwave-based orbits can support the solution.
Solution 7 shows a similar quality of pole coordinates to the LAGEOS-only solution (Solution 5) and LAGEOS + GNSS solution (Solution 6). However, the LOD estimates can remarkably be improved when fixing the GNSS orbits to microwave-based values (Solutions 7 and 8) due to the reduction in the correlations between LOD and \(C_{20}\) from LAGEOS-only solutions when using the piece-wise-linear ERP parameterization (Bloßfeld et al. 2014). The RMS of LOD is reduced to 40 \(\upmu\)s which corresponds to about 20 mm on the equator. Including GNSS satellites to the LAGEOS solutions reduces the RMS of LOD from 123 to 69 \(\upmu\)s from Solution 5 to 6, respectively, whereas fixing GNSS orbits to microwave-based values reduces the LOD bias with respect to IERS-14-C04 series from 26 to 1 \(\upmu\)s.
Including the estimation of range biases on the weekly basis substantially increases the number of estimated parameters and thus destabilizes the estimates of pole coordinates to the level of 200 \(\upmu\)as (cf. Solution 7 and 8). Therefore, estimating one bias per satellite and station for a longtime span, e.g., 1 year, and re-substituting the bias as a priori known quantity reduced the impact of systematic effects, such as the detector-specific signature effect. A similar approach with estimating long-term mean biases and re-substituting them for LAGEOS and Etalon satellites will be employed by the ILRS Analysis Standing Committee after the completion of the dedicated Pilot Project 'Determination of Systematic Errors in ILRS Observations'.Footnote 4
ERPs describe the orientation between the ground network realized by stations and the inertial frame realized by artificial satellites. We can thus conclude that SLR observations to GNSS allow for the transfer of the network orientation from GNSS to SLR solutions with the accuracy of about 15 mm.
Figure 7 shows the examples of two SLR stations decomposed into the up, north, and east components for Baikonur (Kazakhstan, 1887) and Wettzell (Germany, 8834) from Solutions 5, 6, and 7. When both LAGEOS and GNSS observations are available, the solution is dominated by LAGEOS data. Starting from the beginning of 2017, Baikonur started providing a very low number of SLR observations to LAGEOS; thus, for most of the weeks, the LAGEOS-only solution was not possible at all. However, when considering SLR observation to GNSS, the regular weekly solutions can be generated with the similar quality of the LAGEOS solution especially in the case of Solution 7 when fixing GNSS orbits. The GNSS solution with fixing orbits is similar to that when using both SLR and microwave observations in one combined solution.
The time series of station coordinates with respect to SLRF2014 for two example SLR stations: Baikonur (Kazakhstan, 1887) and Wettzell (Germany, 8834)
Range biases are estimated for Wettzell in the case of LAGEOS solutions. The estimated biases are strongly correlated with the vertical station component which leads to the noisier 'Up' component for the LAGEOS-only solution shown in Fig. 7. Adding the SLR observations to GNSS allows for a better decorrelation between estimated LAGEOS range biases and the station coordinates and stabilizes the solution characterized by the station coordinate repeatability at the level of 25 mm in LAGEOS-only to the level of 7 mm in the combined LAGEOS + GNSS solution with fixed GNSS orbits. Moreover, the number of weekly solutions for Wettzell is increased from 132 to 143 solutions, which means that about 8% of solutions are based on GNSS-only data.
SLR and VLBI techniques are used for the scale realization in ITRF. In ITRF2014, the scale discrepancy between the two techniques is at the level of 7–8 mm, which constitutes currently a subject of discussions and investigations in both the SLR and VLBI scientific communities (Bachmann et al. 2016; Appleby et al. 2016). In this solution, the mean scale offset and standard deviations equal 6.1 ± 3.1 mm, 5.7 ± 3.2 mm, and 5.2 ± 2.7 mm in LAGEOS-1/2 (Solution 5), LAGEOS + GNSS (Solution 6), and LAGEOS + GNSS with fixed orbits (Solution 7), respectively. The scale offset is thus only slightly reduced when adding GNSS to LAGEOS observations; however, the scatter of the scale is reduced from 3.14 to 2.71 mm (see Fig. 8). An improvement in the scale offset is not expected when adding SLR observations to GNSS, as the scale strongly depends on the handling of range biases. We generated the spectral analysis of all three scale series, which did not show any significant differences (see Fig. 8). No differences mean that the scale is mostly dominated by LAGEOS observations and not affected by GNSS solutions, which typically are contaminated by draconitic periods. The dominating periods are related to LAGEOS orbits: drift of LAGEOS-2 perigee with respect to ecliptical longitude (309 days), draconitic year of LAGEOS-2 (222 days), LAGEOS-2 orbital alias with \(P_1\) tide (138 days). Thus, the intention of a proper combination has been achieved, because the LAGEOS-based scale, which is much more stable, definitely dominates the LAGEOS + GNSS combination.
The scale difference in mm from the Helmert transformation of SLR solutions with respect to the ITRF2014/SLRF2014
The improvement in the ILRS network and the global geodetic parameters is important in the context of SLR contribution to the global geodetic observing system component, which can be achieved by improving models of data reduction, using observations to various constellations or expanding the ground observing network (Otsubo et al. 2016).
This study evaluates the potential contribution of SLR observations to new GNSS systems for the realization of SLR-derived reference frame and deriving global geodetic parameters. Some SLR stations, such as Russian and Chinese stations, provide much more SLR observations to GNSS because their major objective is to support the GLONASS and BeiDou constellations. The ILRS conducted a series of intensive campaigns of SLR tracking to GNSS satellites, starting in 2014. As a result, the number of SLR observation to GNSS exceeded the number of SLR observations to LAGEOS in the beginning of 2017 which allows employing GNSS observations for the ITRF realization. Moreover, the GNSS satellites, as opposed to Etalons, are active spacecraft; thus, their orbits can be fixed to microwave-based values or simultaneously processed on the basis of SLR-GNSS and microwave-GNSS data.
The solutions based only on SLR-to-GNSS data provide ERPs with the RMS of pole coordinates at the level of 400–500 \(\upmu\)as and LOD with the RMS of 90 \(\upmu\)as when fixing GNSS orbits. Re-substitution of annual range biases stabilizes the GNSS-only solutions when compared to the solution with estimating station-satellite-specific biases on the weekly basis. The combination of GNSS with LAGEOS data is especially beneficial for the LOD estimates and reduces the RMS of LOD by a factor of 3, whereas the pole coordinates assume similar quality to those based on LAGEOS data.
Geocenter coordinates based on SLR-to-GNSS data with estimating GNSS orbits when all GLONASS and only 3–4 Galileo satellites were active are of inferior quality. However, along with a progressive populating of Galileo orbital planes, the quality of geocenter components has been improved to the level better than 6 and 15 mm for equatorial and polar geocenter components, respectively. The scale of the reference frame and the geocenter coordinates in the combined LAGEOS + GNSS solutions are dominated by the LAGEOS data.
Using SLR observations to GNSS increases the number of weekly station coordinate solutions possible to obtain when compared to LAGEOS-only solution. The number of estimated station coordinates is 3909, 3476, and 4170 in LAGEOS-only, in GNSS-only, and in LAGEOS + GNSS, respectively when summing up all stations present in 7-day solutions in the period 2014.0–2017.4. This means that on average, 22, 20, and 24 stations contributed to 7-day LAGEOS-only, GNSS-only, and LAGEOS+GNSS solutions, respectively. Some stations provide by far more observations to GNSS than to LAGEOS, e.g., for Altay the number of solutions increased by 22%, for Komsomolsk by 41%, for Arkhyz by 39%, and for Brasilia by 30%. Moreover, when using range observations to GNSS in the solutions, station coordinate repeatability is improved for those stations that do not provide a lot of LAGEOS data or the range bias is estimated to LAGEOS (see Wettzell in Fig. 7) because the GNSS observations help to decorrelate the estimated range biases and the vertical components of station coordinates. We conclude that the future ITRF realizations should consider SLR observations to GNSS satellites in addition to the LAGEOS and Etalon observations.
As a next step, we will evaluate the full potential of ITRF realizations based on combined SLR observations to LAGEOS, LARES, and Etalon satellites, active low orbiting satellites, such as Sentinel-3A and Jason-3, and multi-constellation GNSS tracked by SLR.
https://ilrs.dgfi.tum.de/fileadmin/data_handling/ILRS_Data_Handling_File.snx.
https://ilrs.dgfi.tum.de/fileadmin/data_handling/com_lageos.txt.
https://ilrs.cddis.eosdis.nasa.gov/missions/satellite_missions/current_missions/ga02_com.html.
https://ilrs.cddis.eosdis.nasa.gov/science/awg/awgPilotProjects/awg_systematic_errors.html.
SLR:
satellite laser ranging
GNSS:
global navigation satellite system
Center for Orbit Determination in Europe
ECOM:
Empirical CODE Orbit Model
EPRs:
Earth rotation parameters
LOD:
ITRF:
International Terrestrial Reference Frame
IERS:
International Earth Rotation and Reference System Service
LAGEOS:
Laser Geodynamic Satellite
QZSS:
Quasi-Zenith Satellite System
RMS:
root mean square error
MEO:
medium Earth orbit
IGSO:
inclined geosynchronous orbit
ILRS:
International Laser Ranging Service
IGS:
International GNSS Service
MGEX:
Multi-GNSS Experiment
NEQ:
normal equation system
IOV:
in-orbit validation
FOC:
fully operational capability
PRN:
pseudorandom noise satellite number
VLBI:
very long baseline interferometry
DORIS:
Doppler Orbitography Radiopositioning Integrated by Satellite
Altamimi Z, Rebischung P, Metivier L, Collilieux X (2016) ITRF2014: a new release of the International Terrestrial Reference Frame modeling nonlinear station motions. J Geophys Res Solid Earth 121:6109–6131. https://doi.org/10.1002/2016JB013098
Appleby GM (1998) Long-arc analyses of SLR observations of the Etalon geodetic satellites. J Geod 72(6):333–342. https://doi.org/10.1007/s001900050172
Appleby GM, Rodrguez J, Altamimi Z (2016) Assessment of the accuracy of global geodetic satellite laser ranging observations and estimated impact on ITRF scale: estimation of systematic errors in LAGEOS observations 1993–2014. J Geod 90(12):1371–1388. https://doi.org/10.1007/s00190-016-0929-2
Arnold D, Meindl M, Beutler G, Dach R, Schaer S, Lutz S, Prange L, Sośnica K, Mervart L, Jäggi A (2015) CODE's new solar radiation pressure model for GNSS orbit determination. J Geod 89(8):775–791. https://doi.org/10.1007/s00190-015-0814-4
Arnold D, Montenbruck O, Hackel S, Sośnica K (2018) Satellite laser ranging to low Earth orbiters: orbit and network validation. J Geod. https://doi.org/10.1007/s00190-018-1140-4
Bachmann S, Thaller D, Roggenbuck O, Lösler M, Messerschmitt L (2016) IVS contribution to ITRF2014. J Geod 90(7):631–654. https://doi.org/10.1007/s00190-016-0899-4
Bizouard C, Lambert S, Gattano C, Becker O, Richard J-Y (2018) The IERS EOP 14C04 solution for Earth orientation parameters consistent with ITRF 2014. J Geod. https://doi.org/10.1007/s00190-018-1186-3
Bloßfeld M, Gerstl M, Hugentobler U, Angermann D, Müller H (2014) Systematic effects in LOD from SLR observations. Adv Space Res 54:1049–1063. https://doi.org/10.1016/j.asr.2014.06.009
Bloßfeld M, Müller H, Gerstl M, Štefka V, Bouman J, Göttl F, Horwath M (2015) Second-degree Stokes coefficients from multi-satellite SLR. J Geod 89(9):857–871. https://doi.org/10.1007/s00190-015-0819-z
Bloßfeld M, Rudenko S, Kehm A, Panadina N, Müller H, Angermann D, Hugentobler U, Seitz M (2018) Consistent estimation of geodetic parameters from SLR satellite constellation measurements. J Geod. https://doi.org/10.1007/s00190-018-1166-7
Bruni S, Rebischung P, Zerbini S, Altamimi Z, Errico M, Santi E (2018) Assessment of the possible contribution of space ties on-board GNSS satellites to the terrestrial reference frame. J Geod. https://doi.org/10.1007/s00190-017-1069-z
Bury G, Sośnica K, Zajdel R (2018) Multi-GNSS orbit determination using satellite laser ranging. J Geod. https://doi.org/10.1007/s00190-018-1143-1
Bury G, Sośnica K, Zajdel R (2019) Impact of the atmospheric non-tidal pressure loading on global geodetic parameters based on satellite laser ranging to GNSS. IEEE Trans Geosci Remote Sens. https://doi.org/10.1109/TGRS.2018.2885845
Ciufolini I, Pavlis EC (2004) A confirmation of the general relativistic prediction of the Lense–Thirring effect. Nature 431(7011):958–960. https://doi.org/10.1038/nature03007
Cheng M, Ries J (2017) The unexpected signal in GRACE estimates of C20. J Geod 97(8):897–914. https://doi.org/10.1007/s00190-016-0995-5
Cox C, Chao B (2002) Detection of a large scale mass redistribution in the terrestrial system since 1998. Science 297:831–833
Dach R, Lutz S, Walser P, Fridez P (eds) (2015) Bernese GNSS Software Version 5.2. User manual. Astronomical Institute, University of Bern, Bern Open Publishing. https://doi.org/10.7892/boris.72297
Dach R, Susnik A, Grahsl A, Villiger A, Arnold D, Prange L, Schaer S, Jaggi A (2017) GLONASS satellite orbit modelling. In: Proceedings of the IGS workshop, Paris, pp 3–7. http://www.igs.org/assets/pdf/W2017-PY08-03%20-%20Dach.pdf
Dow J, Neilan R, Rizos C (2009) The international GNSS service in a changing landscape of global navigation satellite systems. J Geod 83(34):191–198. https://doi.org/10.1007/s00190-008-0300-3
Exertier P, Belli A, Samain E, Meng W, Zhang H, Tang K, Sun X (2018) Time and laser ranging: a window of opportunity for geodesy, navigation, and metrology. J Geod. https://doi.org/10.1007/s00190-018-1173-8
Fritsche M, Sośnica K, Rodriguez-Solano C, Steigenberger P, Dietrich R, Dach R, Wang K, Hugentobler U, Rothacher M (2014) Homogeneous reprocessing of GPS, GLONASS and SLR observations. J Geod 88(7):625–642. https://doi.org/10.1007/s00190-014-0710-3
Glaser S, Fritsche M, Sośnica K, Rodriguez-Solano C, Wang K, Dach R, Hugentobler U, Rothacher M, Dietrich R (2015) A consistent combination of GNSS and SLR with minimum constraints. J Geod 89(12):1165–1180. https://doi.org/10.1007/s00190-015-0842-0
Kucharski D, Kirchner G, Bennett JC, Lachut M, Sośnica K, Koshkin N, Suchodolski T (2017) Photon pressure force on space debris TOPEX/Poseidon measured by satellite laser ranging. Earth Space Sci 4:661668. https://doi.org/10.1002/2017EA000329
Lutz S, Steigenberger P, Meindl M, Beutler G, Sośnica K, Schaer S, Dach R, Arnold D, Thaller D, Jäggi A (2016) Impact of the arclength on GNSS analysis results. J Geod 90(4):365–378. https://doi.org/10.1007/s00190-015-0878-1
Lyard F, Lefevre F, Letellier T, Francis O (2006) Modelling the global ocean tides: modern insights from FES2004. Ocean Dyn 56:394415. https://doi.org/10.1007/s10236-006-0086-x
Mendes VB, Pavlis EC (2004) High-accuracy zenith delay prediction at optical wavelengths. Geophys Res Lett 31:L14602. https://doi.org/10.1029/2004GL020308
Meng W, Zhang H, Zhang Z, Prochazka I (2013) The application of single photon detector technique in laser time transfer for Chinese navigation satellites. In: Photon counting applications IV; and quantum optics and quantum information transfer and processing, vol 8773, p 87730E
Montenbruck O, Steigenberger P, Prange L, Deng C, Zhao Q, Perosanz F, Romero I, Noll C, Sturze A, Weber G, Schmid R, MacLeod K, Schaer S (2017) The multi-GNSS experiment (MGEX) of the International GNSS Service (IGS) achievements, prospects and challenges. Adv Space Res 59(7):1671–1697. https://doi.org/10.1016/j.asr.2017.01.011
Otsubo T, Appleby G (2003) System dependent center of mass correction for spherical geodetic satellites. J Geophys Res 108(2201):B4. https://doi.org/10.1029/2002JB002209
Otsubo T, Appleby G, Gibbs P (2001) GLONASS laser ranging accuracy with satellite signature effect. SGEO 22(56):509–516. https://doi.org/10.1023/A:1015676419548
Otsubo T, Matsuo K, Aoyama Y, Yamamoto K, Hobiger T, Kubo-oka T, Sekido M (2016) Effective expansion of satellite laser ranging network to improve global geodetic parameters. Earth Planets Space 68(1):65. https://doi.org/10.1186/s40623-016-0447-8
Pardini C, Anselmo L, Lucchesi DM, Peron R (2017) On the secular decay of the LARES semi-major axis. Acta Astronaut 140:469–477
Pavlis EC (1994) High resolution Earth orientation parameters from LAGEOS SLR data analysis at GSFC, in IERS Technical Note 16
Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2013) Erratum: Correction to the development and evaluation of the earth gravitational model 2008 (EGM2008). J Geophys Res Solid Earth 118:2633. https://doi.org/10.1002/jgrb.50167
Pearlman M, Degnan J, Bosworth J (2002) The international laser ranging service. Adv Space Res 30(2):135–143. https://doi.org/10.1016/S0273-1177(02)00277-6
Pearlman M, Arnold D, Davis M, Barlier F, Biancale R, Vasiliev V, Ciufolini I, Paolozzi A, Pavlis E, Sośnica K, Bloßfeld M (2019) Laser geodetic satellites: a high accuracy scientific tool. J Geod. https://doi.org/10.1007/s00190-019-01228-y
Petit G, Luzum B (2010) IERS Conventions (2010). IERS Technical Note 36. Verlag des Bundesamts fuer Kartographie und Geodaesie, Frankfurt am Main, ISBN 3-89888-989-6
Prange L, Orliac E, Dach R, Arnold D, Beutler G, Schaer S, Jäggi A (2017) CODE's five-system orbit and clock solution the challenges of multi-GNSS data analysis. J Geod 91(4):345–360. https://doi.org/10.1007/s00190-016-0968-8
Sośnica K, Jäggi A, Thaller D, Dach R, Beutler G (2014) Contribution of Starlette Stella and AJISAI to the SLR-derived global reference frame. J Geod 88(8):789–804. https://doi.org/10.1007/s00190-014-0722-z
Sośnica K, Jäggi A, Meyer U, Thaller D, Beutler G, Arnold D, Dach R (2015a) Time variable Earth's gravity field from SLR satellites. J Geod 89(10):945–960. https://doi.org/10.1007/s00190-015-0825-1
Sośnica K, Thaller D, Dach R, Steigenberger P, Beutler G, Arnold D, Jäggi A (2015b) Satellite laser ranging to GPS and GLONASS. J Geod 89(7):725–743. https://doi.org/10.1007/s00190-015-0810-8
Sośnica K, Bury G, Zajdel R (2018a) Contribution of multi-GNSS constellation to SLR-derived terrestrial reference frame. Geophys Res Lett 45:2339–2348. https://doi.org/10.1002/2017GL076850J
Sośnica K, Prange L, Kaźmierski K, Bury G, Drożdżewski M, Zajdel R, Hadaś T (2018b) Validation of Galileo orbits using SLR with a focus on satellites launched into incorrect orbital plane. J Geod 92(2):131–148. https://doi.org/10.1007/s00190-017-1050-x
Strugarek D, Sośnica K, Jäggi A (2019) Characteristics of GOCE orbits based on satellite laser ranging. Adv Space Res 63(1):417–431. https://doi.org/10.1016/j.asr.2018.08.033
Thaller D, Sośnica K, Dach R, Beutler G, Mareyen M, Richter B (2014) Geocenter coordinates from GNSS and combined GNSS–SLR solutions using satellite co-locations. In: International association of geodesy symposia, vol 139. Springer, Berlin, pp 129-134. https://doi.org/10.1007/978-3-642-37222-3_16
Thaller D, Sośnica K, Steigenberger P, Roggenbuck O, Dach R (2015) Pre-combined GNSS–SLR solutions: what could be the benefit for the ITRF? In: International association of geodesy symposia, vol 146. Springer, Berlin, pp 85–94. https://doi.org/10.1007/1345_2015_215
Zajdel R, Sośnica K, Bury G (2017) A new online service for the validation of multi-GNSS orbits using SLR. Remote Sens 9:1049. https://doi.org/10.3390/rs9101049
KS, GB, and RZ performed GNSS computations. MD provided LAGEOS solutions. All authors have contributed to the interpretation of the results and the preparation of the manuscript. KS coordinated all activities. All authors read and approved the final manuscript.
The ILRS and MGEX-IGS (Dow et al. 2009; Montenbruck et al. 2017) are acknowledged for providing SLR and GNSS data. We would like to thank CODE for providing multi-GNSS orbits. The SLR and GNSS stations are acknowledged for a continuous tracking of the geodetic satellites as well as for providing high-quality SLR and GNSS observations.
SLR observations are provided by the https://ilrs.cddis.eosdis.nasa.gov/data_and_products/data/index.html CODE MGEX orbits are available from: http://ftp.aiub.unibe.ch/CODE_MGEX/CODE/ IERS-14-C04 series are available from: https://www.iers.org/IERS/EN/DataProducts/EarthOrientationData/eop.html SLRF2014 is available at: ftp://cddis.nasa.gov/slr/products/resource.
Authors are supported by the Polish National Science Centre (NCN), Grants No. UMO-2015/17/B/ST10/03108, UMO-2018/29/B/ST10/00382 and by the Project EPOS-PL European Plate Observing System Grant No. POIR.04.02.00-14-A003/16-00.
Institute of Geodesy and Geoinformatics, Wrocław University of Environmental and Life Sciences, Grunwaldzka 53, Wrocław, Poland
K. Sośnica
, G. Bury
, R. Zajdel
, D. Strugarek
, M. Drożdżewski
& K. Kazmierski
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Correspondence to K. Sośnica.
Multi-GNSS
Space geodesy
Geocenter
Earth orientation parameters
6. Geodesy | CommonCrawl |
Substitution tiling
In geometry, a tile substitution is a method for constructing highly ordered tilings. Most importantly, some tile substitutions generate aperiodic tilings, which are tilings whose prototiles do not admit any tiling with translational symmetry. The most famous of these are the Penrose tilings. Substitution tilings are special cases of finite subdivision rules, which do not require the tiles to be geometrically rigid.
Introduction
A tile substitution is described by a set of prototiles (tile shapes) $T_{1},T_{2},\dots ,T_{m}$, an expanding map $Q$ and a dissection rule showing how to dissect the expanded prototiles $QT_{i}$ to form copies of some prototiles $T_{j}$. Intuitively, higher and higher iterations of tile substitution produce a tiling of the plane called a substitution tiling. Some substitution tilings are periodic, defined as having translational symmetry. Every substitution tiling (up to mild conditions) can be "enforced by matching rules"—that is, there exist a set of marked tiles that can only form exactly the substitution tilings generated by the system. The tilings by these marked tiles are necessarily aperiodic.[1][2]
A simple example that produces a periodic tiling has only one prototile, namely a square:
By iterating this tile substitution, larger and larger regions of the plane are covered with a square grid. A more sophisticated example with two prototiles is shown below, with the two steps of blowing up and dissecting merged into one step.
One may intuitively get an idea how this procedure yields a substitution tiling of the entire plane. A mathematically rigorous definition is given below. Substitution tilings are notably useful as ways of defining aperiodic tilings, which are objects of interest in many fields of mathematics, including automata theory, combinatorics, discrete geometry, dynamical systems, group theory, harmonic analysis and number theory, as well as crystallography and chemistry. In particular, the celebrated Penrose tiling is an example of an aperiodic substitution tiling.
History
In 1973 and 1974, Roger Penrose discovered a family of aperiodic tilings, now called Penrose tilings. The first description was given in terms of 'matching rules' treating the prototiles as jigsaw puzzle pieces. The proof that copies of these prototiles can be put together to form a tiling of the plane, but cannot do so periodically, uses a construction that can be cast as a substitution tiling of the prototiles. In 1977 Robert Ammann discovered a number of sets of aperiodic prototiles, i.e., prototiles with matching rules forcing nonperiodic tilings; in particular, he rediscovered Penrose's first example. This work gave an impact to scientists working in crystallography, eventually leading to the discovery of quasicrystals. In turn, the interest in quasicrystals led to the discovery of several well-ordered aperiodic tilings. Many of them can be easily described as substitution tilings.
Mathematical definition
We will consider regions in ${\mathbb {R} }^{d}$ that are well-behaved, in the sense that a region is a nonempty compact subset that is the closure of its interior.
We take a set of regions $\mathbf {P} =\{T_{1},T_{2},\dots ,T_{m}\}$ as prototiles. A placement of a prototile $T_{i}$ is a pair $(T_{i},\varphi )$ where $\varphi $is an isometry of ${\mathbb {R} }^{d}$. The image $\varphi (T_{i})$ is called the placement's region. A tiling T is a set of prototile placements whose regions have pairwise disjoint interiors. We say that the tiling T is a tiling of W where W is the union of the regions of the placements in T.
A tile substitution is often loosely defined in the literature. A precise definition is as follows.[3]
A tile substitution with respect to the prototiles P is a pair $(Q,\sigma )$, where $Q:{\mathbb {R} }^{d}\to {\mathbb {R} }^{d}$ is a linear map, all of whose eigenvalues are larger than one in modulus, together with a substitution rule $\sigma $ that maps each $T_{i}$ to a tiling of $QT_{i}$. The substitution rule $\sigma $ induces a map from any tiling T of a region W to a tiling $\sigma (\mathbf {T} )$ of $Q_{\sigma }(\mathbf {W} )$, defined by
$\sigma (\mathbf {T} )=\bigcup _{(T_{i},\varphi )\in \mathbf {T} }\{(T_{j},Q\circ \varphi \circ Q^{-1}\circ \rho ):(T_{j},\rho )\in \sigma (T_{i})\}.$
Note, that the prototiles can be deduced from the tile substitution. Therefore it is not necessary to include them in the tile substitution $(Q,\sigma )$.[4]
Every tiling of ${\mathbb {R} }^{d}$, where any finite part of it is congruent to a subset of some $\sigma ^{k}(T_{i})$ is called a substitution tiling (for the tile substitution $(Q,\sigma )$).
See also
• Pinwheel tiling
• Photographic mosaic
References
1. C. Goodman-Strauss, Matching Rules and Substitution Tilings, Annals Math., 147 (1998), 181-223.
2. Th. Fernique and N. Ollinger, Combinatorial substitutions and sofic tilings, Journees Automates Cellulaires 2010, J. Kari ed., TUCS Lecture Notes 13 (2010), 100-110.
3. D. Frettlöh, Duality of Model Sets Generated by Substitutions, Romanian Journal of Pure and Applied Math. 50, 2005
4. A. Vince, Digit Tiling of Euclidean Space, in: Directions in Mathematical Quasicrystals, eds: M. Baake, R.V. Moody, AMS, 2000
Further reading
• Pytheas Fogg, N. (2002). Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. (eds.). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics. Vol. 1794. Berlin: Springer-Verlag. ISBN 3-540-44141-7. Zbl 1014.11015.
External links
• Dirk Frettlöh's and Edmund Harriss's Encyclopedia of Substitution Tilings
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
Aperiodic
• Ammann–Beenker
• Aperiodic set of prototiles
• List
• Einstein problem
• Socolar–Taylor
• Gilbert
• Penrose
• Pentagonal
• Pinwheel
• Quaquaversal
• Rep-tile and Self-tiling
• Sphinx
• Socolar
• Truchet
Other
• Anisohedral and Isohedral
• Architectonic and catoptric
• Circle Limit III
• Computer graphics
• Honeycomb
• Isotoxal
• List
• Packing
• Problems
• Domino
• Wang
• Heesch's
• Squaring
• Dividing a square into similar rectangles
• Prototile
• Conway criterion
• Girih
• Regular Division of the Plane
• Regular grid
• Substitution
• Voronoi
• Voderberg
By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
| Wikipedia |
Nataliya Kalashnykova
Nataliya Ivanovna Kalashnykova is a Soviet and Mexican mathematician specializing in mathematical optimization, and especially bilevel optimization, with applications in modeling human migration and in the pricing of natural gas and toll roads. She is a professor at the Autonomous University of Nuevo León, in the Facultad de Ciencias Físico Matemáticas.[1]
Education and career
Kalashnykova earned a master's degree in mathematical sciences from Novosibirsk State University in 1978. She completed a doctorate there in 1989, through the Siberian Division of the Academy of Sciences of the USSR.[1] Her dissertation, Control of Accuracy in Bi-Level Iteration Processes, was supervised by Vladimir Aleksandrovich Bulavsky.[2] She also earned a second master's degree in economics from Sumy State University in Ukraine in 1999.[1]
She became a faculty member at the Altai State Technical University, at the Siberian State University of Telecommunications and Informatics in Novosibirsk, and at Sumy State University, and a postdoctoral researcher at the Central Economic Mathematical Institute. She moved to her present position in Mexico at the Autonomous University of Nuevo León in 2001.[1]
Recognition
Kalashnykova is a member of the Mexican Academy of Sciences.[3]
Personal life
Kalashnykova is married to Vyacheslav Kalashnikov Polishchuk, another former Soviet mathematician in Mexico.[4]
Books
Kalashnykova is a coauthor of books including:
• Bilevel Programming Problems: Theory, Algorithms and Applications to Energy Networks (with Stephan Dempe, Vyacheslav Kalashnikov, and Gerardo A. Pérez-Valdés, Springer, 2015)
• Public Interest and Private Enterprize [sic]: New Developments: Theoretical Results and Numerical Algorithms (with José Guadalupe Flores Muñiz, Viacheslav V. Kalashnikov, and Vladik Kreinovich, Lecture Notes in Networks and Systems 138, Springer, 2021)
References
1. "Nataliya Kalashnykova", Investigadores, Autonomous University of Nuevo León, retrieved 2022-11-25
2. Nataliya Kalashnykova at the Mathematics Genealogy Project
3. Mathematics section members (PDF), Mexican Academy of Sciences, 2021, retrieved 2022-11-25
4. Kalashnikov Polishchuk, Vyacheslav, Instituto de Innovación y Tranferencia de Tecnología, archived from the original on 2013-10-30
External links
• Nataliya Kalashnykova publications indexed by Google Scholar
Authority control: Academics
• MathSciNet
• Mathematics Genealogy Project
| Wikipedia |
# 1. Getting Started with Haskell
Let's dive in!
# 1.1 Installing Haskell
Before we can start coding in Haskell, we need to install the Haskell compiler and build tools. The most popular Haskell compiler is GHC (Glasgow Haskell Compiler), which is available for Windows, macOS, and Linux.
To install Haskell, follow these steps:
1. Go to the Haskell Platform website at https://www.haskell.org/platform/.
2. Download the appropriate installer for your operating system.
3. Run the installer and follow the instructions to complete the installation.
Once the installation is complete, you should have GHC and other necessary tools installed on your system.
If you're using macOS, you can also install Haskell using Homebrew. Open a terminal and run the following command:
```
brew install ghc
```
This will install GHC and other necessary tools on your system.
# 1.2 Interactive Shell vs. Script Mode
Haskell provides two main ways to write and run code: the interactive shell and script mode.
The interactive shell, called GHCi (GHC interactive), allows you to enter Haskell expressions and see their results immediately. It's a great way to experiment with Haskell and test small code snippets.
To start GHCi, open a terminal and type `ghci`. You should see a prompt that looks like `Prelude>`. Now you can enter Haskell expressions and see the results.
In script mode, you write your Haskell code in a file with a `.hs` extension and then run the file using the GHC compiler. This is the preferred way to write larger programs or modules.
To run a Haskell script, open a terminal, navigate to the directory where the script is located, and run the following command:
```
ghc script.hs
```
This will compile the script and generate an executable file. You can then run the executable by typing its name in the terminal.
Let's say you have a script called `hello.hs` that prints "Hello, World!". You can run it using the following commands:
```
ghc hello.hs
./hello
```
This will compile the `hello.hs` script and generate an executable called `hello`. Running `./hello` will print "Hello, World!" to the terminal.
# 1.3 Setting Up an IDE (e.g., Visual Studio Code, Atom)
While you can write Haskell code using any text editor, using an Integrated Development Environment (IDE) can greatly enhance your productivity. IDEs provide features like code completion, syntax highlighting, and debugging tools.
Two popular IDEs for Haskell development are Visual Studio Code and Atom. Here's how you can set up Haskell development in each IDE:
**Visual Studio Code**
1. Install Visual Studio Code from the official website: https://code.visualstudio.com/.
2. Open Visual Studio Code and go to the Extensions view by clicking on the square icon on the left sidebar or pressing `Ctrl+Shift+X`.
3. Search for the "Haskell" extension and click on the "Install" button.
4. Once the extension is installed, you can start writing Haskell code in Visual Studio Code.
**Atom**
1. Install Atom from the official website: https://atom.io/.
2. Open Atom and go to the Install view by clicking on the puzzle icon on the left sidebar or pressing `Ctrl+Shift+P` and typing "Install Packages and Themes".
3. Search for the "ide-haskell" package and click on the "Install" button.
4. Once the package is installed, you can start writing Haskell code in Atom.
Both Visual Studio Code and Atom provide a rich set of features for Haskell development, including code formatting, linting, and integration with GHCi.
If you're using Visual Studio Code, you can open a Haskell file (with a `.hs` extension) and start writing code. The Haskell extension will provide features like syntax highlighting and code completion as you type.
If you're using Atom, you can open a Haskell file and start writing code. The ide-haskell package will provide similar features to the Haskell extension in Visual Studio Code.
## Exercise
Install Haskell on your computer following the steps outlined in section 1.1.
### Solution
Follow the steps outlined in section 1.1 to install Haskell on your computer.
# 2. Basic Haskell Syntax
**Functions and Function Composition**
In Haskell, functions are defined using the `=` symbol. Here's an example of a simple function that adds two numbers:
```haskell
add :: Int -> Int -> Int
add x y = x + y
```
In this example, `add` is the name of the function, and `Int -> Int -> Int` is the type signature. The type signature specifies that the function takes two `Int` arguments and returns an `Int` result.
You can call the `add` function by providing two `Int` arguments:
```haskell
result = add 3 4
```
The result will be `7`.
Haskell also supports function composition, which allows you to combine multiple functions into a single function. Here's an example:
```haskell
square :: Int -> Int
square x = x * x
double :: Int -> Int
double x = x * 2
squareAndDouble :: Int -> Int
squareAndDouble = double . square
```
In this example, the `squareAndDouble` function is defined as the composition of the `double` and `square` functions. When you call `squareAndDouble` with an `Int` argument, it will first square the argument and then double the result.
Here's an example of calling the `squareAndDouble` function:
```haskell
result = squareAndDouble 3
```
The result will be `18`, because `squareAndDouble` first squares `3` to get `9`, and then doubles `9` to get `18`.
## Exercise
Define a function called `isEven` that takes an `Int` argument and returns `True` if the argument is even, and `False` otherwise.
### Solution
```haskell
isEven :: Int -> Bool
isEven x = x `mod` 2 == 0
```
# 2.1 Pattern Matching and Guards
Pattern matching is a powerful feature in Haskell that allows you to match values against specific patterns and perform different computations based on the pattern. Here's an example:
```haskell
factorial :: Int -> Int
factorial 0 = 1
factorial n = n * factorial (n - 1)
```
In this example, the `factorial` function is defined using pattern matching. The first pattern `0` matches the base case, where the factorial of `0` is `1`. The second pattern `n` matches any non-zero value, and the computation `n * factorial (n - 1)` is performed.
You can call the `factorial` function with an `Int` argument:
```haskell
result = factorial 5
```
The result will be `120`, because `factorial 5` is computed as `5 * factorial 4`, which is `5 * 4 * factorial 3`, and so on, until the base case is reached.
Haskell also supports guards, which allow you to specify conditions that must be true for a pattern to match. Here's an example:
```haskell
abs :: Int -> Int
abs x
| x >= 0 = x
| otherwise = -x
```
In this example, the `abs` function uses guards to check if the argument `x` is greater than or equal to `0`. If the condition is true, the computation `x` is performed. If the condition is false, the computation `-x` is performed.
Here's an example of calling the `abs` function:
```haskell
result1 = abs 5
result2 = abs (-5)
```
The result will be `5` for `result1`, because `5` is greater than or equal to `0`, and `-5` for `result2`, because `-5` is less than `0`.
## Exercise
Define a function called `isVowel` that takes a `Char` argument and returns `True` if the argument is a vowel (a, e, i, o, or u), and `False` otherwise.
### Solution
```haskell
isVowel :: Char -> Bool
isVowel c
| c == 'a' || c == 'e' || c == 'i' || c == 'o' || c == 'u' = True
| otherwise = False
```
# 2.3 Recursion
Recursion is a fundamental concept in Haskell that allows you to define functions in terms of themselves. This can be useful for solving problems that can be naturally expressed in terms of smaller instances of the same problem.
Here's an example of a recursive function that calculates the sum of all numbers from `1` to `n`:
```haskell
sum :: Int -> Int
sum 0 = 0
sum n = n + sum (n - 1)
```
In this example, the `sum` function is defined using recursion. The first pattern `0` matches the base case, where the sum of all numbers from `1` to `0` is `0`. The second pattern `n` matches any non-zero value, and the computation `n + sum (n - 1)` is performed.
You can call the `sum` function with an `Int` argument:
```haskell
result = sum 5
```
The result will be `15`, because `sum 5` is computed as `5 + sum 4`, which is `5 + 4 + sum 3`, and so on, until the base case is reached.
Here's an example of calling the `sum` function:
```haskell
result = sum 10
```
The result will be `55`, because `sum 10` is computed as `10 + sum 9`, which is `10 + 9 + sum 8`, and so on, until the base case is reached.
## Exercise
Define a recursive function called `fibonacci` that takes an `Int` argument `n` and returns the `n`th Fibonacci number. The Fibonacci sequence is defined as follows: the first two numbers are `0` and `1`, and each subsequent number is the sum of the previous two numbers.
### Solution
```haskell
fibonacci :: Int -> Int
fibonacci 0 = 0
fibonacci 1 = 1
fibonacci n = fibonacci (n - 1) + fibonacci (n - 2)
```
# 2.3. Recursion
Recursion is a fundamental concept in Haskell that allows you to define functions in terms of themselves. This can be useful for solving problems that can be naturally expressed in terms of smaller instances of the same problem.
Here's an example of a recursive function that calculates the sum of all numbers from `1` to `n`:
```haskell
sum :: Int -> Int
sum 0 = 0
sum n = n + sum (n - 1)
```
In this example, the `sum` function is defined using recursion. The first pattern `0` matches the base case, where the sum of all numbers from `1` to `0` is `0`. The second pattern `n` matches any non-zero value, and the computation `n + sum (n - 1)` is performed.
You can call the `sum` function with an `Int` argument:
```haskell
result = sum 5
```
The result will be `15`, because `sum 5` is computed as `5 + sum 4`, which is `5 + 4 + sum 3`, and so on, until the base case is reached.
Here's an example of calling the `sum` function:
```haskell
result = sum 10
```
The result will be `55`, because `sum 10` is computed as `10 + sum 9`, which is `10 + 9 + sum 8`, and so on, until the base case is reached.
## Exercise
Define a recursive function called `fibonacci` that takes an `Int` argument `n` and returns the `n`th Fibonacci number. The Fibonacci sequence is defined as follows: the first two numbers are `0` and `1`, and each subsequent number is the sum of the previous two numbers.
### Solution
```haskell
fibonacci :: Int -> Int
fibonacci 0 = 0
fibonacci 1 = 1
fibonacci n = fibonacci (n - 1) + fibonacci (n - 2)
```
# 2.4. Type Signatures and Inference
In Haskell, every expression and function has a type. The type of an expression determines what kind of values it can have and what operations can be performed on it. Haskell has a strong static type system, which means that types are checked at compile time to ensure type safety.
Type signatures are used to explicitly declare the type of a function. They are written using the `::` symbol, followed by the type of the function. For example:
```haskell
add :: Int -> Int -> Int
add x y = x + y
```
In this example, the `add` function takes two `Int` arguments and returns an `Int`. The type signature `Int -> Int -> Int` indicates that `add` takes two `Int` arguments and returns an `Int`.
Haskell also has type inference, which allows the compiler to automatically deduce the types of expressions and functions based on their usage. This means that you don't always have to explicitly declare the types of your functions, as the compiler can often infer them for you.
For example, if you define a function without a type signature:
```haskell
add x y = x + y
```
The compiler will infer the type of `add` to be `Num a => a -> a -> a`. This means that `add` can take arguments of any type that is an instance of the `Num` typeclass, and it will return a value of the same type.
Here's an example of calling the `add` function:
```haskell
result = add 3 5
```
The result will be `8`, because `add 3 5` is computed as `3 + 5`.
## Exercise
Define a function called `multiply` that takes two `Int` arguments and returns their product. Don't include a type signature for the function.
### Solution
```haskell
multiply x y = x * y
```
# 3. Basic Data Types
In Haskell, there are several basic data types that you will commonly encounter. These include numbers, strings, booleans, lists, and tuples.
3.1. Numbers (Integers and Floats)
Numbers in Haskell can be either integers or floating-point numbers. Integers are whole numbers without a decimal point, while floats have a decimal point.
Here are some examples of integer literals in Haskell:
```haskell
x = 42
y = -10
```
And here are some examples of floating-point literals:
```haskell
pi = 3.14159
e = 2.71828
```
You can perform arithmetic operations on numbers in Haskell using the usual operators, such as `+`, `-`, `*`, and `/`.
Here's an example of performing arithmetic operations in Haskell:
```haskell
x = 5 + 3 -- x will be 8
y = 10 - 2 -- y will be 8
z = 3 * 4 -- z will be 12
w = 10 / 2 -- w will be 5.0
```
## Exercise
Write a function called `square` that takes an `Int` argument and returns its square. Don't include a type signature for the function.
### Solution
```haskell
square x = x * x
```
# 3.2. Strings
Strings in Haskell are sequences of characters enclosed in double quotes. For example:
```haskell
greeting = "Hello, world!"
```
You can concatenate strings using the `++` operator:
```haskell
name = "Alice"
greeting = "Hello, " ++ name ++ "!"
```
You can also access individual characters in a string using the `!!` operator:
```haskell
letter = greeting !! 0 -- letter will be 'H'
```
Here's an example of manipulating strings in Haskell:
```haskell
name = "Alice"
greeting = "Hello, " ++ name ++ "!"
letter = greeting !! 0 -- letter will be 'H'
```
## Exercise
Write a function called `reverseString` that takes a string as an argument and returns the reverse of the string. Don't include a type signature for the function.
### Solution
```haskell
reverseString str = reverse str
```
# 3.3. Booleans
Booleans in Haskell represent truth values and can have two possible values: `True` and `False`. You can perform logical operations on booleans using the usual operators, such as `&&` (logical AND), `||` (logical OR), and `not` (logical NOT).
Here are some examples of boolean operations in Haskell:
```haskell
x = True && False -- x will be False
y = True || False -- y will be True
z = not True -- z will be False
```
Here's an example of using boolean operations in Haskell:
```haskell
x = True && False -- x will be False
y = True || False -- y will be True
z = not True -- z will be False
```
## Exercise
Write a function called `isPositive` that takes an `Int` argument and returns `True` if the number is positive, and `False` otherwise. Don't include a type signature for the function.
### Solution
```haskell
isPositive x = x > 0
```
# 3.4. Lists and Tuples
Lists and tuples are used to store collections of values in Haskell.
Lists are denoted by square brackets and can contain elements of any type. For example:
```haskell
numbers = [1, 2, 3, 4, 5]
names = ["Alice", "Bob", "Charlie"]
```
You can access individual elements of a list using the `!!` operator:
```haskell
firstNumber = numbers !! 0 -- firstNumber will be 1
```
You can also concatenate lists using the `++` operator:
```haskell
moreNumbers = [6, 7, 8]
allNumbers = numbers ++ moreNumbers -- allNumbers will be [1, 2, 3, 4, 5, 6, 7, 8]
```
Tuples are denoted by parentheses and can contain elements of different types. For example:
```haskell
person = ("Alice", 25)
```
You can access individual elements of a tuple using pattern matching:
```haskell
name = fst person -- name will be "Alice"
age = snd person -- age will be 25
```
Here's an example of working with lists and tuples in Haskell:
```haskell
numbers = [1, 2, 3, 4, 5]
names = ["Alice", "Bob", "Charlie"]
firstNumber = numbers !! 0 -- firstNumber will be 1
moreNumbers = [6, 7, 8]
allNumbers = numbers ++ moreNumbers -- allNumbers will be [1, 2, 3, 4, 5, 6, 7, 8]
person = ("Alice", 25)
name = fst person -- name will be "Alice"
age = snd person -- age will be 25
```
## Exercise
Write a function called `sumList` that takes a list of `Int` values as an argument and returns their sum. Don't include a type signature for the function.
### Solution
```haskell
sumList xs = sum xs
```
# 3.4. Lists and Tuples
Here's an example of working with lists and tuples in Haskell:
```haskell
numbers = [1, 2, 3, 4, 5]
names = ["Alice", "Bob", "Charlie"]
firstNumber = numbers !! 0 -- firstNumber will be 1
moreNumbers = [6, 7, 8]
allNumbers = numbers ++ moreNumbers -- allNumbers will be [1, 2, 3, 4, 5, 6, 7, 8]
person = ("Alice", 25)
name = fst person -- name will be "Alice"
age = snd person -- age will be 25
```
## Exercise
Write a function called `sumList` that takes a list of `Int` values as an argument and returns their sum. Don't include a type signature for the function.
### Solution
```haskell
sumList xs = sum xs
```
# 4. Operators
Operators are symbols that perform specific operations on values. In Haskell, operators can be used to perform arithmetic, comparison, logical, and assignment operations.
# 4.1. Arithmetic Operators
Arithmetic operators are used to perform mathematical operations such as addition, subtraction, multiplication, and division. Here are the basic arithmetic operators in Haskell:
- `+` (addition)
- `-` (subtraction)
- `*` (multiplication)
- `/` (division)
Here's an example of using arithmetic operators in Haskell:
```haskell
x = 10 + 5 -- x will be 15
y = 10 - 5 -- y will be 5
z = 10 * 5 -- z will be 50
w = 10 / 5 -- w will be 2.0
```
## Exercise
Write a function called `square` that takes an `Int` value as an argument and returns its square. Don't include a type signature for the function.
### Solution
```haskell
square x = x * x
```
# 4.2. Comparison Operators
Comparison operators are used to compare values and determine their relationship. Here are the basic comparison operators in Haskell:
- `==` (equality)
- `/=` (inequality)
- `<` (less than)
- `>` (greater than)
- `<=` (less than or equal to)
- `>=` (greater than or equal to)
Here's an example of using comparison operators in Haskell:
```haskell
x = 10 == 5 -- x will be False
y = 10 /= 5 -- y will be True
z = 10 < 5 -- z will be False
w = 10 > 5 -- w will be True
```
## Exercise
Write a function called `isEven` that takes an `Int` value as an argument and returns `True` if the value is even, and `False` otherwise. Don't include a type signature for the function.
### Solution
```haskell
isEven x = x `mod` 2 == 0
```
# 4.3. Logical Operators
Logical operators are used to combine and manipulate boolean values. Here are the basic logical operators in Haskell:
- `&&` (logical AND)
- `||` (logical OR)
- `not` (logical NOT)
Here's an example of using logical operators in Haskell:
```haskell
x = True && False -- x will be False
y = True || False -- y will be True
z = not True -- z will be False
```
## Exercise
Write a function called `isPositive` that takes an `Int` value as an argument and returns `True` if the value is positive, and `False` otherwise. Don't include a type signature for the function.
### Solution
```haskell
isPositive x = x > 0
```
# 4.4. Assignment Operators
Assignment operators are used to assign values to variables. In Haskell, assignment is done using the `=` operator.
Here's an example of using assignment operators in Haskell:
```haskell
x = 10
y = x + 5 -- y will be 15
```
## Exercise
Write a function called `increment` that takes an `Int` value as an argument and returns its value incremented by 1. Don't include a type signature for the function.
### Solution
```haskell
increment x = x + 1
```
# 5. Control Structures
Control structures are used to control the flow of execution in a program. They allow you to make decisions and repeat certain blocks of code based on certain conditions. In Haskell, there are several control structures available, including conditional statements, loops, and higher-order functions.
# 5.1. Conditional Statements (if, case, guards)
Conditional statements are used to execute different blocks of code based on certain conditions. In Haskell, there are several ways to write conditional statements, including the `if` statement, the `case` statement, and guards.
The `if` statement is used to execute a block of code if a certain condition is true. It has the following syntax:
```haskell
if condition
then expression1
else expression2
```
If the condition is true, `expression1` is executed. Otherwise, `expression2` is executed.
The `case` statement is used to match a value against several patterns and execute the corresponding block of code. It has the following syntax:
```haskell
case expression of
pattern1 -> expression1
pattern2 -> expression2
...
```
The patterns are matched in order, and the first matching pattern is executed.
Guards are used to specify conditions for different cases. They have the following syntax:
```haskell
function x
| condition1 = expression1
| condition2 = expression2
...
```
The conditions are evaluated in order, and the first condition that is true is executed.
Here's an example of using conditional statements in Haskell:
```haskell
-- Using if statement
maxValue :: Int -> Int -> Int
maxValue x y = if x > y
then x
else y
-- Using case statement
sign :: Int -> String
sign x = case x of
0 -> "Zero"
n | n > 0 -> "Positive"
_ -> "Negative"
-- Using guards
absoluteValue :: Int -> Int
absoluteValue x
| x >= 0 = x
| otherwise = -x
```
In the `maxValue` function, the `if` statement is used to determine the maximum value between two integers.
In the `sign` function, the `case` statement is used to determine the sign of an integer.
In the `absoluteValue` function, guards are used to determine the absolute value of an integer.
## Exercise
Write a function called `isEven` that takes an `Int` value as an argument and returns `True` if the value is even, and `False` otherwise. Use guards to implement the function.
### Solution
```haskell
isEven :: Int -> Bool
isEven x
| x `mod` 2 == 0 = True
| otherwise = False
```
# 5.2. Loops
In Haskell, loops are typically implemented using recursion. Recursion is a technique where a function calls itself, either directly or indirectly, to solve a problem.
There are two main types of recursion: direct recursion and indirect recursion. In direct recursion, a function calls itself directly. In indirect recursion, two or more functions call each other in a cycle.
To illustrate recursion, let's consider the factorial function. The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n.
The factorial function can be defined recursively as follows:
```haskell
factorial :: Int -> Int
factorial 0 = 1
factorial n = n * factorial (n - 1)
```
The base case of the recursion is when n is 0, in which case the factorial is defined to be 1. For any other value of n, the factorial is computed by multiplying n with the factorial of (n - 1).
Here's an example of using the factorial function:
```haskell
factorial 5
```
The factorial of 5 is computed as follows:
```
factorial 5 = 5 * factorial 4
= 5 * (4 * factorial 3)
= 5 * (4 * (3 * factorial 2))
= 5 * (4 * (3 * (2 * factorial 1)))
= 5 * (4 * (3 * (2 * (1 * factorial 0))))
= 5 * (4 * (3 * (2 * (1 * 1))))
= 5 * (4 * (3 * (2 * 1)))
= 5 * (4 * (3 * 2))
= 5 * (4 * 6)
= 5 * 24
= 120
```
So, the factorial of 5 is 120.
## Exercise
Write a function called `sumOfDigits` that takes a positive integer as an argument and returns the sum of its digits. Use recursion to implement the function.
### Solution
```haskell
sumOfDigits :: Int -> Int
sumOfDigits 0 = 0
sumOfDigits n = (n `mod` 10) + sumOfDigits (n `div` 10)
```
# 5.2.1. Recursion
Recursion is a powerful technique in programming that allows a function to call itself. It is particularly useful when solving problems that can be broken down into smaller subproblems of the same type.
In Haskell, recursion is commonly used to implement loops and iterate over data structures. It allows us to perform repetitive tasks without the need for explicit iteration.
To understand recursion, it's important to understand the concept of a base case. A base case is a condition that, when met, allows the recursion to stop and return a result. Without a base case, the recursion would continue indefinitely, leading to a stack overflow error.
When using recursion, we typically define a recursive function that calls itself with a modified input. Each recursive call brings us closer to the base case, eventually leading to termination.
Let's consider a simple example of recursion: calculating the sum of all numbers from 1 to n. We can define a recursive function called `sumNumbers` that takes an integer `n` as an argument and returns the sum of all numbers from 1 to `n`.
```haskell
sumNumbers :: Int -> Int
sumNumbers 0 = 0
sumNumbers n = n + sumNumbers (n - 1)
```
In this example, the base case is when `n` is 0, in which case the sum is defined to be 0. For any other value of `n`, the sum is computed by adding `n` to the sum of all numbers from 1 to `n-1`.
## Exercise
Write a function called `factorial` that takes a non-negative integer as an argument and returns its factorial. Use recursion to implement the function.
### Solution
```haskell
factorial :: Int -> Int
factorial 0 = 1
factorial n = n * factorial (n - 1)
```
# 5.2.2. List Comprehension
List comprehension is a concise and powerful way to create lists in Haskell. It allows us to define a list by specifying its elements and any conditions or transformations that should be applied to those elements.
List comprehensions are often used in conjunction with recursion to generate lists of values based on some pattern or condition.
The basic syntax of a list comprehension is as follows:
```haskell
[ expression | generator, condition ]
```
The `expression` is the value or transformation that should be applied to each element of the list. The `generator` is used to generate the elements of the list, and the `condition` is an optional filter that can be applied to the elements.
Let's say we want to generate a list of the squares of all even numbers from 1 to 10. We can use a list comprehension to accomplish this:
```haskell
squaresOfEvens = [ x^2 | x <- [1..10], even x ]
```
In this example, the `generator` is `x <- [1..10]`, which generates the elements of the list from 1 to 10. The `condition` is `even x`, which filters out the odd numbers. The `expression` is `x^2`, which calculates the square of each even number.
The resulting list is `[4, 16, 36, 64, 100]`, which contains the squares of the even numbers from 1 to 10.
## Exercise
Write a list comprehension that generates a list of the cubes of all odd numbers from 1 to 10.
### Solution
```haskell
cubesOfOdds = [ x^3 | x <- [1..10], odd x ]
```
# 5.3. Higher-Order Functions
Higher-order functions are functions that can take other functions as arguments or return functions as results. In Haskell, functions are first-class citizens, which means they can be treated just like any other value.
Higher-order functions are a fundamental concept in functional programming and are widely used in Haskell. They allow for more expressive and modular code by enabling functions to be composed, abstracted, and manipulated.
One common higher-order function in Haskell is `map`. The `map` function takes a function and a list as arguments, and applies the function to each element of the list, returning a new list with the results.
The basic syntax of `map` is as follows:
```haskell
map :: (a -> b) -> [a] -> [b]
```
The first argument is a function that takes an element of type `a` and returns an element of type `b`. The second argument is a list of type `[a]`. The result is a new list of type `[b]`.
Let's say we have a list of numbers `[1, 2, 3, 4, 5]`, and we want to double each number in the list. We can use the `map` function to accomplish this:
```haskell
doubleNumbers = map (\x -> x * 2) [1, 2, 3, 4, 5]
```
In this example, the function `(\x -> x * 2)` is applied to each element of the list `[1, 2, 3, 4, 5]`. The result is a new list `[2, 4, 6, 8, 10]`, which contains the doubled numbers.
## Exercise
Write a higher-order function that takes a list of numbers and returns a new list with the square of each number.
### Solution
```haskell
squareNumbers :: [Int] -> [Int]
squareNumbers = map (\x -> x^2)
```
# 5.4. Monads for Control Flow
Monads are a powerful concept in Haskell that allow for precise control over program execution and side effects. They provide a way to encapsulate computations and control the flow of data between them.
In Haskell, monads are used to handle impure computations, such as reading from or writing to external resources, handling exceptions, or performing non-deterministic computations.
One common monad in Haskell is the `Maybe` monad. The `Maybe` monad is used to handle computations that may or may not produce a result. It is often used to handle errors or undefined values.
The `Maybe` monad has two possible values: `Just a`, where `a` is the result of the computation, and `Nothing`, which represents a failed computation or an undefined value.
Let's say we have a function `safeDivide` that takes two numbers as input and returns their division, but it should handle the case where the second number is zero and return `Nothing` instead.
```haskell
safeDivide :: Double -> Double -> Maybe Double
safeDivide x 0 = Nothing
safeDivide x y = Just (x / y)
```
In this example, if the second number is zero, the function returns `Nothing`. Otherwise, it returns `Just (x / y)`, where `x` and `y` are the input numbers.
## Exercise
Write a function `safeSqrt` that takes a number as input and returns its square root if the number is non-negative, and `Nothing` otherwise.
### Solution
```haskell
safeSqrt :: Double -> Maybe Double
safeSqrt x
| x < 0 = Nothing
| otherwise = Just (sqrt x)
```
# 6. Functional Programming Concepts
Functional programming is a programming paradigm that treats computation as the evaluation of mathematical functions and avoids changing state and mutable data. It emphasizes the use of pure functions, immutability, and referential transparency.
One of the key concepts in functional programming is the use of pure functions. A pure function is a function that always produces the same output for the same input and has no side effects. This means that a pure function does not modify any external state or variables and does not rely on any external state or variables.
Here is an example of a pure function in Haskell:
```haskell
add :: Int -> Int -> Int
add x y = x + y
```
The `add` function takes two integers as input and returns their sum. It does not modify any external state or variables and always produces the same output for the same input.
## Exercise
Write a pure function `multiply` that takes two integers as input and returns their product.
### Solution
```haskell
multiply :: Int -> Int -> Int
multiply x y = x * y
```
# 6.1. Pure Functions and Side Effects
Pure functions are an important concept in functional programming because they have several advantages. One of the main advantages is that pure functions are easier to reason about and test. Since pure functions do not modify any external state or variables, their behavior is solely determined by their input parameters.
In contrast, functions that have side effects, such as modifying external state or variables, can be more difficult to reason about and test. The behavior of a function with side effects may depend on the current state of the program, making it harder to predict its outcome.
Here is an example of a function with side effects in Haskell:
```haskell
printHello :: IO ()
printHello = putStrLn "Hello, world!"
```
The `printHello` function has the side effect of printing the string "Hello, world!" to the console. This side effect is not determined by the input parameters of the function, but by the current state of the program (i.e., the console output).
## Exercise
Write a pure function `square` that takes an integer as input and returns its square.
### Solution
```haskell
square :: Int -> Int
square x = x * x
```
# 6.2. Immutability and Referential Transparency
Immutability is another important concept in functional programming. It means that once a value is assigned to a variable, it cannot be changed. Instead of modifying variables, functional programming encourages creating new values based on existing ones.
Referential transparency is closely related to immutability. It means that a function call can be replaced with its result without changing the behavior of the program. This property allows for easier reasoning about and testing of code.
Here is an example that demonstrates immutability and referential transparency in Haskell:
```haskell
addOne :: Int -> Int
addOne x = x + 1
result :: Int
result = addOne 5
```
In this example, the `addOne` function takes an integer as input and returns its successor. The `result` variable is assigned the result of calling `addOne` with the argument `5`. Since `addOne` is a pure function, its result is always the same for the same input, and we can replace the function call with its result:
```haskell
result :: Int
result = 6
```
## Exercise
Write a function `double` that takes an integer as input and returns its double.
### Solution
```haskell
double :: Int -> Int
double x = x * 2
```
# 6.3. Currying and Partial Application
Currying is a technique in functional programming where a function that takes multiple arguments is transformed into a sequence of functions, each taking a single argument. This allows for partial application, where a function is called with fewer arguments than it expects, resulting in a new function that takes the remaining arguments.
In Haskell, all functions are curried by default. This means that a function that appears to take multiple arguments is actually a series of functions, each taking one argument and returning a new function.
Here is an example that demonstrates currying and partial application in Haskell:
```haskell
add :: Int -> Int -> Int
add x y = x + y
addOne :: Int -> Int
addOne = add 1
```
In this example, the `add` function takes two integers as input and returns their sum. The `addOne` function is created by partially applying `add` with the argument `1`. The resulting function takes a single integer as input and returns the sum of that integer and `1`.
## Exercise
Write a function `incrementBy` that takes an integer `n` as input and returns a function that increments its input by `n`.
### Solution
```haskell
incrementBy :: Int -> (Int -> Int)
incrementBy n = add n
```
# 6.4. Lazy Evaluation
Lazy evaluation is a key feature of Haskell that allows for efficient and flexible computation. In lazy evaluation, expressions are not evaluated until their results are actually needed. This can lead to significant performance improvements, as unnecessary computations are avoided.
In Haskell, all computations are expressed as a series of expressions, which are evaluated lazily. This means that the order in which expressions are evaluated can be different from the order in which they are written. Haskell uses a technique called lazy evaluation to determine the order in which expressions are evaluated.
Here is an example that demonstrates lazy evaluation in Haskell:
```haskell
takeFirst :: Int -> [Int] -> [Int]
takeFirst n xs = take n xs
numbers :: [Int]
numbers = [1, 2, 3, 4, 5]
result :: [Int]
result = takeFirst 3 numbers
```
In this example, the `takeFirst` function takes an integer `n` and a list of integers `xs`, and returns the first `n` elements of `xs`. The `numbers` list contains the numbers 1 to 5. The `result` variable is assigned the result of calling `takeFirst` with the arguments `3` and `numbers`.
## Exercise
Write a function `doubleAll` that takes a list of integers as input and returns a new list containing the double of each element in the input list. Use lazy evaluation to avoid unnecessary computations.
### Solution
```haskell
doubleAll :: [Int] -> [Int]
doubleAll xs = map (*2) xs
```
# 7. Music Theory Basics for Programmers
Music is made up of various elements, including pitch, rhythm, scales, and chords. By understanding these elements, you can create melodies and harmonies that are pleasing to the ear.
Let's start with pitch. Pitch refers to the perceived frequency of a sound. In music, pitch is typically represented using a system of notes. The most common system is the Western music notation system, which uses letters A through G to represent different pitches.
The distance between two pitches is called an interval. Intervals can be measured in terms of half steps or whole steps. For example, the interval between C and D is a whole step, while the interval between C and C# is a half step.
In addition to pitch, music is also organized in terms of rhythm. Rhythm refers to the arrangement of sounds and silences in time. It is the pattern of long and short sounds that give music its distinctive feel.
Rhythm is typically represented using a system of time signatures. A time signature consists of two numbers, one on top of the other. The top number represents the number of beats in a measure, while the bottom number represents the type of note that receives one beat.
For example, a time signature of 4/4 indicates that there are four beats in a measure, and a quarter note receives one beat. This is the most common time signature in Western music.
## Exercise
What is the time signature of a waltz?
### Solution
The time signature of a waltz is 3/4, which means there are three beats in a measure and a quarter note receives one beat.
# 7.1. Pitch and Frequency
In music, pitch refers to the perceived frequency of a sound. It is what determines whether a sound is high or low. The pitch of a sound is measured in hertz (Hz), with higher frequencies corresponding to higher pitches and lower frequencies corresponding to lower pitches.
In Haskell, we can represent pitch using numerical values. For example, we can assign the value 440 to represent the pitch A4, which is commonly used as a reference pitch. From there, we can calculate the frequencies of other pitches using the formula:
$$f = 2^{\frac{n}{12}} \times 440$$
where $f$ is the frequency and $n$ is the number of half steps away from A4. This formula allows us to calculate the frequency of any pitch in the equal-tempered scale.
Let's say we want to calculate the frequency of the pitch C5. Since C5 is 9 half steps away from A4, we can substitute $n$ with 9 in the formula:
$$f = 2^{\frac{9}{12}} \times 440$$
Evaluating this expression, we find that the frequency of C5 is approximately 523.25 Hz.
## Exercise
Calculate the frequency of the pitch E4 using the formula provided.
### Solution
To calculate the frequency of E4, we need to determine the number of half steps away from A4. Since E4 is 4 half steps away from A4, we can substitute $n$ with 4 in the formula:
$$f = 2^{\frac{4}{12}} \times 440$$
Evaluating this expression, we find that the frequency of E4 is approximately 329.63 Hz.
# 7.2. Rhythm and Time Signatures
In music, rhythm refers to the pattern of durations and accents in a piece of music. It is what gives music its sense of movement and pulse. Rhythm is typically organized into measures, or bars, which are groups of beats.
Time signatures are used to indicate the organization of beats within a measure. They consist of two numbers written as a fraction. The top number indicates the number of beats in a measure, while the bottom number indicates the note value that receives one beat.
For example, a time signature of 4/4 indicates that there are 4 beats in a measure, and the quarter note receives one beat. This is the most common time signature in Western music and is often referred to as "common time." Another common time signature is 3/4, which indicates 3 beats in a measure, with the quarter note still receiving one beat.
In Haskell, we can represent rhythm using numerical values. For example, we can assign the value 4 to represent a quarter note, and the value 3 to represent a dotted quarter note. We can then use these values to calculate the durations of other notes and rests.
Let's say we want to calculate the duration of an eighth note. Since an eighth note is half the duration of a quarter note, we can divide the value of a quarter note by 2:
$$\text{eighth note duration} = \frac{\text{quarter note duration}}{2}$$
If we assume that the quarter note has a duration of 1, then the eighth note would have a duration of 0.5.
## Exercise
Calculate the duration of a half note using the value of a quarter note as 1.
### Solution
To calculate the duration of a half note, we need to determine how many quarter notes fit into a half note. Since a half note is twice the duration of a quarter note, we can multiply the value of a quarter note by 2:
$$\text{half note duration} = \text{quarter note duration} \times 2$$
If we assume that the quarter note has a duration of 1, then the half note would have a duration of 2.
# 7.3. Scales and Chords
In music theory, scales and chords are fundamental building blocks. They provide the foundation for melodies and harmonies in a piece of music.
A scale is a series of pitches arranged in ascending or descending order. It is used to create melodies and establish a tonal center. There are many different types of scales, including major scales, minor scales, and pentatonic scales.
One of the most common scales in Western music is the major scale. It is a seven-note scale that follows a specific pattern of whole and half steps. For example, the C major scale consists of the pitches C, D, E, F, G, A, and B.
Chords, on the other hand, are groups of pitches played simultaneously. They provide the harmonic structure of a piece of music. Chords are built from scales, and different types of chords can be created depending on the intervals between the pitches.
A major chord, for example, is built from the first, third, and fifth notes of a major scale. In the key of C major, the C major chord consists of the pitches C, E, and G.
In Haskell, we can represent scales and chords using lists of pitches. Each pitch can be represented as a numerical value or a symbolic representation. We can then use functions to manipulate and combine these lists to create melodies and harmonies.
## Exercise
Create a list representing the C major scale using numerical values for the pitches. Assume that the pitch C is represented by the value 0, and each subsequent pitch is represented by an increment of 1.
### Solution
The C major scale can be represented as the list [0, 2, 4, 5, 7, 9, 11]. This list represents the pitches C, D, E, F, G, A, and B, respectively.
# 7.4. Music Notation and Terminology
Music notation is a system of writing music using symbols and markings. It allows musicians to communicate and interpret musical ideas accurately. Understanding music notation is essential for reading and performing music.
The staff is the foundation of music notation. It consists of five horizontal lines and four spaces. Each line and space represents a different pitch. Notes are placed on the staff to indicate the pitch and duration of a sound.
The treble clef is a symbol placed at the beginning of the staff to indicate the pitch range of the notes. It is commonly used for instruments such as the piano, violin, and flute. The notes on the lines of the treble clef staff are E, G, B, D, and F, while the notes in the spaces are F, A, C, and E.
Notes are represented by oval-shaped symbols placed on the lines and spaces of the staff. The position of the note on the staff indicates its pitch, and the shape of the note indicates its duration. The most common note durations are whole notes, half notes, quarter notes, and eighth notes.
A whole note is represented by an open oval shape, while a half note is represented by a filled-in oval shape. A quarter note is represented by a filled-in oval shape with a stem, and an eighth note is represented by a filled-in oval shape with a stem and a flag.
In addition to notes, music notation includes other symbols and markings to indicate dynamics (volume), articulation (how to play the notes), and other musical instructions. These symbols and markings provide important information for performers.
Some common symbols and markings in music notation include:
- Dynamics: p (piano), f (forte), mf (mezzo forte)
- Articulation: staccato (short and detached), legato (smooth and connected)
- Tempo: allegro (fast), adagio (slow)
- Repeat signs: indicating sections of music to be repeated
## Exercise
Create a list of symbols and markings commonly used in music notation. Include at least five symbols or markings and their corresponding meanings.
### Solution
- p (piano): indicates to play softly
- f (forte): indicates to play loudly
- mf (mezzo forte): indicates to play moderately loud
- staccato: indicates to play the notes short and detached
- legato: indicates to play the notes smooth and connected
# 8. Signal Processing Fundamentals
Signal processing is a fundamental concept in music technology. It involves the analysis, manipulation, and synthesis of signals to produce desired audio effects. Understanding signal processing is crucial for creating and modifying sounds in music production.
Sampling is the process of converting continuous-time signals into discrete-time signals. In music, this involves capturing the audio waveform at regular intervals and representing it as a sequence of numbers. The sampling rate determines the number of samples taken per second, and it affects the quality and fidelity of the audio.
For example, consider a digital audio file with a sampling rate of 44.1 kHz. This means that 44,100 samples are taken per second. Each sample represents the amplitude of the audio waveform at a specific point in time.
Quantization is the process of converting the continuous amplitude values of a signal into discrete amplitude values. In digital audio, this involves assigning a specific numerical value to each sample based on its amplitude. The bit depth determines the number of possible amplitude values, and it affects the dynamic range and resolution of the audio.
For example, consider a digital audio file with a bit depth of 16 bits. This means that each sample can have one of 65,536 possible amplitude values. The higher the bit depth, the more accurately the audio waveform can be represented.
The discrete Fourier transform (DFT) is a mathematical algorithm used to analyze the frequency content of a signal. It transforms a signal from the time domain to the frequency domain, revealing the individual frequencies and their amplitudes. The DFT is widely used in music analysis, synthesis, and effects processing.
For example, the DFT can be used to analyze the frequency spectrum of a musical instrument. By applying the DFT to a recorded sound, we can identify the fundamental frequency (pitch) and the harmonic content (overtones) of the instrument.
Filter design and implementation is an important aspect of signal processing in music. Filters are used to modify the frequency content of a signal, allowing us to shape the timbre and character of a sound. There are various types of filters, including low-pass filters, high-pass filters, and band-pass filters.
For example, a low-pass filter allows low-frequency components to pass through while attenuating high-frequency components. This can be used to create a warm and mellow sound. On the other hand, a high-pass filter allows high-frequency components to pass through while attenuating low-frequency components. This can be used to create a bright and sharp sound.
## Exercise
Explain the concept of aliasing in the context of signal processing.
### Solution
Aliasing is a phenomenon that occurs when a signal is sampled at a rate lower than the Nyquist rate, which is twice the highest frequency present in the signal. It leads to the distortion and loss of information in the sampled signal. Aliasing can be avoided by using a higher sampling rate or by applying anti-aliasing filters to remove high-frequency components before sampling.
# 8.1. Sampling and Quantization
Sampling is the process of converting continuous-time signals into discrete-time signals. In music, this involves capturing the audio waveform at regular intervals and representing it as a sequence of numbers. The sampling rate determines the number of samples taken per second, and it affects the quality and fidelity of the audio.
Quantization is the process of converting the continuous amplitude values of a signal into discrete amplitude values. In digital audio, this involves assigning a specific numerical value to each sample based on its amplitude. The bit depth determines the number of possible amplitude values, and it affects the dynamic range and resolution of the audio.
For example, consider a digital audio file with a sampling rate of 44.1 kHz. This means that 44,100 samples are taken per second. Each sample represents the amplitude of the audio waveform at a specific point in time.
For example, consider a digital audio file with a bit depth of 16 bits. This means that each sample can have one of 65,536 possible amplitude values. The higher the bit depth, the more accurately the audio waveform can be represented.
## Exercise
Calculate the total number of samples in a 5-minute audio file with a sampling rate of 48 kHz.
### Solution
The sampling rate is 48,000 samples per second. To calculate the total number of samples in 5 minutes, we multiply the sampling rate by the number of seconds in 5 minutes:
Total number of samples = 48,000 samples/second * 5 minutes * 60 seconds/minute = 14,400,000 samples
# 8.2. Discrete Fourier Transform
The Discrete Fourier Transform (DFT) is a mathematical algorithm used to convert a discrete-time signal from the time domain to the frequency domain. It allows us to analyze the different frequency components present in a signal.
The DFT works by decomposing a signal into a sum of sinusoidal components at different frequencies. Each sinusoidal component is characterized by its frequency, amplitude, and phase. By performing the DFT, we can obtain the frequency spectrum of a signal, which shows the amplitude of each sinusoidal component.
For example, let's say we have a digital audio signal consisting of a single pure tone at 440 Hz. By applying the DFT, we can identify that the frequency component at 440 Hz has a high amplitude, while the other frequency components have low amplitudes.
The DFT is commonly used in music processing for tasks such as pitch detection, harmonic analysis, and audio synthesis. It allows us to extract meaningful information from a signal and manipulate it in the frequency domain.
## Exercise
Calculate the DFT of a signal consisting of a single pure tone at 1000 Hz sampled at a rate of 44100 samples per second. Assume a duration of 1 second and an amplitude of 1.
### Solution
To calculate the DFT of a signal, we need to apply the DFT algorithm to each sample of the signal. The DFT algorithm involves summing sinusoidal components at different frequencies, so we need to choose the frequencies at which we want to evaluate the DFT.
In this case, we want to evaluate the DFT at frequencies ranging from 0 Hz to the Nyquist frequency, which is half the sampling rate (22050 Hz in this case). We can choose a set of equally spaced frequencies within this range.
Once we have the frequencies, we can calculate the DFT by summing sinusoidal components at each frequency. The amplitude of each sinusoidal component can be obtained by taking the absolute value of the complex-valued DFT result.
The DFT of a signal can be represented as a complex-valued array, where each element corresponds to a frequency component. The magnitude of each complex-valued element represents the amplitude of the corresponding frequency component.
The DFT of a signal can be calculated using the Fast Fourier Transform (FFT) algorithm, which is an efficient implementation of the DFT. The FFT algorithm is widely used in practice due to its computational efficiency.
To calculate the DFT of a signal in Haskell, we can use the `fft` function from the `Data.Complex` module. The `fft` function takes a list of complex numbers as input and returns a list of complex numbers representing the DFT.
Here's an example code snippet that calculates the DFT of a signal consisting of a single pure tone at 1000 Hz sampled at a rate of 44100 samples per second:
```haskell
import Data.Complex
-- Define the sampling rate and duration
samplingRate = 44100
duration = 1
-- Define the frequency of the pure tone
frequency = 1000
-- Calculate the number of samples
numSamples = round (samplingRate * duration)
-- Generate the time values
timeValues = map (\n -> fromIntegral n / samplingRate) [0..numSamples-1]
-- Generate the signal
signal = map (\t -> cis (2 * pi * frequency * t)) timeValues
-- Calculate the DFT
dft = fft signal
-- Print the DFT result
main = print dft
```
This code calculates the DFT of a signal consisting of a single pure tone at 1000 Hz sampled at a rate of 44100 samples per second. The DFT result is printed to the console.
Note that the `fft` function returns a list of complex numbers, where each element represents a frequency component. The magnitude of each complex number can be obtained using the `magnitude` function from the `Data.Complex` module.
You can run this code in a Haskell interpreter or compiler to see the DFT result.
# 8.3. Filter Design and Implementation
Filtering is a fundamental operation in signal processing that allows us to modify the frequency content of a signal. Filters can be used to remove unwanted noise, enhance certain frequency components, or extract specific features from a signal.
In Haskell, we can design and implement filters using various techniques. One common approach is to use digital filter design methods, such as the windowing method or the frequency sampling method. These methods allow us to design filters with specific frequency response characteristics, such as low-pass, high-pass, band-pass, or band-stop.
For example, let's say we have a noisy audio signal and we want to remove the high-frequency noise. We can design and implement a low-pass filter that attenuates frequencies above a certain cutoff frequency. This will effectively remove the high-frequency noise from the signal.
Once we have designed a filter, we can implement it using techniques such as convolution or recursive filtering. Convolution involves convolving the filter's impulse response with the input signal to obtain the filtered output. Recursive filtering, on the other hand, involves recursively applying the filter's transfer function to the input signal.
## Exercise
Design and implement a low-pass filter with a cutoff frequency of 1000 Hz. Apply the filter to a noisy audio signal and compare the filtered signal with the original signal.
### Solution
To design a low-pass filter, we can use the `fir1` function from the `Data.Filter` module. The `fir1` function takes the filter order and the cutoff frequency as input and returns the filter coefficients.
Here's an example code snippet that designs and implements a low-pass filter with a cutoff frequency of 1000 Hz:
```haskell
import Data.Filter
-- Define the filter order and cutoff frequency
order = 100
cutoffFrequency = 1000
-- Design the filter
filterCoefficients = fir1 order (cutoffFrequency / samplingRate)
-- Implement the filter
filteredSignal = filter filterCoefficients inputSignal
-- Compare the filtered signal with the original signal
main = do
print "Original signal:"
print inputSignal
print "Filtered signal:"
print filteredSignal
```
This code designs and implements a low-pass filter with a cutoff frequency of 1000 Hz. The `fir1` function is used to design the filter, and the `filter` function is used to implement the filter.
Note that the `fir1` function returns a list of filter coefficients, which are used by the `filter` function to implement the filter. The `inputSignal` variable represents the noisy audio signal that we want to filter.
You can run this code in a Haskell interpreter or compiler to see the filtered signal.
# 8.4. Applications in Music Synthesis and Analysis
The Discrete Fourier Transform (DFT) and digital filters have various applications in music synthesis and analysis. They allow us to manipulate and analyze the frequency content of musical signals, which is essential for tasks such as sound synthesis, pitch detection, and harmonic analysis.
In music synthesis, the DFT can be used to generate musical tones with specific frequency components. By controlling the amplitude and phase of the sinusoidal components in the DFT, we can create complex sounds with different timbres and textures.
For example, let's say we want to synthesize a piano sound. We can analyze the frequency content of a real piano sound using the DFT, and then use this information to generate a synthetic piano sound with similar frequency components.
In music analysis, the DFT and digital filters can be used to extract meaningful information from musical signals. For example, the DFT can be used to detect the pitch of a musical note, or to identify the harmonic components of a musical sound.
## Exercise
Choose a musical signal of your choice and analyze its frequency content using the DFT. Identify the main frequency components and their amplitudes.
### Solution
To analyze the frequency content of a musical signal, we can use the `fft` function from the `Data.Complex` module. The `fft` function takes a list of complex numbers as input and returns a list of complex numbers representing the DFT.
Here's an example code snippet that analyzes the frequency content of a musical signal:
```haskell
import Data.Complex
-- Define the musical signal
signal = [0, 0, 0, 1, 0, 0, 0]
-- Calculate the DFT
dft = fft signal
-- Print the DFT result
main = print dft
```
This code analyzes the frequency content of a musical signal consisting of a single tone. The `fft` function is used to calculate the DFT of the signal, and the result is printed to the console.
Note that the `fft` function returns a list of complex numbers, where each element represents a frequency component. The magnitude of each complex number can be obtained using the `magnitude` function from the `Data.Complex` module.
You can run this code in a Haskell interpreter or compiler to see the frequency components of the musical signal.
# 9. Composition with Haskell
To represent musical structures in Haskell, we can use algebraic data types. For example, we can define a `Note` data type that represents a single note with properties such as pitch, duration, and velocity. We can also define a `Music` data type that represents a sequence of notes or other musical events.
Here's an example of how we can define the `Note` and `Music` data types in Haskell:
```haskell
data Pitch = C | D | E | F | G | A | B
data Duration = Whole | Half | Quarter | Eighth | Sixteenth
data Velocity = Soft | Medium | Loud
data Note = Note Pitch Duration Velocity
data Music = Music [Note]
```
With these data types, we can create musical structures by combining notes and other musical events. For example, we can define a melody as a sequence of notes:
```haskell
melody :: Music
melody = Music [Note C Quarter Medium, Note D Quarter Medium, Note E Quarter Medium, Note F Quarter Medium]
```
Once we have defined musical structures, we can generate music with functions. For example, we can define a function that takes a musical structure and repeats it a certain number of times, or a function that transposes a melody to a different key.
Here's an example of how we can define a function that repeats a musical structure:
```haskell
repeatMusic :: Int -> Music -> Music
repeatMusic n (Music notes) = Music (concat (replicate n notes))
```
With this function, we can repeat a melody three times:
```haskell
repeatedMelody :: Music
repeatedMelody = repeatMusic 3 melody
```
In addition to generating music with functions, we can use music theory algorithms to create interesting compositions. For example, we can use algorithms for chord progression, melody generation, and rhythm generation to create complex and harmonically rich compositions.
Here's an example of how we can use a music theory algorithm to generate a melody:
```haskell
generateMelody :: Music
generateMelody = -- use a music theory algorithm to generate a melody
```
With this code, we can generate a melody using a music theory algorithm of our choice.
## Exercise
Using the `Note` and `Music` data types defined earlier, create a function that transposes a melody to a different key. Test your function by transposing the `melody` defined earlier to a different key.
### Solution
To transpose a melody to a different key, we can define a function that takes a melody and a key as input, and returns a new melody with all the notes transposed to the new key.
Here's an example implementation of the `transposeMelody` function:
```haskell
transposeMelody :: Music -> Pitch -> Music
transposeMelody (Music notes) newKey = Music (map (\(Note pitch duration velocity) -> Note (transposePitch pitch newKey) duration velocity) notes)
transposePitch :: Pitch -> Pitch -> Pitch
transposePitch pitch newKey = -- implement the pitch transposition logic here
transposedMelody :: Music
transposedMelody = transposeMelody melody D
```
In this code, the `transposeMelody` function takes a melody and a new key as input, and uses the `map` function to apply the `transposePitch` function to each note in the melody. The `transposePitch` function takes a pitch and a new key as input, and implements the logic for transposing the pitch to the new key.
You can test this code by transposing the `melody` defined earlier to a different key, such as `D`.
# 9.1. Representing Musical Structures in Haskell
In Haskell, we can represent musical structures using algebraic data types. Algebraic data types allow us to define custom data structures with multiple constructors.
To represent musical structures, we can define data types for notes, chords, melodies, and other musical elements. Each data type can have properties such as pitch, duration, velocity, and more.
Here's an example of how we can define a data type for notes in Haskell:
```haskell
data Note = Note { pitch :: Pitch, duration :: Duration, velocity :: Velocity }
```
In this code, `Note` is the name of the data type, and `Pitch`, `Duration`, and `Velocity` are properties of a note. We can access these properties using record syntax, like `pitch note`, `duration note`, and `velocity note`.
We can also define data types for chords and melodies by combining multiple notes. For example, a chord can be represented as a list of notes, and a melody can be represented as a sequence of chords or notes.
Here's an example of how we can define a data type for chords in Haskell:
```haskell
data Chord = Chord { notes :: [Note] }
```
In this code, `Chord` is the name of the data type, and `notes` is a list of notes that make up the chord.
With these data types, we can create complex musical structures by combining notes, chords, and melodies. We can use functions to manipulate and transform these structures, such as transposing a melody to a different key or adding a note to a chord.
## Exercise
Define a data type for melodies in Haskell. A melody can be represented as a list of chords or notes. Include properties such as key and tempo.
### Solution
Here's an example of how we can define a data type for melodies in Haskell:
```haskell
data Melody = Melody { chords :: [Chord], key :: Key, tempo :: Tempo }
```
In this code, `Melody` is the name of the data type, and `chords` is a list of chords that make up the melody. `Key` represents the key of the melody, and `Tempo` represents the tempo or speed of the melody.
# 9.2. Generating Music with Functions
In Haskell, we can generate music using functions. Functions allow us to define patterns, transformations, and algorithms that can be applied to musical structures.
To generate music, we can define functions that take inputs such as key, tempo, and duration, and produce outputs such as melodies, chords, or notes. These functions can be composed and combined to create complex musical patterns.
Here's an example of how we can define a function to generate a simple melody in Haskell:
```haskell
generateMelody :: Key -> Tempo -> Duration -> Melody
generateMelody key tempo duration = Melody [Chord [Note (pitch key) duration 100]] key tempo
```
In this code, `generateMelody` is the name of the function. It takes inputs such as `key`, `tempo`, and `duration`, and produces a melody as output. The melody consists of a single chord with a single note, where the pitch is determined by the input key, the duration is determined by the input duration, and the velocity is set to 100.
We can also use higher-order functions to generate more complex musical patterns. Higher-order functions are functions that take other functions as arguments or return functions as results. They allow us to create reusable patterns and transformations.
Here's an example of how we can use a higher-order function to generate a repeating pattern of chords in Haskell:
```haskell
generatePattern :: (Int -> Chord) -> Int -> [Chord]
generatePattern chordFunction n = take n (map chordFunction [1..])
```
In this code, `generatePattern` is the name of the function. It takes inputs such as `chordFunction`, which is a function that takes an integer as input and produces a chord as output, and `n`, which is the number of chords to generate. The function uses `map` to apply the `chordFunction` to a range of integers, and `take` to limit the number of chords to `n`.
With these functions, we can generate music by applying different patterns, transformations, and algorithms. We can experiment with different inputs and combinations to create unique and interesting musical compositions.
## Exercise
Define a function to generate a chord progression in Haskell. The function should take inputs such as key, tempo, and number of chords, and produce a melody as output. Use the `generatePattern` function from the previous example to generate a repeating pattern of chords.
### Solution
Here's an example of how we can define a function to generate a chord progression in Haskell:
```haskell
generateChordProgression :: Key -> Tempo -> Int -> Melody
generateChordProgression key tempo n = Melody (generatePattern chordFunction n) key tempo
where chordFunction = \x -> Chord [Note (pitch key + x) 1 100]
```
In this code, `generateChordProgression` is the name of the function. It takes inputs such as `key`, `tempo`, and `n`, and produces a melody as output. The melody consists of a repeating pattern of chords, where each chord is a single note with a pitch that is determined by the input key plus the current index. The duration is set to 1, and the velocity is set to 100. The `chordFunction` is defined using a lambda function.
# 9.3. Music Theory Algorithms
Music theory algorithms are mathematical algorithms that can be used to generate and manipulate musical structures. These algorithms can be implemented in Haskell to create complex and interesting musical compositions.
One example of a music theory algorithm is the algorithm for generating chord progressions. A chord progression is a sequence of chords that are played in a specific order. There are many different algorithms that can be used to generate chord progressions, such as the circle of fifths algorithm or the Roman numeral analysis algorithm.
Here's an example of how we can implement the Roman numeral analysis algorithm for generating chord progressions in Haskell:
```haskell
generateChordProgression :: Key -> Mode -> [Chord]
generateChordProgression key mode = map (getChord key mode) [I, IV, V, I]
where getChord key mode degree = Chord [Note (getPitch key mode degree) 1 100]
getPitch key mode degree = pitch key + getInterval mode degree
getInterval mode degree = case degree of
I -> 0
II -> 2
III -> 4
IV -> 5
V -> 7
VI -> 9
VII -> 11
```
In this code, `generateChordProgression` is the name of the function. It takes inputs such as `key` and `mode`, and produces a list of chords as output. The function uses `map` to apply the `getChord` function to a list of degrees, which are represented using Roman numerals. The `getChord` function takes inputs such as `key`, `mode`, and `degree`, and produces a chord as output. The `getPitch` function calculates the pitch of the chord based on the input `key`, `mode`, and `degree`. The `getInterval` function calculates the interval between the input `mode` and `degree` using a `case` statement.
There are many other music theory algorithms that can be implemented in Haskell, such as algorithms for generating melodies, harmonizing melodies, and creating musical variations. These algorithms can be combined and modified to create unique and interesting musical compositions.
## Exercise
Define a function to generate a melody using a music theory algorithm of your choice. The function should take inputs such as key, mode, and length, and produce a melody as output.
### Solution
Here's an example of how we can define a function to generate a melody using a music theory algorithm in Haskell:
```haskell
generateMelody :: Key -> Mode -> Int -> Melody
generateMelody key mode length = Melody (map (getNote key mode) [1..length]) key 120
where getNote key mode degree = Note (getPitch key mode degree) 1 100
getPitch key mode degree = pitch key + getInterval mode degree
getInterval mode degree = case degree of
1 -> 0
2 -> 2
3 -> 4
4 -> 5
5 -> 7
6 -> 9
7 -> 11
```
In this code, `generateMelody` is the name of the function. It takes inputs such as `key`, `mode`, and `length`, and produces a melody as output. The melody consists of a sequence of notes, where each note is generated using the `getNote` function. The `getNote` function takes inputs such as `key`, `mode`, and `degree`, and produces a note as output. The `getPitch` function calculates the pitch of the note based on the input `key`, `mode`, and `degree`. The `getInterval` function calculates the interval between the input `mode` and `degree` using a `case` statement. The melody has a fixed tempo of 120 beats per minute.
# 9.4. Combining Music and Signal Processing in Haskell
Combining music and signal processing in Haskell allows us to create complex and unique sounds. By applying signal processing techniques to musical data, we can manipulate and transform the sound in various ways.
One common technique is to apply filters to the audio signal. Filters can be used to remove unwanted frequencies, enhance certain frequencies, or create special effects. In Haskell, we can use libraries such as `hsc3` to apply filters to audio signals.
Here's an example of how we can apply a low-pass filter to an audio signal in Haskell:
```haskell
import Sound.SC3
main :: IO ()
main = audition $ out 0 $ lpf (saw AR 440) 1000
```
In this code, we first import the necessary functions from the `Sound.SC3` library. We then define the `main` function, which is the entry point of our program. Inside the `main` function, we use the `audition` function to play the audio signal. The `out` function is used to specify the output bus and the audio signal. The `lpf` function is used to apply a low-pass filter to the audio signal. In this example, we apply the filter to a sawtooth wave with a frequency of 440 Hz and a cutoff frequency of 1000 Hz.
Another technique is to apply effects to the audio signal. Effects can be used to add reverb, delay, distortion, or other creative effects to the sound. In Haskell, we can use libraries such as `hsc3` to apply effects to audio signals.
Here's an example of how we can apply a delay effect to an audio signal in Haskell:
```haskell
import Sound.SC3
main :: IO ()
main = audition $ out 0 $ delayN (saw AR 440) 0.5 0.5
```
In this code, we again import the necessary functions from the `Sound.SC3` library. We define the `main` function, which is the entry point of our program. Inside the `main` function, we use the `audition` function to play the audio signal. The `out` function is used to specify the output bus and the audio signal. The `delayN` function is used to apply a delay effect to the audio signal. In this example, we apply the delay effect to a sawtooth wave with a frequency of 440 Hz, a delay time of 0.5 seconds, and a feedback of 0.5.
## Exercise
Using the `hsc3` library, implement a chorus effect in Haskell. The chorus effect should take an audio signal as input and produce a modified audio signal as output. Experiment with different parameters to achieve the desired chorus effect.
### Solution
Here's an example of how we can implement a chorus effect in Haskell using the `hsc3` library:
```haskell
import Sound.SC3
main :: IO ()
main = audition $ out 0 $ chorus (saw AR 440) 0.1 0.05 0.2
```
In this code, we import the necessary functions from the `Sound.SC3` library. We define the `main` function, which is the entry point of our program. Inside the `main` function, we use the `audition` function to play the audio signal. The `out` function is used to specify the output bus and the audio signal. The `chorus` function is used to apply a chorus effect to the audio signal. In this example, we apply the chorus effect to a sawtooth wave with a frequency of 440 Hz, a depth of 0.1, a rate of 0.05, and a feedback of 0.2.
# 10. Advanced Topics in Haskell and Music
Now that we have covered the basics of combining music and signal processing in Haskell, let's explore some advanced topics in this area. These topics will allow us to further enhance our musical creations and push the boundaries of what is possible with Haskell.
One advanced topic is concurrency and parallelism. Haskell provides powerful features for concurrent and parallel programming, which can be applied to music composition and performance. By leveraging multiple threads or distributed computing, we can create complex and dynamic musical compositions that take advantage of the full capabilities of modern hardware.
Another advanced topic is the use of advanced data structures and algorithms for music processing. Haskell's strong type system and functional programming paradigm make it well-suited for implementing efficient and elegant data structures and algorithms. By using these advanced techniques, we can optimize our music processing code and improve its performance.
Sound design and audio effects are also important aspects of music production. Haskell provides libraries and tools for creating and manipulating sound, allowing us to design unique and immersive sonic experiences. By exploring these tools and techniques, we can add depth and richness to our musical compositions.
Finally, we will explore the development of interactive music applications. Haskell's interactive and reactive programming capabilities make it an ideal language for creating interactive music applications, such as music games or live coding environments. By combining Haskell's expressive power with real-time user interaction, we can create engaging and interactive musical experiences.
## Exercise
Think about a real-world application of Haskell in music that you find interesting. Describe the application and explain why Haskell would be a good choice for implementing it.
### Solution
One interesting real-world application of Haskell in music is live coding and performance. Live coding involves writing and modifying code in real-time to generate and manipulate music. Haskell's expressive syntax and powerful type system make it well-suited for live coding, as it allows for rapid prototyping and experimentation. Additionally, Haskell's lazy evaluation and functional programming paradigm enable the creation of complex and dynamic musical compositions. By using Haskell for live coding, musicians can create unique and evolving performances that push the boundaries of traditional music composition and performance.
# 10.1. Concurrency and Parallelism
Concurrency refers to the ability of a program to execute multiple tasks simultaneously. In Haskell, concurrency can be achieved through lightweight threads called "green threads". These threads can be created and managed using the `forkIO` function, allowing different parts of a music application to run concurrently.
Parallelism, on the other hand, refers to the ability of a program to execute multiple tasks simultaneously on multiple processors or cores. Haskell provides powerful abstractions for parallel programming, such as the `par` and `pseq` combinators, which allow us to express parallelism in a declarative manner.
By leveraging concurrency and parallelism in Haskell, we can create music applications that take full advantage of modern hardware. For example, we can use concurrency to play multiple musical phrases simultaneously, or to perform real-time audio processing while generating new musical material. Parallelism can be used to speed up computationally intensive tasks, such as audio synthesis or signal processing.
```haskell
import Control.Concurrent
-- Play two musical phrases concurrently
playConcurrently :: IO ()
playConcurrently = do
forkIO (playPhrase "C D E F")
forkIO (playPhrase "G A B C")
-- Perform real-time audio processing while generating new music
processAudio :: IO ()
processAudio = do
forkIO audioProcessing
generateMusic
-- Speed up audio synthesis using parallelism
synthesizeAudio :: [Note] -> [Note] -> [Note]
synthesizeAudio notes1 notes2 = runEval $ do
let result1 = rpar (synthesize notes1)
result2 = rpar (synthesize notes2)
rseq result1
rseq result2
return (result1 ++ result2)
```
In the `playConcurrently` example, we use `forkIO` to create two threads that play different musical phrases simultaneously. This allows us to create complex and layered musical compositions.
In the `processAudio` example, we use `forkIO` to perform real-time audio processing while generating new music. This enables us to create interactive and dynamic musical experiences.
In the `synthesizeAudio` example, we use parallelism to speed up audio synthesis by synthesizing two sets of notes in parallel. This can significantly improve the performance of computationally intensive tasks.
## Exercise
Think of a scenario where concurrency or parallelism could be applied in a music application. Describe the scenario and explain how concurrency or parallelism could enhance the application.
### Solution
One scenario where concurrency could be applied in a music application is in a live performance setting. In a live performance, musicians often need to synchronize their playing with other musicians or with pre-recorded tracks. By using concurrency, musicians can create separate threads for each musical part and ensure that they play in sync with each other. This can greatly enhance the overall performance and allow for more complex and dynamic musical compositions.
Parallelism could be applied in a music application that involves real-time audio processing, such as audio effects or synthesis. By using parallelism, the application can distribute the processing load across multiple processors or cores, allowing for faster and more efficient audio processing. This can result in a smoother and more responsive user experience, especially in applications that require real-time interaction with the audio output.
# 10.2. Data Structures and Algorithms for Music Processing
One important data structure used in music processing is the musical score. A musical score represents a piece of music and consists of various elements such as notes, chords, and rests. In Haskell, we can represent a musical score using custom data types and algebraic data types. For example, we can define a `Note` data type to represent a single note, and a `Score` data type to represent a sequence of notes.
```haskell
data Note = Note Pitch Duration
data Score = Score [Note]
```
Another common data structure used in music processing is the musical waveform. A waveform represents the shape of a sound wave over time and is typically used for audio synthesis and analysis. In Haskell, we can represent a waveform using lists or arrays of samples, where each sample represents the amplitude of the sound wave at a specific point in time.
```haskell
type Sample = Double
type Waveform = [Sample]
```
In addition to data structures, algorithms play a crucial role in music processing. One common algorithm used in music applications is the Fast Fourier Transform (FFT), which is used for spectral analysis and pitch detection. The FFT can be implemented in Haskell using existing libraries such as `hmatrix` or `accelerate`.
```haskell
import Numeric.FFT
-- Perform FFT on a waveform
fft :: Waveform -> Waveform
fft waveform = runST $ do
let n = length waveform
let input = fromList $ map (:+ 0) waveform
output <- thaw input
dft output
result <- freeze output
return $ map realPart $ toList result
```
```haskell
-- Calculate the pitch of a waveform using FFT
pitchDetection :: Waveform -> Pitch
pitchDetection waveform = do
let spectrum = fft waveform
-- Perform pitch detection algorithm
...
```
In the `fft` example, we use the `Numeric.FFT` library to perform the FFT on a waveform. This allows us to analyze the frequency content of the waveform and extract useful information for music processing.
In the `pitchDetection` example, we use the FFT to calculate the pitch of a waveform. By analyzing the frequency spectrum of the waveform, we can determine the dominant frequency and map it to a musical pitch.
## Exercise
Think of a scenario where a data structure or algorithm could be applied in a music application. Describe the scenario and explain how the data structure or algorithm could enhance the application.
### Solution
One scenario where a data structure could be applied in a music application is in a music recommendation system. In a music recommendation system, the application needs to store and organize a large collection of music tracks and associated metadata. By using an efficient data structure such as a balanced search tree or a hash table, the application can quickly search and retrieve music tracks based on various criteria such as genre, artist, or popularity. This can enhance the user experience by providing personalized music recommendations and facilitating efficient music browsing and discovery.
Another scenario where an algorithm could be applied in a music application is in automatic music transcription. Automatic music transcription is the process of converting an audio recording of music into a symbolic representation such as sheet music. By using algorithms such as pitch detection, onset detection, and rhythm analysis, the application can analyze the audio recording and extract the musical elements such as notes, chords, and rhythms. This can enable various music analysis and processing tasks, such as music transcription, music information retrieval, and music synthesis.
# 10.3. Sound Design and Audio Effects
One important aspect of sound design is synthesis, which involves creating sounds from scratch using mathematical algorithms and signal processing techniques. In Haskell, we can implement various synthesis techniques such as additive synthesis, subtractive synthesis, and frequency modulation synthesis using libraries like `Euterpea` or `HSC3`.
Additive synthesis involves combining multiple sine waves with different frequencies and amplitudes to create complex sounds. In Haskell, we can define a function that generates a sine wave with a given frequency and amplitude, and then combine multiple sine waves to create a more complex sound.
```haskell
import Euterpea
-- Generate a sine wave with a given frequency and amplitude
sineWave :: Double -> Double -> Music Pitch
sineWave freq amp = note qn (pitch (hzToPitch freq)) :=: rest qn
-- Combine multiple sine waves to create a complex sound
complexSound :: Music Pitch
complexSound = foldr1 (:=:) [sineWave freq amp | (freq, amp) <- [(440, 0.5), (660, 0.3), (880, 0.2)]]
```
Subtractive synthesis involves filtering and shaping a complex sound using filters and envelopes. In Haskell, we can implement various filters such as low-pass filters, high-pass filters, and band-pass filters using libraries like `Euterpea` or `HSC3`.
Audio effects are another important aspect of sound design. They allow musicians and producers to modify the sound of individual instruments and recordings in creative and expressive ways. Some common audio effects used in music include reverb, delay, chorus, and distortion. In Haskell, we can implement these effects using signal processing techniques such as convolution, feedback delay networks, and waveshaping.
```haskell
import Euterpea
import HSC3
-- Apply reverb effect to a sound
reverb :: Music Pitch -> Music Pitch
reverb sound = sound :+: sound
-- Apply delay effect to a sound
delay :: Music Pitch -> Music Pitch
delay sound = sound :+: rest wn :+: sound
-- Apply chorus effect to a sound
chorus :: Music Pitch -> Music Pitch
chorus sound = sound :=: transpose 3 sound :=: transpose (-3) sound
-- Apply distortion effect to a sound
distortion :: Music Pitch -> Music Pitch
distortion sound = sound :=: (sound * 2) :=: (sound * 0.5)
```
In the `reverb` example, we apply a reverb effect to a sound by simply concatenating the sound with itself. This creates the illusion of a larger and more spacious sound.
In the `delay` example, we apply a delay effect to a sound by adding a rest followed by the sound itself. This creates the illusion of a repeating echo or reflection.
In the `chorus` example, we apply a chorus effect to a sound by combining the sound with two transposed versions of itself. This creates the illusion of multiple voices or instruments playing together.
In the `distortion` example, we apply a distortion effect to a sound by multiplying the sound with different factors. This creates the illusion of a more aggressive and distorted sound.
## Exercise
Think of a scenario where a sound design technique or audio effect could be applied in a music application. Describe the scenario and explain how the technique or effect could enhance the application.
### Solution
One scenario where a sound design technique could be applied in a music application is in a virtual reality (VR) game. In a VR game, the application needs to create immersive and realistic sound environments that match the visual and interactive elements of the game. By using sound design techniques such as spatialization, reverberation, and dynamic sound mixing, the application can enhance the sense of presence and realism for the player. For example, the application can use spatialization techniques to position sounds in 3D space, creating the illusion of sound sources coming from different directions and distances. The application can also use reverberation techniques to simulate the acoustic properties of different environments, such as a concert hall or a cave. Finally, the application can use dynamic sound mixing techniques to adapt the sound environment in real-time based on the player's actions and the game's events, creating a more interactive and immersive experience.
# 10.4. Interactive Music Applications
One common feature of interactive music applications is the ability to generate music in real-time based on user input or other external factors. In Haskell, we can use libraries like `Euterpea` or `HSC3` to generate music dynamically based on user actions or events. For example, we can create a simple interactive drum machine that allows users to create drum patterns by clicking on different buttons.
```haskell
import Euterpea
-- Define a drum sound
drumSound :: Dur -> Music Pitch
drumSound dur = perc AcousticBassDrum dur
-- Define a drum machine
drumMachine :: [Dur -> Music Pitch] -> Music Pitch
drumMachine patterns = line [pattern dur | (pattern, dur) <- zip patterns [qn, en, sn, sn]]
-- Define drum patterns
pattern1 :: Dur -> Music Pitch
pattern1 dur = times 4 (drumSound dur)
pattern2 :: Dur -> Music Pitch
pattern2 dur = times 2 (drumSound dur) :+: times 2 (rest dur)
-- Create a drum machine with the patterns
machine :: Music Pitch
machine = drumMachine [pattern1, pattern2]
-- Play the drum machine
main :: IO ()
main = play machine
```
In this example, we define two drum patterns (`pattern1` and `pattern2`) that generate drum sounds of different durations. We then create a drum machine (`drumMachine`) that combines these patterns into a single music piece. Finally, we play the drum machine using the `play` function.
Another feature of interactive music applications is the ability to manipulate and control music in real-time. In Haskell, we can use libraries like `Euterpea` or `HSC3` to create interactive interfaces that allow users to control various aspects of the music, such as tempo, volume, and effects. For example, we can create a simple interactive synthesizer that allows users to play different notes using a MIDI controller.
```haskell
import Euterpea
-- Define a synthesizer sound
synthSound :: Pitch -> Music Pitch
synthSound pitch = instrument AcousticGrandPiano (note qn pitch)
-- Define a synthesizer
synthesizer :: Pitch -> Music Pitch
synthesizer pitch = line [synthSound (trans i pitch) | i <- [0, 2, 4, 5, 7, 9, 11]]
-- Play the synthesizer
main :: IO ()
main = play (forever (synthesizer (C, 4)))
```
In this example, we define a simple synthesizer (`synthesizer`) that plays a sequence of notes using an acoustic grand piano instrument. We then use the `play` function to play the synthesizer indefinitely, allowing users to play different notes using a MIDI controller.
## Exercise
Think of an interactive music application that you would like to create. Describe the main features and functionalities of the application, and explain how Haskell can be used to implement these features.
### Solution
One interactive music application that I would like to create is a live looping station. The application would allow users to record and loop different musical phrases in real-time, creating layered and evolving compositions. Users would be able to record and loop multiple tracks, adjust the volume and panning of each track, and apply effects such as reverb and delay. The application would also provide a live performance mode, where users can manipulate and control the loops in real-time using MIDI controllers or other input devices. Haskell can be used to implement this application by providing libraries for audio recording and playback, real-time signal processing, and MIDI input/output. Haskell's functional programming paradigm and strong type system would also make it easier to manage and manipulate the complex data structures required for live looping, such as audio buffers and MIDI events.
# 11. Real-World Applications of Haskell in Music
One area where Haskell is being used is live coding and performance. Live coding refers to the practice of writing and modifying code in real-time to create and manipulate music. Haskell provides a powerful and expressive language for live coding, allowing musicians to create complex and dynamic musical compositions on the fly. Tools like Tidal and Conductive are examples of Haskell libraries that enable live coding and performance.
Tidal is a Haskell library that allows musicians to create and manipulate patterns of musical events using concise and expressive code. It provides a domain-specific language for live coding, allowing users to define musical patterns and transformations in a concise and readable manner. Here is an example of a simple Tidal pattern:
```
d1 $ sound "bd sn"
```
This code defines a pattern that plays a bass drum followed by a snare drum. By modifying and combining patterns like this, musicians can create intricate and evolving musical compositions in real-time.
Another application of Haskell in music is music education and pedagogy. Haskell provides a powerful and flexible platform for teaching and learning music theory and composition. Libraries like Euterpea and Haskore provide tools for creating and manipulating musical structures, allowing students to experiment with different musical concepts and techniques.
Euterpea is a Haskell library for computer music that provides a high-level interface for creating and manipulating musical structures. It allows students to experiment with different musical concepts, such as melody, harmony, and rhythm, and provides tools for composing and performing music. Here is an example of a simple melody created using Euterpea:
```haskell
import Euterpea
melody :: Music Pitch
melody = line [c 4 qn, d 4 qn, e 4 qn, f 4 qn]
main :: IO ()
main = play melody
```
This code defines a melody that plays four quarter notes: C, D, E, and F. By modifying and combining musical structures like this, students can explore different musical ideas and develop their composition skills.
## Exercise
Think of a real-world application of Haskell in music that you find interesting. Describe the application and explain how Haskell can be used to implement it.
### Solution
One real-world application of Haskell in music that I find interesting is music information retrieval. Music information retrieval (MIR) is the field of study that focuses on developing algorithms and techniques for automatically analyzing and organizing music data. Haskell can be used to implement MIR systems by providing a powerful and expressive language for manipulating and processing music data. Haskell's functional programming paradigm and strong type system make it well-suited for developing efficient and scalable algorithms for tasks such as music classification, recommendation, and similarity analysis. Haskell libraries like Haskore and Euterpea provide tools for representing and manipulating music data, making it easier to implement MIR algorithms and systems.
# 11.1. Live Coding and Performance
Live coding and performance is a growing trend in the music industry. It involves creating and manipulating music in real-time by writing and modifying code. Haskell provides a powerful and expressive language for live coding, allowing musicians to create complex and dynamic musical compositions on the fly.
One popular Haskell library for live coding is Tidal. Tidal allows musicians to create and manipulate patterns of musical events using concise and expressive code. Musicians can define musical patterns and transformations in a readable manner. For example, the following code defines a pattern that plays a bass drum followed by a snare drum:
```haskell
d1 $ sound "bd sn"
```
By modifying and combining patterns like this, musicians can create intricate and evolving musical compositions in real-time.
Here's another example of a Tidal pattern:
```haskell
d1 $ sound "bd [sn bd]*4"
```
This code defines a pattern that plays a bass drum followed by a snare drum, and then repeats this pattern four times. Tidal allows for complex and dynamic patterns like this to be easily created and manipulated.
## Exercise
Try creating your own Tidal pattern. Define a pattern that plays a melody of your choice. Experiment with different musical events and transformations to create a unique composition.
### Solution
```haskell
d1 $ sound "c e g"
```
This code defines a pattern that plays a simple melody consisting of the notes C, E, and G. You can modify this pattern by adding more notes or applying transformations to create your own unique composition.
# 11.2. Music Education and Pedagogy
Haskell can also be used as a tool for music education and pedagogy. By teaching programming concepts through music, students can develop a deeper understanding of both subjects. Haskell's functional programming paradigm aligns well with the principles of music theory, making it a natural fit for teaching music-related concepts.
One way to use Haskell in music education is to create interactive music theory lessons. Students can write Haskell code to explore different musical concepts, such as scales, chords, and harmonies. By experimenting with code, students can gain a hands-on understanding of how these concepts work and how they can be applied in composition.
Here's an example of a Haskell program that generates a major scale:
```haskell
majorScale :: Pitch -> [Pitch]
majorScale root = [root, root + 2, root + 4, root + 5, root + 7, root + 9, root + 11, root + 12]
-- Example usage:
-- majorScale C => [C, D, E, F, G, A, B, C]
```
In this code, the `majorScale` function takes a root pitch as input and generates a list of pitches representing the major scale starting from that root. Students can experiment with different root pitches to generate different scales and explore the patterns and intervals within them.
## Exercise
Write a Haskell function that generates a major chord given a root pitch. Test your function with different root pitches to generate different major chords.
### Solution
```haskell
majorChord :: Pitch -> [Pitch]
majorChord root = [root, root + 4, root + 7]
-- Example usage:
-- majorChord C => [C, E, G]
```
Students can modify this code to generate other types of chords, such as minor chords or seventh chords, and explore the relationships between different chords and scales.
# 11.3. Music Information Retrieval
Music information retrieval (MIR) is a field of study that focuses on extracting meaningful information from music. This can include tasks such as music transcription, genre classification, and similarity analysis. Haskell can be a powerful tool for performing MIR tasks, thanks to its expressive and composable nature.
One popular Haskell library for MIR is Haskore. Haskore provides a set of functions and data types for representing and manipulating musical information. It allows users to parse MIDI files, extract musical features, and perform various analysis tasks.
Here's an example of how Haskore can be used to parse a MIDI file and extract the tempo:
```haskell
import Sound.MIDI.File
import Sound.MIDI.Message.Channel
getTempo :: Midi -> Maybe Int
getTempo midi = case filter isTempoEvent (events midi) of
[] -> Nothing
(TempoEvent _ tempo : _) -> Just tempo
-- Example usage:
-- let midi = importFile "song.mid"
-- getTempo midi => Just 120
```
In this code, the `getTempo` function takes a MIDI file as input and extracts the tempo information from it. Students can use this function to analyze different MIDI files and explore the relationship between tempo and musical characteristics.
## Exercise
Write a Haskell function that calculates the duration of a MIDI file in seconds. Test your function with different MIDI files to calculate their durations.
### Solution
```haskell
import Sound.MIDI.File
getDuration :: Midi -> Double
getDuration midi = fromIntegral (ticks midi) / fromIntegral (division midi) * (60 / getTempo midi)
-- Example usage:
-- let midi = importFile "song.mid"
-- getDuration midi => 240.0
```
Students can modify this code to calculate other musical features, such as the average pitch or the number of notes per second, and explore the relationships between these features and musical characteristics.
# 11.4. Collaborative Music Composition and Production
Collaborative music composition and production is an area where Haskell can shine. Haskell's strong type system and functional programming paradigm make it well-suited for building complex music production systems that can handle multiple contributors and real-time collaboration.
One popular Haskell library for collaborative music composition is TidalCycles. TidalCycles is a live coding environment for music that allows users to create and manipulate patterns in real-time. It provides a rich set of functions and operators for sequencing and transforming musical events.
Here's an example of how TidalCycles can be used to create a simple drum pattern:
```haskell
d1 $ sound "bd sn" # gain "1.5"
```
In this code, the `d1` function represents a drum pattern, and the `sound` function specifies the sounds to be played (in this case, a bass drum and a snare drum). The `gain` function controls the volume of the sounds.
## Exercise
Using TidalCycles, create a pattern that plays a melody using different synthesizer sounds. Experiment with different notes, durations, and effects to create your own unique composition.
### Solution
```haskell
d1 $ s "superpiano" $ n "c e g" # legato 0.5 # room 0.5
```
Students can modify this code to create their own melodies and explore the different sounds and effects available in TidalCycles. They can also experiment with combining multiple patterns and controlling parameters in real-time to create dynamic and interactive compositions. | Textbooks |
\begin{document}
\maketitle \vspace*{-0.5cm} \begin{abstract}
We establish a weighted pointwise Jacobian determinant inequality on corank 1 Carnot groups related to optimal mass transportation akin to the work of Cordero-Erausquin, McCann and Schmuckenschl\"ager. In this setting, the presence of abnormal geodesics does not allow the application of the general sub-Riemannian optimal mass transportation theory developed by Figalli and Rifford and we need to work with a weaker notion of Jacobian determinant. Nevertheless, our result achieves a transition between Euclidean and sub-Riemannian structures, corresponding to the mass transportation along abnormal and strictly normal geodesics, respectively. The weights appearing in our expression are distortion coefficients that reflect the delicate sub-Riemannian structure of our space. As applications, entropy, Brunn-Minkowski and Borell-Brascamp-Lieb inequalities are established on Carnot groups. \end{abstract}
\vspace*{0.5cm}
\noindent {\it Keywords}: Carnot group; Jacobian determinant inequality; optimal mass transportation; abnormal and normal geodesics; entropy inequality; Brunn-Minkowski inequality; Borell-Bras\-camp-Lieb inequality.\\
\noindent {\it MSC}: 53C17, 35R03, 49Q20.
\section{Introduction} As a general framework of our results, let $(X,d,\textsf{m})$ be a suitably regular geodesic metric measure space with topological dimension $N\in \mathbb N$ where the theory of optimal mass transportation can be successfully developed. Examples for such spaces include Riemannian and Finsler manifolds, see McCann \cite{McCann-GAFA} and Ohta \cite{Ohta}, the Heisenberg group $\mathbb H^n$, see Ambrosio and Rigot \cite{AR}, or even more general sub-Riemannian structures with 'well-behaved' cut locus, see Figalli and Rifford \cite{FR}. Let $\mu_0$ and $\mu_1$ be two probability measures on $X$ which are absolutely continuous w.r.t. the reference measure $\textsf{m}$, and let $\mu_s=(\psi_s)_\#\mu_0,$ $s\in [0,1],$ be the unique displacement interpolation measure joining $\mu_0$ and $\mu_1$ throughout the so-called $s$-intermediate optimal transport map $\psi_s:X\to X$. Roughly speaking, for $s\in (0,1)$ fixed, the Jacobian determinant inequality reads as \begin{equation}\label{Jacobian-000} \left({\rm Jac}(\psi_s)(x)\right)^\frac{1}{N}\geq \tau_{1-s}^{N}(\theta_x)+\tau_{s}^{N}(\theta_x)\left({\rm Jac}(\psi)(x)\right)^\frac{1}{N}\mbox{ for } \mu_0 \mbox{-a.e. } x \in X. \end{equation}
Here, {and in the sequel} ${\rm Jac}(\psi_s)(x)$ and ${\rm Jac}(\psi)(x)$ are interpreted as {densities, or} the Radon-Nikodym derivatives of $\mu_s$ and of $\mu_1$ w.r.t. the reference measure $\textsf{m}.$ {Note that in case when $X= \mathbb{R}^n$ and $\psi_s$ is differentiable at $x$ the term ${\rm Jac}(\psi_s)(x)$ can be computed as ${\rm Jac}(\psi_s)(x)= |\det D \psi_s(x) |$. On the other hand, the Jacobian determinant in the above sense might exist as density even in the case when $\psi_s$ is not differentiable.} The expression $\tau_{s}^{N}$ is the distortion coefficient which encodes information on the geometric structure of the space $X$. Expressions of $\tau_{s}^{N}$ can be calculated in terms of the Jacobian of the exponential map or estimated in terms of a curvature condition. The expression $\theta_x$ can be given as a function of $d(x,\psi(x))$ or its derivatives.
The Jacobian determinant inequality (\ref{Jacobian-000}) in the above general form has been considered first in the setting of complete Riemannian manifolds (endowed with the natural Riemaniann distance and volume form) in the pioneering work of Cordero-Erausquin, McCann and Schmuckenschl{\"a}ger \cite{CMS}. This result constituted the starting point of an extensive study of the geometry of metric measure spaces, while relation (\ref{Jacobian-000}) became an equivalent formulation of the famous curvature-dimension condition $CD(K,N)$, due to Lott and Villani \cite{LV}, and Sturm \cite{Sturm1, Sturm2}, where $\tau_{s}^{N}$ is replaced by explicit expressions $\tau_{s}^{N, K}$, $K$ being the lower bound of the Ricci curvatures in the Riemannian setting. Namely, $\tau_{s}^{N, K}$ is given by {\small
$$\tau_s^{K,N}(\theta)=\left\{
\begin{array}{lll}
s^\frac{1}{N}\left(\sinh\left(\sqrt{-\frac{K}{N-1}}s\theta\right)\big/\sinh\left(\sqrt{-\frac{K}{N-1}}\theta\right)\right)^{1-\frac{1}{N}}&
{\rm if} & K\theta^2<0;\\
s & {\rm if} & K\theta^2=0;\\
s^\frac{1}{N}\left(\sin\left(\sqrt{\frac{K}{N-1}}s\theta\right)\big/\sin\left(\sqrt{\frac{K}{N-1}}\theta\right)\right)^{1-\frac{1}{N}}& {\rm if} &
0<K\theta^2<(N-1)\pi^2;\\
+\infty & {\rm if} & K\theta^2\geq (N-1)\pi^2,
\end{array}
\right.$$} and $\theta=\theta_x$ is precisely the Riemannian distance $ d(x,\psi(x))$.
Juillet \cite{Jui} proved that the Lott-Sturm-Villani curvature-dimension condition does not hold for any pair of parameters $(N,K)$ on the Heisenberg group $\mathbb H^n$ (endowed with its usual Carnot-Carath\'eo\-do\-ry metric $d_{CC}$ and $\mathcal L^{2n+1}$-measure), which is the simplest sub-Riemannian structure. Accordingly, there were strong doubts on the validity of a sub-Riemannian version of the Jacobian determinant inequality in the sub-Riemannian context. However, by using a natural Riemannian approximation of the Heisenberg group as in Ambrosio and Rigot \cite{AR}, the authors of the present paper proved (\ref{Jacobian-000}) on $\mathbb H^n$, see \cite{BKS1,BKS}, where the Heisenberg distortion coefficient
$\tau_s^{2n+1}:[0,2\pi]\to [0,\infty]$ is defined by
\begin{eqnarray}\label{concentration}
\tau_s^{2n+1}(\theta) = \left\{
\begin{array}{lll}
{s^\frac{1}{2n+1}}
\left(\frac{\sin\frac{\theta s}{2}}{\sin\frac{\theta }{2}}\right)^\frac{2n-1}{2n+1}\left(\frac{\sin\frac{\theta s}{2}-\frac{\theta s}{2}\cos\frac{\theta s}{2}}{\sin\frac{\theta }{2}-\frac{\theta }{2}\cos\frac{\theta
}{2}}\right)^\frac{1}{2n+1}
\ &\mbox{if} & \theta\in( 0,2\pi); \\
s^\frac{2n+3}{2n+1} &\mbox{if} & \theta=0;\\%\ {\rm or} \theta=2\pi.
+\infty &\mbox{if} & \theta=2\pi,\\
\end{array}\right.
\end{eqnarray}
and $\theta = \theta_x$ is the 'vertical' derivative of $\frac{d_{CC}^2(\psi(x),\cdot)}{2}$ at the point $x$.
In the present paper we prove a Jacobian determinant inequality on corank 1 Carnot groups where the sub-Riemannian geometry is more complicated than the one of the model Heisenberg group $\mathbb H^n$ due to the presence of {abnormal} geodesics and the 'anisotropic' structure of the cut locus. Our method is different from the one in \cite{BKS1, BKS} as we obtain the Jacobian determinant inequality by an intrinsic approach, without using a Riemannian approximation. As in \cite{BKS1, BKS}, we apply our Jacobian determinant inequality to establish various functional and geometric inequalities in the present setting including entropy, Brunn-Minkowski and Borell-Brascamp-Lieb inequalities. These results should open up the way to considering the above inequalities in a broader context outside the realm of $CD(K,N)$-type conditions by replacing the coefficients $\tau_{s}^{N, K}$ by expressions that are suitable for sub-Riemannian geometries. {In this way, our results motivate the so-called "grande unification" of the three main geometries (Riemannian, Finslerian and sub-Riemannian), suggested by C. Villani in \cite[p. 43]{Villani-2}.}
In order to present our main result, let us fix some notation. We denote by $G$ a $k+1$ dimensional corank 1 Carnot group with its Lie algebra $\mathfrak g=\mathfrak g_1\oplus \mathfrak g_2$, where dim$\mathfrak g_1=k\geq 2$ and dim$\mathfrak g_2=1$. The operation on $\mathfrak g$ (in exponential coordinates on $\mathbb R^k\times \mathbb R$) can be given by $$x\circ y=\left(x_1+y_1,...,x_k+y_k,x_z+y_z-\frac{1}{2}\sum_{i,j=1}^k \mathcal A_{ij}x_jy_i\right ),$$ where $x=(x_1,...,x_k,x_z)$, $y=(y_1,...,y_k,y_z)$, and $\mathcal A=[\mathcal A_{ij}]$ is a $k\times k$ real skew-symmetric matrix. Let $e=(0_{\mathbb R^k},0)\in \mathbb R^k\times \mathbb R$ be the neutral element in $(G,\circ).$ The layers $\mathfrak g_1$ and $\mathfrak g_2$ are generated by the left-invariant vector fields \begin{equation}\label{vector-field} X_i=\partial_{x_i}-\frac{1}{2}\sum_{j=1}^k \mathcal A_{ij}x_j\partial_{z},\ \ i=1,...,k. \end{equation} Moreover, $[X_i,X_j]=\mathcal A_{ij}\partial_{ z}.$
By the spectral theorem for skew-symmetric matrices one can consider the diagonalized representation of $\mathcal A$ given by \begin{equation}\label{matrix-representation} \mathcal A=\left[ \begin{array}{cccc} \mathbb{0}_{k-2d}& & & \bigzero\\ & \alpha_1 J & & \\
\bigzero & & \ddots \\ & & & \alpha_d J \end{array} \right],\ \ \ J=\left[\begin{matrix} 0 & 1\\ -1& 0 \end{matrix}\right], \end{equation} where $0<\alpha_1\leq ...\leq \alpha_d,$ and $\mathbb{0}_{k-2d}$ is the $(k-2d)\times (k-2d)$ square null-matrix; {from now on, we assume the matrix $\mathcal A$ has this representation. }
For further use, let us introduce the functions $\mathbb d_1, \mathbb d_2:[0,2\pi]\times(0,1)\to \mathbb R$ given by $$\mathbb d_1(t,s)=\frac{\sin(ts/2)}{s}\ \ {\rm and}\ \ \mathbb d_2(t,s)=\frac{\sin(ts/2)-ts/2\cos(ts/2)}{s}.$$ To define the distortion coefficient, we introduce the set
$$D = \left\{p = (p_x,p_z) \in \mathbb{R}^{k+1} : |p_z| < \frac{2\pi}{\alpha_d} \mbox{ and } \mathcal Ap_x \neq 0_{\mathbb R^k}\right\} \subset T_e^* G,$$
where $p_x=(p_x^0,p_x^1,...,p_x^d)\in \mathbb R^{k-2d}\times \mathbb R^2\times...\times \mathbb R^2$, and let
$\overline D$ be the closure of $D$. The {\it distortion coefficient $\tau_s^{k,\alpha}:\overline D\to \mathbb R$ on the Carnot group} $(G,\circ)$ is defined by {\small \begin{eqnarray*}
\tau_s^{k,\alpha}(p)= \left\{
\begin{array}{lll}
s\left(\frac{\displaystyle\sum_{i=1}^d\|p_x^i\|^2
\displaystyle\prod_{j\neq i}\mathbb d_1^2(\alpha_j p_z,s)\mathbb d_1(\alpha_i p_z,s)\mathbb d_2(\alpha_i p_z,s)}{\displaystyle\sum_{i=1}^d\|p_x^i\|^2
\displaystyle\prod_{j\neq i}\mathbb d_1^2(\alpha_j p_z,1)\mathbb d_1(\alpha_i p_z,1)\mathbb d_2(\alpha_i p_z,1)}\right)^\frac{1}{k+1}
\ &\mbox{if} & p\in D \ \& \ p_z\neq 0; \\
s^\frac{k+3}{k+1} &\mbox{if} & p\in D \ {\rm \&}\ p_z= 0;\\
+\infty&\mbox{if} & \mathcal Ap_x\neq 0_{\mathbb R^k} \ {\rm \&}\ |p_z|=\frac{2\pi}{\alpha_d};\\
s&\mbox{if} & \mathcal Ap_x=0_{\mathbb R^k},
\end{array}\right.
\end{eqnarray*} }
where $p=(p_x,p_z)$ and $\alpha=(\alpha_1,...,\alpha_d).$ {The functions $\mathbb d_1$ and $\mathbb d_2$ appear explicitly in the Jacobian of the exponential map, see (\ref{Jacobian-Juillet}) below. In fact, $\mathbb d_2$ is a typical sub-Riemannian function appearing once after differentiating the exponential map along the 'vertical' direction, while $\mathbb d_1$ appears on the diagonal of the Jacobian matrix with multiplicity $2d-1$, see also Rizzi \cite{Rizzi}. }
Let us consider two compactly supported probability measures $\mu_0$ and $\mu_1$ on $G$ which are absolutely continuous w.r.t. $\mathcal{L}^{k+1}$. Since the distribution $\Delta=\{X_1,...,X_k\}$ on the corank 1 Carnot group $G$ is two-generating, there exists a unique map realizing the optimal transportation between the measures $\mu_0$ and $\mu_1$ w.r.t. the cost function ${d^2_{CC}}/{2}$, see Figalli and Rifford \cite[Proposition 4.2 and Theorem 3.2]{FR}; this map can be defined $\mu_0$-a.e. through a $d_{CC}^2/2$-concave function $\varphi:G\to \mathbb R$ as {\begin{eqnarray} \label{DefIntMapH}
\psi(x):=
\left\{
\begin{array}{lll}
\exp_x(-\nabla\varphi(x))
\ &\mbox{if} & x\in \mathcal M_\varphi\cap {\rm supp}(\mu_0); \\
x &\mbox{if} & x\in \mathcal S_\varphi\cap {\rm supp}(\mu_0).
\end{array}\right.
\end{eqnarray} Hereafter, $d_{CC}$ is the Carnot-Carath\'eo\-do\-ry metric on $G$ and the sets $\mathcal M_\varphi$ and $\mathcal S_\varphi$ denote the moving and static sets of the transportation, respectively; see Section \ref{SecPrelim} for details. For $s\in (0,1)$ fixed, we also introduce the $s$-interpolant optimal transport map as {\begin{eqnarray} \label{DefIntMapH-2}
\psi_s(x):=
\left\{
\begin{array}{lll}
\exp_x(-s\nabla\varphi(x))
\ &\mbox{if} & x\in \mathcal M_\varphi\cap {\rm supp}(\mu_0); \\
x &\mbox{if} & x\in \mathcal S_\varphi\cap {\rm supp}(\mu_0).
\end{array}\right.
\end{eqnarray}
Our main result reads as follows.
\begin{theorem}\label{TJacobianDetIneq}{\bf (Jacobian determinant inequality on Carnot groups)} Let $(G,\circ)$ be a $k+1$ dimensional corank 1 Carnot group, and assume that $\mu_0$ and $\mu_1$ are two compactly supported Borel probability measures on $G$, both absolutely continuous w.r.t. $\mathcal L^{k+1}$. Let $s \in (0,1)$ be fixed, $\psi:G\to G$ be the unique optimal transport map transporting $\mu_0$ to $\mu_1$ associated to the cost function $\frac{d^2_{CC}}{2}$ and $\psi_s$ its $s$-interpolant map. Then the following Jacobian determinant inequality holds
\begin{equation}\label{Jacobi-inequality-elso}
\left({\rm Jac}(\psi_s)(x)\right)^\frac{1}{k+1}\geq \tau_{1-s}^{k,\alpha}(\theta_x)+\tau_{s}^{k,\alpha}(\theta_x)\left({\rm Jac}(\psi)(x)\right)^\frac{1}{k+1}\mbox{ for } \mu_0 \mbox{-a.e. } x \in G,
\end{equation}
where $\theta_x=(p_x,p_z)\in T_e^*G$ is given by $\exp_e(\theta_x)=x^{-1}\circ \psi(x).$
\end{theorem}
Let us notice that if $p=(p_x,p_z)\in D$, we have that
$$\lim_{p_z\to 0}\tau^{k,\alpha}_s(p)=s^{\frac{k+3}{k+1}}\ {\rm and}\ \lim_{p_z\to \pm2\pi/\alpha_d}\tau^{k,\alpha}_s(p)=+\infty.$$ Furthermore, monotonicity properties of the functions $\mathbb d_1$ and $\mathbb d_2$ {(cf. \cite[Lemma 2.1]{BKS})} show that
\begin{eqnarray}\label{tau-lower-bound}
\tau^{k,\alpha}_s(p)\geq s^{\frac{k+3}{k+1}}\ {\rm for\ all}\ s\in (0,1),\ p\in \overline D.
\end{eqnarray} Therefore, the {\it measure contraction property} {\rm{\textsf{ MCP}}}$(0,k+3)$ proved by Rizzi \cite{Rizzi} is formally a consequence of (\ref{Jacobi-inequality-elso}). Notice, however that we use Rizzi's result to prove the absolute continuity of the interpolant measure $\mu_s=(\psi_s)_\#\mu_0$ (see Proposition \ref{interpolant-absolute-cont}), needed in the proof of the Jacobian determinant inequality.
In our next remark we consider the situation when $G=\mathbb H^n$ is the $n$-dimensional Heisenberg group. In this case we have $k=2n=2d$ and $\alpha_i=4$ for every $i\in \{1,...,d\}$. Moreover, no abnormal geodesics appear in $\mathbb H^n$ and the Carnot distortion coefficient $\tau_s^{2n,\alpha}(p_x,p_z)$ reduces to the Heisenberg distortion coefficient $\tau_s^{2n+1}(4p_z),$ which is nothing but relation \eqref{concentration} (introduced in \cite{BKS}). Thus, most of the results of \cite{BKS} will be covered in the present work.
Let us notice furthermore, that in general corank 1 Carnot groups, the coefficients $\tau^{k,\alpha}_s$ and $\tau^{k,\alpha}_{1-s}$ depend not only on the parameter $p_z$ (as in the Heisenberg group) but also on $\|p_x^i\|$, $i\in \{1,...,d\}$, showing a more anisotropic character of the present geometric setting as compared to the Heisenberg group. As we shall see later, $\|p_x^i\|$ and $p_z$ can be obtained by differentiating $\frac{d_{CC}^2(\psi(x),\cdot)}{2}$ at the point $x$ w.r.t. the horizontal vector fields from the distribution $\Delta$ and the vertical vector field $\partial_z$, respectively (see Lemma \ref{LemmaFirstDeriv} below).
Our final remark is of technical nature, but the details will be clear by reading the proof of Theorem \ref{TJacobianDetIneq}. In this proof, we shall distinguish the cases when the mass is transported along \textit{{abnormal}} and \textit{{strictly normal}} geodesics, respectively. On one hand, when the mass transport is realized along {\it abnormal} geodesics, it turns out that the Jacobian determinant inequality reduces to an {\it Euclidean-type} determinant inequality thus the distortion coefficient can be $\tau^{k,\alpha}_s=s$ as in the Euclidean framework. We notice that in this case the full Jacobian matrix of $\psi_s$ might not exist; however, since the matrix has a triangular structure, the Jacobian can be reduced to two parts of the diagonal which are well defined and inequality (\ref{Jacobi-inequality-elso}) makes sense. Furthermore, the triangular structure of the Jacobi matrix will allow us to perform the necessary changes of variable in order to provide important applications (see e.g. the entropy and Borell-Brascamp-Lieb inequalities via a suitable Monge-Amp\`ere equation). On the other hand, once the mass transport is along {\it strictly normal} geodesics, the distortion coefficient $\tau^{k,\alpha}_s$ encodes information on the genuine {\it sub-Riemannian} character of the Carnot group obtained by a careful analysis of the Jacobian for the exponential map. It could also happen that a positive part of the mass is transported along abnormal geodesics while the complementary mass is transported by strictly normal geodesics, so different formulas for $\tau^{k,\alpha}_s$ will be used in the same instance of the mass transportation; such a scenario will be presented in Example \ref{example} (see also Figure \ref{abra-elso}). {In conclusion, our results can be applied also in the presence of both abnormal and strictly normal geodesics in the so-called \textit{non-ideal} sub-Riemannian setting. Similar result in the case of general \textit{ideal} sub-Riemannian geometries have been recently obtained by Barilari and Rizzi \cite{Barilari-Rizzi}.}
The organization of the paper is as follows. The proof of Theorem \ref{TJacobianDetIneq} will be provided in Section \ref{SecProof} after a self-contained presentation of the needed technical details in Section \ref{SecPrelim}, i.e., properties of the Carnot-Carath\'eodory metric $d_{CC}$, exponential map and its Jacobian, the cut locus, and the optimal mass transportation on corank 1 Carnot groups. We emphasize that the optimal mass transportation developed by Figalli and Rifford \cite{FR} for large classes of sub-Riemannian manifolds cannot be directly applied since the squared distance function $d_{CC}^2$ is not necessarily locally semiconcave outside of the diagonal of $G\times G$ which is crucial in \cite{FR} (e.g. the regularity of optimal mass transport maps $\psi$ and $\psi_s$, or the validity of the Monge-Amp\`ere equation).
Section \ref{SecApps} is devoted to applications, i.e., by the Jacobian determinant inequality we shall derive entropy inequalities, the Brunn-Minkovski inequality and the Borell-Brascamp-Lieb inequality on corank 1 Carnot groups.
\noindent {\bf Acknowledgements.} We express our gratitude to Luca Rizzi for motivating conversations about the subject of this paper. A. Krist\'aly is grateful to the Mathematisches Institute of Bern for the warm hospitality where this work has been developed. We also wish to thank the anonymous referees for their detailed reports and valuable comments that greatly improved the presentation of the manuscript.
\section{Preliminaries} \label{SecPrelim}
\subsection{Carnot-Carath\'eodory metric and energy functional on {corank 1} Carnot groups}
{We shall consider a corank 1 Carnot group $(G,\circ)$, and make use of the notations already introduced in the previous section.} A horizontal curve on $(G,\circ)$ is an absolutely continuous curve $\gamma: [0, r] \to G$ for which there exist {bounded} measurable functions ${u}_j: [0,r] \to \mathbb{R}$ ($j = 1, ..., k$) such that \begin{equation} \label{horiz} \dot{\gamma}(s) = \sum\limits_{j=1}^k {u_j}(s) X_j(\gamma(s)) \quad \mbox{a.e. } s \in [0,r]. \end{equation} In the sequel we denote by $\gamma_u$ such a horizontal curve. The length of this curve is given by
$$l(u) = l(\gamma_u) = \int\limits_{0}^{r} \|\dot{\gamma_u}(s)\| {\rm d} s= \int\limits_{0}^{r} \sqrt{\sum\limits_{j = 1}^{k} {u^2_j}(s) } {\rm d} s.$$ The classical Chow-Rashewsky theorem assures that any two points from the Carnot group can be joined by a horizontal curve. Thus we can equip the Carnot group $G$ with its natural Carnot-Carath\'eodory metric by $$d_{CC}(x,y) = \inf \{l(\gamma): \gamma \mbox{ is a horizontal curve joining } x \mbox{ and } y\},$$ where $x,y\in G$ are arbitrarily fixed.
Let $e=(0_{\mathbb R^k},0) \in \mathbb{R}^k \times \mathbb{R}$ be the neutral element in $(G,\circ)$. The left invariance of the vector fields in the distribution $\Delta=\{X_1, \dots, X_k\}$ is inherited by the distance $d_{CC}$, thus $$d_{CC}(x, y) = d_{CC}(e, x^{-1} \circ y) \quad {\rm for\ every}\ x,y \in G.$$
{Beside the length function $u\mapsto l(\gamma_u)$ we also consider the energy functional
$$J(u) =\frac{1}{2} \int\limits_{0}^{r} \|\dot{\gamma_u}(s)\|^2 {\rm d} s= \int\limits_{0}^{r} {\sum\limits_{j = 1}^{k} {u^2_j}(s) } {\rm d} s.$$ It is well-known that the minimisers of $J$ induce {up to a reparametrisation} length minimising horizontal curves with constant speed between two fixed endpoints.}
\subsection{Geodesics, exponential map and its Jacobian}
{Geodesics are horizontal curves that are locally energy minimizers between their endpoints. Let $\mathcal U\subset L^\infty([0,r],\mathbb R^k)$ be an open set and for a fixed $x\in G$, let $E_x:\mathcal U\to G$ be the usual end-point map, $E_x(u)=\gamma_u(r)$, where $\gamma_u$ is the unique curve with the property that $\gamma_u(0)= x$ and satisfying \eqref{horiz} see e.g. Figalli and Rifford \cite[\S 2.1]{FR}. A minimizing geodesic $\gamma_u $ for $ u\in \mathcal U$ is a solution of the problem $$J(v)\to\min,\ \ E_x(v)=y,\ \ v\in \mathcal U.$$ According to the Lagrange multipliers rule, there is $(\lambda,\mu)\in T^*_yG\times \{0,1\}\setminus \{ (0,0)\}$ such that $$\lambda (D_uE_x)=\mu D_uJ.$$ The associated curve $\gamma_u$ is normal if $\mu=1$ and abnormal if $\mu=0$ (the latter being equivalent to the fact that $u$ is a critical point of $E_x$). We notice that on any corank 1 Carnot group all minimizing geodesics are normal. Following Rizzi \cite{Rizzi}, the explicit form of such {normal} minimal geodesics can be described as follows.}
\begin{proposition} {\rm (Rizzi \cite{Rizzi})}\label{proposition-geodetikus} On a corank $1$ Carnot group $(G,\circ)$ the geodesic $s\mapsto \exp_e(sp)\in G$ starting from $e=(0_{\mathbb R^k},0)$, with {initial covector} $$p=(\underbrace{p_x^0, p_x^1, ..., p_x^d}_{p_x},p_z) \in \left(\mathbb{R}^{k-2d} \times \mathbb{R}^2 \times ... \times \mathbb{R}^2\right) \times \mathbb{R}= T_e^*G $$ has the following equation \begin{eqnarray}\label{GeodesicEqOnCarnotGroup} \exp_e(sp): \left\{ \begin{array}{l} \gamma^0(s) = p_x^0 s,\\ \gamma^i(s) = \left(\frac{\sin(\alpha_ip_z s)}{\alpha_i p_z}I + \frac{\cos(\alpha_i p_z s) -1}{\alpha_i p_z}J\right) p_x^i,\\
\gamma_z(s) = \sum_{i=1}^d \|p_x^i\|^2 \frac{\alpha_i p_z s - \sin(\alpha_i p_z s)}{2 \alpha_i p_z^2}, \end{array} \right. \quad s \in [0,1], \end{eqnarray} when $p_z\neq 0$. When $p_z = 0$, the geodesic is \begin{eqnarray}\label{GeodesicEqNullpz} s\mapsto \exp_e(sp)=(p_x^0s, p_x^1s, ..., p_x^d s,0), \quad s \in [0,1]. \end{eqnarray} Hereafter, $I$ denotes the $2\times 2$ unit matrix { and $J=\left[\begin{matrix}
0 & 1\\
-1& 0 \end{matrix}\right]. $} \end{proposition}
{Once $\mathcal A$ has a non-trivial kernel, every nonzero covector $(p_x,p_z)$ with $\mathcal Ap_x=0$, corresponds to an abnormal geodesic; more precisely, for every choice of $p_z\in \mathbb R$ one has \begin{eqnarray}\label{AbnGeodesicEq} s\mapsto \exp_e(p_x^0s,0,...,0,p_zs)=(p_x^0s,0_{\mathbb R^{2d+1}}),\ s\in [0,1]. \end{eqnarray} Note that the image of such a geodesic can be also obtained by (\ref{GeodesicEqNullpz}), letting $p_z = 0$ and $p_x^1 = ... = p_x^d = 0_{\mathbb R^2}$. {These type of geodesics are normal and also abnormal at the same time}. It turns out that all abnormal geodesics have this representation.}
We recall from Rizzi \cite{Rizzi} that the Jacobian determinant of the exponential map is \begin{eqnarray}\label{Jacobian-Juillet} {\rm Jac}(\exp_e)(p) = \left\{ \begin{array}{lll}
\displaystyle\frac{2^{2d}}{\prod_{i=1}^d \alpha_i^2 p_z^{2d+2}} \displaystyle\sum_{i=1}^d\|p_x^i\|^2 \displaystyle\prod_{j\neq i}\left({\sin\frac{\alpha_j p_z }{2}}\right)^2\sin\frac{\alpha_i p_z }{2}\times\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times\left({\sin\frac{\alpha_i p_z }{2}-\frac{\alpha_i p_z }{2}\cos\frac{\alpha_i p_z }{2}}\right) \ &\mbox{if} & p_z\neq 0; \\ & & \\
\frac{1}{12}{\displaystyle\sum_{i=1}^d\|p_x^i\|^2}\alpha_i^2 &\mbox{if} & p_z=0. \end{array}\right. \end{eqnarray}
\noindent By left-invariance, the minimal geodesics on $G$ starting from an arbitrary point $x\in G$ are represented by $s\mapsto \exp_x(sp)=x\circ \exp_e(s\tilde p)$, $s\in [0,1],$ where the two covectors $p\in T_x^*G$ and $\tilde p\in T_e^*G$ can be identified. Moreover, since for every $x\in G$ the left-translation $L_x(y)=x\circ y$, $y\in G,$ is a volume-preserving map, it follows that \begin{equation}\label{Jacobi-left} {\rm Jac}(\exp_x)(p)={\rm Jac}(\exp_e)(p)\ {\rm for\ every}\ p\in T_x^*G. \end{equation}
Given $x,y\in G$ and assume that $x=\exp_y(p)$ for some $p=(p_x,p_z)=({p_x^0, p_x^1, ..., p_x^d},p_z)\in T_y^*G$. Then $y=\exp_x(\overline p)$, where $\overline p=(\overline p^0_x,\overline p^1_x,...,\overline p^d_x,\overline p_z)$ is given by \begin{equation}\label{reverse-repres} \left\{ \begin{array}{lll} \overline p^0_x=- p_x^0;\\ \overline p^i_x=\left(-\cos(\alpha_i p_z)I+\sin(\alpha_ip_z)J\right) p_x^i,\ \ i\in \{1,...,d\};\\ \overline p_z=- p_z. \end{array} \right. \end{equation}
We notice that $\Delta=\{X_1,...,X_{k}\}$ is {\it not} a fat distribution whenever the kernel of $\mathcal A$ is non-trivial. Indeed, in this case we have $T_xG\neq \Delta(x)+[X_j,\Delta](x)$ for every $x\in G$ and $j\in \{1,...,k-2d\}$. However, $\Delta$ is two-generating, i.e., $$T_xG= \Delta(x)+[\Delta,\Delta](x)\ {\rm for\ every}\ x\in G.$$
For simplicity of notation, we reorganize the vector fields in $T_xG$ as
\begin{equation}\label{vector-fields}
\left\{
\begin{array}{lll}
X^0=(X_1,..., X_{k-2d});\\
X^i=(X_{k-2d+2i-1}, X_{k-2d+2i}),\ \ i\in \{1,...,d\};\\
Z=\partial_z.
\end{array}
\right.
\end{equation}
We split the distribution $\Delta$ on $G$ into two types of vector fields; namely, $\Delta_0=\{X^0\}$ and $\tilde \Delta=\{X^{1},...,X^{d}\}$. This splitting gives the following trivial representation of the distance function $d_{CC}$:
\begin{lemma}{\rm (Pythagorean rule)} \label{LemmaPythagorean}
For every $(\xi,\eta,z), (\overline \xi,\overline \eta,\overline z)\in \mathbb R^{k-2d}\times \mathbb R^{2d}\times \mathbb R$ , we have
$$d_{CC}^2((\xi,\eta,z), (\overline \xi,\overline \eta,\overline z))=d^2_{\mathbb R^{k-2d}}(\xi,\overline \xi)+\tilde d_{CC}^2((\eta,z), (\overline \eta,\overline z)),$$ where $d_{\mathbb R^{k-2d}}$ is the Euclidean metric in $\mathbb R^{k-2d}$ while $\tilde d_{CC} $ is the Carnot-Carath\'eodory distance on $\mathbb R^{2d}\times \mathbb R$ w.r.t. to the distribution $\tilde \Delta$ inherited from the original sub-Riemannian structure.
\end{lemma} {\it Proof.}
By the left-invariance of the metric $d_{CC}$, we have
$$d_{CC}^2((\xi,\eta,z), (\overline \xi,\overline \eta,\overline z))=d_{CC}^2(e, (-\xi, -\eta, -z) \circ (\overline \xi, \overline \eta,\overline z)).$$
Let $\gamma=(\gamma^0,\gamma^1,...,\gamma^d,\gamma_z):[0,1]\to G$ be the geodesic given by (\ref{GeodesicEqOnCarnotGroup}) or (\ref{GeodesicEqNullpz}) joining $e$ and the element $(-\xi, -\eta, -z) \circ (\overline \xi, \overline \eta,\overline z)$, having its initial vector $p=(p_x^0,p_x^1,...,p_x^d,p_z)\in \mathbb R^{k-2d}\times \mathbb R^{2}\times \cdots \times \mathbb{R}^2 \mathbb \times \mathbb R$. We have that $d_{CC}^2((\xi,\eta,z), (\overline \xi,\overline \eta,\overline z))=\sum_{i=0}^d \|p_x^i\|^2.$
Note that $\|p_x^0\|_{\mathbb R^{k-2d}}=d_{\mathbb R^{k-2d}}(\xi,\overline \xi)$ and $$\sum_{i=1}^d \|p_x^i\|^2= d_{CC}^2(e, (0_{\mathbb R^{k-2d}}, -\eta, -z) \circ (0_{\mathbb R^{k-2d}}, \overline \eta,\overline z))=\tilde d_{CC}^2( (\eta, z), (\overline \eta,\overline z))$$ which is realized precisely by the geodesic $\tilde \gamma=(\gamma^1,...,\gamma^d,\gamma_z)$, concluding the proof.
$\square$
\subsection{Cut locus} \label{SubseqCut}
Let us consider the set
$$D = \left\{p = (p_x,p_z) \in \mathbb{R}^{k+1} : |p_z| < \frac{2\pi}{\alpha_d} \mbox{ and } \mathcal Ap_x \neq 0_{\mathbb R^k}\right\} \subset T_e^* G.$$
Rizzi \cite[Lemma 16]{Rizzi} proved that $D$ is precisely the injectivity domain of parameters associated to geodesics joining the origin $e$ {to almost all points of} $G$. We know that all points in the corank 1 Carnot group $G$ can be reached by a minimal normal geodesic; namely, for every $x \in G$ there exists a parametrization $p$ in the closure of $D$, i.e.,
$$\overline{D} = \left\{p = (p_x,p_z) \in \mathbb{R}^{k+1} : |p_z| \leq \frac{2\pi}{\alpha_d}\right\},$$
which defines a minimal normal geodesic joining $e$ and $x$.
The {\it cut locus} of the origin $e$ in $G$ is
\begin{eqnarray*}
{\rm cut}_G(e) &=& \exp_e(\overline{D} \setminus D)=G\setminus \exp_e(D)\\
&=&\left(\mathbb R^{k-2d}\times \{0_{\mathbb R^{2d+1}}\}\right)\cup \left\{\exp_e\left(p_x,\pm\frac{2\pi}{\alpha_d}\right):\mathcal Ap_x\neq 0_{\mathbb R^k}\right\}.
\end{eqnarray*}
The set $\mathbb R^{k-2d}\times \{0_{\mathbb R^{2d+1}}\}$ in the above representation corresponds to the image of abnormal geodesics while the latter set contains the conjugate points to $e$, see (\ref{Jacobian-Juillet}).
{Corank 1 Carnot groups have negligible cut loci, see Rizzi \cite[Section 1.4]{Rizzi}; alternatively, due to (\ref{GeodesicEqOnCarnotGroup}), one has that ${\rm cut}_G(e)\subset \mathbb R^{k-2}\times 0_{\mathbb R^2}\times \mathbb R$ , thus $\mathcal L^{k+1}({\rm cut}_G(e))=0$.} By left-invariance, the cut locus of the point $x\in G$ is $${\rm cut}_G(x) = L_x( {\rm cut}_G(e)),$$
thus ${\rm cut}_G(x)$ is closed and $\mathcal L^{k+1}({\rm cut}_G(x))=0$ for every $x\in G;$ moreover, by (\ref{reverse-repres}) it follows that $y\in {\rm cut}_G(x)$ if and only if $x\in {\rm cut}_G(y)$.
{The following two results are specifications to the case of corank 1 Carnot groups of the well-known fact $f:=\frac{1}{2}d_{CC}^2(y,\cdot)$ is smooth in a neighborhood
of $x\in G$ whenever $x\notin {\rm cut}_G(y)$, and one can recover the initial covector $\lambda_x\in T_xG$ of the unique
geodesic joining $x$ with $y$ by $\lambda_x=-\nabla f(x)$. }
\begin{lemma}\label{LemmaFirstDeriv}
Fix $y\in G$ and let $x=(x^0,x^1,...,x^d,z)\notin {\rm cut}_G(y)$. If $x=\exp_y(p_x^0,p_x^1,...,p_x^d,p_z)$ then we have
\begin{itemize}
\item[(i)] $X^0\frac{d_{CC}^2(y,\cdot)}{2}\big|_{x}=p_x^0 $ and $Z \frac{d_{CC}^2(y,\cdot)}{2}\big|_{x}=p_z;$
\item[(ii)] for every $i\in \{1,...,d\}$,
\begin{equation}\label{derivalt-1}
X^{i} \frac{d_{CC}^2(y,\cdot)}{2}\big|_x=[\cos(\alpha_i p_z)I-\sin(\alpha_i p_z)J]p_x^i.
\end{equation} \end{itemize}
\end{lemma}
{\it Proof.} By exploring the left-invariance, it is enough to consider the case when $y=e$. Let us introduce the auxiliary functions $f,g:(-2\pi,2\pi)\setminus\{0\}\to \mathbb R$ defined by
\begin{equation}\label{fesg}
f(t)=\frac{\sin^2\left(\frac{t}{2}\right)}{\left(\frac{t}{2}\right)^2}\ \ {\rm and}\ \ g(t)=\frac{t-\sin(t)}{\sin^2\left(\frac{t}{2}\right)},\ t\in (-2\pi,2\pi) \setminus\{0\}.
\end{equation}
We consider the case when $p_z\neq 0;$ the case $p_z=0$ can be obtained by a limiting procedure, {i.e., one must consider the limit $p_z\to 0$.} Since $x\notin {\rm cut}_G(e)$ and the cut locus is closed, there exists a small neighborhood $V_x$ of $x$ such that $V_x\cap {\rm cut}_G(e)=\emptyset$. Let $w=({x_w^0,x_w^1,...,x_w^d},z_w)={\exp_e\left((p_w)^0_x,(p_w)^1_x,...,(p_w)^d_x,(p_w)_z\right)}\in V_x$ be arbitrarily fixed. By (\ref{GeodesicEqOnCarnotGroup}) (for $s=1$) we have that
$$\|{x_w^i}\|^2={\|{(p_w)^i_x}\|^2}{f(\alpha_i {(p_w)_z})},\ i\in \{1,...,d\}.$$
Thus, one has
\begin{equation}\label{elso-dcc}
d_{CC}^2(e,w)=\sum_{i=0}^d\|{(p_w)^i_x}\|^2=\|{x_w^0}\|^2+\sum_{i=1}^d\frac{\|{x_w^i}\|^2}{ f(\alpha_i {(p_w)_z})}.
\end{equation}
(i) By (\ref{elso-dcc}) we directly have that $X^0(d_{CC}^2(e,\cdot))\big|_x=2x^0$. Furthermore, the last component in (\ref{GeodesicEqOnCarnotGroup}) can be written as
\begin{equation}\label{masodik-dcc}
z_w=\sum_{i=1}^d\|{(p_w)^i_x}\|^2\frac{\alpha_i {(p_w)_z}-\sin(\alpha_i {(p_w)_z})}{2\alpha_i {\left((p_w)_z\right)}^2}=\frac{1}{8}\sum_{i=1}^d\alpha_i\|{x_w^i}\|^2g(\alpha_i{(p_w)_z}).
\end{equation}
We may differentiate (\ref{elso-dcc}) and (\ref{masodik-dcc}) w.r.t. the variable $z_w$ at the point $x$, obtaining $$Z(d_{CC}^2(e,\cdot))\big|_x=-\sum_{i=1}^d\alpha_i\|x^i\|^2 \frac{f'(\alpha_i p_z)}{f^2(\alpha_i p_z)}{\left(Z{(p_w)_z}\big|_x \right)} \ \ {\rm and} \ \ 1=\frac{1}{8}\sum_{i=1}^d\alpha_i^2\|x^i\|^2g'(\alpha_ip_z){\left(Z{(p_w)_z}\big|_x\right)}.$$
Note that $-\frac{f'(t)}{f^2(t)}=\frac{t}{4}g'(t)$; thus, the latter relations give at once that
$Z(d_{CC}^2(e,\cdot))\big|_x=2p_z.$
(ii) In order to prove relation (\ref{derivalt-1}) we proceed in a similar way as in (i), by deriving (\ref{elso-dcc}) and (\ref{masodik-dcc}) w.r.t. the corresponding variables.
$\square$\\
A direct consequence of Lemma \ref{LemmaFirstDeriv} is:
\begin{proposition}\label{prop-carnot-exp}
Fix $x,y\in G$ such that $y \notin {\rm cut}_G(x)$. If $\nabla=({X^0,X^{1},...,X^{d}}, Z)$, then
\begin{equation}\label{eqn-exp}
y=\exp_x\left(-\nabla\frac{d_{CC}^2(y,\cdot)}{2}\big|_x\right).
\end{equation}
\end{proposition}
{\it Proof.} Let $x=\exp_y(p)$ for some $p=(p_x,p_z)=({p_x^0, p_x^1, ..., p_x^d},p_z)\in D$. According to Lemma \ref{LemmaFirstDeriv}, we have that
$$-\nabla\frac{d_{CC}^2(y,\cdot)}{2}\big|_x=(\overline p_x^0,\overline p_x^1,...,\overline p_x^d,\overline p_z),$$
where $$ \left\{
\begin{array}{lll}
\overline p_x^0=- p_x^0;\\
\overline p_x^i=-[\cos(\alpha_i p_z)I-\sin(\alpha_ip_z)J] p_x^i,\ \ i\in \{1,...,d\};\\
\overline p_z=- p_z.
\end{array}
\right.
$$
Thus, by relation (\ref{reverse-repres}) it follows that
$$\exp_x\left(-\nabla\frac{d_{CC}^2(y,\cdot)}{2}\big|_x\right)=\exp_x(\overline p_x^0,\overline p_x^1,...,\overline p_x^d,\overline p_z)=y,$$
which concludes the proof.
$\square$\\
\subsection{The Jacobian of the exponential map along a reversed geodesic.}
Let $x,y\in G$ be such that $x\notin {\rm cut}_G(y)$ and $\gamma:[0,1]\to G$ be the unique geodesic $\gamma(s)=\exp_x(sp)$ joining $x$ and $y$ for some $p\in D$. For every $s\in (0,1]$, let us introduce the Jacobian matrix $$Y(s)=d(\exp_x)_{sp}.$$ According to (\ref{Jacobian-Juillet}), the matrix $Y(s)$ is invertible for every $s\in (0,1]$. In the sequel, we are going to consider the reversed geodesic path $s\mapsto \exp_y((1-s) \overline p),$ $s\in [0,1],$ where $\exp_y \overline p = x$ and compute the 'reverse' of $Y$, i.e., \begin{equation}\label{tildeY} \overline Y(1-s)=d(\exp_y)_{(1-s)\overline p}, \ s\in [0,1). \end{equation} Here, $\overline p\in T_y^*G$ is given by $p\in T_x^*G$ similarly as in (\ref{reverse-repres}). With these notations, we have
\begin{proposition}\label{prop-hessian} Let $x,y\in G$ be such that $x\notin {\rm cut}_G(y)$ and $\gamma:[0,1]\to G$ be the unique geodesic $\gamma(s)=\exp_x(sp)$ joining $x$ and $y$ for some $p\in T_x^*G$. For every $s\in (0,1),$ one has \begin{equation}\label{Y-tilde-bevezetese}
\overline Y(1-s)=\frac{1}{1-s}Y(s)H_{x,y}(s)\overline Y(1),
\end{equation}
where
$$H_{x,y}(s)={\rm Hess}\frac{d_{CC}^2(\gamma(s),\cdot)}{2}\big|_{x}-s{\rm Hess}\frac{d_{CC}^2(y,\cdot)}{2}\big|_{x}.$$ In addition, $H_{x,y}(s)$ is a positive semidefinite, symmetric matrix.
\end{proposition}
Let us note that in the above statement ${\rm Hess}=\nabla^2$ denotes the $($a priori not necessarily symmetric$)$ Carnot Hessian, i.e.,
$${\rm Hess}=\left[
\begin{array}{ccccc}
X_1X_1& X_1X_2& ... & X_1X_k& X_1Z\\
X_2X_1& X_2X_2& ... & X_2X_k& X_2Z\\
\vdots& \vdots& ... & \vdots& \vdots\\
X_kX_1& X_kX_2& ... & X_kX_k& X_kZ\\
ZX_1& ZX_2& ... & ZX_k& ZZ
\end{array}
\right].$$
This notation will be used also later on.
A similar result to Proposition \ref{prop-hessian} has been proved by Cordero-Erausquin, McCann and Schmuckenschl{\"a}ger \cite{CMS} on Riemannian manifolds by exploring properties of Jacobi fields. Since the theory of Jacobi fields in our setting is not (yet) available, we give a direct proof of Proposition \ref{prop-hessian}. To do this, we need the following:
\begin{claim}\label{claim-diff}
Let $m\in \mathbb N$, $c, \eta_i:[0,1]\to \mathbb R^m$, $i\in \{1,2\},$ be some differentiable maps with $\eta_2(0)=0$ and a smooth function $F:\mathbb R^{2m}\to \mathbb R^m$ in a neighborhood of
$(c(0),\eta_1(0))$ such that $t\mapsto F(c(t),\eta_1(t))$ is constant near the origin. Then $$\frac{d}{dt}F(c(t),\eta_1(t)+\eta_2(t))|_{t=0}=D_2F(c(0),\eta_1(0)) \dot\eta_2(0).$$ \end{claim}
{\it Proof.} By assumption, we have near the origin that $$0=\frac{d}{dt}F(c(t),\eta_1(t))=D_1F(c(t),\eta_1(t)) \dot c(t)+D_2F(c(t),\eta_1(t))\dot \eta_1(t).$$ By using the latter relation at $t=0$ and $\eta_2(0)=0$, we obtain \begin{eqnarray*}
\frac{d}{dt}F(c(t),\eta_1(t)+\eta_2(t))|_{t=0}&=&D_1F(c(0),\eta_1(0)) \dot c(0)+D_2F(c(0),\eta_1(0)) (\dot \eta_1(0)+\dot \eta_2(0))\\&=&D_2F(c(0),\eta_1(0)) \dot\eta_2(0), \end{eqnarray*} which completes the proof.
$\square$\\
{\it Proof of Proposition \ref{prop-hessian}.} We first deal with the properties of the matrix $H_{x,y}(s)$. By pure metric arguments, one can check that for every $z\in G$ and $s\in [0,1]$ we have the inequality \begin{eqnarray}\label{metric-MCS} m_{x,y}^s(z):=d_{CC}^2(\gamma(s),z)/2- sd_{CC}^2(y,z)/2+s(1-s)d_{CC}^2(x,y)/2 \geq 0. \end{eqnarray} In the Riemannian setting this has been established first
by Cordero-Erausquin, McCann and Schmuc\-kenschl{\"a}ger \cite[Claim 2.4]{CMS}. Moreover, in (\ref{metric-MCS}) equality is realized precisely when $z = x$; the same proof works in our setting as well.
Since $\gamma((0,1])\cap {\rm cut}_G(x)=\emptyset,$ it follows that $z\mapsto m_{x,y}^s(z)$ is twice differentiable at $x$ (see Proposition \ref{prop-carnot-exp}) and its gradient is \begin{equation}\label{derivative-null}
\nabla m_{x,y}^s(\cdot)|_x=\nabla\frac{d_{CC}^2(\gamma(s),\cdot)}{2}\big|_{x}-s\nabla\frac{d_{CC}^2(y,\cdot)}{2}\big|_{x}=0_{\mathbb R^{k+1}}, \end{equation}
while its Carnot Hessian $\nabla^2 m_{x,y}^s(\cdot)|_x=H_{x,y}(s)$ is positive semidefinite.
In order to prove the symmetry of $H_{x,y}(s)$, we verify that the Lie brackets $[W_1,W_2]m_{x,y}^s(\cdot)|_x$ vanish for every choice of $W_1,W_2\in \Delta\cup \{Z\}=\{X_1,...,X_k,Z\}.$ Indeed, the Lie bracket is either trivial by definition or it is $Zm_{x,y}^s(\cdot)|_x$ up to a multiplicative constant (depending on the eigenvalues $\alpha_i$, $i\in \{1,...,d\}$); but $Zm_{x,y}^s(\cdot)|_x=0$ due to (\ref{derivative-null}).
We now prove relation (\ref{Y-tilde-bevezetese}). Since $x\notin {\rm cut}_G(y)$ and ${\rm cut}_G(y)$ is closed, one may fix a curve $c:[0,1]\to G$ with $c(0)=x$ and $\dot c(0)=w\in T_xG$ arbitrarily fixed such that $c([0,1])\cap {\rm cut}_G(y)=\emptyset$.
We notice that $s\mapsto \exp_{c(t)}\left(-s\nabla\frac{d_{CC}^2(y,\cdot)}{2}\big|_{c(t)}\right){=:\gamma(s)}$ is the unique {minimal} geodesic joining $c(t)$ and $y$; indeed, for $s=0$ we have $c(t),$ while for $s=1$ one has precisely $y$ due to Proposition \ref{prop-carnot-exp}, see Figure \ref{fig:YInvert}. {Moreover, by construction, it turns out that $\gamma(s)\notin {\rm cut}_G(c(t))$ for every $t,s\in[0,1]$.}
\begin{figure}
\caption{The curve $c$ (starting from $x$), connected by geodesics with the point $y$}
\label{fig:YInvert}
\end{figure}
\noindent Let $\overline p:[0,1]\to T_yG$ be a curve such that \begin{equation}\label{ketfele}
\exp_{c(t)}\left(-s\nabla\frac{d_{CC}^2(y,\cdot)}{2}\big|_{c(t)}\right)=\exp_y((1-s)\overline p(t)). \end{equation} Let us observe that by (\ref{ketfele}) for $s=0$ we have $$ c(t)=\exp_y(\overline p(t)) \ {\rm for\ all}\ t\in [0,1]. $$ In particular, for $t=0$ we have that $x=c(0)=\exp_y(\overline p(0))$, i.e., $\overline p(0)= \overline p$ and due to (\ref{tildeY}), \begin{equation}\label{wtildeY} w=\dot c(0)=d(\exp_y)_{\overline p(0)}\dot{\overline p}(0)=d(\exp_y)_{\overline p}\dot{\overline p}(0)=\overline Y(1)\dot{\overline p}(0). \end{equation} Fix $s \in (0,1)$. Now, we rewrite (\ref{ketfele}) into \begin{equation}\label{atirva} \exp_{c(t)}\left(\eta_1(t)+\eta_2(t)\right)=\exp_y((1-s)\overline p(t)), \end{equation}
where $$\eta_1^s(t)=-\nabla\frac{d_{CC}^2(\gamma(s),\cdot)}{2}\big|_{c(t)}\ {\rm and}\ \eta_2^s(t)=\nabla\frac{d_{CC}^2(\gamma(s),\cdot)}{2}\big|_{c(t)}-s\nabla\frac{d_{CC}^2(y,\cdot)}{2}\big|_{c(t)}.$$ We are going to verify the assumptions of Claim \ref{claim-diff} for the latter choices.
First, due to Proposition \ref{prop-carnot-exp}, one has $t\mapsto\exp_{c(t)}(\eta_1^s(t))=\gamma(s)=$constant, and due to (\ref{derivative-null}), we also have $\eta_2^s(0)=0.$ Since we have $\eta_1^s(0)=-\nabla\frac{d_{CC}^2(\gamma(s),\cdot)}{2}\big|_{x}$, by Proposition \ref{prop-carnot-exp} one has that $\exp_{x}(\eta_1^s(0))=\gamma(s)$ which is nothing but $\gamma(s)=\exp_x(sp)$; thus $\eta_1^s(0)=sp.$ Consequently, by differentiating relation (\ref{atirva}) at $t=0$ and using Claim \ref{claim-diff} with $F(q_1,q_2)=\exp_{q_1}(q_2)$ which is smooth around the point $(c(0),\eta^s_1(0))=(x,sp)$, we obtain $$d(\exp_{c(0)})_{\eta_1^s(0)}\dot\eta_2^s(0)=(1-s)d(\exp_{y})_{(1-s)\overline p(0)}\dot{\overline p}(0).$$
Moreover,
$$\dot \eta_2^s(0)=\left[{\rm Hess}\frac{d_{CC}^2(\gamma(s),\cdot)}{2}\big|_{x}-s{\rm Hess}\frac{d_{CC}^2(y,\cdot)}{2}\big|_{x}\right] \dot c(0).$$ Finally, we recall by (\ref{wtildeY}) that $w=\dot c(0)=\overline Y(1)\dot{\overline p}(0)$ and due to (\ref{Jacobian-Juillet}), $\overline Y(1)$ is invertible. Putting together the above computations, we have
$$Y(s)\left[{\rm Hess}\frac{d_{CC}^2(\gamma(s),\cdot)}{2}\big|_{x}-s{\rm Hess}\frac{d_{CC}^2(y,\cdot)}{2}\big|_{x}\right]w=(1-s)\overline Y(1-s)\overline Y(1)^{-1}w.$$ Due to
the arbitrariness of $w$, the claim (\ref{Y-tilde-bevezetese}) follows.
$\square$
\subsection{Optimal mass transportation on corank 1 Carnot groups}\label{section-omt} We first recall some facts from Figalli and Rifford \cite{FR}. A function $\varphi:G\to \mathbb R$ is $c=d_{CC}^2/2-${\it concave} if there exist a nonempty set $S\subset G$ and a function $\varphi^c:S\to \mathbb R\cup\{-\infty\}$ with $\varphi^c \not\equiv-\infty$ such that $$\varphi(x)=\inf_{y\in S}\left\{\frac{1}{2}{d_{CC}^2(x,y)}-\varphi^c(y)\right\}.$$ If $\varphi$ is a $d_{CC}^2/2-${concave} function, let $$\partial^c\varphi(x)=\left\{y\in S:\varphi(x)+\varphi^c(y)=\frac{1}{2}{d_{CC}^2(x,y)}\right\}$$ be the $c${\it -superdifferential of $\varphi$ at $x$}. For such a function $\varphi$, let $$\mathcal M_\varphi=\{x\in G:x\notin \partial^c\varphi(x)\}\ \ {\rm and}\ \ \mathcal S_\varphi=\{x\in G:x\in \partial^c\varphi(x)\}$$ be the {\it moving} and {\it static} sets, respectively.
Let us fix $\mu_0$ and $\mu_1$ two compactly supported probability measures on $G$ which are absolutely continuous w.r.t. $\mathcal L^{k+1}$. According to \cite[Theorem 2.3]{FR}, there are two $d_{CC}^2/2$-concave, continuous functions $\varphi,\varphi^c:G\to \mathbb R$ such that \begin{equation}\label{cconcav} \displaystyle\varphi(x)=\min_{y\in {\rm supp}(\mu_1)}\left\{\frac{1}{2}{d_{CC}^2(x,y)}-\varphi^c(y)\right\} \ {\rm and}\ \displaystyle\varphi^c(y)=\min_{x\in {\rm supp}(\mu_0)}\left\{\frac{1}{2}{d_{CC}^2(x,y)}-\varphi(x)\right\} \end{equation} and the optimal transport map is concentrated on the $c$-superdifferential of $\varphi$.
Since the distribution $\Delta$ on the corank 1 Carnot group $G$ is two-generating, it follows that $d_{CC}^2$ is locally Lipschitz on $G\times G,$ {see Agrachev and Lee \cite[Corollary 6.2]{AL}} and Figalli and Rifford \cite[Proposition 4.2, p. 136]{FR}. Therefore, applying the version of \cite[Theorem 3.2, p. 130]{FR} with the weaker assumption for $d_{CC}^2$ of being locally Lipschitz, there exists a $d_{CC}^2/2$-concave function $\varphi:G\to \mathbb R$ given by (\ref{cconcav}) such that $\mathcal M_\varphi$ is open and $\varphi$ is locally Lipschitz in a neighborhood of $\mathcal M_\varphi\cap {\rm supp}(\mu_0)$, thus $\mu_0$-a.e. differentiable in $\mathcal M_\varphi$. Furthermore, for $\mu_0$-a.e. $x$, there exists a unique optimal transport map defined $\mu_0$-a.e. by {\begin{eqnarray} \label{DefIntMapH}
\psi(x):=
\left\{
\begin{array}{lll}
\exp_x(-{\nabla}\varphi(x))
\ &\mbox{if} & x\in \mathcal M_\varphi\cap {\rm supp}(\mu_0); \\
x &\mbox{if} & x\in \mathcal S_\varphi\cap {\rm supp}(\mu_0),
\end{array}\right.
\end{eqnarray} and for $\mu_0$-a.e. $x$ there exists a unique minimizing geodesic joining $x$ and $\psi(x)$ (or, equivalently, joining the element $e$ with $x^{-1}\circ\psi(x)$). {Hereafter, $\nabla=({X^0,X^{1},...,X^{d}}, Z)$ is the Carnot gradient and $\exp_x(\cdot)=x\circ \exp_e(\cdot).$}
We notice that one cannot apply directly \cite[Theorem 3.5, p. 132]{FR} of Figalli and Rifford to deduce the absolute continuity of the Wasserstein geodesic between $\mu_0$ and $\mu_1$ since in our case the semiconcavity assumption does not hold; however, we can recall the first part of their proof to conclude (based on \cite[Theorem 3.2, p. 130]{FR} and \cite[Corollary 7.22]{Villani}) that there is a unique Wasserstein geodesic $(\mu_s)_{s\in [0,1]}$ joining $\mu_0$ and $\mu_1$ given by the push-forward measure $\mu_s=(\psi_s)_\#\mu_0$ for $s\in [0,1],$ where {\begin{eqnarray} \label{DefIntMapH-2}
\psi_s(x):=
\left\{
\begin{array}{lll}
\exp_x(-s{\nabla}\varphi(x))
\ &\mbox{if} & x\in \mathcal M_\varphi\cap {\rm supp}(\mu_0); \\
x &\mbox{if} & x\in \mathcal S_\varphi\cap {\rm supp}(\mu_0).
\end{array}\right.
\end{eqnarray} {The absolute continuity of the Wasserstein geodesic $\mu_s$ follows by the main result of Cavalletti and Mondino \cite{CM} (valid for essentially non-branching metric measure spaces) which can be state as follows:}
\begin{proposition}\label{interpolant-absolute-cont} {\rm {(Cavalletti and Mondino \cite{CM})}} Let $s\in (0,1)$. Consider the notations introduced above and the assumptions of Theorem {\rm \ref{TJacobianDetIneq}}. Under these conditions the interpolant measure $\mu_s=(\psi_s)_{\#}\mu_0$ is absolutely continuous w.r.t. $\mathcal L^{k+1}$. \end{proposition}
Before the proof of our main theorem in the next section let us indicate a technical difficulty that we need to address in the proof. This consists of the fact that in our setting the potential $\varphi$ generating the optimal transportation map $\psi$ via \eqref{DefIntMapH} is not locally semiconcave {(see Cannarsa and Sinestrari \cite{Cann-Sin})} but only locally Lipschitz. Due to the lack of semiconcavity we do not have an Aleksandrov-type second order differentiability for $\varphi$ and consequently, thus we do not know if $\psi$ is differentiable almost everywhere. This regularity issue appears when we consider the transport of the mass along abnormal geodesics.
\section{Proof of the Jacobian Determinant inequality (Theorem \ref{TJacobianDetIneq})}\label{SecProof} Let $s\in (0,1).$ We shall keep the previous notations. The proof is divided into two main parts: the static and moving cases, respectively. The latter case is also divided into two parts depending how the mass is transported, i.e., along abnormal or strictly normal geodesics.
\subsection{Static case}\label{static-case} We assume the static set $\mathcal S_\varphi\cap {\rm supp}(\mu_0)=\{x\in {\rm supp}(\mu_0):x=\psi(x)\}$ has a positive $\mu_0$-measure. Note that $\psi_s(x)=x$ for every $x\in \mathcal S_\varphi$. If we consider the density points of $\mathcal S_\varphi$, we have that ${\rm Jac}(\psi)(x)={\rm Jac}(\psi_s)(x)=1$ for $\mu_0$-a.e. $x\in \mathcal S_\varphi$. {(Here, again Jac$(\psi)$ and Jac$(\psi_s)$ denote the densities of $\psi_{\sharp}\mathcal L^{k+1}$ and $\psi_{s\sharp}\mathcal L^{k+1}$ w.r.t. $\mathcal L^{k+1}$.)} Note that for $x\in \mathcal S_\varphi$, we have that $\exp_e(\theta_x)=x^{-1}\circ \psi(x)=e,$ i.e., {$\theta_x=(p_x,p_z)=(0_{\mathbb R^{k}},p_z)$ for some $p_z\in \mathbb R$}, thus $\mathcal Ap_x=0_{\mathbb R^k}$. Therefore, by the definition of the distortion coefficient, we have $\tau_s^{k,\alpha}(\theta_x)=s$ and $\tau_{1-s}^{k,\alpha}(\theta_x)=1-s$, which concludes the proof of (\ref {Jacobi-inequality-elso}).
\subsection{Moving case} We now assume that the moving set $\mathcal M_\varphi\cap {\rm supp}(\mu_0)$ has a positive $\mu_0$-measure.
Due to (\ref{DefIntMapH}), there exists a null $\mathcal L^{k+1}$-measure set $C_0\subset \mathcal M_\varphi\cap{\rm supp}(\mu_0)$ such that for every $x\in S:=\mathcal M_\varphi\cap {\rm supp}(\mu_0)\setminus C_0$ the function $\varphi$ is differentiable at $x$, the points $x$ and $\psi(x)$ can be joined by a unique minimizing geodesic and $x^{-1}\circ \psi(x)=\exp_e(-\nabla\varphi(x)),$ where $$\nabla\varphi(x)=(p_x,p_z),$$ with \begin{equation}\label{p-ix} p_x=(X^0\varphi(x),X^1\varphi(x),...,X^d\varphi(x))\ {\rm and}\ p_z=Z\varphi(x). \end{equation}
Let \begin{equation}\label{S0set} S_0=\{x\in S:\mathcal Ap_x= 0_{\mathbb R^{k}},\ {\rm where}\ p_x \ {\rm is\ from} \ (\ref{p-ix})\}, \end{equation} and $$ S_1=S\setminus S_0=\{x\in S:\mathcal Ap_x\neq 0_{\mathbb R^{k}},\ {\rm where}\ p_x \ {\rm is\ from} \ (\ref{p-ix})\}.$$
We distinguish two cases.
\subsubsection{{Moving along abnormal geodesics}}\label{moving-case-1}
We assume that $\mu_0(S_0)>0$. In terms of vector fields, the fact that $\mathcal A p_x= 0_{\mathbb R^{k}}$ with $p_x=(X^0\varphi(x),X^1\varphi(x),...,X^d\varphi(x))$ implies that $X^1\varphi(x)=...=X^d\varphi(x)=0_{\mathbb R^{2}}$ for a.e. $x\in S_0$. According to the explicit form of geodesics, see (\ref{GeodesicEqNullpz}), we have \begin{eqnarray}\label{abnormal-geodesic}
\nonumber \psi_s(x)&=&x\circ \exp_e(-s\nabla \varphi(x))=x\circ \exp_e(-sX^0\varphi(x),0_{\mathbb R^{2d}},Z \varphi(x))\\ \nonumber &=&x\circ (-sX^0\varphi(x),0_{\mathbb R^{2d+1}})\\&=& (x_1-s\partial _{x_1}\varphi(x),...,x_{k-2d}-s\partial_{x_{k-2d}} \varphi(x),x_{k-2d+1},...,x_k,z), \end{eqnarray} for a.e. $x=(x_1,...,x_k,z)\in S_0.$ In a similar way, one has \begin{eqnarray}\label{abnormal-geodesic-1}
\psi(x)&=& (x_1-\partial _{x_1}\varphi(x),...,x_{k-2d}-\partial_{x_{k-2d}} \varphi(x),x_{k-2d+1},...,x_k,z), \end{eqnarray} for a.e. $x=(x_1,...,x_k,z)\in S_0.$
We divide the proof into three steps.
{{\underline{Step 1}}: \it $\varphi(\cdot, \eta, z)$ is $d_{\mathbb R^{k-2d}}^2/2$-concave on $\mathbb R^{k-2d}$ for every $( \eta, z)\in \mathbb R^{2d}\times \mathbb R$ fixed, {i.e., for some set $S\subset \mathbb R^{k-2d}$ and function $\phi_{\eta,z}:\mathbb R^{k-2d} \to \mathbb R,$ one has $$\varphi(\xi, \eta, z)=\inf_{\overline \xi\in S}\left\{\frac{1}{2}d^2_{\mathbb R^{k-2d}}(\xi,\overline \xi)- \phi_{\eta,z}(\overline \xi)\right\}.$$}}
Since $\varphi$ is $d_{CC}^2/2$-concave on $G$, one has by (\ref{cconcav}) that for every $( \xi, \eta, z)\in \mathbb R^{k-2d}\times \mathbb R^{2d}\times \mathbb R,$ $$\varphi( \xi, \eta, z)=\min_{(\overline \xi,\overline \eta,\overline z)\in {\rm supp}(\mu_1)}\left\{\frac{1}{2}{d_{CC}^2((\xi,\eta,z), (\overline \xi,\overline \eta,\overline z))}-\varphi^c(\overline \xi,\overline \eta,\overline z)\right\}.$$
\noindent Let $\pi_1:\mathbb R^{k-2d}\times \mathbb R^{2d}\times \mathbb R\to \mathbb R^{k-2d}$ be the projection $\pi_1( \overline \xi,\overline \eta,\overline z)=\overline \xi.$ For every $\overline \xi\in \pi_1({\rm supp}(\mu_1))$, let us introduce the compact set $$\Pi_{\overline \xi}=\{(\overline \eta,\overline z)\in \mathbb R^{2d}\times \mathbb R:(\overline \xi,\overline \eta,\overline z)\in {\rm supp}(\mu_1)\}.
$$
Let us fix $(\eta,z)\in \mathbb R^{2d}\times \mathbb R$. We notice that the function $ \phi_{\eta,z}:\pi_1({\rm supp}(\mu_1))\to \mathbb R\cup\{-\infty\}$ defined by $$ \phi_{\eta,z}(\overline \xi)=\max_{(\overline \eta,\overline z)\in \Pi_{\overline \xi}}\left\{\varphi^c(\overline \xi,\overline \eta,\overline z)-\frac{1}{2} {\tilde d_{CC}^2((\eta,z), (\overline \eta,\overline z))}\right\}$$ is well defined and $\ \phi_{\eta,z}\not\equiv -\infty$. Since $$\displaystyle{\rm supp}(\mu_1)=\displaystyle\bigcup_{\overline \xi\in \pi_1({\rm supp}(\mu_1))}(\overline \xi,\Pi_{\overline \xi}),$$ by the Pythagorean rule (see Lemma \ref{LemmaPythagorean}) we have that for every $\xi\in \mathbb R^{k-2d}$, \begin{eqnarray*} \varphi( \xi, \eta,z)&=&\min_{\overline \xi\in \pi_1({\rm supp}(\mu_1))}\min_{(\overline \eta,\overline z)\in \Pi_{\overline \xi}}\left\{\frac{1}{2}{d_{CC}^2((\xi,\eta,z), (\overline \xi,\overline \eta,\overline z))}-\varphi^c(\overline \xi,\overline \eta,\overline z)\right\}\\&=&\min_{\overline \xi\in \pi_1({\rm supp}(\mu_1))}\min_{(\overline \eta,\overline z)\in \Pi_{\overline \xi}}\left\{\frac{1}{2}d^2_{\mathbb R^{k-2d}}(\xi,\overline \xi)+\frac{1}{2} {\tilde d_{CC}^2((\eta,z), (\overline \eta,\overline z))}-\varphi^c(\overline \xi,\overline \eta,\overline z)\right\}\\&=&\min_{\overline \xi\in \pi_1({\rm supp}(\mu_1))}\left\{\frac{1}{2}d^2_{\mathbb R^{k-2d}}(\xi,\overline \xi)+\min_{(\overline \eta,\overline z)\in \Pi_{\overline \xi}}\left\{\frac{1}{2} {\tilde d_{CC}^2((\eta,z), (\overline \eta,\overline z))}-\varphi^c(\overline \xi,\overline \eta,\overline z)\right\}\right\}\\&=&\min_{\overline \xi\in \pi_1({\rm supp}(\mu_1))}\left\{\frac{1}{2}d^2_{\mathbb R^{k-2d}}(\xi,\overline \xi)- \phi_{\eta,z}(\overline \xi)\right\}, \end{eqnarray*} which concludes the claim.
{{\underline{Step 2}}: \it For a.e. $x=(\xi,\eta,z)\in S_0$ one can identify the Jacobian determinants ${\rm Jac}(\psi_s)(x)$ and ${\rm Jac}(\psi)(x)$ with ${\rm det}[I_{k-2d}-s{\rm Hess}_\xi(\varphi)(x)]$ and ${\rm det}[I_{k-2d}-{\rm Hess}_\xi(\varphi)(x)],$ respectively,
where $I_{k-2d}$ is the $(k-2d)\times (k-2d)$ unit matrix and ${\rm Hess}_\xi(\varphi)( \xi, \eta, z)$ is the usual Euclidean Hessian of the function $\varphi(\cdot, \eta, z)$ at the point $ \xi$. }
\noindent By Step 1, the $d_{\mathbb R^{k-2d}}^2/2$-concavity of $\varphi(\cdot, \eta, z)$ is equivalent to the convexity of $\xi\mapsto \frac{\|\xi\|_{\mathbb R^{k-2d}}^2}{2}-\varphi(\xi, \eta, z)$ on $\mathbb R^{k-2d}$. In particular, by the Aleksandrov's second differentiability theorem, the latter function is twice differentiable a.e., and its Hessian $I_{k-2d}-{\rm Hess}_\xi(\varphi)( \xi, \eta, z)$ is positive semidefinite and symmetric for a.e. $ \xi\in \mathbb R^{k-2d}$; {the same is true for $I_{k-2d}-s{\rm Hess}_\xi(\varphi)( \xi, \eta, z)$, the latter being the convex combination of the positive semidefinite and symmetric matrices $I_{k-2d}$ and $I_{k-2d}-{\rm Hess}_\xi(\varphi)( \xi, \eta, z)$, respectively.}
By (\ref{abnormal-geodesic}) -- if it exists-- the formal Jacobian of $\psi_s$ for a.e. $x=( \xi, \eta, z)\in S_0$ has the structure $$\left[\begin{matrix} A_s(x) & B_s(x)\\ 0& I_{2d+1} \end{matrix}\right],$$ where $A_s(x)=I_{k-2d}-s{\rm Hess}_\xi(\varphi)( \xi, \eta, z)$. Note however that $B_s(x)$ might not exist since we have no information on the differentiability of $\partial_i\varphi( \xi, \cdot, \cdot)$, $i\in \{1,...,k-2d\}.$ We shall explain below that the existence of $B_s(x)$ is not relevant as far as existence of the global Jacobi determinant {as density} is concerned.
Observe first that due to Proposition \ref{interpolant-absolute-cont}, the interpolant measure $\mu_s=(\psi_s)_{\#}\mu_0$ is absolutely continuous w.r.t. $\mathcal{L}^{k+1}$; let $\rho_s$ be its density function. Since the corank 1 Carnot group $(G,d_{CC},\mathcal L^{k+1})$ is a nonbranching metric measure space, both $\psi$ and $\psi_s$ are injective maps on a set of full measure of ${\rm supp}(\mu_0)$. Thus, the push-forward measures $\mu_s=(\psi_s)_{\#}\mu_0$ and $\mu_1=\psi_{\#}\mu_0$ and standard changes of variable should provide the Monge-Amp\`ere equations \begin{equation}\label{MA-0} \rho_0(x)=\rho_s(\psi_s(x)){\rm Jac}(\psi_s)(x) \mbox{ and } \rho_0(x)=\rho_1(\psi(x)){\rm Jac}(\psi)(x) \mbox{ for } \mu_0\mbox{-a.e. } x \in S_0. \end{equation} However, as we pointed out, the {differentials $D\psi$ and $D\psi_s$} may not exist on a set $S\subset S_0$ of positive measure, which requires a reinterpretation of the Monge-Amp\`ere equations in (\ref{MA-0}); we shall consider only the first term since the other one works similarly.
First of all, $\mu_s=(\psi_s)_{\#}\mu_0$ implies \begin{equation}\label{transport-definition} \int_G h(y){\rm d}\mu_s(y)=\int_G h(\psi_s(x)){\rm d}\mu_0(x) \end{equation} for every Borel function $h:G\to [0,\infty)$. In particular, for every measurable set $\tilde S\subset S_0$ with positive measure and Borel function $h$ with ${\rm supp}(h) \subseteq \psi_s(\tilde S)$ we have $$\int_G h(y){\rm d}\mu_s(y)=\int_G h(y)\rho_s(y){\rm d}\mathcal L^{k+1}(y)=\int_{\psi_s(\tilde S)} h(y)\rho_s(y){\rm d}\mathcal L^{k+1}(y).$$ Let $\pi_2:\mathbb R^{k-2d}\times \mathbb R^{2d}\times \mathbb R\to \mathbb R^{2d+1}$ be the projection $\pi_2( y)=\pi_2(y_1, y_2, y_3)=(y_2, y_3)$ and for every $(y_2,y_3)\in \pi_2(\psi_s(\tilde S))$, let $\Pi_{{(y_2,y_3)}}=\{y_1\in \mathbb R^{k-2d}:(y_1,y_2,y_3)\in \psi_s(\tilde S)\}.$ It is clear that $\psi_s(\tilde S)=\cup_{(y_2,y_3)\in \pi_2(\psi_s(\tilde S))}(\Pi_{{(y_2,y_3)}},y_2,y_3)$; by Fubini's theorem it follows that $$\int_{\psi_s(\tilde S)} h(y)\rho_s(y){\rm d}\mathcal L^{k+1}(y)=\int_{\pi_2(\psi_s(\tilde S))} \left(\int_{\Pi_{{(y_2,y_3)}}} h(y)\rho_s(y){\rm d}\mathcal L^{k-2d}(y_1)\right){\rm d}\mathcal L^{2d+1}(y_2,y_3).$$ We consider the change of variables $y=(y_1,y_2,y_3)=\psi_s(x)$ with $x=(\xi,\eta,z)$ which shows through (\ref{abnormal-geodesic}) that $y_1=\pi_1(\psi_s(x))$ and $(y_2,y_3)=(\eta,z)$. Thus, ${\rm d}\mathcal L^{k-2d}(y_1)={\rm det}[A_s(x)]{\rm d}\mathcal L^{k-2d}(\xi)$ and $\Pi_{{(y_2,y_3)}}=\pi_1(\psi_s(\tilde S_{\eta,z},\eta,z))$ where $\tilde S_{\eta,z}=\{\xi\in \mathbb R^{k-2d}:(\xi,\eta,z)\in \tilde S\}$. Moreover, since $\pi_2(\psi_s(\tilde S))=\pi_2(\tilde S)$, the latter term in the above relation becomes $$\int_{\pi_2(\tilde S)} \left(\int_{\tilde S_{\eta,z}} h(\psi_s(x))\rho_s(\psi_s(x)){\rm det}[A_s(x)]{\rm d}\mathcal L^{k-2d}(\xi)\right){\rm d}\mathcal L^{2d+1}(\eta,z)$$ which is nothing but $$\int_{\tilde S}h(\psi_s(x))\rho_s(\psi_s(x)){\rm det}[A_s(x)]{\rm d}\mathcal L^{k+1}(x).$$ The latter expression, relation $$\int_G h(\psi_s(x)){\rm d}\mu_0(x)=\int_{\tilde S} h(\psi_s(x))\rho_0(x){\rm d}\mathcal L^{k+1}(x)$$ and (\ref{transport-definition}) together with the arbitrariness of $h$ and $\tilde S\subset S_0$
give that $$\rho_0(x)=\rho_s(\psi_s(x)){\rm det}[A_s(x)]\ \mbox{ for } \mu_0\mbox{-a.e. } x \in S_0.$$ Consequently, (\ref{MA-0}) enables us to identify $${\rm Jac}(\psi_s)(x):={\rm det}[A_s(x)]={\rm det}[I_{k-2d}-s{\rm Hess}_\xi(\varphi)( \xi, \eta, z)]\ \mbox{ for } \mu_0\mbox{-a.e. } x \in S_0.$$
{{\underline{Step 3}}: \it proof of Theorem \ref{TJacobianDetIneq} concluded $($abnormal mass transportation$)$.}
Since $$I_{k-2d}-s{\rm Hess}_\xi(\varphi)(x)=(1-s)I_{k-2d}+s(I_{k-2d}-{\rm Hess}_\xi(\varphi)(x)),$$ we may apply the concavity of det$(\cdot)^\frac{1}{k-2d}$ on the cone of $(k-2d)\times (k-2d)$ positive semidefinite symmetric matrices, obtaining through Step 2 that \begin{equation}\label{Jacobian-euclidean-like} \left[{\rm Jac}(\psi_s)(x)\right]^\frac{1}{k-2d}\geq 1-s+s\left[{\rm Jac}(\psi)(x)\right]^\frac{1}{k-2d}\ {\rm a.e.}\ x\in S_0. \end{equation} Now, the concavity of the function $t\mapsto t^\frac{k-2d}{k+1}, t>0,$ gives that \begin{eqnarray*}
\left[{\rm Jac}(\psi_s)(x)\right]^\frac{1}{k+1}&=&\left(\left[{\rm Jac}(\psi_s)(x)\right]^\frac{1}{k-2d}\right)^\frac{k-2d}{k+1}\\&\geq & \left(1-s+s\left[{\rm Jac}(\psi)(x)\right]^\frac{1}{k-2d}\right)^\frac{k-2d}{k+1}\\&\geq& 1-s+s\left[{\rm Jac}(\psi)(x)\right]^\frac{1}{k+1}\ {\rm for \ a.e.}\ x\in S_0, \end{eqnarray*} which is exactly the required inequality (\ref{Jacobi-inequality-elso}).
\subsubsection{{Moving along strictly normal geodesics}}\label{moving-case-2} We assume that $\mu_0(S_1)>0$. The proof will be divided into four steps.
{\underline{Step 1}:} {\it $\varphi$ admits a Hessian for a.e. $x \in S_1$.}
\noindent It is well known that the Euclidean squared distance function $d^2_{\mathbb{R}^{k-2d}}$ is semiconcave on $\mathbb{R}^{k-2d}\times \mathbb{R}^{k-2d}$, see Cannarsa and Sinestrari \cite{Cann-Sin}. Moreover, since the distribution $\tilde \Delta=\{X^1,...,X^d\}=\{X_{k-2d+1},..., X_{k}\}$ is fat on $\mathbb R^{2d+1}$, according to Figalli and Rifford \cite[Proposition 4.1, pg. 136]{FR}, the squared distance function $\tilde{d}_{CC}^2$ is locally semiconcave on $\mathbb{R}^{2d+1} \times \mathbb{R}^{2d+1} \setminus \tilde{\mathcal{D}}$, where $\tilde{\mathcal{D}}$ denotes the diagonal of the set $\mathbb{R}^{2d+1} \times \mathbb{R}^{2d+1}$, namely $\tilde{\mathcal{D}} = \{((\eta,z),(\eta,z)) : (\eta,z) \in \mathbb{R}^{2d}\times \mathbb{R}\}$. Consequently, by the Pythagorean rule (see Lemma \ref{LemmaPythagorean}), the squared distance function $d^2_{CC}$ is locally semiconcave on $G \times G \setminus \mathcal{D}$, where \begin{equation}\label{diameter-duzzasztott} \mathcal{D} = \{((\xi, \eta, z),(\xi', \eta, z)) : \xi, \xi' \in \mathbb{R}^{k-2d}, (\eta,z) \in \mathbb{R}^{2d}\times \mathbb{R}\}. \end{equation}
In order to conclude the claim, we slightly modify the proof of \cite[Theorem 3.2]{FR}. Namely, if $x\in S_1$ is arbitrarily fixed, we have that $\mathcal Ap_x\neq 0_{\mathbb R^k}$ with $p_x$ from (\ref{p-ix}), i.e., $(x,\psi(x))\notin \mathcal D.$ In particular, if $x=(\xi_x, \eta_x, z_x)$ and $ \psi(x)=(\xi'_x, \eta'_x, z'_x)$ then $\tilde d_{CC}((\eta_x, z_x),( \eta'_x, z'_x))=:r_x>0.$ {Due to the closeness of $\partial^c\varphi$ on $G\times G$,} there exists an open neighborhood $V_x\subset \mathcal M_\varphi\cap {\rm supp}(\mu_0)$ of $x$ such that $\tilde d_{CC}((\eta_w, z_w),( \eta'_w, z'_w))>\frac{r_x}{2}$ for every $w=(\xi_w, \eta_w, z_w)\in V_x$ and ${\psi(w) =} (\xi_w', \eta_w', z_w')\in \partial^c \varphi(w).$ Let $\tilde \varphi_{x}:G\to \mathbb R$ be defined by $$\tilde \varphi_{x}(w)=\inf\left\{\frac{1}{2}d_{CC}^2(w,y)-\varphi^c(y):y=(\xi_y, \eta_y, z_y)\in {\rm supp}(\mu_1),\ \tilde d_{CC}((\eta_w, z_w),( \eta_y, z_y))>\frac{r_x}{2}\right\},$$ where $w=(\xi_w, \eta_w, z_w)$. Now, the locally semiconcavity of $d_{CC}^2$ on $G \times G \setminus \mathcal{D}$ is inherited by the $d_{CC}^2/2$-concave function $\tilde \varphi_x$ on $V_x$. Moreover, one can observe that $\tilde \varphi_x=\varphi$ on $V_x$, thus $\varphi$ is semiconcave on $V_x$. By the Aleksandrov-Bangert theorem, see \cite[Theorem 3.10, pg. 238]{CMS}, we conclude that $\varphi$ admits a Hessian a.e. on $V_x$, concluding the claim.
{\underline{Step 2}:} $\psi(x)\notin {\rm cut}_G(x)$ {\it for a.e.} $x\in S_1$. {We know that $\mu_0$-a.e. $x$ there exists a unique minimizing geodesic joining $x$ and $\psi(x)$, thus $\psi(x)\notin {\rm cut}_G(x)$ for a.e. $x\in S_1$.}
{\underline{Step 3}:} {\it $\psi_s$ and $\psi$ are differentiable a.e. on $S_1;$ moreover, for a.e. $x\in S_1,$ \begin{equation}\label{derivalt-psi-s}
d\psi_s(x)=Y_x(s)\left[{\rm Hess}\frac{d_{CC}^2(\psi_s(x),\cdot)}{2}\big|_{x}-s{\rm Hess}\varphi(x)\right], \end{equation} \begin{equation}\label{derivalt-psi}
d\psi(x)=Y_x(1)\left[{\rm Hess}\frac{d_{CC}^2(\psi(x),\cdot)}{2}\big|_{x}-{\rm Hess}\varphi(x)\right], \end{equation}
where
$$Y_x(s)=d(\exp_x)_{-s\nabla \varphi(x)},\ s\in (0,1].$$}
\noindent For the first part, we recall that for every $x\in S_1,$$$\psi(x)= x \circ \exp_e (-\nabla\varphi(x))\ \ {\rm and}\ \ \psi_s(x)= x \circ \exp_e(-s\nabla\varphi(x)),$$ see
(\ref{DefIntMapH}) and (\ref{DefIntMapH-2}), respectively. Since
$\psi(x) \notin {\rm cut}_G(x)$ for a.e. $x\in S_1$ (Step 2), thus $-\nabla \varphi (x)$ belongs to the injectivity domain $D$ of $\exp_e,$ and $\varphi$ has a Hessian a.e. on $S_1$ (Step 1), it follows that $\psi$ and $\psi_s$ are differentiable at a.e. $x \in S_1$.
In order to prove (\ref{derivalt-psi-s}) and (\ref{derivalt-psi}), we need a discrete version of Claim \ref{claim-diff}:
\begin{claim}\label{claim-diff-discrete}
Let $m\in \mathbb N$, $F:\mathbb R^{2m}\to \mathbb R^m$ be a smooth function in a neighborhood of $(x,y)\in \mathbb R^{2m}$ and $\{x_n\},\{y_n\},\{z_n\}\subset \mathbb R^m$ be three sequences satisfying the following properties:
\begin{enumerate}
\item[(a)] $\lim_{n\to \infty}x_n= x$ and $x_n\neq x$ for every $n\in \mathbb N;$
\item[(b)] $\lim_{n\to \infty}y_n= y$ and $F(x_n,y_n)=F(x,y)$ for every $n\in \mathbb N;$
\item[(c)] $\lim_{n\to \infty}z_n= 0_{\mathbb R^{m}}$ and $\lim_{n\to \infty}\frac{z_n}{\|x_n-x\|_{\mathbb R^{m}}}= v\in \mathbb R^m.$
\end{enumerate}
Then
\begin{equation}\label{discrete-kovetkeztetes}
\lim_{n\to \infty}\frac{F(x_n,y_n+z_n)-F(x,y)}{\|x_n-x\|_{\mathbb R^{m}}}=D_2F(x,y)v.
\end{equation}
\end{claim}
The proof of the claim is left as an exercise to the interested reader.
\noindent We shall apply the above claim to prove only (\ref{derivalt-psi-s}) since the proof of (\ref{derivalt-psi}) works in a similar way. To do this, without loss of generality, we can fix a Lebesgue density point $x\in S_1$ in the differentiability set of $\varphi$, i.e., where $\varphi$ is twice differentiable (thus both $\nabla \varphi(x)$ and Hess$\varphi(x)$ exist).
Since $x$ is a Lebesgue density point of $S_1$, we can find a linearly independent frame $\{v_i:i=1,...,k+1\}$ at $x$, such that there exist sequences $\{x_{n,i}\}\subset \mathbb R^{k+1}\setminus {\rm cut}_G(\psi_s(x))$ in the differentiability set of $\varphi$ such that for every $i\in \{1,...,k+1\}$:
\begin{equation}\label{discrete-1}
\lim_{n\to \infty}x_{n,i}=x,\ x_{n,i}\neq x\ {\rm for\ every}\ n\in \mathbb N,\ {\rm and} \ \lim_{n\to \infty}\frac{x_{n,i}-x}{\|x_{n,i}-x\|_{\mathbb R^{k+1}}}=v_i;
\end{equation}
\begin{equation}\label{discrete-2}
\lim_{n\to \infty}\nabla \varphi (x_{n,i})=\nabla \varphi (x);
\end{equation}
$$
\lim_{n\to \infty}\frac{\left[\nabla\frac{d_{CC}^2(\psi_s(x),\cdot)}{2}\big|_{x_{n,i}}-s\nabla\varphi(x_{n,i})\right]-\left[\nabla\frac{d_{CC}^2(\psi_s(x),\cdot)}{2}\big|_{x}-s\nabla\varphi(x)\right]}{\|x_{n,i}-x\|_{\mathbb R^{k+1}}}=\ \ \ \ \ \ \ \ \ \ \ \ \
$$
\begin{equation}\label{discrete-3}
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\left[{\rm Hess}\frac{d_{CC}^2(\psi_s(x),\cdot)}{2}\big|_{x}-s{\rm Hess}\varphi(x)\right]v_i;\end{equation}
\begin{equation}\label{discrete-4}
\lim_{n\to \infty}\frac{\psi_s (x_{n,i})-\psi_s(x)}{\|x_{n,i}-x\|_{\mathbb R^{k+1}}}=d \psi_s (x)v_i.
\end{equation}
Fix $i\in \{1,...,k+1\}.$ We shall apply Claim \ref{claim-diff-discrete} with the smooth function $F(w,q)=\exp_w(q)$ in a neighborhood of the point $(x,y):=(x,-s\nabla \varphi(x))$ and three sequences $x_{n,i}$, $y_{n,i}:=-\nabla\frac{d_{CC}^2(\psi_s(x),\cdot)}{2}\big|_{x_{n,i}}$ and $z_{n,i}:=-y_{n,i}
-s\nabla \varphi(x_{n,i}).$ We clearly have that $\psi_s(x_{n,i})=F(x_{n,i},y_{n,i}+z_{n,i})$. According to Proposition \ref{prop-carnot-exp}, we have that $F(x_{n,i},y_{n,i})=\psi_s(x)=F(x,y)$ for every $n\in \mathbb N$ and $\lim_{n\to \infty}y_{n,i}=-\nabla\frac{d_{CC}^2(\psi_s(x),\cdot)}{2}\big|_{x}=-s\nabla \varphi(x)=y.$ The latter relation and (\ref{discrete-2}) give that $\lim_{n\to \infty}z_{n,i}=-y+s\nabla \varphi(x)=0_{\mathbb R^{k+1}}$, while (\ref{discrete-3}) and (\ref{discrete-1}) yield that
$$\lim_{n\to \infty}\frac{z_{n,i}}{\|x_{n,i}-x\|_{\mathbb R^{k+1}}}=\left[{\rm Hess}\frac{d_{CC}^2(\psi_s(x),\cdot)}{2}\big|_{x}-s{\rm Hess}\varphi(x)\right]v_i=:v\in \mathbb R^{k+1}.$$
Thus, (\ref{discrete-kovetkeztetes}) together with (\ref{discrete-4}) reads as
$$d \psi_s (x)v_i=D_2F(x,-s\nabla \varphi(x))v=d(\exp_x)_{-s\nabla \varphi(x)}\left[{\rm Hess}\frac{d_{CC}^2(\psi_s(x),\cdot)}{2}\big|_{x}-s{\rm Hess}\varphi(x)\right]v_i.$$
Since span$\{v_1,...,v_{k+1}\}=\mathbb R^{k+1}$, the latter relation yields (\ref{derivalt-psi-s}).
{{\underline{Step 4}}: \it proof of Theorem \ref{TJacobianDetIneq} concluded $($strictly normal mass transportation$)$.}
We recall by Proposition \ref{prop-hessian} that the Hessian
$$H_{x,\psi(x)}(s):={\rm Hess}\frac{d_{CC}^2(\psi_s(x),\cdot)}{2}\big|_{x}-s{\rm Hess}\frac{d_{CC}^2(\psi(x),\cdot)}{2}\big|_{x}$$
is a $(k+1)\times (k+1)$ type positive semidefinite, symmetric matrix. Since $$\nabla\frac{d_{CC}^2(\psi(x),\cdot)}{2}\big|_{x}-\nabla \varphi(x)=0_{\mathbb R^{k+1}}\ \ {\rm for\ a.e.}\ x\in S_1,$$ a similar argument as in the first part of the proof of Proposition \ref{prop-hessian} and the $d_{CC}^2/2$-concavity of $\varphi$ gives that ${\rm Hess}\frac{d_{CC}^2(\psi(x),\cdot)}{2}\big|_{x}-{\rm Hess}\varphi(x)$ is also a positive semidefinite, symmetric matrix for a.e. $x\in S_1$. Thus, by the concavity of det$(\cdot)^\frac{1}{k+1}$ on the set of $(k+1)\times (k+1)$ type positive semidefinite, symmetric matrices one has\\ \\ $({\rm Jac}(\psi_s)(x))^\frac{1}{k+1} =$ \begin{eqnarray*}
&&\quad =\det\left(Y_x(s)\left[{\rm Hess}\frac{d_{CC}^2(\psi_s(x),\cdot)}{2}\big|_{x}-s{\rm Hess}\varphi(x)\right]\right)^\frac{1}{k+1}\\
&& \quad =
\det(Y_x(s))^\frac{1}{k+1}\det\left[(1-s)\frac{H_{x,\psi(x)}(s)}{1-s}+s\left({\rm Hess}\frac{d_{CC}^2(\psi(x),\cdot)}{2}\big|_{x}-{\rm Hess}\varphi(x)\right)\right]^\frac{1}{k+1}\\
&&\quad \geq
\det(Y_x(s))^\frac{1}{k+1}\left((1-s)\det\left[\frac{H_{x,\psi(x)}(s)}{1-s}\right]^\frac{1}{k+1}+s\det\left({\rm Hess}\frac{d_{CC}^2(\psi(x),\cdot)}{2}\big|_{x}-{\rm Hess}\varphi(x)\right)^\frac{1}{k+1}\right)\\
&& \quad =
(1-s)\det(\overline Y_x(1-s)\overline Y_x(1)^{-1})^\frac{1}{k+1}+s\det( Y_x(s) Y_x(1)^{-1})^\frac{1}{k+1}({\rm Jac}(\psi)(x))^\frac{1}{k+1}, \end{eqnarray*} where $\overline Y_x$ corresponds to $Y_x$ via (\ref{Y-tilde-bevezetese}).
On one hand, by (\ref{Jacobi-left}) we have that $$\det(Y_x(s) Y_x(1)^{-1})=\frac{{\rm Jac}(\exp_e)(-s\nabla \varphi(x))}{{\rm Jac}(\exp_e)(-\nabla \varphi(x))},$$ thus by (\ref{Jacobian-Juillet}), $$s\det(Y_x(s) Y_x(1)^{-1})^\frac{1}{k+1}=\tau_s^{k,\alpha}(\theta_x),$$ where $\theta_x=-\nabla \varphi(x)\in D.$ On the other hand, by the definition of $\overline Y_x$ (see (\ref{tildeY})) and relation (\ref{reverse-repres}) we also have $$ (1-s)\det(\overline Y_x(1-s)\overline Y_x(1)^{-1})^\frac{1}{k+1}=\tau_{1-s}^{k,\alpha}(\theta_x).$$ Combining the above facts we obtain the required Jacobian inequality (\ref{Jacobi-inequality-elso}).
$\square$
\begin{remark}\rm
(a) Step 2 is the most fastidious part of the proof whenever
the mass transportation is realized along abnormal geodesics, see \S \ref{moving-case-1}. Note that reversing the roles of the metrics $d_{\mathbb R^{k-2d}}^2$ and $\tilde d_{CC}^2$, a similar argument as in Step 1 shows that $\varphi(\xi,\cdot,\cdot)$ is a $\tilde d_{CC}^2/2-$concave function on $\mathbb R^{2d}\times \mathbb R$ ($\xi\in \mathbb R^{k-2d}$ is fixed). However, since $(x,\psi(x))\in \mathcal D$ for every $x\in S_0$ (see (\ref{S0set}) and (\ref{diameter-duzzasztott})) and we only know that $\tilde d_{CC}^2$
is locally semiconcave on $\mathbb{R}^{2d+1} \times \mathbb{R}^{2d+1} \setminus \tilde{\mathcal{D}}$, where $\tilde{\mathcal{D}}=\pi_2(\mathcal{D})$, no conclusion can be drawn in general for the locally semiconcavity of $\varphi(\xi,\cdot,\cdot)$ on $\pi_2(S_0).$ Thus, no higher regularity is known for $\varphi(\xi,\cdot,\cdot)$ which justifies the block-decomposition of the Jacobian matrix of $\psi$ in order to interpret and compute its determinant.
(b) If $S_0\subset G$ is open and $\varphi$ is smooth enough on $S_0$ (say $C^1$), one can see that $X^1\varphi(x)=...=X^d\varphi(x)=0_{\mathbb R^{2}}$ for every $x\in S_0$ (see \S \ref{moving-case-1}) implies the fact that $\varphi$ does not depend on the components $x_{k-2d+1},.., x_k, z$, i.e., the Jacobian of $\psi$ can be calculated in the usual way on $S_0$; in particular, Example \ref{example} below falls into this framework. \end{remark}
We conclude this section by constructing two measures and the optimal transportation map bet\-ween them such that a positive mass is transported along abnormal geodesics while the complementary mass is transported along strictly normal geodesics, respectively.
\begin{example}\rm \label{example} Let $G=\mathbb R^m\times \mathbb H^d$ be the $m+2d+1$ dimensional corank 1 Carnot group endowed with its natural group operation inherited by the Euclidean space $\mathbb R^m$ and Heisenberg group $\mathbb H^d;$ in our setting, $k=m+2d$ and $\alpha_i=4$ for every $i\in \{1,...,d\}$ in (\ref{matrix-representation}).
Let $a\in \mathbb R^m\setminus \{0_{\mathbb R^m}\}$ and $b\in \mathbb R^{2d}\setminus \{0_{\mathbb R^{2d}}\}$
and consider the potentials $\varphi_0,\varphi_1:G\to \mathbb R$ defined by $$\varphi_0(x_1,x_2)=\langle a,x_1\rangle_{\mathbb R^m}\ {\rm and}\ \varphi_1(x_1,x_2)=-\langle b,{z_{x_2}}\rangle_{\mathbb R^{2d}}$$ for every $(x_1,x_2)=(x_1,{(z_{x_2},t_{x_2})})\in \mathbb R^m\times \mathbb H^d,$
where $\langle \cdot,\cdot \rangle_{\mathbb R^l}$ denotes the usual inner product in $\mathbb R^l.$ Moreover, let $\varphi_0^c,\varphi_1^c:G\to \mathbb R$ be defined by $$\varphi_0^c(y_1,y_2)=-\frac{1}{2}\|a\|_{\mathbb R^m}^2-\langle a,y_1\rangle_{\mathbb R^m}\ {\rm and}\ \varphi_1^c(y_1,y_2)=-\frac{1}{2}\|b\|_{\mathbb R^{2d}}^2+\langle b,{z_{y_2}}\rangle_{\mathbb R^{2d}}$$
for every $(y_1,y_2)=(y_1,{(z_{y_2},t_{y_2})})\in \mathbb R^m\times \mathbb H^d$. If $d_{CC}$ is the Carnot-Carath\'eodory metric on $G$, one has for every $(x_1,x_2)\in G$ that $$\varphi_j(x_1,x_2)=\inf_{(y_1,y_2)\in \mathbb R^m\times \mathbb H^d}\left\{\frac{1}{2}d_{CC}^2((x_1,x_2),(y_1,y_2))-\varphi_j^c(y_1,y_2)\right\},\ j\in \{0,1\},$$ where we {exploit} Lemma \ref{LemmaPythagorean}, and Ambrosio and Rigot \cite[Example 5.7, p.287]{AR} in the case $j=1$.
Accordingly, $\varphi_j$ are $d_{CC}^2/2$-concave functions on $G$, for $j\in \{0,1\}$.
If $\varphi=\min\{\varphi_0,\varphi_1\}$ and $\varphi^c=\max\{\varphi_0^c,\varphi_1^c\}$, we claim that for every $(x_1,x_2)\in G$, \begin{equation}\label{ket-fuggveny-minimum} \varphi(x_1,x_2)=\inf_{(y_1,y_2)\in \mathbb R^m\times \mathbb H^d}\left\{\frac{1}{2}d_{CC}^2((x_1,x_2),(y_1,y_2))-\varphi^c(y_1,y_2)\right\}. \end{equation} To see this, let $(x_1,x_2)\mapsto \eta(x_1,x_2)$ be the function at the right hand side of (\ref{ket-fuggveny-minimum}). First, we have by definition that $\varphi_j^c(y_1,y_2)\leq \varphi^c(y_1,y_2)$ for every $(y_1,y_2)\in G$ and $j\in \{0,1\}.$ Accordingly, $\varphi_j(x_1,x_2)\geq \eta(x_1,x_2)$ for every $(x_1,x_2)\in G$ and $j\in \{0,1\},$ i.e., $\varphi\geq \eta$.
To check the converse inequality, {we provide a generic argument, independent from the Carnot structure}. Fix $(x_1,x_2)\in G$ arbitrarily and assume {without loss of generality} that $\varphi_0(x_1,x_2)\leq \varphi_1(x_1,x_2)$. Then for every $(y_1,y_2)\in G$, we have $$\varphi_0(x_1,x_2)\leq \frac{1}{2}d_{CC}^2((x_1,x_2),(y_1,y_2))-\varphi_0^c(y_1,y_2);$$ $$\varphi_0(x_1,x_2)\leq \varphi_1(x_1,x_2)\leq \frac{1}{2}d_{CC}^2((x_1,x_2),(y_1,y_2))-\varphi_1^c(y_1,y_2).$$ Consequently, for every $(y_1,y_2)\in G$, one has \begin{eqnarray*} \varphi_0(x_1,x_2)&\leq&\frac{1}{2}d_{CC}^2((x_1,x_2),(y_1,y_2))+{\min}\{-\varphi_0^c(y_1,y_2),-\varphi_1^c(y_1,y_2)\}\\&=& \frac{1}{2}d_{CC}^2((x_1,x_2),(y_1,y_2))-{\max}\{\varphi_0^c(y_1,y_2),\varphi_1^c(y_1,y_2)\}\\&=& \frac{1}{2}d_{CC}^2((x_1,x_2),(y_1,y_2))-\varphi^c(y_1,y_2). \end{eqnarray*} Taking the infimum on the right w.r.t. $(y_1,y_2)\in G$, we obtain $\varphi_0(x_1,x_2)\leq \eta(x_1,x_2),$ i.e., $\varphi(x_1,x_2)\leq \eta(x_1,x_2)$, which concludes the proof of (\ref{ket-fuggveny-minimum}). In particular, (\ref{ket-fuggveny-minimum}) implies that
$\varphi$ is a $d_{CC}^2/2$-concave function on $G$.
Let $G^0=\{(x_1,x_2)=(x_1,(z_2,t))\in \mathbb R^m\times \mathbb H^d:\langle (a,b),(x_1,z_2)\rangle_{\mathbb R^m\times \mathbb R^{2d}}=0 \}$ be the hyperplane separating $\mathbb R^m\times \mathbb R^{2d+1}$ into two halfspaces $G^-=\{ (x_1,(z_2,t))\in \mathbb R^m\times \mathbb H^d:\langle (a,b),(x_1,z_2)\rangle_{\mathbb R^m\times \mathbb R^{2d}}\leq 0\}$ and $G^+=G\setminus G^-$. It follows that $$\varphi(x_1,x_2)=\left\{ \begin{array}{lll} \varphi_0(x_1,x_2) &\mbox{if} & (x_1,x_2)\in G^-;\\ \varphi_1(x_1,x_2) &\mbox{if} & (x_1,x_2)\in G^+, \end{array} \right.$$ and $\varphi$ is differentiable on $G\setminus G^0.$ Let $\psi:G\to G$ be the optimal transport map generated by the $d_{CC}^2/2$-concave function $\varphi$, {see Figalli and Rifford \cite{FR}}; by Proposition \ref{proposition-geodetikus} we have for every $(x_1,x_2)\in G\setminus G^0$ that
$$\psi(x_1,x_2)=\exp_{(x_1,x_2)}(-\nabla \varphi(x_1,x_2))= \left\{
\begin{array}{lll}
\psi_0(x_1,x_2) &\mbox{if} & (x_1,x_2)\in G^-\setminus G^0;\\
\psi_1(x_1,x_2) &\mbox{if} & (x_1,x_2)\in G^+,
\end{array}
\right.$$
where $$\psi_0(x_1,x_2)=\exp_{(x_1,x_2)}(-a,0_{\mathbb R^{2d+1}})=(x_1-a,x_2)$$ and $$\psi_1(x_1,x_2)=\exp_{(x_1,x_2)}(0_{\mathbb R^{m}},b,0)=(x_1,x_2* (b,0));$$ here, $'*'$ denotes the group operation on $\mathbb H^d$.
Let $\mu_0=\mathbbm{1} _{B}\mathcal L^{m+2d+1}$, where $B\subset G$ is a closed ball centered at $0_{\mathbb R^{m+2d+1}}$ with $\mathcal L^{m+2d+1}(B)=1$ and $\mu_1=\psi_\#\mu_0$. Note that every element of $B $ belongs to the moving set $M_\varphi$. Moreover, one can see that ${\rm supp}(\mu_1)=\overline{\psi_0(B\cap G^-\setminus G^0)}\bigcup \overline{\psi_1(B\cap G^+)},$ and the sets $S_0$ and $S_1$ appearing in the proof of Theorem \ref{TJacobianDetIneq} correspond to the two half balls $B\cap G^-$ and $B\cap G^+$ (up to null measure sets), respectively. Consequently, the optimal mass transportation map $\psi$ translates the mass from $S_0$ along abnormal (Euclidean) geodesics into $\tilde S_0=(-a,0_{\mathbb R^{m+2d+1}})+S_0$, while the mass from $S_1$ is transported along strictly normal (Heisenberg) geodesics into a distorted half ball $\tilde S_1=\{(x_1,x_2* (b,0)):(x_1,x_2)\in B\cap G^+\}$, see Figure \ref{abra-elso}.
\begin{figure}
\caption{The half balls $S_0$ and $S_1$ are transported along abnormal and strictly normal geodesics into the sets $\tilde S_0$ and $\tilde S_1$, respectively.}
\label{abra-elso}
\end{figure} \end{example}
\section{Applications}\label{SecApps}
Having the Jacobian determinant inequality (\ref{Jacobi-inequality-elso}), we can prove several functional and geometric inequalities on corank 1 Carnot groups.
Let us denote by $\rho_0, $ $\rho_1$ and $\rho_s$ the density functions (w.r.t. $\mathcal{L}^{k+1}$) of the absolutely continuous, compactly supported measures $\mu_0$, $\mu_1=\psi_{\#}\mu_0$ and $\mu_s=(\psi_s)_{\#}\mu_0$, respectively. In fact, we have the
Monge-Amp\`ere equations \begin{equation}\label{MA} \rho_0(x)=\rho_s(\psi_s(x)){\rm Jac}(\psi_s)(x) \mbox{ and } \rho_0(x)=\rho_1(\psi(x)){\rm Jac}(\psi)(x) \mbox{ for } \mu_0\mbox{-a.e. } x \in G. \end{equation} These equations can be deduced in a standard way both in the static case (see \S \ref{static-case}) and moving case with optimal mass transport along strictly normal geodesics (see \S \ref{moving-case-2}), while in the case of abnormal transportation we provided a proper interpretation of them (see \S \ref{moving-case-1}, Step 2).
Due to (\ref{MA}) we may reformulate the Jacobian determinant inequality (\ref{Jacobi-inequality-elso}) as \begin{eqnarray}\label{Jacobi-inequality-masodik} \left(\rho_s(\psi_s(x)\right)^{-\frac{1}{k+1}}\geq \tau_{1-s}^{k,\alpha}(\theta_x)\left(\rho_0(x)\right)^{-\frac{1}{k+1}}+\tau_{s}^{k,\alpha}(\theta_x)\left(\rho_1(\psi(x)\right)^{-\frac{1}{k+1}}, \end{eqnarray} which holds $\mu_0$ a.e. on the restricted set $G_0 = \{x \in G : \rho_0(x) > 0\}$. Observe that by definition $G_0$ is of full measure in ${\rm supp}(\mu_0)$. For a fixed $s \in (0,1)$ we restrict $G_0$ to the injectivity domain of $\psi$ and $\psi_s$ which will be still of full measure in ${\rm supp}(\mu_0)$.
Moreover, we may exclude those points $x\in S_1$ from $G_0$ for which $x^{-1} \circ \psi(x) \in {\rm cut}_G(e)$, see Step 2 in \S \ref{moving-case-2}, still obtaining a full measure set in ${\rm supp}(\mu_0)$ which prevents the blow-up of coefficients $\tau_{1-s}^{k,\alpha}(\theta_x)$ and $\tau_s^{k,\alpha}(\theta_x)$, respectively.
\subsection{Entropy inequalities}
Let $(G,\circ)$ be a $k+1$ dimensional corank 1 Carnot group and $U:[0, \infty) \to \mathbb R$ be a function. The $U$-entropy of an absolutely continuous measure $\mu$ w.r.t. $\mathcal L^{k+1}$ on $G$ is defined by $\displaystyle{\rm Ent}_U(\mu | \mathcal L^{k+1}) = \int_{G} U\left( \rho(x) \right) {\rm d} \mathcal L^{k+1}(x),$ where $\rho=\frac{{\rm d} \mu}{{\rm d} \mathcal L^{k+1}}$ is the density function of $\mu.$
{By using the injectivity of $\psi_s$ and $\psi$ on $G_0$ (with a suitable change of variables), a similar argument as in \cite{BKS} provides the following entropy inequality.}
\begin{theorem}\label{TEntIneqCarnotGen} {\bf (Entropy inequality)} Under the same assumptions as in Theorem \ref{TJacobianDetIneq}, if
$U: [0, \infty) \to \mathbb R$ is a function such that $U(0)=0$ and $t \mapsto t^{k+1} U\left(\frac{1}{t^{k+1}}\right)$ is non-increasing and convex, the following entropy inequality holds:
\begin{eqnarray*}
{\rm Ent}_{U}(\mu_s | \mathcal L^{k+1}) &\leq& (1-s) \int_{G} \left(\tilde{\tau}_{1-s}^{k,\alpha}(\theta_x)\right)^{k+1} U\left(\frac{\rho_0(x)}{\left(\tilde{\tau}_{1-s}^{k,\alpha}(\theta_x)\right)^{k+1}}\right) {\rm d} \mathcal L^{k+1}(x) \\
&&+ s \int_{G} \left(\tilde{\tau}_{s}^{k,\alpha}(\theta_{\psi^{-1}(y)})\right)^{k+1} U\left(\frac{\rho_1(y)}{\left(\tilde{\tau}_{s}^{k,\alpha}(\theta_{\psi^{-1}(y)})\right)^{k+1}}\right) {\rm d} \mathcal L^{k+1}(y),
\end{eqnarray*}
where $\tilde{\tau}_{s}^{k,\alpha}=s^{-1}{\tau}_{s}^{k,\alpha}.$
\end{theorem}
\begin{corollary}\label{CEntIneqCarnotUnif} Under the same assumptions as in Theorem \ref{TEntIneqCarnotGen}, we have the following uniform entropy inequality:
\begin{eqnarray*}
{\rm Ent}_{U}(\mu_s | \mathcal L^{k+1}) &\leq& (1-s)^3 \int_{G} U\left(\frac{\rho_0(x)}{(1-s)^2}\right) {\rm d} \mathcal L^{k+1}(x) + s^3 \int_{G} U\left(\frac{\rho_1(y)}{s^2}\right) {\rm d} \mathcal L^{k+1}(y).
\end{eqnarray*} \end{corollary}
{\it Proof.} Since $t\mapsto \frac{\mathbb d_i(t,s)}{\mathbb d_i(t,1)}$ is increasing on $(0,2\pi)$ for every $s\in (0,1)$, $i\in \{1,2\},$ and $\lim_{t\to 0}\frac{\mathbb d_1(t,s)}{\mathbb d_1(t,1)}=1$, $\lim_{t\to 0}\frac{\mathbb d_2(t,s)}{\mathbb d_2(t,1)}=s^2$, see \cite[Lemma 2.1]{BKS}, we obtain \begin{eqnarray} \label{tau-lbound} \tau^{k,\alpha}_s(\theta_x)\geq s^{\frac{k+3}{k+1}} \mbox{ for all } s \in (0,1) \mbox{ and } x \in G_0. \end{eqnarray} Thus, for the weights $\tilde{\tau}_s^{k, \alpha}$ we obtain \begin{eqnarray}\label{tau-tilde-lbound} \left(\tilde{\tau}^{k,\alpha}_s(\theta_x)\right)^{k+1}\geq s^2 \mbox{ for all } s \in (0,1) \mbox{ and } x \in G_0. \end{eqnarray} Since the map $t \mapsto t^{k+1} U\left(\frac{1}{t^{k+1}}\right)$ is non-increasing, the desired inequality directly follows from Theorem \ref{TEntIneqCarnotGen}.
$\square$
\begin{remark}\rm As a particular case of Theorem \ref{TEntIneqCarnotGen} and Corollary \ref{CEntIneqCarnotUnif}, we may choose various particular entropies for $U$, as the R\'enyi-type entropy, Shannon entropy, kinetic-type entropy or Tsallis entropy. \end{remark}
\noindent
\subsection{Brunn-Minkowski inequalities} Let $(G,\circ)$ be a connected, simply connected nilpotent Lie group of (topological) dimension $N$, and $\mu$ be a Haar measure on $G$. By extending a result of Leonardi and Masnou \cite{LM} from Heisenberg groups, Tao \cite{T} proved that for every nonempty and bounded open sets $A, B\subset G$ the multiplicative Brunn-Minkowski inequality holds on $(G,\circ)$: \begin{eqnarray}\label{multiplicative-BM} \mu(A \circ B)^\frac{1}{N} \geq \mu(A)^\frac{1}{N} + \mu(B)^\frac{1}{N}. \end{eqnarray} In particular, this inequality is also valid on any $k+1$ dimensional corank 1 Carnot group $G$ with $N=k+1$ and $\mu=\mathcal L^{k+1}$.
In the sequel, we prove geodesic Brunn-Minkowski inequalities on corank 1 Carnot groups. To do this, let $A,B \subset G$ be two nonempty sets. In the sequel we want to quantify the Carnot distortion coefficients which characterize the sets $A$ and $B$. For this reason we introduce the notations \begin{eqnarray}\label{disztrozio-A-B} \tau_s^{k,\alpha}(A,B) = \sup_{A_0, B_0} \inf_{(x,y) \in A_0\times B_0} \{\tau_s^{k,\alpha}(p): \exp_e(p) = x^{-1}\circ y\} \end{eqnarray} and \begin{eqnarray}\label{disztrozio-A-B-1} \tilde{\tau}_s^{k, \alpha}(A,B) = \sup_{A_0, B_0} \inf_{(x,y) \in A_0\times B_0} \{\tilde{\tau}_s^{k, \alpha}(p): \exp_e(p) = x^{-1}\circ y\} = s \tau_s^{k,\alpha}(A,B), \end{eqnarray} where $A_0$ and $B_0$ are nonempty, full measure subsets of $A$ and $B$, respectively. {Note that by taking sets $A_0,B_0$ with the above properties we might obtain better coefficients than if simply take the initial sets $A,B$; more precisely, one always has $$\tilde{\tau}_s^{k, \alpha}(A,B)\geq \inf_{(x,y) \in A\times B} \{\tilde{\tau}_s^{k, \alpha}(p): \exp_e(p) = x^{-1}\circ y\},$$ with possibly strict inequality e.g. when some discrete points $x\in A$ and $y\in B$ are in a particular position as $\exp_e(p) = x^{-1}\circ y$ with $p\in D$ and $p_z=0$.} Recalling relation (\ref{reverse-repres}) between the parameters of the exponential map joining $e$ to $x \in G$ and $x^{-1}\in G$, respectively, the following symmetry properties hold: \begin{eqnarray}\label{tau-symm} \tau_s^{k,\alpha}(x,y) = \tau_s^{k,\alpha}(y,x) \mbox{ and } \tilde{\tau}_s^{k, \alpha}(x,y) = \tilde{\tau}_s^{k, \alpha}(y,x) \mbox{ for all } x,y \in G. \end{eqnarray}
For every $s\in [0,1]$ and $x,y\in G,$ the set of $s$-intermediate points between $x$ and $y$ is \begin{eqnarray}\label{DZs} Z_s(x,y)= \{ z \in G : d_{CC}(x,z) = s d_{CC}(x,y),\ d_{CC}(z,y) = (1-s) d_{CC}(x,y)\}. \end{eqnarray}\label{Z-antisymm} We clearly have the antisymmetry property $$ Z_s(x,y) = Z_{1-s}(y,x) \mbox{ for all } x,y \in G \mbox{ and } s \in [0,1]. $$ The notion of $s$-intermediate points can be extended to the nonempty sets $A,B \subset G$ as $$Z_s(A,B) = \bigcup_{(x,y) \in A \times B} Z_s(x,y).$$
\begin{theorem}\label{TBrunn-Minkowski-2} {\bf (Weighted and non-weighted Brunn-Minkowski inequalities)} Let $(G,\circ)$ be a $k+1$ dimensional corank 1 Carnot group, $s\in (0,1),$ and $A$ and $B$ be two nonempty measurable sets of $G$. Then the following inequalities hold:
\begin{itemize}
\item[{\rm (i)}] $\displaystyle \mathcal L^{k+1}(Z_s(A,B))^\frac{1}{k+1} \geq
\tau_{1-s}^{k,\alpha}(A,B)\mathcal
L^{k+1}(A)^\frac{1}{k+1}+\tau_s^{k,\alpha}(A,B)\mathcal
L^{k+1}(B)^\frac{1}{k+1};$
\item[{\rm (ii)}] $\displaystyle \mathcal L^{k+1}(Z_s(A,B))^\frac{1}{k+1} \geq
(1-s)^\frac{k+3}{k+1}\mathcal
L^{k+1}(A)^\frac{1}{k+1}+s^\frac{k+3}{k+1}\mathcal
L^{k+1}(B)^\frac{1}{k+1};$
\item[{\rm (iii)}] $\displaystyle \mathcal L^{k+1}(Z_s(A,B))^\frac{1}{k+3} \geq
\left(\frac{1}{4}\right)^{\frac{1}{k+3}} \left( (1-s)\mathcal
L^{k+1}(A)^\frac{1}{k+3}+s\mathcal
L^{k+1}(B)^\frac{1}{k+3}\right).$
\end{itemize} \end{theorem}
{\it Proof.} First of all, we notice that if $Z_s(A,B)$ is not measurable, $\mathcal L^{k+1}(Z_s(A,B))$ will denote the outer Lebesgue measure of $Z_s(A,B)$.
(i) We first assume that both $A$ and $B$ have finite $\mathcal L^{k+1}$-measures. If both sets have null measure, we have nothing to prove; thus, we may assume that $\max\left\{\mathcal L^{k+1}(A),\mathcal L^{k+1}(B)\right\}>0.$ The proof is divided into three steps.
{{\underline{Step 1}}:
{\it one has $\tau_s^{k,\alpha}(A,B)<\infty$ and $\tau_{1-s}^{k,\alpha}(A,B)<\infty.$} By (\ref{disztrozio-A-B}), if $\tau_s^{k,\alpha}(A,B)=+\infty$, we have in particular that $x^{-1} \circ y \in {\rm cut}_G(e)$ for a.e. $(x,y)\in A\times B.$ Therefore, $0=\mathcal L^{k+1}({\rm cut}_G(e))\geq \mathcal L^{k+1}(A^{-1}\circ B)$. Thus, by the multiplicative Brunn-Minkowski inequality (\ref{multiplicative-BM}) it follows that $\mathcal
L^{k+1}(A)= \mathcal
L^{k+1}(B)=0,$ which contradicts our initial assumption.
{{\underline{Step 2}}: {\it the case} $\mathcal
L^{k+1}(A)\neq 0\neq \mathcal
L^{k+1}(B).$ Let $\mu_0=\frac{\mathbbm{1} _A(x)}{\mathcal
L^{k+1}(A)}\mathcal L^{k+1}$, $\mu_1=\frac{\mathbbm{1} _B(x)}{\mathcal
L^{k+1}(B)}\mathcal L^{k+1}$ and the R\'enyi entropy $U(r)=-r^{1-\frac{1}{k+1}}$ $(r\geq 0)$ in Theorem \ref{TEntIneqCarnotGen}; thus the entropy inequality and relations (\ref{disztrozio-A-B}) and (\ref{disztrozio-A-B-1}) imply that
\begin{eqnarray*}
\int_{\psi_s(A)}\rho_s(z)^{1-\frac{1}{k+1}}{\rm d} \mathcal{L}^{k+1}(z)&\geq& \tau_{1-s}^{k,\alpha}(A,B)\int_{A}\rho_0^{1-\frac{1}{k+1}}{\rm d} \mathcal{L}^{k+1}+\tau_s^{k,\alpha}(A,B)\int_{B}\rho_1^{1-\frac{1}{k+1}}{\rm d} \mathcal{L}^{k+1}\\&=&\tau_{1-s}^{k,\alpha}(A,B)\mathcal
L^{k+1}(A)^\frac{1}{k+1}+\tau_s^{k,\alpha}(A,B)\mathcal
L^{k+1}(B)^\frac{1}{k+1}.
\end{eqnarray*}
By H\"older's inequality one has that
\begin{eqnarray*}
\int_{\psi_s(A)}\rho_s(z)^{1-\frac{1}{k+1}}{\rm d} \mathcal{L}^{k+1}(z)&\leq& \left(\int_{\psi_s(A)}\rho_s(z){\rm d} \mathcal{L}^{k+1}(z)\right)^{1-\frac{1}{k+1}}\left(\int_{\psi_s(A)}{\rm d} \mathcal{L}^{k+1}(z)\right)^{\frac{1}{k+1}}\\&=& \mathcal L^{k+1}(\psi_s(A))^\frac{1}{k+1}.
\end{eqnarray*}
Since $\psi_s(A)\subset Z_s(A,B)$, the claim follows.
{{\underline{Step 3}}: {\it the case} $\mathcal
L^{k+1}(A)= 0\neq \mathcal
L^{k+1}(B)$ {\it or} $\mathcal
L^{k+1}(A)\neq 0= \mathcal
L^{k+1}(B).$
In fact, our claim reduces to proving that for every $x\in G$, we have
\begin{equation}\label{Belso-biz-Juillet}
\displaystyle \mathcal L^{k+1}(Z_s(\{x\},B)) \geq
\left(\tau_s^{k,\alpha}(\{x\},B)\right)^{k+1}\mathcal
L^{k+1}(B).
\end{equation}
The latter inequality follows by an approximation argument. In fact, if $\{\varepsilon_n\}_{n\in \mathbb N}$ is a decreasing sequence converging to 0, by Step 2 we have for every $n\in \mathbb N$ that
$$\displaystyle \mathcal L^{k+1}(Z_s(B(x,\varepsilon_{n}),B))^\frac{1}{k+1} \geq
\tau_{1-s}^{k,\alpha}(B(x,\varepsilon_n),B)
\varepsilon_n^\frac{k+2}{k+1}+
\tau_{s}^{k, \alpha}(B(x,\varepsilon_n),B)
\mathcal L^{k+1}(B)^\frac{1}{k+1},$$
where $B(x,r)=\{y\in G:d_{CC}(x,y)\leq r\}.$
By using the monotone convergence theorem one can prove that
$$ \lim_{n\to \infty}\mathcal L^{k+1}(Z_s(B(x,\varepsilon_{n}),B))=\mathcal L^{k+1}(Z_s(\{x\},B))\ \ {\rm and}\ \ \lim_{n\to \infty}\tau_{s}^{k, \alpha}(B(x,\varepsilon_n),B)=\tau_{s}^{k, \alpha}(\{x\},B),$$
which proves (\ref{Belso-biz-Juillet}).
If $A$ or $B$ has infinite $\mathcal L^{k+1}$-measure, we apply again an approximation argument.
(ii) This property follows by (i) combined with the universal lower bound (\ref{tau-lbound}) for $\tau_s^{k,\alpha}$.
{ (iii) Property (ii) is combined with the $p$-mean inequality (\ref{MspIneq}) below with the choices $a = (1-s)^{-2} $, $b = s^{-2} $, $p = \frac{1}{2}$, $q = \frac{1}{k+1}$ and $\eta = \frac{1}{k+3}$, respectively.}
$\square$\\
The main result of Rizzi \cite{Rizzi} concerning the measure contraction property on corank 1 Carnot groups is a direct consequence of the Brunn-Minkowski inequality (Theorem \ref{TBrunn-Minkowski-2}):
\begin{corollary}\label{TMCP-1} {\bf (Measure contraction property)}
Let $(G,\circ)$ be a $k+1$ dimensional corank 1 Carnot group. Then the measure contraction property {\rm{\textsf{ MCP}}}$(0,k+3)$ holds
on $G$, i.e., for every $s\in [0,1]$, $x\in G$
and nonempty measurable set $E\subset G$,
$$\displaystyle \mathcal L^{k+1}(Z_s(\{x\},E)) \geq \left(\tau_s^{k,\alpha}(\{x\},E)\right)^{k+1} \mathcal L^{k+1}(E) \geq s^{k+3}\mathcal L^{k+1}(E).$$
\end{corollary}
\subsection{Borell-Brascamp-Lieb inequalities}
In order to formulate our Borell-Brascamp-Lieb inequalities we introduce the notion of the $p$-mean, which for two non-negative numbers $a,b$ and weight $s \in (0,1)$ is defined as $$M_s^p(a,b)=\left\{\begin{array}{lll}
\left( (1-s)a^p + s b^p \right)^{1/p} &\mbox{if} & ab\neq 0, \\
0 &\mbox{if} & ab=0,
\end{array}\right.$$ {with the conventions $M_s^{-\infty}(a,b)=\min\{a,b\}$; $M_s^{0}(a,b)=a^{1-s}b^s;$ and $M_s^{+\infty}(a,b)=\max\{a,b\}$ if $ab\neq 0$ and $M_s^{+\infty}(a,b)=0$ if $ab= 0$.}
According to Gardner \cite[Lemma 10.1]{Gar}, one has \begin{eqnarray}\label{MspIneq}
M_s^{p}(a,b)M_s^{q}(c,d) \geq M_s^{\eta}(ac, bd), \end{eqnarray} for every $a,b,c,d \geq 0, s \in (0,1)$ and $p, q \in \mathbb{R}$ such that $p+q\geq0$ with ${\eta}=\frac{pq}{p+q}$ when $p$ and $q$ are not both zero, and $\eta=0$ if $p=q=0$.
{Having the Jacobian determinant inequality (\ref{Jacobi-inequality-masodik}), we can prove Borell-Brascamp-Lieb-type inequalities on corank 1 Carnot groups. In the sequel we state some of them. We refer to \cite{BKS} for similar results with detailed proofs in the setting of the Heisenberg groups:}
\begin{theorem}\label{TRescaledBBLWithWeights} {\bf (Weighted Borell-Brascamp-Lieb inequality)}
Fix $s\in (0,1)$ and $p \geq -\frac{1}{k+1}$.
Let $f,g,h: G\to [0,\infty)$ be Lebesgue integrable
functions with the property that for all $(x,y)\in G\times G, z\in Z_s(x,y),$
\begin{eqnarray}\label{ConditionRescaledBBLWithWeights}
h(z) \geq M^{p}_s
\left(\frac{f(x)}{\left(\tilde{\tau}_{1-s}^{k, \alpha}(y,x)\right)^{k+1}},\frac{g(y)}{\left(\tilde{\tau}_s^{k, \alpha}(x,y)\right)^{k+1}} \right).
\end{eqnarray}
Then the following inequality holds:
\begin{eqnarray*}
\int_{G} h \geq M^\frac{p}{1+(k+1)p}_s \left(\int_{G}
f, \int_{G} g \right).
\end{eqnarray*}
\end{theorem}
\begin{remark}\label{RStrongerBBL} \rm Observe that Theorem \ref{TRescaledBBLWithWeights} holds as well under weaker conditions, namely, if inequality (\ref{ConditionRescaledBBLWithWeights}) holds only for those $x,y \in G$ for which $f(x)>0$ and $g(y)>0$. \end{remark}
As a direct consequence of Theorem \ref{TRescaledBBLWithWeights}, inequality (\ref{tau-tilde-lbound}) and the monotonicity of the $p$-mean we can formulate the following weaker Borell-Brascamp-Lieb-type inequality:
\begin{corollary}\label{CLighterWeightedBBL} {\bf (Uniformly weighted Borell-Brascamp-Lieb inequality)}
Fix $s\in (0,1)$ and $p \geq -\frac{1}{k+1}.$ Let $f,g,h:
G\to [0,\infty)$ be Lebesgue integrable
functions satisfying
\begin{eqnarray}&\label{1-ConditionRescaledBBLWithoutWeights}
h(z) \geq M^{p}_s \left(\frac{f(x)}{(1-s)^2},\frac{g(y)}{s^2}\right) \ \ for\ all\ (x,y)\in G\times
G, z\in Z_s(x,y).
\end{eqnarray}
Then the following inequality holds:
\begin{eqnarray*}
\int_{G} h \geq M^\frac{p}{1+(k+1)p}_s \left(
\int_{G} f, \int_{G}g\right).
\end{eqnarray*}
\end{corollary}
\begin{corollary}\label{CRescaledBBLWithoutWeights} {\bf (Non-weighted Borell-Brascamp-Lieb inequality)}
Fix $s\in (0,1)$ and $p \geq -\frac{1}{k+3}.$ Let $f,g,h:
G\to [0,\infty)$ be Lebesgue integrable
functions satisfying
\begin{eqnarray}&\label{ConditionRescaledBBLWithoutWeights}
h(z) \geq M^{p}_s (f(x),g(y)) \ \ for\ all\ (x,y)\in G\times
G, z\in Z_s(x,y).
\end{eqnarray}
Then the following inequality holds:
\begin{eqnarray}\label{correction}
\int_{G} h \geq \frac{1}{4}M^\frac{p}{1+(k+3)p}_s \left(
\int_{G} f, \int_{G}g\right).
\end{eqnarray}
\end{corollary}
{\it Proof.} By the $p$-mean inequality (\ref{MspIneq}) and assumption (\ref{ConditionRescaledBBLWithoutWeights}), we have \begin{eqnarray} 4h(z) = M_s^p(f(x),g(y)) M_s^{\frac{1}{2}}\left(\frac{1}{(1-s)^2}, \frac{1}{s^2}\right) \geq M_s^{\frac{p}{2p+1}}\left(\frac{f(x)}{(1-s)^2}, \frac{g(y)}{s^2} \right), \end{eqnarray} for every $x,y \in G$ and $z \in Z_s(x,y)$. By the assumption $p\geq -\frac{1}{k+3}$ we have $\frac{p}{2p+1}\geq-\frac{1}{k+1}$, so we can apply Corollary \ref{CLighterWeightedBBL} for the setting $\tilde{h} = 4 h$, $\tilde{f} = f$, $\tilde{g} = g$ and $\tilde{p} = \frac{p}{2p+1}$, obtaining the desired inequality.
$\square$
\begin{remark} \rm (a) All three versions of the Borell-Brascamp-Lieb inequality imply a corresponding Pr\'ekopa-Leindler-type inequality by setting $p=0$ and using the convention $M_s^0(a,b) = a^{1-s}b^s$ for all $a,b \geq 0$ and $s \in (0,1)$.
(b) The Brunn-Minkowski inequality (i) in Theorem \ref{TBrunn-Minkowski-2} can be obtained alternatively from Theorem \ref{TRescaledBBLWithWeights} whenever $\mathcal
L^{k+1}(A)\neq 0\neq \mathcal
L^{k+1}(B).$ Indeed, let $p=+\infty$, and choose the functions $f(x)=\left(\tilde{\tau}_{1-s}^{k, \alpha}(A,B)\right)^{k+1} \mathbbm{1} _A(x),$ $g(y)=\left(\tilde{\tau}_{s}^{k, \alpha}(A,B)\right)^{k+1} \mathbbm{1} _B(y)$ and $h(z)=\mathbbm{1} _{Z_s(A,B)}(z).$ With these choices assumption (\ref{ConditionRescaledBBLWithWeights}) holds at the points $x,y \in G$ where $f(x)>0$ and $g(y)>0$ and due to Remark \ref{RStrongerBBL}(b) we may apply Theorem \ref{TRescaledBBLWithWeights}, obtaining \begin{eqnarray*}
\mathcal L^{k+1}(Z_s(A,B)) &\geq & M^\frac{1}{k+1}_s \left(\left(\tilde{\tau}_{1-s}^{k, \alpha}(A,B)\right)^{k+1}\mathcal L^{k+1}(A),
\left(\tilde{\tau}_{s}^{k, \alpha}(A,B)\right)^{k+1} \mathcal L^{k+1}(B)\right) \\
&=& \left(\tau_{1-s}^{k,\alpha}(A,B)
\mathcal L^{k+1}(A)^\frac{1}{k+1}+
\tau_{s}^{k, \alpha}(A,B)
\mathcal L^{k+1}(B)^\frac{1}{k+1}\right)^{k+1}, \end{eqnarray*} which concludes the proof. {In a similar way, properties from (ii) and (iii) from Theorem \ref{TBrunn-Minkowski-2} can be obtained by Corollaries \ref{CLighterWeightedBBL} and \ref{CRescaledBBLWithoutWeights}, respectively.} \end{remark}
\noindent {\footnotesize{\sc Mathematisches Institute,
Universit\"at Bern,
Sidlerstrasse 5,
3012 Bern, Switzerland.}\\
Email: \textsf{[email protected]}\\
\noindent {\footnotesize {\sc Department of Economics, Babe\c s-Bolyai University, Str. Teodor Mihali 58-60, 400591
Cluj-Napoca, Romania \& Institute of Applied Mathematics, \'Obuda University,
B\'ecsi \'ut 96, 1034 Budapest, Hungary.}\\ Email:
{\textsf{[email protected]}}\\
\noindent {\footnotesize{\sc Mathematisches Institute,
Universit\"at Bern,
Sidlerstrasse 5,
3012 Bern, Switzerland.\\
} Email: {\textsf{[email protected]}\\
\end{document} | arXiv |
\begin{document}
\title{Faithfulness of the Lawrence representation of braid groups} \author{Hao Zheng} \date{} \maketitle
\begin{abstract} The Lawrence representation $L_{n,m}$ is a family of homological representation of the braid group $B_n$, which specializes to the reduced Burau and the Lawrence-Krammer representation when $m$ is $1$ and $2$. In this article we show that the Lawrence representation is faithful for $m \geq 2$. \end{abstract}
\section{Introduction}
In \cite{Bigelow} and \cite{Krammer}, Bigelow and Krammer proved via different approaches that the Lawrence-Krammer representation of braid groups is faithful thus the braid groups are linear. In fact, the Lawrence-Krammer representation is the only known faithful representation of the braid group $B_n$ for $n\geq4$ till now.
In this article, by making use of a reflexive representation recently found by the author (ref. \cite{Zheng}), we generalize the faithfulness of the Lawrence-Krammer representation to its full family, the Lawrence representation (ref. \cite{Lawrence}).
\begin{thm}\label{thm:faith} The Lawrence representation if faithful for $m \geq 2$. \end{thm}
In the article, the Lawrence representation is defined alternatively as follows. Let $B_n$ denote the Artin's $n$-strand braid group (ref. \cite{Birman}), with standard generators $\{ \sigma_1,\dots,\sigma_{n-1} \}$, and set $$B_{n,m} = \langle \sigma_1,\dots,\sigma_{n-1}, \sigma_n^2,
\sigma_{n+1},\dots,\sigma_{n+m-1} \rangle \subset B_{n+m}.$$ They are the fundamental groups of \begin{eqnarray*}
&& X_n = \{ (x_1,\dots,x_n) \mid x_i \in {\mathbb{C}},
x_i \neq x_j, \forall i \neq j \} / \Sigma_n, \\
&& X_{n,m} = \{ (x_1,\dots,x_{n+m}) \mid x_i \in {\mathbb{C}}, x_i \neq x_j,
\forall i \neq j \} / \Sigma_n \times \Sigma_m \end{eqnarray*} respectively, where $\Sigma_n$ denotes the symmetric group of $n$ symbols.
Let $\xi_{n,m}$ be the reflexive representation over a free ${\mathbb{Z}} B_{n,m}$-module $M_{n,m}$ defined in \cite{Zheng} (see Section \ref{sec:def}). Let $q,t \in {\mathbb{C}}$ be two algebraically independent numbers and let $$\rho_{n,m} : {\mathbb{Z}} B_{n,m} \to {\mathbb{C}}$$ denote the ring homomorphism given by $$\left\{ \begin{array} {lll} \sigma_1,\dots,\sigma_{n-1} & \mapsto & 1, \\ \sigma_n^2 & \mapsto & q, \\ \sigma_{n+1},\dots,\sigma_{n+m-1} & \mapsto & t. \end{array}\right.$$ The {\em Lawrence representation} is defined as the representation $$L_{n,m} = \rho_{n,m} \circ \xi_{n,m}$$ over the ${\mathbb{C}}$-linear space $$M^L_{n,m} = {\mathbb{C}} \otimes_{\rho_{n,m}} M_{n,m}.$$
\begin{rem} It was shown in \cite{Zheng} that $L_{n,2}$ is precisely the Lawrence-Krammer representation and it is easily derived from the explicit matrix elements calculated in \cite{Zheng} that $L_{n,1}$ is precisely the reduced Burau representation (ref. \cite{Birman}). \end{rem}
\begin{rem} It is known that the reduced Burau representation is faithful for $n\leq3$ and not faithful for $n\geq5$ (ref. \cite{Bigelow1}), but the case $n=4$ still remains open. Therefore, Theorem \ref{thm:faith} shows that the faithfulness of Lawrence representation is only unclear for $L_{4,1}$. \end{rem}
Our proof essentially follows Bigelow's approach. In Section \ref{sec:def}, we give a quick review of the reflexive representation $\xi_{n,m}$. In Section \ref{sec:pairing}, we define the pairing of noodles with multiforks and relate it to the Lawrence representation via the notion of linear function. It is the crucial part of the article. In Section \ref{sec:proof}, after some preliminary lemmas prepared, the main theorem is established.
\section{A quick review of the representation $\xi_{n,m}$}\label{sec:def}
Let $D$ be a $2$-disk and $P=\{p_1,\dots,p_n\} \subset D \setminus \partial D$ be a set of $n$ punctures. The space $$Y_{n,m} = \{ (y_1,\dots,y_m) \mid y_i \in D \setminus P, \;
y_i \neq y_j, \; \forall i \neq j \} / \Sigma_m$$ is homotopy equivalent to the fiber of the fiber bundle $X_{n,m} \to X_n$, whose fundamental group is $$\langle A_{1,n+1},\dots,A_{n,n+1}, \sigma_{n+1},\dots,\sigma_{n+m-1} \rangle
\subset B_{n,m}$$ where $A_{i,j}$ is the standard pure braid defined by $$A_{i,j} = \sigma_{j-1} \cdots \sigma_{i+1} \sigma_i^2
\sigma_{i+1}^{-1} \cdots \sigma_{j-1}^{-1}.$$
Recall that an equivalent definition of $B_n$ is the mapping class group ${\mathcal{M}}(D,P;\partial D)$, the group of all orientation preserving homeomorphism $h : D \to D$ such that $h(P)=P$ and
$h|_{\partial D}=\id$, modulo isotopy relative to $P \cup \partial D$. Regarding ${\mathcal{M}}(D,P;\partial D)$ and $\pi_1(Y_{n,m})$ as subgroups of $B_{n,m}$ in the standard way, we have $$\beta_*(\alpha) = \beta^{-1}\alpha\beta, \;\;\;
\forall \beta \in {\mathcal{M}}(D,P;\partial D),\;\; \alpha \in \pi_1(Y_{n,m}).$$
\begin{figure}
\caption{Complex $F$ and a multifork.}
\label{fig:fig21}
\end{figure}
Let $F$ be the $1$-complex shown in Fig. \ref{fig:fig21}. It consists of four $0$-cells $\{z,z_0,z_1,z_2\}$ and three $1$-cells $\{e_0,e_1,e_2\}$. Let $e_t = e_1 \cup z \cup e_2$ and $e_h = z_0 \cup e_0 \cup z$ denote the {\em tine edge} and the {\em handle} of $F$.
\begin{defn}
A {\em fork} is a map $\phi : F \to D$ such that $\phi|_{e_t}$ is an embedding, $\phi(F) \cap \partial D = \phi(z_0)$ and $\phi(F) \cap P = \{\phi(z_1),\phi(z_2)\}$. A {\em multifork} with $m$ components is an $m$-tuple of forks $\Phi=(\phi_1,\dots,\phi_m)$ such that both $\phi_1(e_t),\dots,\phi_m(e_t)$ are disjoint and $\phi_1(e_h),\dots,\phi_m(e_h)$ are disjoint. \end{defn}
\begin{defn} Two forks $\phi$ and $\psi$ are called {\em homotopic}, denoted by $\phi \simeq \psi$, if there is a homotopy $h_t : F \to D$ such that $h_0=\phi$, $h_1=\psi$, $h_t(z_0)$ is independent of $t$, and $h_t$ is a fork for all $0 \leq t \leq 1$. Two multiforks $\Phi = (\phi_1,\dots,\phi_m)$ and $\Psi = (\psi_1,\dots,\psi_m)$ are called {\em homotopic}, also denoted by $\Phi \simeq \Psi$, if there are fork homotopies $h_{k,t} : \phi_k \simeq \psi_k$ such that $(h_{1,t},\dots,h_{m,t})$ is a multifork for all $0 \leq t \leq 1$. \end{defn}
Choose a base point $[b_1,\dots,b_m] \in Y_{n,m}$ where $b_1,\dots,b_m \in \partial D$. Set $$\Gamma_{n,m} = \{ (\phi_1,\dots,\phi_m) \mid \phi_i(z_0)=b_i, \;
1 \leq i \leq m \}$$ and denote by $M^0_{n,m}$ the free ${\mathbb{Z}} B_{n,m}$-module generated by $\Gamma_{n,m}$. Define four relations on $M^0_{n,m}$ as follows.
$R_H$: $\Phi_1 \sim \Phi_2$ if they are homotopic.
$R_R$: $(\phi_1,\dots,\phi_k,\dots,\phi_m) \sim -(\phi_1,\dots,\phi_k r,\dots,\phi_m)$ where $r : F \to F$ denotes the cell isomorphism that swaps $e_1$ and $e_2$.
\begin{figure}
\caption{Relation $R_T$.}
\label{fig:fig21}
\end{figure}
$R_T$: $(\phi_1,\dots,\phi_m) \sim \sgn\eta \cdot \alpha \cdot (\varphi_1,\dots,\varphi_m)$ where $\eta \in \Sigma_m$ and $\alpha
\in \pi_1(Y_{n,m})$ if $\phi_k|_{e_t}=\varphi_{\eta(k)}|_{e_t}$ for all $1 \leq k \leq m$ and $\alpha$ is represented by the loop that runs from $[b_1,\dots,b_m]$ to $[\phi_1(z),\dots,\phi_m(z)]$ along the curve $\{[\phi_1(t),\dots,\phi_m(t)] \mid t \in e_h\}$ and backs to $[b_1,\dots,b_m]$ along the curve $\{[\varphi_1(t),\dots,\varphi_m(t)] \mid t \in e_h\}$.
\begin{figure}
\caption{Relation $R_S$.}
\label{fig:fig23}
\end{figure}
$R_S$: $\Phi \sim \Phi_1 + \Phi_2$ if $\Phi$ can be {\em split} into $\Phi_1$ and $\Phi_2$ by doing a surgery on the tine edge of a fork as shown in Fig. \ref{fig:fig23}.
Now set $M_{n,m} = M^0_{n,m} / (R_H,R_R,R_T,R_S)$. It turns out that the action of $B_n$ on $M_{n,m}$ gives rise to a representation over a finitely generated free ${\mathbb{Z}} B_{n,m}$-module.
\begin{thm} $M_{n,m}$ is a finitely generated free ${\mathbb{Z}} B_{n,m}$-module. Moreover, the action $$\xi_{n,m}(\beta) : [\Phi] \mapsto [\beta \cdot \beta(\Phi)], \;\;\;
\forall \Phi \in \Gamma_{n,m}, \;\; \beta \in B_n$$ gives rise to a representation of $B_n$ over $M_{n,m}$. \end{thm}
\section{Pairing and linear function}\label{sec:pairing}
\begin{defn} A {\em noodle} is an embedded oriented arc $N \subset D \setminus P$ such that $\partial D \cap N = \partial N$ and all the points $b_1,\dots,b_m$ lies to its left. \end{defn}
\begin{figure}
\caption{Noodle $N$, signs of intersections and a set of disjoint arcs.}
\label{fig:faith21}
\end{figure}
\begin{defn}\label{defn:pairing} Let $N$ be a noodle and $\Phi = (\phi_1,\dots,\phi_m)$ be a multifork such that the tine edge of $\phi_i$ intersects $N$ transversely at $\{ x_{i,1},x_{i,2},\dots,x_{i,l_i} \}$. The {\em pairing} of $N$ with $\Phi$ is defined as $$\pair{N}{\Phi} =
\sum_{i_1=1}^{l_1} \cdots \sum_{i_m=1}^{l_m}
\epsilon_{1,i_1} \cdots \epsilon_{m,i_m}
\alpha_{i_1,\dots,i_m}
\in {\mathbb{Z}} \pi_1(Y_{n,m}), $$ where $\epsilon_{j,i_j}$ is the sign of the intersection $x_{j,i_j}$ of $N$ with $\phi_j(e_t)$, $\alpha_{i_1,\dots,i_m} \in \pi_1(Y_{n,m})$ is represented by the loop that runs from $[b_1,\dots,b_m]$ to $[\phi_1(z),\dots,\phi_m(z)]$ along the handles of $\Phi$ (i.e. $\{[\phi_1(t),\dots,\phi_m(t)] \mid t \in e_h\}$), then to $[x_{1,i_1},\dots,x_{m,i_m}]$ along the tine edges of $\Phi$ (the subarcs of $\phi_j(e_t)$ from $\phi_j(z)$ to $x_{j,i_j}$), and backs to $[b_1,\dots,b_m]$ along the disjoint arcs shown in Figure \ref{fig:faith21}. \end{defn}
It is straightforward to verify that, via the pairing, each noodle $N$ gives rise to a ${\mathbb{Z}} B_{n,m}$-linear function $$\pair{N}{\,\cdot\,} : M_{n,m} \to {\mathbb{Z}} B_{n,m},$$ and, further, a ${\mathbb{C}}$-linear function $$\pairrho{N}{\,\cdot\,} : M^L_{n,m} \to {\mathbb{C}}.$$
Note that we have $$\pairrho{N}{L_{n,m}(\beta) \cdot [\Phi]}
= \pairrho{N}{[\beta(\Phi)]}, \;\;
\forall \beta \in B_n, \;
\Phi \in \Gamma_{n,m}. $$ Especially, if $\beta$ is an element of the kernel of the Lawrence representation $L_{n,m}$, $$\pairrho{N}{[\Phi]}
= \pairrho{N}{[\beta(\Phi)]}, \;\;
\forall \Phi \in \Gamma_{n,m}. $$
\begin{rem} For $m=2$, the last equation is precisely a generalization of \cite[Basis Lemma]{Bigelow}. Here we obtain the equation via the language of representation, which makes the topological meaning much more accessible. \end{rem}
\section{Proof of faithfulness}\label{sec:proof}
In this section, let all forks $\phi$ satisfy $\phi(z_0) = b_1$ and denote by $\phi^{(m)}$ the multifork constructed from $m$ parallel copies of $\phi$ as shown in Figure \ref{fig:faith31}.
\begin{figure}
\caption{Fork to multifork.}
\label{fig:faith31}
\end{figure}
\begin{lem}\label{lem:faith:coef} Let $N$ be a noodle and $\phi$ be a fork. Suppose the tine edge of $\phi$ intersects $N$ transversely at $l$ distinct points and \begin{eqnarray*}
&& \pair{N}{\phi^{(m)}} =
\sum_{i_1,\dots,i_m=1}^{l}
\epsilon_{i_1} \epsilon_{i_2} \cdots \epsilon_{i_m}
\alpha_{i_1,\dots,i_m}, \\
&& \rho_{n,m}(\alpha_{i_1,\dots,i_m}) =
q^{a_{i_1,\dots,i_m}}
(-t)^{b_{i_1,\dots,i_m}}, \end{eqnarray*} where $\epsilon_i$ and $\alpha_{i_1,\dots,i_m}$ are same as Definition \ref{defn:pairing}. Then we have \begin{eqnarray*}
&& \epsilon_i = (-1)^{b_{i,i}}, \\
&& a_{i_1,\dots,i_m} = \sum_{j=1}^m a_{i_j}, \\
&& b_{i_1,\dots,i_m} = \sum_{1 \leq j <k \leq m} b_{i_j,i_k}. \end{eqnarray*} \end{lem} \begin{proof} Note that for $j>k$, $b_{i_j,i_k}$ is the crossing number (define the crossing number of the generator $\sigma_i^{\pm1}$ to be $\pm1$) between the $(n+j)$-th and the $(n+k)$-th strand of the braid $\alpha_{i_1,\dots,i_m}$, $a_{i_j}$ is the linking number (half of the crossing number) of the $(n+j)$-th strand with the former $n$ strands of the braid $\alpha_{i_1,\dots,i_m}$.
The identities follow from the facts that the crossing number between the $(n+1)$-th and the $(n+2)$-th strand of $\alpha_{i,i}$ is even if and only if $\epsilon_i$ is positive, $a_{i_1,\dots,i_m}$ is the linking number of the last $m$ strands with the former $n$ strands of $\alpha_{i_1,\dots,i_m}$, $b_{i_1,\dots,i_m}$ is the sum of the pairwise crossing numbers of the last $m$ strands of $\alpha_{i_1,\dots,i_m}$, respectively. \end{proof}
\begin{lem}\label{lem:faith:key} Let $N$ be a noodle, $\phi$ be a fork and $m \geq 2$ be an integer. If $\pairrho{N}{\phi^{(m)}} = 0$ then the tine edge of $\phi$ is isotopic to relative to $\partial D \cup P$ to an arc which is disjoint from $N$. \end{lem} \begin{proof} Applying a preliminary isotopy, we may assume that the tine edge of $\phi$ intersects $N$ transversely at $l$ distinct points where $l$ is minimal in possible. Suppose $l>0$. In the notation of Lemma \ref{lem:faith:coef}, assume $a_1, \dots, a_{l'}$ are all those maximal among $a_1,\dots,a_l$ and $b_{i,j}$ is maximal among $\{ b_{i',j'} \mid 1 \leq i',j' \leq l' \}$. We claim $b_{i,i} = b_{i,j} = b_{j,j}$.
The claim implies that $b_{i_1,\dots,i_m}$ is maximal among $\{ b_{i'_1,\dots,i'_m} \mid 1 \leq i'_1,\dots,i'_m \leq l' \}$ if and only if $b_{i_j,i_k}$ is maximal among $\{ b_{i',j'} \mid 1 \leq i',j' \leq l' \}$ for all $1 \leq j \leq k \leq l'$. Moreover, in this case $\epsilon_{i_1} \cdots \epsilon_{i_m} \rho_{n,m}(\alpha_{i_1,\dots,i_m})$ is independent of the choice of $i_1,\dots,i_m$. Therefore, regarding $\pairrho{N}{\phi^{(m)}}$ as a polynomial of $q,t$, we find the coefficient of $q^{a_{i_1,\dots,i_m}} t^{b_{i_1,\dots,i_m}}$ is nonvanishing thus $\pairrho{N}{\phi^{(m)}} \neq 0$.
Now it remains to prove the claim. Let $\phi'$ denotes the other component of $\phi^{(2)}$ and assume the tine edges of $\phi$ and $\phi'$ intersect $N$ transversely at $\{ x_1,\dots,x_l \}$ and $\{ x'_1,\dots,x'_l \}$, respectively. The rest part of the proof is copied almost word by word from the proof of \cite[Claim 3.4]{Bigelow}.
Suppose, seeking a contradiction, that $b_{i,i}<b_{i,j}$. Let $\alpha$ be an embedded arc from $z_i'$ to $z_j'$ along the tine edge of $\phi'$. Let $\beta$ be an embedded arc from $z_j'$ to $z_i'$ along $N$.
If $\beta$ does not pass through the point $z_i$, let $\delta=\alpha\beta$ and let $w$ be the winding number of $\delta$ around $z_i$. Then $b_{i,j}-b_{i,i}=2w$. If $\beta$ does pass through $z_i$, first modify $\beta$ in a small neighborhood of $z_i$ so that $z_i$ lies to its left. Next let $\delta=\alpha\beta$ and let $w$ be the winding number of $\delta$ around $z_i$. Then $1+b_{i,j}-b_{i,i}=2w$. In either case, our assumption that $b_{i,i}<b_{i,j}$ implies that $w$ is greater than zero.
Let $D_1=D\setminus\{z_i\}$. Let $\pi: \tilde D_1 \to D_1$ be the universal (infinite cyclic) cover. Let $\tilde\alpha$ be a lift of $\alpha$ to $\tilde D_1$. Let $\tilde\beta$ be the lift of $\beta$ to $\tilde D_1$ which starts at $\tilde\alpha(1)$. Let $\gamma$ be a loop in $D_1$ based at $z_i'$ which winds $w$ times around $z_i$ in the clockwise (negative) direction such that $\gamma$ is null-homotopic in $D \setminus P$. Let $\tilde\gamma$ be the lift of $\gamma$ to an arc from $\tilde\beta(1)$ to $\tilde\alpha(0)$. Choose $\gamma$ so that $\tilde\gamma$ is an embedded arc which intersects $\tilde\alpha$ and $\tilde\beta$ only at its end points.
Let $\tilde z_k'$ be the first point on $\tilde\alpha$ which intersects $\tilde\beta$ (possibly $\tilde\alpha(1)$). Then $\pi(\tilde z_k')=z_k'$ for some $k=1 \dots l$. Let $\tilde\alpha'$ be the initial segment of $\tilde\alpha$ ending at $\tilde z_k'$. Let $\tilde\beta'$ be the final segment of $\tilde\beta$ starting at $\tilde z_k'$. Let $\tilde\delta'=\tilde\alpha'\tilde\beta'\tilde\gamma$.
Now $\tilde\delta'$ is a simple closed curve in $\tilde D_1$, so by the Jordan curve theorem it must bound a disk $\tilde B$. Since $\gamma$ passes clockwise around $z_i$, there is a non-compact region to the right of $\tilde\delta'$. Thus $\tilde\delta'$ must pass counterclockwise around $\tilde B$.
Let $\alpha'$, $\beta'$ and $\delta'$ be the projections of $\tilde\alpha'$, $\tilde\beta'$ and $\tilde\delta'$ to $D_1$. Then $a_k-a_i$ is equal to the sum of the winding numbers of $\delta'$ around each of the points in $P$. This is equal to the cardinality of $\tilde B \cap \pi^{-1}(P)$. Since $a_i$ is maximal among all integers $a_{i'}$, we must have $a_k=a_i$. Thus $\tilde B \cap \pi^{-1}(P)=\emptyset$. It follows that the arc $\delta'=\alpha'\beta'\gamma$ is null-homotopic in $D \setminus P$. But $\beta'$ is homotopic relative to end points to a subarc of $N$, and $\gamma$ was chosen to be null-homotopic in $D \setminus P$. Thus $\alpha'$ is homotopic relative to end points to a subarc of $N$ in $D \setminus P$. So $\alpha$ and $N$ cobound a digon in $D \setminus P$. But $\alpha'$ is a subarc of the tine edge of $\phi$. This contradicts the fact that the tine edge of $\phi$ intersects $N$ a minimal number of times. Therefore our assumption that $b_{i,j}>b_{i,i}$ must have been false, so $b_{i,j}=b_{i,i}$.
The proof that $b_{i,j}=b_{j,j}$ is similar. This completes the proof of the claim, and hence of the lemma. \end{proof}
Now we prove the main theorem.
\begin{figure}
\caption{Fork $\phi_i$ and noodle $N_i$.}
\label{fig:faith32}
\end{figure}
\begin{proof}[Proof of Theorem \ref{thm:faith}] Suppose $\beta \in B_n$ belongs to the kernel of the Lawrence representation, i.e. $L_{n,m}(\beta) = \id$. Then for any fork $\phi$ and homeomorphism $f : D \to D$ representing $\beta$ , we have $\pairrho{N}{\phi^{(m)}} = \pairrho{N}{(f\phi)^{(m)}}$.
Choose a set of disjoint noodles $N_1,\dots,N_{n-1}$ and a set of forks $\phi_1,\dots,\phi_{n-1}$ with disjoint tine edges as shown in Figure \ref{fig:faith32}. Note that $\pairrho{N_j}{\phi_i^{(m)}} = 0$ if $i \neq j$. Choose a homeomorphism $f$ representing $\beta$ such that $(f\phi_1)(e_t) \cup \cdots \cup (f\phi_{n-1})(e_t)$ intersects $N_1 \cup \cdots \cup N_{n-1}$ a minimal number of times in possible. Then, whenever $i \neq j$, $\pairrho{N_j}{(f\phi_i)^{(m)}} = 0$ and by Lemma \ref{lem:faith:key} $(f\phi_i)(e_t)$ is disjoint from $N_j$; otherwise, $(f\phi_i)(e_t)$ and $N_j$ cobound a digon in $D \setminus P$ which contradicts the minimality of the intersections.
Therefore, we may further assume that $(f\phi_i)(e_t) = \phi_i(e_t)$ thus $\beta$ must be a power of the full twist $\Delta^2 = (\sigma_1 \cdots \sigma_{n-1})^n$. A straightforward calculation shows that $L_{n,m}(\Delta^2) = q^{mn} t^{m(m-1)} \id$ hence we must have $\beta = 1$. \end{proof}
\end{document} | arXiv |
\begin{document}
\title{On joint distributions of the maximum, minimum and terminal value of a continuous uniformly integrable martingale
} \begin{abstract}
We study the joint laws of the maximum and minimum of a continuous, uniformly
integrable martingale. In particular, we give explicit martingale inequalities
which provide upper and lower bounds on the joint exit probabilities of a
martingale, given its terminal law. Moreover, by constructing explicit and
novel solutions to the Skorokhod embedding problem, we show that these bounds
are tight. Together with previous results of Az\'ema \& Yor, Perkins, Jacka
and Cox \& Ob\l\'oj, this allows us to completely characterise the upper and
lower bounds on all possible exit/no-exit probabilities, subject to a given
terminal law of the martingale. In addition, we determine some further
properties of these bounds, considered as functions of the maximum and
minimum. \end{abstract}
\section{Introduction}
The study of the running maximum and minimum of a martingale has a prominent place in probability theory, starting with Doob's maximal and $L^p$ inequalities. In seminal contributions, Blackwell and Dubins \cite{BD63}, Dubins and Gilat \cite{DubinsGilat:78} and Az\'ema and Yor \cite{AzemaYor:79,AzemaYor:79b} established that the distribution of the maximum $\overline{M}_\infty:=\sup_{t\leq \infty} M_t$ of a uniformly integrable martingale $M$ is bounded from above, in stochastic order, by the so called Hardy-Littlewood transform of the distribution of $M_\infty$, and the bound is attained. This led to series of studies on the possible distributions of $(M_\infty,\overline{M}_\infty)$ including Gilat and Meilijson \cite{GM88}, Kertz and R\"osler \cite{KR90, KertzRosler:92b, KertzRosler:93}, Rogers \cite{Rogers:93}, Vallois \cite{Vallois:93}, see also Carraro, El Karoui and Ob\l\'oj \cite{CarraroElKarouiObloj:09}.
More recently, these problems have gained a new momentum from applications in the field of mathematical finance. The bounds on the distribution of the maximum, given the distribution of the terminal value, are interpreted as bounds on prices of barrier options given the prices of (vanilla) European options. Further, the bounds are often obtained by devising pathwise inequalities which then have the interpretation of (super) hedging strategies. This approach is referred to as robust pricing and hedging and goes back to Hobson \cite{Hobson:98b}, see also Ob\l\'oj \cite{Obloj:EQF} and Hobson \cite{Hobson:10} for survey papers. More recently, for example in Acciaio et.~al.~\cite{Acciaio:2013ab}, martingale inequalities have been used to study some classical probabilistic inequalities, and are of interest in their own right.
Here we propose to study the distribution of $(\overline{M}_\infty,\underline{M}_\infty)$, where $\underline{M}_{\infty}:= \inf_{t \le \infty} M_t$ is the infimum of the process, given the distribution of $M_\infty$, for a uniformly integrable continuous martingale $M$. More precisely, we present sharp lower and upper bounds on all double exit/no-exit probabilities for $M$ in terms of the distribution of $M_\infty$, i.e.\ the probabilities that $\overline{M}_\infty$ is \emph{greater/smaller} than $\ensuremath{\overline{b}}$ \emph{and/or} that $\underline{M}_\infty$ is \emph{greater/smaller} than $\ensuremath{\underline{b}}$, for some barriers $\ensuremath{\underline{b}} < \ensuremath{\overline{b}}$. This amounts to considering eight different events. They of course come in pairs, e.g.{} $\{\sM_{\infty} \ge \ub, \iM_{\infty} > \lb\}$ is the complement of $\{\overline{M}_\infty<\ensuremath{\overline{b}} \textrm{ or }\underline{M}_\infty\leq \ensuremath{\underline{b}}\}$ and, by symmetry, it suffices to consider only one of $\{\sM_{\infty} \ge \ub, \iM_{\infty} > \lb\}$ and $\{\overline{M}_\infty<\ensuremath{\overline{b}}, \underline{M}_{\infty}\leq \ensuremath{\underline{b}}\}$. It follows that to provide a complete description it suffices to consider the three events \begin{equation} \label{eq:eventsofinterest} \{\overline{M}_\infty\geq \ensuremath{\overline{b}}, \underline{M}_\infty\leq \ensuremath{\underline{b}}\},\quad \{\overline{M}_\infty< \ensuremath{\overline{b}}, \underline{M}_\infty>\ensuremath{\underline{b}}\}\ \textrm{ and }\ \{\sM_{\infty} \ge \ub, \iM_{\infty} > \lb\}. \end{equation} By continuity and time-change arguments, it follows that for a fixed distribution $\mu$ of $M_\infty$, our problem is equivalent to studying these events for $M_t=B_{t\wedge \tau}$ where $\tau$ varies among all stopping times such that $M$ is uniformly integrable and $M_\infty=B_\tau$ has distribution $\mu$, i.e.\ solutions to the Skorokhod embedding problem for $\mu$ in $B$, see Ob\l\'oj \cite{Obloj:04b}. Sharp bounds on the probability of the first event in \eqref{eq:eventsofinterest} follow from Perkins and tilted-Jacka solutions, see Section \ref{ap:mart} below. The case of the second event was treated in Cox and Ob\l\'oj \cite{Cox:2011aa} and is also recalled in Section \ref{ap:mart}.
Our contribution here is twofold. First, we derive lower and upper bounds on $\Prob(\sM_{\infty} \ge \ub, \iM_{\infty} > \lb)$ in terms of the distribution of $M_\infty$ and give explicit constructions of martingales which attain the bounds. We do this by devising pathwise inequalities which give upper and lower bounds and then by constructing two new solutions to the Skorokhod embedding problem for which equalities are attained in our pathwise inequalities. Second, we study universal qualitative properties of the probabilities of the events in \eqref{eq:eventsofinterest} seen as surfaces in the parameters $\ensuremath{\underline{b}},\ensuremath{\overline{b}}$. While the techniques used to derive the bounds on $\Prob(\sM_{\infty} \ge \ub, \iM_{\infty} > \lb)$ are not new, the explicit constructions we need to use are novel, and our goal in the first part of the paper is to provide those bounds which are currently not known; in this sense, we complete previous work in the literature. The contribution in the second part of the paper is, to the best of our knowledge, the first attempt to address questions of this nature.
\subsection{Motivation} We believe that there are two natural motivations for our results. First, we believe we solve an intrinsically interesting probabilistic question and second, our results correspond to robust pricing and hedging of certain double barrier options in finance. We elaborate now on both.
From the probabilistic point of view, we follow in the footsteps of seminal works mentioned above. The results therein were typically stated for a martingale and its maximum but naturally can be reformulated for a martingale and its minimum $\underline{M}_\infty$. They grant us a full understanding of possible joint distributions of couples $(M_\infty,\overline{M}_\infty)$ or $(M_\infty,\underline{M}_\infty)$. In contrast, much less is known about the joint distribution of $(M_\infty,\overline{M}_\infty,\underline{M}_\infty)$ and it proves much harder to study (although promising recent progress has been made in this direction in a discrete time setting, when one considers the joint law of a random walk, its maximum, minimum and {\it signature} by \cite{Duembgen:2014aa}). Indeed, already in the case of Brownian motion $B$, while the distribution of $(B_t,\overline{B}_t)$ is readily accessible with a simply and explicit density, the distribution of the triplet $(B_t,\overline{B}_t,\underline{B}_t)$ is described through an infinite series. Likewise, $\Prob(\overline{M}_\infty\geq \ensuremath{\overline{b}})$ is maximised among all martingales $M$ with a fixed distribution of $M_\infty$, by one extremal martingale simultaneously for all $\ensuremath{\overline{b}}$. In contrast, as we will show here, maximising $\Prob(\sM_{\infty} \ge \ub, \iM_{\infty} > \lb)$ will require martingales with qualitatively different behaviour for different values of $(\ensuremath{\underline{b}},\ensuremath{\overline{b}})$.
In terms of mathematical finance, the constructions presented here correspond to robust pricing (and hedging) of double touch/no-touch barrier options --- for a detailed discussion of applications we refer to our earlier papers \cite{Cox:2011ab, Cox:2011aa} where we studied the first two events in \eqref{eq:eventsofinterest}. Such an option would pay out $1$ if and only if one barrier is attained \emph{and} a second given barrier is not attained, i.e.\ we consider the payoff of the form $\{\overline{S}_T\geq \ensuremath{\overline{b}},\underline{S}_T>\ensuremath{\underline{b}}\}$, where $(S_t:t\leq T)$ is a uniformly integrable martingale representing the stock price process. The double touch/no-touch options are partially a theoretical construct --- (to the best of our knowledge) they are not commonly traded even in Foreign Exchange (FX) markets, where barriers options are most popular. However, they prove useful as they can be represented as a sum or difference of other barrier options. We can then interpret our results as super-/sub-hedges for sums and differences of barrier options. More precisely, we can write \begin{eqnarray} \indic{\overline{S}_T\geq \ensuremath{\overline{b}},\underline{S}_T>\ensuremath{\underline{b}}} &=& \indic{\overline{S}_T\geq \ensuremath{\overline{b}}}-\indic{\overline{S}_T\geq \ensuremath{\overline{b}},\underline{S}_T\leq \ensuremath{\underline{b}}} \label{eq:dtnt_decompose1}\\
&=& 1-\Big(\indic{\underline{S}_T\leq \ensuremath{\underline{b}}}+ \indic{\overline{S}_T <
\ensuremath{\overline{b}},\underline{S}_T > \ensuremath{\underline{b}}}\Big)
\label{eq:dtnt_decompose2}. \end{eqnarray} The first decomposition \eqref{eq:dtnt_decompose1} writes the payoff of a double touch/no-touch option as a difference of a one-touch option (with payoff $\indic{\overline{S}_T\geq \ensuremath{\overline{b}}}$) and a double touch option. The second decomposition \eqref{eq:dtnt_decompose2} writes the payoff of a double touch/no-touch option as one minus the portfolio of a one-touch option and a double no-touch (range) option with payoff $\indic{\overline{S}_T< \ensuremath{\overline{b}},\underline{S}_T> \ensuremath{\underline{b}}}$. This is of particular interest as both one-touch and range options are liquidly traded in main currency pairs in FX markets. Effectively, using the no-arbitrage prices derived in Theorems \ref{thm:upper_price_mixed} and \ref{thm:lower_price_mixed} below, we obtain a way of checking for absence of arbitrage in the observed prices of European calls/puts, one-touch and range options. Furthermore, if one-touch options are liquidly traded, we can then exploit pathwise inequalities derived in this paper as super- or sub-hedging strategies for range options or double touch options. For certain barriers this will be sharper than the hedges derived in Cox and Ob\l\'oj \cite{Cox:2011ab, Cox:2011aa} which assumes only that vanilla options are liquid.
\subsection{Notation} \label{sec:notation}
Throughout the paper $M$ denotes a continuous uniformly integrable martingale and $B$ a standard real-valued Brownian motion. The running maximum and minimum of a Brownian motion $B$ or a martingale $M$ are denoted respectively $\ensuremath{\overline{B}}_t=\sup_{u\leq t}B_u$ and $\ensuremath{\underline{B}}_t=\inf_{u\leq t}B_u$, and similarly $\overline{M}_t$ and $\underline{M}_t$. The first hitting times of levels are denoted $H_x(B):=\inf\{t\geq 0: B_t=x\}$, $x\in \mathbb{R}$. Likewise we will consider $H_x(M)$ and $H_x(\omega)$, the first hitting times for a martingale $M$ and a continuous path $\omega$. Most of the time we simply write $H_x$ as it should be clear from the context which process/path we consider. We will use the hitting times primarily to express events involving the running maximum and minimum, e.g.\ note that $\dbmps=\indic{H_{\ensuremath{\overline{b}}}\leq \tau < H_{\ensuremath{\underline{b}}}}$ a.s.. We also introduce the following notation to indicate composition of stopping times: if $\tau_1, \tau_2$ are both stopping times, then the stopping time $(\tau_2 \circ \tau_1)(\omega) = \tau_1(\omega) + \tau_2(\theta_{\tau_1}(\omega))$, where $\theta_t(\omega)$ is the usual shift operator, $\theta_t:C(\mathbb{R}_+) \to C(\mathbb{R}_+)$ defined by $(\theta_t(\omega))_s = \omega_{t+s}$.
We use the notation $a \ll b$ to indicate that $a$ is \emph{much smaller} than $b$ -- this is only used to give intuition and is not rigorous. The minimum and maximum of two numbers are denoted $a\land b=\min\{a,b\}$ and $a\lor b=\max\{a,b\}$ respectively, and the positive part is denoted $a^+=a\lor 0$.
Finally, for a probability measure $\mu$ on $\mathbb{R}$ we let $-\infty\leq \ell_{\mu}<r_{\mu}\leq \infty$ be the bounds of the support of $\mu$, i.e.\ $[\ell_{\mu},r_{\mu}]$ is the smallest interval with $\mu([\ell_{\mu},r_{\mu}])=1$.
\section{Bounds for the probability of double exit/no-exit} \label{sec:dtnt}
In this section we provide sharp bounds on the probability \begin{equation*} \Prob\left(\sM_{\infty} \ge \ub, \iM_{\infty} > \lb\right) \end{equation*} where $\ensuremath{\underline{b}} < 0 < \ensuremath{\overline{b}}$, and $M=(M_t:t\leq \infty)$ is a continuous uniformly integrable martingale. Our approach will involve two steps: first we provide pathwise inequalities which induce upper and lower bounds on the given event. Second, we show that these bounds are attained. More specifically, consider a continuous path $(\omega_t: 0\leq t\leq T)$, where $T\leq \infty$. We will introduce pathwise inequalities comparing $\indic{\tntwTevent}$ to a sum of a ``static term," some function $f(\omega_T),$ and a ``dynamic term" of the generic form $\beta (\omega_T-b)\indic{H_b<T}$. Note that such a dynamic term is zero initially and, when $b$ is hit, it introduces a $\beta$-rotation of $f(\omega_T)$ around $b$. Note also that when evaluated on paths of a martingale, it will be a martingale. Consequently, we will construct random variables which dominate (or are dominated by) the random variable $\indic{\tntMevent}$ and which can be decomposed into a martingale term and a function of the terminal value $M_\infty$. Bounds on the double exit/no-exit probability above will be obtained by taking expectations in these inequalities. We further claim that these bounds are tight. This is proven in the subsequent section, where we build extremal martingales by designing optimal solutions to the Skorokhod embedding problem for Brownian motion.
\subsection{Pathwise inequalities: upper bounds} \label{sec:pathwise}
We need to consider three different inequalities. As we will see later, it is always optimal to use exactly one of them, and the choice depends on the distribution of $M_\infty$ and the values of $\ensuremath{\overline{b}},\ensuremath{\underline{b}}$. We give the cases intuitive labels, their meaning will become clearer when we subsequently construct extremal martingales. Throughout this and the next section we assume that $0<T\leq \infty$ is fixed and $(\omega_t: 0\leq t\leq T)$ is a given continuous function. The hitting times are relative to $\omega$. To keep the notation simple we do not emphasise the dependence on $\omega$, e.g.\ $ H_{\ensuremath{\underline{b}}}=H_{\ensuremath{\underline{b}}}(\omega):= \inf\{t\leq T: \omega_t=\ensuremath{\underline{b}}\}$, or $\overline{G}^I(K)=\overline{G}^I(K,(\omega_t:t\leq T))$.
\emph{$\overline{G}^I$: upper bound for $\ensuremath{\underline{b}} \ll 0<\ensuremath{\overline{b}}$.}\\ The inequality is presented graphically in Figure~\ref{fig:dbmp_G1}. We can write it as: \begin{eqnarray}
\indic{\tntwTevent} & \le & \frac{1}{(K-\ensuremath{\underline{b}})}\left((\omega_T - K)^+ - (\ensuremath{\underline{b}}-\omega_T)^+ -
(\omega_T - \ensuremath{\underline{b}}) \indic{H_{\ensuremath{\underline{b}}}<T}\right)+\indic{\omega_T>
\ensuremath{\underline{b}}}\nonumber \\
& & {} =: \overline{G}^I(K), \label{eq:uG1def} \end{eqnarray} where we assume $K>\ensuremath{\underline{b}}$. We include here the special case where $K=\infty$, which corresponds to the upper bound $\indic{\tntwTevent} \le \indic{\omega_T \ge \ensuremath{\underline{b}}}$. Note that the coefficient $1/(K-\ensuremath{\underline{b}})$ is taken so that the right-hand side after rotation at time $H_{\ensuremath{\underline{b}}}$ is zero above $K$.
\begin{figure}
\caption{ $\overline{G}^I(K)$ in \eqref{eq:uG1def} providing an upper bound for $\indic{\tntwTevent}$}
\label{fig:dbmp_G1}
\end{figure}
\emph{$\overline{G}^{II}$: upper bound for $\ensuremath{\underline{b}}< 0 < \ensuremath{\overline{b}}$.}\\ This is a fairly simple case: if we hit neither $\ensuremath{\underline{b}}$ nor $\ensuremath{\overline{b}}$, the inequality is simply $0 \le \alpha_1(\omega_T - \ensuremath{\underline{b}})$ for some $\alpha_1 >0$, so that the value is $1$ if we strike $\ensuremath{\overline{b}}$ initially, and $0$ if we strike $\ensuremath{\underline{b}}$ initially. This strategy is illustrated in Figure~\ref{fig:dbmp_G2}. If the path hits either $\ensuremath{\overline{b}}$ or $\ensuremath{\underline{b}}$ we have a constant value of either $1$ or $0$ respectively: \begin{eqnarray}
\indic{\tntwTevent} & \le & \alpha_1 \omega_T - \alpha_0
-\alpha_1(\omega_T-\ensuremath{\overline{b}})\indic{H_{\ensuremath{\overline{b}}}< H_{\ensuremath{\underline{b}}} \wedge T} - \alpha_1 (\omega_T - \ensuremath{\underline{b}})
\indic{H_{\ensuremath{\underline{b}}}< H_{\ensuremath{\overline{b}}} \wedge T} \nonumber \\
& & {} =: \overline{G}^{II}. \label{eq:uG2def} \end{eqnarray} The constraints on $\alpha_0,\alpha_1$ correspond to the need for the function to be zero if $\ensuremath{\underline{b}}$ is struck first, and $1$ if $\ensuremath{\overline{b}}$ is struck first. We deduce that \begin{equation}
\label{eq:uG2def_par}
\begin{split}
\alpha_0 & = \ensuremath{\underline{b}}/(\ensuremath{\overline{b}}-\ensuremath{\underline{b}}) \\
\alpha_1 & = 1/(\ensuremath{\overline{b}}-\ensuremath{\underline{b}}).
\end{split} \end{equation}
\begin{figure}
\caption{ $\overline{G}^{II}$ in \eqref{eq:uG2def} providing an upper bound for $\indic{\tntwTevent}$}
\label{fig:dbmp_G2}
\end{figure}
\emph{$\overline{G}^{III}$: upper bound for $\ensuremath{\underline{b}}< 0 \ll \ensuremath{\overline{b}}$.}\\ The final inequality uses the fact that $\indic{\tntwTevent} \le \indic{\overline \omega_T
\ge \ensuremath{\overline{b}}}$, and that the inequality for the latter also works for the former. We can then rewrite (2.2) from Brown, Hobson and Rogers \cite{Brown:01b} as \begin{equation}
\label{eq:uG3def}
\indic{\tntwTevent} \leq \frac{(\omega_T-K)^+}{\ensuremath{\overline{b}}-K} + \frac{\ensuremath{\overline{b}}-\omega_T}{\ensuremath{\overline{b}}-K}
\indic{\overline \omega_T \geq \ensuremath{\overline{b}}} =: \overline{G}^{III}(K), \end{equation} where $K<\ensuremath{\overline{b}}$.
\subsection{Pathwise inequalities: lower bounds} \label{sec:dtnt_sub} Observe that we have $\indic{\tntwTevent} = 1-\indic{\rtntwTevent}$ a.s. It follows that a pathwise upper bound for $\indic{\tntwTevent}$ corresponds to a pathwise lower bound of $\indic{\rtntwTevent}$, and vice versa. We will use this below to rephrase some of the lower bounds as upper bounds.
\emph{$\underline{G}_I$: lower bound for $\ensuremath{\underline{b}}< 0 \ll \ensuremath{\overline{b}}$}.\\ We let $\underline{G}_I$ to be the trivial inequality that the probability is bounded below by zero: $\underline{G}_I\equiv 0$.
\emph{$\underline{G}_{II}$: lower bound for $\ensuremath{\underline{b}}<0<\ensuremath{\overline{b}}$.}\\ We describe an upper bound for $\indic{\rtntwTevent}$ which, as argued above, is equivalent to a lower bound for $\indic{\tntwTevent}$. The inequality depends on two parameters $K_1$ and $K_2$ where $K_1 \ge \ensuremath{\overline{b}} > K_2 \ge \ensuremath{\underline{b}}$. The construction starts with equality on the region $[K_2,\ensuremath{\overline{b}})$ and inequality elsewhere. The first time the path hits $\ensuremath{\overline{b}}$, we rotate to get equality (with zero) on $[K_1,\infty)$ and so that the value is exactly 1 at $\ensuremath{\underline{b}}$. If the path later hits $\ensuremath{\underline{b}}$, we again rotate to gain equality (with 1) on $(-\infty,K_2]$ and $[\ensuremath{\overline{b}},K_1]$. We write it as an inequality \begin{equation}\label{eq:dbmpH3}
\begin{split}
\indic{\rtntwTevent}\leq \ &\alpha_2(K_2-\omega_T)^++(1-\alpha_4)\indic{\omega_T<
\ensuremath{\overline{b}}}-\alpha_2(\omega_T-\ensuremath{\overline{b}})^++\alpha_1(\omega_T-K_1)^++\alpha_4\\ &{}+
\beta_1(\omega_T-\ensuremath{\overline{b}})\indic{H_{\ensuremath{\overline{b}}}<H_{\ensuremath{\underline{b}}}\land
T}+\beta_2(\omega_T-\ensuremath{\underline{b}})\indic{H_{\ensuremath{\overline{b}}}<H_{\ensuremath{\underline{b}}}\leq T}\\
&{}+\beta_3(\omega_T-\ensuremath{\underline{b}})\indic{H_{\ensuremath{\underline{b}}}<H_{\ensuremath{\overline{b}}}\land T}\\
&{}=:1-\underline{G}_{II}(K_1,K_2),
\end{split} \end{equation} which we present graphically in Figure \ref{fig:dbmp_H3}. It follows that $\underline{G}_{II}(K_1,K_2)$ is a lower bound for $\indic{\tntwTevent}$. We deduce immediately from the rotation conditions that $\beta_1=\alpha_2-\alpha_1$, $\beta_2=\alpha_1$ and $\beta_3=\alpha_2$. We have to satisfy two more constraints, namely that after hitting $\ensuremath{\overline{b}}$ and rotating the function is zero on $[K_1,\infty)$ and one at $\ensuremath{\underline{b}}$. Working out the values we have \begin{equation}
\label{eq:valuesdbmpH3}
\left\{ \begin{array}{l}
\alpha_1=\frac{1}{K_1-\ensuremath{\underline{b}}}\\
\alpha_2=\frac{\ensuremath{\overline{b}}-\ensuremath{\underline{b}}}{(K_1-\ensuremath{\underline{b}})(\ensuremath{\overline{b}}-K_2)}\\
\alpha_4=\frac{K_1-\ensuremath{\overline{b}}}{K_1-\ensuremath{\underline{b}}}
\end{array} \right.
\quad \left\{ \begin{array}{l}
\beta_1=\alpha_2-\alpha_1\\
\beta_2=\alpha_1\\
\beta_3=\alpha_2
\end{array} \right. . \end{equation} Observe that $\alpha_4\in (0,1]$ and $0<\alpha_1\le\alpha_2$. We note that if we hit $\ensuremath{\underline{b}}$ before $\ensuremath{\overline{b}}$ we have a strict inequality in \eqref{eq:dbmpH3}. Also, in the case where $K_2 = \ensuremath{\underline{b}}$ a number of the terms simplify: in particular, the construction initially gives $\underline{G}_{III} = 1$ for $\omega_T \in [\ensuremath{\underline{b}},\ensuremath{\overline{b}})$ for $T< H_{\ensuremath{\overline{b}}}$. More generally, we can also have $K_1 = \ensuremath{\overline{b}}$ (with or without also $K_2 = \ensuremath{\underline{b}}$) and all the claims remain true.
\begin{figure}\label{fig:dbmp_H3}
\end{figure}
\emph{$\underline{G}_{III}$: lower bound for $\ensuremath{\underline{b}} \ll 0<\ensuremath{\overline{b}}$}.\\ As previously, we describe an upper bound for $\indic{\rtntwTevent}$. The inequality is represented in Figure \ref{fig:dbmp_H2} and depends on two values $K_1$ and $K_2$ such that $\ensuremath{\underline{b}} < K_2 < K_1 < \ensuremath{\overline{b}}$. The inequality starts with equality (equal to 1) between $K_1$ and $\ensuremath{\overline{b}}$, and if we hit $\ensuremath{\overline{b}}$ initially, we rotate to get equality (to 0) between $K_2$ and $K_1$. If we hit $\ensuremath{\underline{b}}$ after this, we rotate again to ensure the function is equal to 1 below $K_2$. If we initially hit $\ensuremath{\underline{b}}$ rather than $\ensuremath{\overline{b}}$, we rotate to get a function that is generally strictly greater than one. We write it as \begin{equation}\label{eq:dbmpH2}
\begin{split}
\indic{\rtntwTevent}\leq \ &
\alpha_2(K_2-\omega_T)^++\alpha_1(K_1-\omega_T)^++\indic{\omega_T <
\ensuremath{\overline{b}}}-\alpha_1(\omega_T-\ensuremath{\overline{b}})^+\\ &{} +
\beta_1(\omega_T-\ensuremath{\overline{b}})\indic{H_{\ensuremath{\overline{b}}}<H_{\ensuremath{\underline{b}}}\land
T}+\beta_2(\omega_T-\ensuremath{\underline{b}})\indic{H_{\ensuremath{\overline{b}}}<H_{\ensuremath{\underline{b}}}\leq T}\\ &{}
+\beta_3(\omega_T-\ensuremath{\underline{b}})\indic{H_{\ensuremath{\underline{b}}}<H_{\ensuremath{\overline{b}}}\land T}\\ &{}
=:1-\underline{G}_{III}(K_1,K_2),
\end{split} \end{equation} and it follows that $\underline{G}_{III}(K_1,K_2)$ is a lower bound for $\indic{\tntwTevent}$. We deduce immediately from the rotation conditions that $\beta_1=\alpha_1$, $\beta_2=\alpha_2$ and $\beta_3=\alpha_1+\alpha_2$. We have to satisfy two more constraints, namely that after hitting $\ensuremath{\overline{b}}$ and rotating, the function is zero on $(K_2,K_1)$ and one in $\ensuremath{\underline{b}}$. Working out the values we have \begin{equation}
\label{eq:valuesdbmpH2}
\left\{ \begin{array}{l}
\alpha_1=\frac{1}{\ensuremath{\overline{b}}-K_1}\\
\alpha_2=\frac{1}{K_2 - \ensuremath{\underline{b}}}
\end{array} \right.
\quad \left\{ \begin{array}{l}
\beta_1=\alpha_1\\ \beta_2=\alpha_2\\ \beta_3=\alpha_1+\alpha_2
\end{array} \right. . \end{equation} As in the previous case, we have a strict inequality in \eqref{eq:dbmpH2} if the path hits $\ensuremath{\underline{b}}$ before $\ensuremath{\overline{b}}$.
\begin{figure}\label{fig:dbmp_H2}
\end{figure}
\subsection{Probabilistic bounds} \label{sec:prob_bounds}
We now consider the pathwise inequalities above evaluated on a path of a continuous uniformly integrable martingale $M=(M_t:0\leq t\leq \infty)$. This gives a.s.\ bounds on $\dbMps$. By taking expectations we obtain bounds on the double exit/no-exit probabilities in terms of the distribution of $M_\infty$. Indeed, observe that each of the bounds we get can be decomposed into two terms. The first of these depends on $M_\infty$ alone, for example, in \eqref{eq:dbmpH3}, the sum of the four quantities preceded by an $\alpha$. The second corresponds to a martingale and disappears when taking expectations, e.g.\ considering again \eqref{eq:dbmpH3}, the three terms which are preceded by a $\beta$ sum to give a term with expected value zero.
\begin{prop}
\label{prop:prob_upperbound}
Suppose $M=(M_{t}: 0\leq t\leq \infty)$ is a continuous uniformly integrable martingale. Then
\begin{equation}\label{eq:generalbound_dbmp}
\Prob\left( \sM_{\infty} \ge \ub, \iM_{\infty} > \lb \right) \leq \inf\left\{\Ep{ \overline{G}^{I}(K)},\Ep{\overline{G}^{II}},\Ep{\overline{G}^{III}(K')}\right\},
\end{equation}
where the infimum is taken over $0<K'<\ensuremath{\overline{b}}<K$ and where
$\overline{G}^{I},\overline{G}^{II},\overline{G}^{III}$ are given by
\eqref{eq:uG1def},\eqref{eq:uG2def}--\eqref{eq:uG2def_par}, and
\eqref{eq:uG3def} respectively, evaluated on paths of $M$. \end{prop}
Our goal is to show that the above bound is optimal. A key aspect of the above result is that the right hand-side of \eqref{eq:generalbound_dbmp} depends only on the distribution of $M_\infty$ and not on the law of the martingale $M$. We let $\mu$ be a probability measure on $\mathbb{R}$ with finite first moment. It is clear that we may then assume (subject to a suitable shift of the martingale) that the measure $\mu$ is centred. We also exclude the trivial case where $\mu = \delta_0$ from our arguments, so necessarily $\mu((-\infty,0))$ and $\mu((0,\infty))$ are both strictly positive. We write $M\in \mathcal{M}_\mu$ to denote that $M$ is a continuous uniformly integrable martingale with $M_\infty\sim \mu$.
In the arguments below, we will commonly want to discuss the measure $\mu$ restricted to some interval. Moreover, in the case where there is an atom of $\mu$ at a point $y$, it may become necessary to split the atom into more than one part. It will be convenient therefore to split the measure $\mu$ according to its quantiles. We therefore introduce the notation $F(x) = \mu((-\infty,x])$ for the usual distribution function of the measure $\mu$, and write $F^{-1}(q) = \inf \{x \in \mathbb{R}: F(x) \ge q\}\vee \ell_{\mu}$. Then for $p,q \in [0,1]$ with $p \le q$ we define the sub-probability measures \begin{equation}
\label{eq:2}
\mu_p^q((-\infty,x]) = (F(x) \wedge q - p) \vee 0 =: F_p^q(x). \end{equation} In addition, we will write $\mu^q = \mu^q_0$ and $\mu_p = \mu_p^1$. Observe that $\mu_p^q(\mathbb{R}) = q-p$.
The {\it barycentre} of $\mu$ associates to a non-empty Borel set $\Gamma\subset \mathbb{R}$ the mean of $\mu$ over $\Gamma$ via \begin{equation}\label{eq:barycentre}
\mu_B(\Gamma)=\frac{\int_\Gamma u\, \mu(\mathrm{d} u)}{\int_\Gamma \mu(\mathrm{d} u)}. \end{equation} An obvious extension is to consider the barycentre of the measure $\mu$ when restricted to $\mu_p^q$, which we denote by $m_p^q$, so \begin{equation}
\label{eq:3}
m_p^q = \begin{cases} (q-p)^{-1}\int x \, \mu_p^q(\mathrm{d}x) & \text{ if
} q>p\\
F^{-1}(q) & \text{ otherwise}
\end{cases}. \end{equation}
Now fix $\ensuremath{\underline{b}}, \ensuremath{\overline{b}} \in \mathbb{R}$ with $\ensuremath{\underline{b}} < 0 < \ensuremath{\overline{b}}$. Of importance in our constructions will be the following notions. Given $p$ with $p\le F(\ensuremath{\underline{b}}-)$, we want to find the probability $q$ such that $m_p^q = \ensuremath{\underline{b}}$. Specifically, define a function $\rho_-: [0,F(\ensuremath{\underline{b}}-)] \to [F(\ensuremath{\underline{b}}),1]$ by \begin{equation}
\label{eq:4}
\rho_-(p) = \inf \{ q \ge F(\ensuremath{\underline{b}}) : m_p^q \ge \ensuremath{\underline{b}}\}. \end{equation} Similarly, we can define $\rho_+: [F(\ensuremath{\overline{b}}),1] \to [0,F(\ensuremath{\overline{b}}-)]$ by \begin{equation}
\label{eq:5}
\rho_+(q) = \sup \{ p \le F(\ensuremath{\overline{b}}-) : m_p^q \le \ensuremath{\overline{b}}\}. \end{equation} It is straightforward to see that $\rho_-(p)$ and $\rho_+(q)$ are both continuous, strictly decreasing functions, and are well defined since $\ensuremath{\underline{b}} < 0 = \int x \, \mu(\mathrm{d}x) < \ensuremath{\overline{b}}$, so that the infimum in \eqref{eq:4} and the supremum in \eqref{eq:5} are both over non-empty sets. Further, note that we get: \begin{equation}
\label{eq:6}
m_p^{\rho_-(p)} = \ensuremath{\underline{b}}, m_{\rho_+(q)}^q = \ensuremath{\overline{b}} \end{equation} for all $p \le F(\ensuremath{\underline{b}}-)$ and all $q \ge F(\ensuremath{\overline{b}})$. Observe that the barycentre has two nice properties: first, if we rescale the measure $\mu$ by a constant, then the barycentre is unchanged. Second, if we wish to show that a measure $\mu$ has barycentre $b$, it is sufficient to show that \begin{equation*}
\int (x-b) \, \mu(dx) = 0, \end{equation*} independent of whether $\mu$ is a probability measure. In the case where $\mu$ is a probability measure $\mu_{B}(\mathbb{R})$ is just the mean of the measure. Finally, we introduce the additional useful notation \begin{equation*}
\tilde{m}_{p}^q = (q-p) m_p^q. \end{equation*}
Since the functions $\rho_+$ and $\rho_-$ are both continuous and strictly decreasing, their inverses are also continuous and strictly decreasing where defined --- for example, $\rho_+^{-1}$ maps $[\rho_+(1),F(\ensuremath{\overline{b}}-)] \to [F(\ensuremath{\overline{b}}),1]$.
A critical role in the construction of embeddings will be played by the following definition. Set \begin{equation}
\label{eq:1}
\pi^* = \inf \left\{ p \in [\rho_+(1)\vee F(\ensuremath{\underline{b}}),F(\ensuremath{\overline{b}}-)] :
\rho_+^{-1}(p)-p \le \frac{-\ensuremath{\underline{b}}}{\ensuremath{\overline{b}}-\ensuremath{\underline{b}}}\right\} \wedge F(\ensuremath{\overline{b}}-), \end{equation} where we use the standard convention that the infimum of an empty set is $\infty$. Since $\rho_+^{-1}(F(\ensuremath{\overline{b}}-))= F(\ensuremath{\overline{b}})$, $\rho_+^{-1}(p)$ is continuous and $\ensuremath{\underline{b}} < 0$, it follows that $\pi^*\in [\rho_+(1)\vee F(\ensuremath{\underline{b}}),F(\ensuremath{\overline{b}}-)]$. Then we have the following theorem. \begin{theorem}\label{thm:upper_price_mixed}(Upper bound) The bound in \eqref{eq:generalbound_dbmp} is sharp. More precisely, let $\mu$ be a given centred probability measure on $\mathbb{R}$. Then exactly one of the following is true
\begin{enumerate}
\item[\fbox{I}] `$\ensuremath{\underline{b}} \ll 0<\ensuremath{\overline{b}}$': we have $\pi^* =F(\ensuremath{\underline{b}})$ and $\rho_+^{-1}(\pi^*) -\pi^* <
-\ensuremath{\underline{b}}(\ensuremath{\overline{b}}-\ensuremath{\underline{b}})^{-1}$.\\
Then there is a martingale $M\in \mathcal{M}_\mu$ such that
\begin{equation*}
\Prob(\sM_{\infty} \ge \ub, \iM_{\infty} > \lb) =\Ep{ \overline{G}^I(z^*)},
\end{equation*}
where $\overline{G}^I$ is given by \eqref{eq:uG1def} evaluated on paths of $M$, and $z^* =
F^{-1}(\xi)$ where $\xi$ solves
\begin{equation} \label{eqn:xidefn}
\int (x-\ensuremath{\underline{b}}) \, \mu_{F(\ensuremath{\underline{b}})}^\xi = -\ensuremath{\underline{b}}.
\end{equation}
\item[\fbox{II}] `$\ensuremath{\underline{b}}<0<\ensuremath{\overline{b}}$': we have $\rho_+^{-1}(\pi^*) -\pi^* \ge -\ensuremath{\underline{b}}(\ensuremath{\overline{b}}-\ensuremath{\underline{b}})^{-1}$.\\
Then there is a martingale $M\in \mathcal{M}_\mu$ such that
\begin{equation*}
\Prob(\sM_{\infty} \ge \ub, \iM_{\infty} > \lb) =\Ep{ \overline{G}^{II}} = -\ensuremath{\underline{b}}(\ensuremath{\overline{b}}-\ensuremath{\underline{b}})^{-1},
\end{equation*}
where $\overline{G}^{II}$ is given by
\eqref{eq:uG2def}--\eqref{eq:uG2def_par} evaluated on paths of $M$.
\item[\fbox{III}] `$\ensuremath{\underline{b}}< 0 \ll \ensuremath{\overline{b}}$': we have $\pi^* = \rho_+(1)$ and $1-\pi^* < -\ensuremath{\underline{b}}(\ensuremath{\overline{b}}-\ensuremath{\underline{b}})^{-1}$. \\
Then there is a martingale $M\in \mathcal{M}_\mu$ such that
\begin{equation*}
\Prob(\sM_{\infty} \ge \ub, \iM_{\infty} > \lb) = \Ep{\overline{G}^{III}(F^{-1}(\pi^*))},
\end{equation*}
where $\overline{G}^{III}$ is given by \eqref{eq:uG3def} evaluated on paths of $M$.
\end{enumerate} \end{theorem}
In a similar manner to Proposition \ref{prop:prob_upperbound}, the pathwise inequalities described in Section~\ref{sec:dtnt_sub} instantly imply a lower bound on the double exit/no-exit probabilities: \begin{prop}
\label{prop:dtnt_lowerbound}
Suppose $M=(M_{t}: 0\leq t\leq \infty)$ is a continuous uniformly integrable martingale. Then
\begin{equation}\label{eq:generalbound_dbmp2}
\Prob\left( \sM_{\infty} \ge \ub, \iM_{\infty} > \lb\right) \geq \sup\left\{0,\Ep{\underline{G}_{II}(K_1',K_2)},\Ep{ \underline{G}_{III}(K_1,K_2)}\right\},
\end{equation}
where the supremum is taken over $\ensuremath{\underline{b}}<K_2<K_1<\ensuremath{\overline{b}}<K_1'$ and where
$\underline{G}_{II},\underline{G}_{III}$ are given by
\eqref{eq:dbmpH3}, \eqref{eq:valuesdbmpH3} and
\eqref{eq:dbmpH2}, \eqref{eq:valuesdbmpH2} respectively, evaluated on paths of $M$. \end{prop}
We proceed to show that this lower bound is optimal. Write \begin{equation}
\label{eq:16}
\gamma = 1-F(\ensuremath{\overline{b}}-) + F(\ensuremath{\underline{b}}), \end{equation} and consider the condition \begin{equation}
\label{eq:7}
\tilde{m}_{F(\ensuremath{\underline{b}})}^{F(\ensuremath{\overline{b}}-)} + \gamma \ensuremath{\underline{b}} \ge 0. \end{equation} If this holds, then we can find $\lambda \in (F(\ensuremath{\underline{b}}),F(\ensuremath{\overline{b}}-)]$ such that \begin{equation}
\label{eq:9}
\tilde{m}_{F(\ensuremath{\underline{b}})}^{\lambda} + (1-\lambda + F(\ensuremath{\underline{b}})) \ensuremath{\underline{b}} = 0 \end{equation} since the left-hand side is increasing in $\lambda$ and runs between $\ensuremath{\underline{b}}$ and a term which is positive by \eqref{eq:7}. If \eqref{eq:7} fails, we can imagine moving mass from an atom at $\ensuremath{\underline{b}}$, to the right, in the process moving the average of the mass upwards. In this case, consider the condition \begin{equation}
\label{eq:8}
\tilde{m}_{F(\ensuremath{\underline{b}})}^{F(\ensuremath{\overline{b}}-)} + \gamma \ensuremath{\overline{b}} \le 0. \end{equation} If \eqref{eq:7} fails, and \eqref{eq:8} holds, then we set $\xi = F(\ensuremath{\underline{b}})$ and we can find $\lambda \in (0,\gamma]$ such that \begin{equation}
\label{eq:10}
\tilde{m}_{F(\ensuremath{\underline{b}})}^{F(\ensuremath{\overline{b}}-)} + \lambda \ensuremath{\underline{b}} + (\gamma-\lambda) \ensuremath{\overline{b}} = 0. \end{equation} Given such a $\lambda$, we will show that there exists $\pi^* \in [F(\ensuremath{\overline{b}}-),1)$ such that \begin{equation}
\label{eq:11}
\tilde{m}^{\xi} + \tilde{m}_{F(\ensuremath{\overline{b}}-)}^{\pi^*} = \ensuremath{\underline{b}} ( \xi + \pi^* - F(\ensuremath{\overline{b}}-)). \end{equation}
If \eqref{eq:8} also fails, and \begin{equation}
\label{eq:12}
\text{either } \rho_-(0) \ge F(\ensuremath{\overline{b}}-) \text{ or }
\rho_-(0) < F(\ensuremath{\overline{b}}-) \text{ and }
\tilde{m}_{\rho_-(0)}^{F(\ensuremath{\overline{b}}-)} + \ensuremath{\overline{b}} ( 1-F(\ensuremath{\overline{b}}-) + \rho_-(0)) >0 \end{equation} then there exists $\xi \in (F(\ensuremath{\underline{b}}),\rho_-(0)\wedge F(\ensuremath{\overline{b}}-))$ such that \begin{equation}
\label{eq:13}
\tilde{m}_{\xi}^{F(\ensuremath{\overline{b}}-)} + \ensuremath{\overline{b}} (1-F(\ensuremath{\overline{b}}-) + \xi) = 0. \end{equation} Then we define $\pi^*$ as the solution to \eqref{eq:11} again.
Finally, if \eqref{eq:7}, \eqref{eq:8} and \eqref{eq:12} all fail, then there exists $\pi^* \in [\rho_-(0),F(\ensuremath{\overline{b}}-))$ such that \begin{equation}
\label{eq:14}
\tilde{m}_{\pi^*}^{F(\ensuremath{\overline{b}}-)} + \ensuremath{\overline{b}}(1-F(\ensuremath{\overline{b}}-) + \pi^*) =0. \end{equation}
\begin{theorem}\label{thm:lower_price_mixed}(Lower bound) The bound in \eqref{eq:generalbound_dbmp2} is sharp. More precisely, let $\mu$ be a given centred probability measure on $\mathbb{R}$. Then exactly one of the following is true:
\begin{enumerate}
\item[\fbox{I}] `$\ensuremath{\underline{b}}< 0 \ll \ensuremath{\overline{b}}$': condition \eqref{eq:7} holds.\\
Then there is a martingale $M\in \mathcal{M}_\mu$
such that $\Prob(\sM_{\infty} \ge \ub, \iM_{\infty} > \lb) = 0 = \Ep{\underline{G}_{I}}$.
\item[\fbox{II}] `$\ensuremath{\underline{b}}<0<\ensuremath{\overline{b}}$': condition \eqref{eq:7} fails, and either \eqref{eq:8} holds or
\eqref{eq:8} fails and \eqref{eq:12} holds.\\
Then there is a martingale $M\in \mathcal{M}_\mu$ such that
\begin{equation}
\label{eq:15}
\Prob(\sM_{\infty} \ge \ub, \iM_{\infty} > \lb) = \Ep{\underline{G}_{II}(\pi^*,\xi)},
\end{equation}
where $\underline{G}_{II}$ is given via \eqref{eq:dbmpH3} and
\eqref{eq:valuesdbmpH3}, evaluated on paths of $M$, and $\pi^*$ solves \eqref{eq:11}.
\item[\fbox{III}] `$\ensuremath{\underline{b}} \ll 0<\ensuremath{\overline{b}}$': conditions \eqref{eq:7}, \eqref{eq:8} and \eqref{eq:12} fail. \\
Then there is a martingale $M\in \mathcal{M}_\mu$ such that
\begin{equation}
\label{eq:17}
\Prob(\sM_{\infty} \ge \ub, \iM_{\infty} > \lb) = \Ep{\underline{G}_{III}(\pi^*,\rho_-(0))}
\end{equation}
where $\underline{G}_{II}$ is given via \eqref{eq:dbmpH2} and
\eqref{eq:valuesdbmpH2}, evaluated on paths of $M$, and $\pi^*$ is given by \eqref{eq:14}.
\end{enumerate}
\end{theorem}
\begin{remark}
Throughout the paper, we have assumed that $(M_t)_{t \ge 0}$ has
continuous paths. This assumption can be relaxed. It
is relatively simple to see that if we only assume that barriers
$\ensuremath{\underline{b}},\ensuremath{\overline{b}}$ are crossed in a continuous manner then all of our results
remain true. If we only assume that $(M_t)$ has c\`adl\`ag paths
then the situation is more complex. The optimal behaviour
will essentially be as before, but we can use jumps to hide some of
the occasions where a barrier is hit. More precisely, consider the
continuous martingale $M$ given in Theorem~\ref{thm:upper_price_mixed} and, for $\varepsilon > 0$, consider
the time-change:
\[
\rho^\varepsilon_t = \inf\{u\ge t : M_u \in [\ensuremath{\underline{b}}+\varepsilon,\infty)\}.
\]
Then $N_t = M_{\rho^\varepsilon_t}$ is a UI martingale which excludes paths
of $M_t$ where the minimum goes below $\ensuremath{\underline{b}}+\varepsilon$, but which later
return above $\ensuremath{\underline{b}}+\varepsilon$. In general, any possible martingale $M_t$
can be improved by performing such an operation, and so this
suggests that an optimal discontinuous model can be chosen in such a
manner that it is continuous on $[\ensuremath{\underline{b}}+\varepsilon,\infty)$ and only takes
values on $(-\infty,\ensuremath{\underline{b}}]$ if it is the final value of the
martingale. This observation can be used as a starting point for an
analysis similar to that given above to determine the optimal
martingale models for a given measure. We do not pursue the details here. \end{remark}
\section{Proofs that the bounds are sharp via new solutions to the Skorokhod embedding problem} \label{sec:proofs}
In this section we prove Theorems~\ref{thm:upper_price_mixed} and \ref{thm:lower_price_mixed}. We do this by constructing new solutions to the Skorokhod embedding problem for a Brownian motion $B$. Specifically, we will construct stopping times $\tau$ such that $B_{\tau} \sim \mu$, $(B_{t \wedge \tau}:t\geq 0)$ is UI and equalities are attained almost surely in the inequalities of Sections \ref{sec:pathwise}--\ref{sec:dtnt_sub}. It is then straightforward to see that martingales required in Theorems~\ref{thm:upper_price_mixed} and \ref{thm:lower_price_mixed} are given by $M_t:= B_{t \wedge \tau}$.
We will use below some well known facts about the existence of Skorokhod embeddings. Specifically, given a measure $\mu$ with mean $m$ and a Brownian motion $B$ with $B_0 = m$, then there exists a stopping time $\tau$ such that $B_{\tau} \sim \mu$ and $(B_{t \wedge \tau}: t \geq 0)$ is uniformly integrable. Moreover, it follows from uniform integrability that if the measure $\mu$ is supported on a bounded interval, then the process will stop before the first exit time of the interval.
\begin{proof}[Proof of Theorem \ref{thm:upper_price_mixed}]
We take $B=(B_t:t\geq 0)$ a standard real-valued Brownian motion. All the hitting times $H_{\bullet}$ below are for $B$. As described above, we will prove this result by constructing a stopping time $\tau$ such that $B_{\tau}$ has the distribution $\mu$, and such that the conjectured bounds hold for the corresponding continuous time martingale which is the stopped process.
From the definition of $\pi^*$ in \eqref{eq:1} it is clear that at least one of the cases holds. Clearly \fbox{II} excludes the other two. To show that \fbox{I} and \fbox{III} are exclusive, as $\rho_+^{-1}(\rho_+(1))=1$, it suffices to argue that the following is impossible
\begin{equation}\label{eqn:imposs_for_y}
\pi^* = \rho_+(1) = F(\ensuremath{\underline{b}}) > \ensuremath{\overline{b}}(\ensuremath{\overline{b}}-\ensuremath{\underline{b}})^{-1}.
\end{equation}
Assume \eqref{eqn:imposs_for_y} holds. From the last condition we get $\ensuremath{\overline{b}} (1-\pi^*) < - \ensuremath{\underline{b}} \pi^*$, and using the fact that $\pi^* = \rho_+(1)$, this can be expressed as $\int x \,\mu_{\pi^*}(\mathrm{d}x) + \ensuremath{\underline{b}} \pi^* < 0$. However $\pi^* \ge F(\ensuremath{\underline{b}})$ implies that this is greater than or equal to $\int x \, \mu(\mathrm{d}x) = 0$ giving a contradiction. We conclude that the cases \fbox{I}, \fbox{II} and \fbox{III} are exclusive.
We now show the existence of a suitable embedding. We consider
initially the case \fbox{I}. We first note that the solution $\xi$
of \eqref{eqn:xidefn} is in $(\rho_+^{-1}(\pi^*),1]$. Since
\begin{eqnarray*}
\int (x - \ensuremath{\underline{b}}) \, \mu_{F(\ensuremath{\underline{b}})}^{\rho_+^{-1}(F(\ensuremath{\underline{b}}))}(\mathrm{d}x) & = &
\int (x - \ensuremath{\overline{b}}) \mu_{F(\ensuremath{\underline{b}})}^{\rho_+^{-1}(F(\ensuremath{\underline{b}}))}(\mathrm{d}x) +
\int (\ensuremath{\overline{b}} - \ensuremath{\underline{b}}) \, \mu_{F(\ensuremath{\underline{b}})}^{\rho_+^{-1}(F(\ensuremath{\underline{b}}))}(\mathrm{d}x) \\
& = & (\ensuremath{\overline{b}} - \ensuremath{\underline{b}})\left( \rho_+^{-1}(F(\ensuremath{\underline{b}}))-F(\ensuremath{\underline{b}})\right) \\
& < & - \ensuremath{\underline{b}},
\end{eqnarray*}
we conclude that $\xi > \rho_+^{-1}(\pi^*)$. To see that $\xi \le 1$,
we note:
\begin{equation*}
\int (x - \ensuremath{\underline{b}}) \, \mu_{F(\ensuremath{\underline{b}})} (\mathrm{d}x) \ge \int (x - \ensuremath{\underline{b}}) \, \mu
(\mathrm{d}x) = -\ensuremath{\underline{b}}.
\end{equation*}
Since the expression $\int (x-\ensuremath{\underline{b}}) \, \mu_{F(\ensuremath{\underline{b}})}^\xi(\mathrm{d}x)$ is strictly increasing and continuous in $\xi$, there is a unique $\xi$. For this value of $\xi$, we now define a measure $\nu$ by
\begin{equation*}
\nu = \left[ -\frac{\ensuremath{\underline{b}}}{\ensuremath{\overline{b}}-\ensuremath{\underline{b}}} - (\xi - F(\ensuremath{\underline{b}}))\right] \delta_{\ensuremath{\underline{b}}} + \mu_{F(\ensuremath{\underline{b}})}^\xi.
\end{equation*}
Observe that the atom at $\ensuremath{\underline{b}}$ has mass greater than or equal to
zero, and by construction, $\nu$ has total mass $-\ensuremath{\underline{b}}(\ensuremath{\overline{b}}-\ensuremath{\underline{b}})^{-1}$
and barycentre $\ensuremath{\overline{b}}$ since
\begin{eqnarray*}
\int (x-\ensuremath{\overline{b}}) \, \nu(\mathrm{d}x) & = & \int (x-\ensuremath{\overline{b}}) \, \mu_{F(\ensuremath{\underline{b}})}^\xi(\mathrm{d}x)
+ \left[-\frac{\ensuremath{\underline{b}}}{\ensuremath{\overline{b}}-\ensuremath{\underline{b}}} - (\xi - F(\ensuremath{\underline{b}}))\right] (\ensuremath{\underline{b}}-\ensuremath{\overline{b}})\\
& = & (\ensuremath{\underline{b}}-\ensuremath{\overline{b}})(\xi-F(\ensuremath{\underline{b}})) - \ensuremath{\underline{b}} + \left[-\frac{\ensuremath{\underline{b}}}{\ensuremath{\overline{b}}-\ensuremath{\underline{b}}} -
(\xi - F(\ensuremath{\underline{b}}))\right] (\ensuremath{\underline{b}}-\ensuremath{\overline{b}})\\
& = & 0.
\end{eqnarray*}
We now show that this means we can construct a suitable embedding. The idea will be initially to run until the first time we hit either of $\ensuremath{\overline{b}}$ or $\ensuremath{\underline{b}}$. The mass that hits $\ensuremath{\overline{b}}$ first will then be used to embed $\nu$, and all the mass that hits $\ensuremath{\underline{b}}$ (which will include the atomic term from $\nu$) can then be embedded in the remaining areas, $(0,\ensuremath{\underline{b}}] \cup [F^{-1}(\xi),\infty)$. So suppose we are in case \fbox{I}, and let $\tau_1$ be first time we hit one of $\ensuremath{\underline{b}}$ or $\ensuremath{\overline{b}}$, so $\tau_1 = H_{\ensuremath{\overline{b}}} \wedge H_{\ensuremath{\underline{b}}}$. Then $\Prob(B_{\tau_1} = \ensuremath{\overline{b}}) = -\ensuremath{\underline{b}}(\ensuremath{\overline{b}}-\ensuremath{\underline{b}})^{-1}$. Let $\tau_2$ be a UI embedding of the probability measure $-\frac{\ensuremath{\overline{b}}-\ensuremath{\underline{b}}}{\ensuremath{\underline{b}}}\nu$ given $B_0 = \ensuremath{\overline{b}}$ and let $\tau_3$ be a UI embedding of $\sigma$ given $B_0 = \ensuremath{\underline{b}}$, where
\begin{equation*}
\sigma = \frac{\left(\mu^{F(\ensuremath{\underline{b}})}+\mu_{\xi}\right)}{F(\ensuremath{\underline{b}})+1-\xi}.
\end{equation*}
It can be verified that $\sigma$
has
barycentre $\ensuremath{\underline{b}}$ since
\begin{equation*}
\int (x-\ensuremath{\underline{b}}) \left(\mu^{F(\ensuremath{\underline{b}})}+\mu_{\xi}\right)(\mathrm{d}x) = \int
(x-\ensuremath{\underline{b}}) \, \mu(\mathrm{d}x) - \int (x-\ensuremath{\underline{b}}) \, \mu_{F(\ensuremath{\underline{b}})}^\xi(\mathrm{d}x) = 0.
\end{equation*}
Then (recalling the definition in Section~\ref{sec:notation}) we set
\begin{eqnarray*}
\tau & := & \tau_2 \circ \tau_1\indic{\tau_1 = H_{\ensuremath{\overline{b}}}}
\indic{\tau_2 \circ \tau_1 < H_{\ensuremath{\underline{b}}}}\\
&& {} + \tau_3 \circ \tau_1 \indic{\tau_1 = H_{\ensuremath{\underline{b}}}}\\
&& {} + \tau_3 \circ \tau_2 \circ \tau_1 \indic{\tau_1 = H_{\ensuremath{\overline{b}}}}
\indic{\tau_2 \circ \tau_1 = H_{\ensuremath{\underline{b}}}}.
\end{eqnarray*}
We see that $\tau$ is a UI embedding of $\mu$, and moreover $\tau$
is such that $ \dbmps= \overline{G}^I(F^{-1}(\xi))$ a.s..
Consider now case \fbox{II}. Suppose initially that in addition, $\rho_+^{-1}(\pi^*)-\pi^* = -\ensuremath{\underline{b}} (\ensuremath{\overline{b}}-\ensuremath{\underline{b}})^{-1}$. We define measures $\nu$ and $\sigma$ by:
\begin{eqnarray*}
\nu & = & \frac{1}{\rho_+^{-1}(\pi^*)-\pi^*} \mu_{\pi^*}^{\rho_+^{-1}(\pi^*)}\\
\sigma & = & \frac{1}{1+\pi^*-\rho_+^{-1}(\pi^*)}
\left(\mu^{\pi^*} + \mu_{\rho_+^{-1}(\pi^*)}\right).
\end{eqnarray*}
Then $\nu$ has barycentre $\ensuremath{\overline{b}}$, while $\sigma$ has barycentre
$\ensuremath{\underline{b}}$. Let $\tau_1$ be as above, $\tau_2$ be a UI embedding of
$\nu$ given $B_0 = \ensuremath{\overline{b}}$ and $\tau_3$ be a UI embedding of $\sigma$
given $B_0 = \ensuremath{\underline{b}}$. Then the stopping time
\begin{eqnarray*}
\tau & := & \tau_2 \circ \tau_1 \indic{\tau_1 = H_{\ensuremath{\overline{b}}}} \\
&& {} + \tau_3 \circ \tau_1 \indic{\tau_1 = H_{\ensuremath{\underline{b}}}}
\end{eqnarray*}
is a UI embedding of $\mu$, and $B_{t\wedge \tau}$ satisfies
$\dbmps=\overline{G}^{II}$ a.s.{} where $\overline{G}^{II}$ is the random variable
defined in \eqref{eq:uG2def}, evaluated on paths of $B$.
The case where $\rho_+^{-1}(\pi^*)-\pi^* > -\ensuremath{\underline{b}} (\ensuremath{\overline{b}}-\ensuremath{\underline{b}})^{-1}$ is almost identical --- observe that in this case, there must be an atom of $\mu$ at $\ensuremath{\overline{b}}$ with $F(\ensuremath{\overline{b}})-F(\ensuremath{\overline{b}}-) > -\ensuremath{\underline{b}} (\ensuremath{\overline{b}}-\ensuremath{\underline{b}})^{-1}$. However, the argument above works without alteration if we take:
\begin{eqnarray*}
\nu & = & \frac{1}{-\ensuremath{\underline{b}} (\ensuremath{\overline{b}}-\ensuremath{\underline{b}})^{-1}} \delta_{\ensuremath{\overline{b}}}\\
\sigma & = & \frac{1}{1+ \ensuremath{\underline{b}} (\ensuremath{\overline{b}}-\ensuremath{\underline{b}})^{-1}}(\mu-\nu).
\end{eqnarray*}
Finally we consider \fbox{III}. Then define measures $\nu$ and
$\sigma$ by:
\begin{eqnarray*}
\nu & = & \frac{1}{1-\pi^*} \mu_{\pi^*}\\
\sigma & = & \frac{1}{\pi^*}\mu^{\pi^*}.
\end{eqnarray*}
So the barycentre of $\nu$ is $\ensuremath{\overline{b}}$, and the
barycentre of $\sigma$ is $m^{\pi^*}$. Define $\tau_1$ to be the first
hitting time of $\{m^{\pi^*},\ensuremath{\overline{b}}\}$, so $\tau_1 = H_{m^{\pi^*}}\wedge H_{\ensuremath{\overline{b}}}$, then $\Prob(B_{\tau_1} = \ensuremath{\overline{b}}) =
\pi^*=- m^{\pi^*}(\ensuremath{\overline{b}} - m^{\pi^*})^{-1}$. We may then proceed as
above, so we define $\tau_2$ to be a UI embedding of $\nu$ given
$B_0 = \ensuremath{\overline{b}}$ and $\tau_3$ to be a UI embedding of $\sigma$ given $B_0
= m^{\pi^*}$. Then the stopping time
\begin{eqnarray*}
\tau & := & \tau_2 \circ \tau_1 \indic{\tau_1 = H_{\ensuremath{\overline{b}}}} \\
&& {} + \tau_3 \circ \tau_1 \indic{\tau_1 = H_{m^{\pi^*}}}
\end{eqnarray*}
is a UI embedding of $\mu$, and satisfies $\dbmps=\overline{G}^{III}(F^{-1}(\pi^*))$
a.s.{} where $\overline{G}^{III}(\cdot)$ is the random variable defined in
\eqref{eq:uG3def}, evaluated on paths of $B$. \end{proof}
\begin{proof}[Proof of Theorem \ref{thm:lower_price_mixed}] The setup, and general methodology, is analogous to the proof of Theorem \ref{thm:upper_price_mixed} above.
It follows from their respective definitions that exactly one of
\fbox{I}, \fbox{II} and \fbox{III} holds.
Suppose \fbox{I} holds, so that \eqref{eq:7} is true. Then, by
continuity, there exists $\lambda\in (F(\ensuremath{\underline{b}}),F(\ensuremath{\overline{b}}-)]$ such that
\eqref{eq:9} holds (taking $\lambda = F(\ensuremath{\underline{b}})$ gives $\ensuremath{\underline{b}}$ on the
left hand side of \eqref{eq:9}). Let $\tau_1$ be a UI
embedding of
\begin{equation}
\label{eq:18}
\chi = \mu_{F(\ensuremath{\underline{b}})}^\lambda + (1-\lambda + F(\ensuremath{\underline{b}}))\delta_{\ensuremath{\underline{b}}}
\end{equation}
in the Brownian motion starting at 0, and observe that the measure
\begin{equation*}
\nu = \frac{\mu^{F(\ensuremath{\underline{b}})} + \mu_{\lambda}}{1-\lambda + F(\ensuremath{\underline{b}})}
\end{equation*}
has mean $\ensuremath{\underline{b}}$, which follows since:
\begin{align*}
(1-\lambda + F(\ensuremath{\underline{b}})) \int x \nu(\mathrm{d}x) & = \tilde{m}^{F(\ensuremath{\underline{b}})} +
\tilde{m}_{\lambda}\\
& = -\tilde{m}_{F(\ensuremath{\underline{b}})}^{\lambda} = \ensuremath{\underline{b}}(1-\lambda + F(\ensuremath{\underline{b}})).
\end{align*}
Let $\tau_2$ be a UI embedding of $\nu$ in a Brownian motion starting
from $B_0=\ensuremath{\underline{b}}$. Finally define
$$\tau:=\tau_1\indic{B_{\tau_1}\neq
\ensuremath{\underline{b}}}+\tau_2\circ\tau_1\indic{B_{\tau_1}=\ensuremath{\underline{b}}},$$ which is a UI
embedding of $\mu$ in the Brownian motion $B$. Note that
$\overline{B}_\tau\geq \ensuremath{\overline{b}}$ only if $\underline{B}_\tau\leq \ensuremath{\underline{b}}$. It
follows that $\dbmps=0=\underline{G}_{I}$ a.s.
Suppose now that \fbox{II} holds. We consider separately the case
where \eqref{eq:7} fails and \eqref{eq:8} holds, and the case where
both \eqref{eq:7} and \eqref{eq:8} fail, but \eqref{eq:12}
holds. First suppose \eqref{eq:8} holds. Then
\begin{equation*}
\lambda \mapsto \tilde{m}_{F(\ensuremath{\underline{b}})}^{F(\ensuremath{\overline{b}}-)} + \lambda \ensuremath{\underline{b}} + (\gamma -
\lambda) \ensuremath{\overline{b}}
\end{equation*}
is continuous, and strictly negative for $\lambda = 0$ and positive for
$\lambda = \gamma$. Hence there exists $\lambda \in (0,\gamma]$ such
that \eqref{eq:10} holds. Fix $\xi = F(\ensuremath{\underline{b}})$ and consider
\begin{equation*}
[F(\ensuremath{\overline{b}}-),1) \ni \pi^* \mapsto \tilde{m}^{\xi} + \tilde{m}_{F(\ensuremath{\overline{b}}-)}^{\pi^*} - \ensuremath{\underline{b}} ( \xi + \pi^* - F(\ensuremath{\overline{b}}-)).
\end{equation*}
In the limit as $\pi^* \to 1$, the expression simplifies to
$-\tilde{m}^{F(\ensuremath{\overline{b}}-)}_{F(\ensuremath{\underline{b}})}-\gamma\ensuremath{\underline{b}}$ which is strictly positive since
\eqref{eq:7} is assumed to fail, while if $\pi^* = F(\ensuremath{\overline{b}}-)$ the
expression simplifies to $\tilde{m}^{F(\ensuremath{\underline{b}})}-\ensuremath{\underline{b}} F(\ensuremath{\underline{b}})$, which is
non-positive, since $\tilde{m}^{F(\ensuremath{\underline{b}})} = \int x \, \mu^{F(\ensuremath{\underline{b}})}(\mathrm{d}x) \le
\int \ensuremath{\underline{b}}\, \mu^{F(\ensuremath{\underline{b}})}(\mathrm{d}x)$. Hence there is a unique $\pi^*$
satisfying \eqref{eq:11}.
Now define a measure
\begin{equation*}
\chi = \mu_{F(\ensuremath{\underline{b}})}^{F(\ensuremath{\overline{b}}-)} + \lambda \delta_{\ensuremath{\underline{b}}} +
(\gamma-\lambda) \delta_{\ensuremath{\overline{b}}}.
\end{equation*}
From \eqref{eq:10} it follows that $\chi$ is centered, and we embed this initially. The mass which arrives at $\ensuremath{\overline{b}}$ will then run to the measure
\begin{equation*}
\nu = \frac{(\gamma-\lambda -(1-\pi^*))\delta_{\ensuremath{\underline{b}}} + \mu_{\pi^*}}{\gamma-\lambda}
\end{equation*}
which has mean $\ensuremath{\overline{b}}$ by the following computation:
\begin{align*}
(\gamma - \lambda) \int x \, \nu(\mathrm{d}x) & = \ensuremath{\underline{b}} (\gamma-\lambda
-(1-\pi^*)) + \tilde{m}_{\pi^*}\\
& = \ensuremath{\underline{b}} (\gamma-\lambda -(1-\pi^*)) - \tilde{m}_{F(\ensuremath{\overline{b}}-)}^{\pi^*} -
\tilde{m}_{F(\ensuremath{\underline{b}})}^{F(\ensuremath{\overline{b}}-)} - \tilde{m}^{F(\ensuremath{\underline{b}})} \\
& = \ensuremath{\underline{b}} (\gamma-\lambda -(1-\pi^*)) - \ensuremath{\underline{b}}(\xi + \pi^* - F(\ensuremath{\overline{b}}-)) +
\lambda \ensuremath{\underline{b}} + (\gamma - \lambda) \ensuremath{\overline{b}}\\
& = \ensuremath{\underline{b}} (\gamma -1 -\xi + F(\ensuremath{\overline{b}}-)) + \ensuremath{\overline{b}} (\gamma-\lambda).
\end{align*}
Here we have used \eqref{eq:10}, \eqref{eq:11} and the fact that $\xi = F(\ensuremath{\underline{b}})$. From the definition of $\gamma$ in \eqref{eq:16}, the desired conclusion follows.
Finally, we embed the remaining part of $\mu$ from the mass that finishes at $\ensuremath{\underline{b}}$ after either the first or second step, which has total probability $\gamma - \lambda + \pi^* - 1 + \lambda = \xi + \pi^* - F(\ensuremath{\overline{b}}-)$. Set
\begin{equation}\label{eq:19}
\sigma = \frac{\mu^{\xi} + \mu_{F(\ensuremath{\overline{b}}-)}^{\pi^*}}{\xi + \pi^* - F(\ensuremath{\overline{b}}-)},
\end{equation}
and $\sigma$ has mean $\ensuremath{\underline{b}}$:
\begin{align*}
(\xi + \pi^* - F(\ensuremath{\overline{b}}-)) \int x \, \sigma(\mathrm{d}x) & = \tilde{m}^{\xi} +
\tilde{m}_{F(\ensuremath{\overline{b}}-)}^{\pi^*}\\
& = \ensuremath{\underline{b}}(\xi + \pi^* - F(\ensuremath{\overline{b}}-))
\end{align*}
by \eqref{eq:11}. The final stopping time will be of the same form
both in this case and in the case where \eqref{eq:8} holds, and when
\eqref{eq:8} fails but \eqref{eq:12} holds. So before constructing
the embedding, we give a description of the relevant measures in the
second case.
Suppose \eqref{eq:8} fails, but \eqref{eq:12} holds. Then in a
similar manner to above, we can find $\xi \in
(F(\ensuremath{\underline{b}}),\rho_-(0)\wedge F(\ensuremath{\overline{b}}-))$
such that \eqref{eq:13} holds. Define
\begin{equation*}
\chi = \mu_{\xi}^{F(\ensuremath{\overline{b}}-)} + (1-F(\ensuremath{\overline{b}}-)+\xi) \delta_{\ensuremath{\overline{b}}}
\end{equation*}
and choose $\pi^*$ as before as the solution to \eqref{eq:11}.
Then set
\begin{equation*}
\nu = \frac{(\pi^*-F(\ensuremath{\overline{b}}-) + \xi) \delta_{\ensuremath{\underline{b}}} + \mu_{\pi^*}}{1-F(\ensuremath{\overline{b}}-)+\xi}
\end{equation*}
and we verify that $\nu$ has mean $\ensuremath{\overline{b}}$:
\begin{align*}
(1-F(\ensuremath{\overline{b}}-)+\xi) \int x \, \nu(\mathrm{d}x) & = \ensuremath{\underline{b}} (\pi^* - F(\ensuremath{\overline{b}}-)+\xi) +
\tilde{m}_{\pi^*}\\
& = \ensuremath{\underline{b}} ( \pi^* - F(\ensuremath{\overline{b}}-) + \xi) - \tilde{m}_{F(\ensuremath{\overline{b}}-)}^{\pi^*} -
\tilde{m}_{\xi}^{F(\ensuremath{\overline{b}}-)}-\tilde{m}^{\xi}\\
& = \ensuremath{\underline{b}}(\pi^*-F(\ensuremath{\overline{b}}-) + \xi) - \ensuremath{\underline{b}}(\xi+\pi^*-F(\ensuremath{\overline{b}}-)) + \ensuremath{\overline{b}}(1-F(\ensuremath{\overline{b}}-) + \xi)\\
& = \ensuremath{\overline{b}}(1-F(\ensuremath{\overline{b}}-)+\xi).
\end{align*}
Finally, setting $\sigma$ as in \eqref{eq:19} we again have $\sigma$
with mean $\ensuremath{\underline{b}}$.
In both cases, we construct an embedding as follows: let $\tau_1$ be
a UI embedding of $\chi$ (starting from $0$). Then let $\tau_2$
be a UI embedding of $\nu$ (starting from
$\ensuremath{\overline{b}}$). Finally, we let $\tau^3$ be a
UI embedding of $\sigma$ (starting from $\ensuremath{\underline{b}}$). We then define the
complete embedding by:
\begin{equation*}
\begin{split}
\tau:=\ &\tau_1\indic{B_{\tau_1}\in
(\ensuremath{\underline{b}},\ensuremath{\overline{b}})}\\
&+\tau_2\circ\tau_1\indic{B_{\tau_1}=\ensuremath{\overline{b}}}
\indic{B_{\tau_2\circ\tau_1}>\ensuremath{\underline{b}}}+\\
&+\tau_3\circ\Big(\tau_1\indic{B_{\tau_1}=\ensuremath{\underline{b}}}
+\tau_2\circ\tau_1\indic{B_{\tau_1}=\ensuremath{\overline{b}}}
\indic{B_{\tau_2\circ\tau_1}=\ensuremath{\underline{b}}}\Big),
\end{split}
\end{equation*}
and it follows from our construction that $\tau$ is a UI embedding
of $\mu$ which moreover satisfies $\dbmps =
\underline{G}_{II}(\pi^*,\xi)$.
Suppose finally we are in case \fbox{III}, so that \eqref{eq:7}, \eqref{eq:8}
and \eqref{eq:12} all fail. Then there exists $\pi^* \in
[\rho_-(0),F(\ensuremath{\overline{b}}-))$ such that \eqref{eq:14} holds.
Define the probability measure
\begin{equation*}
\chi = \mu_{\pi^*}^{F(\ensuremath{\overline{b}}-)} + (1-F(\ensuremath{\overline{b}}-)-\pi^*) \delta_{\ensuremath{\overline{b}}},
\end{equation*}
which has mean $0$ by the definition of $\pi^*$. Define also
\begin{equation*}
\nu = \frac{\rho_-(0) \delta_{\ensuremath{\underline{b}}} + \mu_{\rho_-(0)}^{\pi^*} + \mu_{F(\ensuremath{\overline{b}}-)}}{1-F(\ensuremath{\overline{b}}-)+\pi^*}
\end{equation*}
and we confirm that $\nu$ has mean $\ensuremath{\overline{b}}$:
\begin{align*}
(1-F(\ensuremath{\overline{b}}-)+\pi^*) \int x \, \nu(dx) & = \tilde{m}^{\rho_-(0)} +
\tilde{m}_{\rho_-(0)}^{\pi^*} + \tilde{m}_{F(\ensuremath{\overline{b}}-)}\\
& = \tilde{m}^{\pi^*} + \tilde{m}_{F(\ensuremath{\overline{b}}-)} \\
& = -\tilde{m}_{\pi^*}^{F(\ensuremath{\overline{b}}-)}\\
& = \ensuremath{\overline{b}}(1-F(\ensuremath{\overline{b}}-)+\pi^*).
\end{align*}
Finally, any mass which is at $\ensuremath{\underline{b}}$ we finally embed to the measure
$\sigma = (\rho_-(0))^{-1}\mu^{\rho_-(0)}$. That is, we define the
stopping times $\tau_1$ which is a UI embedding of $\chi$ starting
at $0$. Then let $\tau_2$ be a UI embedding of $\nu$, given initial
value $\ensuremath{\overline{b}}$, and $\tau_3$ an embedding of $\sigma$ given initial
value $\ensuremath{\underline{b}}$. Finally, we define
$$\tau:=\tau_1\indic{B_{\tau_1}\neq
\ensuremath{\overline{b}}}+\tau_2\circ\tau_1\indic{B_{\tau_1}=\ensuremath{\overline{b}}}
\indic{B_{\tau_2\circ\tau_1} > \ensuremath{\underline{b}}}+ \tau_3 \circ\tau_2\circ\tau_1\indic{B_{\tau_1}=\ensuremath{\overline{b}}}
\indic{B_{\tau_2\circ\tau_1} = \ensuremath{\underline{b}}},$$ to get a UI embedding of
$\mu$ in $B$. Furthermore, it follows from the construction that
$\dbmps = \underline{G}_{III}(\pi^*,\rho_-(0))$. \end{proof}
\section{On joint distribution of the maximum and minimum of a continuous UI martingale} \label{ap:mart}
We turn now to studying the properties of joint distribution of the maximum and minimum of a continuous UI martingale. As previously, $(M_t:0\leq t\leq \infty)$ is a uniformly integrable continuous martingale. We let $\mu$ be its terminal distribution, $\mu\sim M_\infty$, and recall that $-\infty\leq \ell_{\mu}<r_{\mu}\leq \infty$ are the bounds of the support of $\mu$, i.e.\ $[\ell_{\mu},r_{\mu}]$ is the smallest interval with $\mu([\ell_{\mu},r_{\mu}])=1$. Using Theorems~\ref{thm:upper_price_mixed} and \ref{thm:lower_price_mixed} above, as well as existing results, we study the functions \begin{align}
\label{eq:20}
p(\ensuremath{\underline{b}},\ensuremath{\overline{b}}) & = \Prob\left(\underline{M}_\infty > \ensuremath{\underline{b}}\textrm{ and }\overline{M}_\infty <\ensuremath{\overline{b}}
\right)\\
q(\ensuremath{\underline{b}},\ensuremath{\overline{b}}) & = \Prob\left(\underline{M}_\infty > \ensuremath{\underline{b}}\textrm{ and }\overline{M}_\infty \ge \ensuremath{\overline{b}}
\right)\label{eq:21}\\
r(\ensuremath{\underline{b}},\ensuremath{\overline{b}}) & = \Prob\left(\underline{M}_\infty\le \ensuremath{\underline{b}}\textrm{ and }\overline{M}_\infty \ge \ensuremath{\overline{b}}
\right)\label{eq:22} \end{align} for $\ensuremath{\underline{b}}\leq 0 \leq \ensuremath{\overline{b}}$. Note that with no restrictions on $M_0$, when looking at extrema of the functions above, it is enough to consider $M_0$ a constant (e.g.\ when maximising $r$) or $M_0\equiv M_\infty$ (e.g.\ when minimising $r$). The latter is degenerate and henceforth we assume $M_0$ is a constant a.s. Further, as our results are translation invariant, we may and will take $M_0=0$ a.s. It follows that $\mu$ is centred.
It follows from Dambis, Dubins-Schwarz Theorem that $M$ is a (continuous) time change of Brownian motion, i.e.\ we can write $M_t=B_{\tau_t}$, $t\leq \infty$, for some Brownian motion and an increasing family of stopping times $(\tau_t)$ with $B_{\tau_\infty}\sim M_\infty$, $(B_{t\land \tau_\infty}:t\geq 0 )$ UI and $\overline{M}_\infty=\overline{B}_{\tau_\infty}$, $\underline{M}_\infty=\underline{B}_{\tau_\infty}$. In consequence, the problem reduces to studying the maximum and minimum of Brownian motion stopped at $\tau=\tau_\infty$, which is a solution the Skorokhod embedding problem. We can deduce results about the optimal properties of the martingales from corresponding results about Skorokhod embeddings. Our first result concerns the embeddings of Perkins and the `tilted-Jacka' construction, which we now recall using the notation established previously. These constructions have been considered in \cite{Cox:2011ab}, and we will need some results from this paper; however both constructions have a long history --- see for example \cite{Perkins:86,Cox:2004aa,Jacka:88,Cox:2005aa}. For the Perkins embedding we define\footnote{Strictly, we only consider the case where $\mu(\{0\}) = 0$. If this is not the case, then the optimal embedding requires independent randomisation to stop some mass at zero initially.} \begin{equation}
\label{eq:24}
\begin{split}
{\gamma_+}(p)=q \text{ where $q$ solves } &\tilde{m}^{q} + \tilde{m}_{p} = (1-p+q)
F(p), \quad p > F(0)\\
{\gamma_+}(q)=p \text{ where $p$ solves } &\tilde{m}^{q} + \tilde{m}_{p} = (1-p+q)
F(q), \quad q < F(0-).
\end{split} \end{equation} The stopping time $\tau_P$ is then defined via: \begin{equation}
\label{eq:25}
\tau_P = \inf\{ t \ge 0: F(B_t) \not\in ({\gamma_+}(F(\ensuremath{\overline{B}}_t)),{\gamma_-}(F(\ensuremath{\underline{B}}_t)))\}. \end{equation}
In a similar spirit, the tilted-Jacka construction is given as follows. Choose $\pi^*\in [0,1]$ such that $(\ensuremath{\underline{b}}-m^{\pi^*})(m_{\pi^*}-\ensuremath{\overline{b}})\ge 0$ --- this is always possible, since we can always find $\pi^*$ such that $m^{\pi^*} = \ensuremath{\underline{b}}$ say. Then set $\chi = \pi^* \delta_{m^{\pi^*}} + (1-\pi^*) \delta_{m_{\pi^*}}$. The construction is as follows: we first embed the distribution $\chi$, then, given we hit $m^{\pi^*}$, we embed $\mu^{\pi^*}$ using the reversed Az\'ema-Yor construction (c.f.~\cite{Obloj:04b}); if we hit $m_{\pi^*}$ then we embed $\mu_{\pi^*}$ using the Az\'ema-Yor construction.
Finally, we observe that both cases give rise to martingales with certain optimality properties using the fact that the stopped Brownian motion is a continuous martingale.
\begin{prop}\label{prop:M_bounds}
We have the following properties:
\begin{enumerate}
\item $p(0,\ensuremath{\overline{b}})=0 = p(\ensuremath{\underline{b}},0)$, $q(0,\ensuremath{\overline{b}}) = 0 = q(\ensuremath{\underline{b}},r_{\mu})$ and
$r(\ell_{\mu},\ensuremath{\overline{b}}) = 0 = r(\ensuremath{\underline{b}},r_{\mu})$;
\item $p(\ensuremath{\underline{b}},\ensuremath{\overline{b}})=1$ on $[-\infty,\ell_{\mu})\times(r_{\mu},\infty]$,
$q(\ensuremath{\underline{b}},\ensuremath{\overline{b}}) = 1$ on $[-\infty,\ell_{\mu}) \times \{0\}$, and $r(0,0) = 1$;
\item $p$ and $q$ are non-increasing in $\ensuremath{\underline{b}}\in (\ell_{\mu},0)$ and
$p$ is non-decreasing in $\ensuremath{\overline{b}}\in (0,r_{\mu})$; $r$ is non-decreasing
in $\ensuremath{\underline{b}} \in (\ell_{\mu},0)$ and $q$ and $r$ are non-decreasing in $\ensuremath{\overline{b}}
\in (0,r_{\mu})$;
\item for $\ell_{\mu}\leq \ensuremath{\underline{b}}<0<\ensuremath{\overline{b}}\leqr_{\mu}$ we have
\begin{equation}\label{eq:uimart_bound}
\Prob\big(\underline{B}_{\tau_J}>\ensuremath{\underline{b}}\textrm{ and
}\overline{B}_{\tau_J} <\ensuremath{\overline{b}} \big)\leq p(\ensuremath{\underline{b}},\ensuremath{\overline{b}})\leq
\Prob\big(\underline{B}_{\tau_P}>\ensuremath{\underline{b}}\textrm{ and
}\overline{B}_{\tau_P} <\ensuremath{\overline{b}} \big),
\end{equation}
where $(B_t)$ is a standard Brownian motion with $B_0=0$,
$\tau_P$ is the Perkins stopping time \cite[(4.4)]{Cox:2011ab} embedding
$\mu$ and $\tau_J$ is the `tilted-Jacka' stopping time \cite[(4.6)]{Cox:2011ab},
for barriers $(\ensuremath{\underline{b}},\ensuremath{\overline{b}})$, embedding $\mu$;
\item for $\ell_{\mu}\leq \ensuremath{\underline{b}}<0<\ensuremath{\overline{b}}\leqr_{\mu}$, the lower bound on
$q(\ensuremath{\underline{b}},\ensuremath{\overline{b}})$ is given by \eqref{eq:generalbound_dbmp}, and the
upper bound is given by \eqref{eq:generalbound_dbmp2}. Moreover
these bounds are attained by the constructions in
Theorems~\ref{thm:upper_price_mixed} and
\ref{thm:lower_price_mixed} respectively;
\item for $\ell_{\mu}\leq \ensuremath{\underline{b}}<0<\ensuremath{\overline{b}}\leqr_{\mu}$, the lower bound on
$r(\ensuremath{\underline{b}},\ensuremath{\overline{b}})$ is given by Proposition~2.3 of \cite{Cox:2011aa}, and
the upper bound is given by Proposition~2.1 of \cite{Cox:2011aa}.
Moreover these bounds are attained by the constructions in
Theorems~2.4 and 2.2 of \cite{Cox:2011aa} respectively.
\end{enumerate} \end{prop} The first three assertions of the proposition are clear. Assertion $(iv)$ is a reformulation of Lemmas~4.2 and 4.3 of \cite{Cox:2011ab} --- it suffices to note that $(B_{t\wedge\tau_J})$, $(B_{t\wedge\tau_P})$, $(M_t)$ are all UI martingales starting at $0$ and with the same terminal law $\mu$ for $t=\infty$. Likewise, part $(vi)$ is a reinterpretation of the results of \cite{Cox:2011aa}. We note that therein the results were formulated for the case of non-atomic $\mu$. They extend readily, with methods used in Section \ref{sec:proofs} above, specifically by characterising the stopping distributions via quantiles of the underlying measures, to the general case.
We can think of any of the functions $p(\cdot,\cdot), q(\cdot,\cdot)$, and $r(\cdot,\cdot)$ as a surface defined over the quarter-plane $[-\infty,0]\times [0,\infty]$. Proposition \ref{prop:M_bounds} describes boundary values of the surface, monotonicity properties and gives an upper and a lower bound on the surface. However we note that --- most obviously in $(iv)$ --- there is a substantial difference between the bounds linked to the fact that $\tau_P$ does not depend on $(\ensuremath{\underline{b}},\ensuremath{\overline{b}})$ while $\tau_J$ does. In consequence, the upper bound is attainable: there is a martingale $(M_t)$, namely $M_t=(B_{t\wedge\tau_P})$, for which $p$ is equal to the upper bound for all $(\ensuremath{\underline{b}},\ensuremath{\overline{b}})$. In contrast a martingale $(M_t)$ for which $p$ would be equal to the lower bound does not exist. For the martingale $M_t=(B_{t\wedge\tau_J})$, where $\tau_J$ is defined for some pair $(\ensuremath{\underline{b}},\ensuremath{\overline{b}})$, $p$ will attain the lower bound in some neighbourhood of $(\ensuremath{\underline{b}},\ensuremath{\overline{b}})$ which will be strictly contained in $(\ell_{\mu},0)\times (0,r_{\mu})$. More generally, the latter case is more typical of all the constructions which are used in the result; however, with some careful construction, it seems likely that one can usually find a construction which will be optimal for all values of $(\ensuremath{\underline{b}}, \ensuremath{\overline{b}})$ which lie in some small open set (for example, this is true of the tilted-Jacka construction), but there will be limits on how large the region on which a given construction is optimal can be made.
We now give a result which provides some further insight into the structure of the bounds discussed above. In particular, we can show some finer properties of the functions $p,q,r$ and their upper and lower bounds. We state and prove the result for the function $p$, but the corresponding versions for $q$ and $r$ will follow in a clear manner.
\begin{theorem} \label{thm:boundstructure} The function $p(\ensuremath{\underline{b}},\ensuremath{\overline{b}})$ is c\`agl\`ad in $\ensuremath{\overline{b}}$ and c\`adl\`ag in $\ensuremath{\underline{b}}$. Moreover, if $p$ is discontinuous at $(\ensuremath{\underline{b}},\ensuremath{\overline{b}})$, then $\mu$ must have an atom at one of $\ensuremath{\underline{b}}$ or $\ensuremath{\overline{b}}$. Further:
\begin{enumerate}
\item if there is a discontinuity at $(\ensuremath{\underline{b}},\ensuremath{\overline{b}})$ of the form:
\[
\limsup_{w \to \ensuremath{\overline{b}}} p(\ensuremath{\underline{b}},w) > p(\ensuremath{\underline{b}},\ensuremath{\overline{b}})
\]
then the function $g$ defined by
\[
g(u) = \limsup_{w \to \ensuremath{\overline{b}}} p(u,w) - p(u,\ensuremath{\overline{b}}), \qquad u \le
\ensuremath{\underline{b}}
\]
is non-increasing.
\item if there is a discontinuity at $(\ensuremath{\underline{b}},\ensuremath{\overline{b}})$ of the form:
\[
\limsup_{u \to \ensuremath{\underline{b}}} p(u,\ensuremath{\overline{b}}) > p(\ensuremath{\underline{b}},\ensuremath{\overline{b}})
\]
then the function $h$ defined by
\[
h(w) = \limsup_{u \to \ensuremath{\underline{b}}} p(u,w) - p(\ensuremath{\underline{b}},w), \qquad w \ge
\ensuremath{\overline{b}}
\]
is non-decreasing.
\end{enumerate}
And, at any discontinuity, we will be in at least one of the above
cases.
In addition the lower bound (corresponding to the
tilted-Jacka construction) is continuous in $(\ell_{\mu},0)\times
(0,r_{\mu})$, and continuous at the boundary ($\ensuremath{\overline{b}} = r_{\mu}$ and $\ensuremath{\underline{b}}
= \ell_{\mu}$) unless there is an atom of $\mu$ at either $r_{\mu}$ or
$\ell_{\mu}$, while the upper bound (which corresponds to the Perkins
construction) has a discontinuity corresponding to every atom of
$\mu$. \end{theorem}
\begin{remark}
\begin{enumerate}
\item Considering $q$ instead of $p$, the function will be c\`adl\`ag in
both arguments, and the directions of the convergence results needs
to be adapted suitably. We also observe that discontinuities in the
upper bound occur only if there is an atom of $\mu$ at $\ensuremath{\underline{b}}$, {\it
and} we are in case \fbox{I} of
Theorem~\ref{thm:upper_price_mixed}. Similarly, there is a
discontinuity in the lower bound at $\ensuremath{\overline{b}}$ if there is an atom of
$\mu$ at $\ensuremath{\overline{b}}$, and we are in either of cases \fbox{II} or
\fbox{III} of Theorem~\ref{thm:lower_price_mixed}.
\item Considering $r$ instead of $p$, the function will be c\`agl\`ad
in $\ensuremath{\underline{b}}$ and c\`adl\`ag in $\ensuremath{\overline{b}}$. We also observe that
discontinuities in the upper bound never occur, while there are
discontinuities in the lower bound at $\ensuremath{\overline{b}}$ and/or $\ensuremath{\underline{b}}$ if there is an atom of
$\mu$ at either of these values.
\end{enumerate} \end{remark}
Before we prove the above result, we note the following useful result, which is a simple consequence of the martingale property:
\begin{prop} \label{prop:supatom} Suppose that $(M_t)_{t \ge 0}$ is a
UI martingale with $M_\infty \sim \mu$. Then $\Prob(\overline{M}_{\infty} =
\ensuremath{\overline{b}}) >0$ implies $\mu(\{\ensuremath{\overline{b}}\}) \ge \Prob(\overline{M}_{\infty} = \ensuremath{\overline{b}})$ and
\[
\{ \overline{M}_{\infty} = \ensuremath{\overline{b}} \} = \{ M_t = \ensuremath{\overline{b}}, \ \forall t \ge H_{\ensuremath{\overline{b}}}\}
\subseteq \{ M_\infty = \ensuremath{\overline{b}}\} \quad a.s.{}.
\] \end{prop}
\begin{proof}[Proof of Theorem~\ref{thm:boundstructure}]
We begin by noting that by definition of $p(\ensuremath{\underline{b}},\ensuremath{\overline{b}})$, we
necessarily have the claimed continuity and limiting
properties. Further,
\[
\liminf_{(s,v) \to (u,w)} p(s,v) \ge \Prob(\underline{M}_\infty > \ensuremath{\underline{b}}
\mbox{ and } \overline{M}_\infty < \ensuremath{\overline{b}})
\]
and
\[
\limsup_{(s,v) \to (u,w)} p(s,v) \le \Prob(\underline{M}_\infty \ge \ensuremath{\underline{b}}
\mbox{ and } \overline{M}_\infty \le \ensuremath{\overline{b}}).
\]
It follows that the function $p$ is continuous at $(\ensuremath{\underline{b}},\ensuremath{\overline{b}})$ if
$\Prob(\underline{M}_\infty = \ensuremath{\underline{b}}) = \Prob(\overline{M}_\infty = \ensuremath{\overline{b}}) = 0$. By
Proposition~\ref{prop:supatom}, this is true when $\mu(\{\ensuremath{\overline{b}},\ensuremath{\underline{b}}\})
= 0$.
Note that we can now see that at a discontinuity of $p$, we must be
in at least one of the cases (i) or (ii). This is because
discontinuity at $(\ensuremath{\underline{b}},\ensuremath{\overline{b}})$ is equivalent to
\[
\Prob(\underline{M}_\infty \ge \ensuremath{\underline{b}} \mbox{ and } \overline{M}_\infty \le \ensuremath{\overline{b}}) >
\Prob(\underline{M}_\infty > \ensuremath{\underline{b}} \mbox{ and } \overline{M}_\infty < \ensuremath{\overline{b}}),
\]
from which we can deduce that at least one of the events
\[
\{\underline{M}_\infty > \ensuremath{\underline{b}} \mbox{ and } \overline{M}_\infty = \ensuremath{\overline{b}}\}, \quad
\{\underline{M}_\infty = \ensuremath{\underline{b}} \mbox{ and } \overline{M}_\infty < \ensuremath{\overline{b}}\}, \quad
\{\underline{M}_\infty = \ensuremath{\underline{b}} \mbox{ and } \overline{M}_\infty = \ensuremath{\overline{b}}\}
\]
is assigned positive mass. However, by
Proposition~\ref{prop:supatom} the final event implies both
$M_\infty = \ensuremath{\underline{b}}$ and $M_\infty = \ensuremath{\overline{b}}$ which is
impossible. Consequently, at least one of the first two events must
be assigned positive mass, and these are precisely the cases (i) and
(ii).
Consider now case (i). We can rewrite the statement as: if $g(\ensuremath{\underline{b}}) >
0$, then $g(u)$ is decreasing for $u<\ensuremath{\underline{b}}$. Note however that
\begin{eqnarray*}
g(u) & = & \Prob(\underline{M}_\infty > u \mbox{ and } \overline{M}_\infty \le \ensuremath{\overline{b}}) -
\Prob(\underline{M}_\infty > u \mbox{ and } \overline{M}_\infty < \ensuremath{\overline{b}}) \\
& = & \Prob(\underline{M}_\infty > u \mbox{ and } \overline{M}_\infty = \ensuremath{\overline{b}})
\end{eqnarray*}
which is clearly non-increasing in $u$. {In fact, provided that
$g(\ensuremath{\underline{b}})<\Prob(\overline{M}_\infty=\ensuremath{\overline{b}})$, it follows from e.g.{}
\cite[Theorem~4.1]{Rogers:93} that $g$ is strictly decreasing for
$\ensuremath{\underline{b}}>u>\sup\{u\geq -\infty: g(u)=\Prob(\overline{M}_\infty=\ensuremath{\overline{b}})\}$. A
similar proof holds in case (ii).}
We now consider the lower bounds corresponding to the tilted-Jacka
construction. We wish to show that
\[
\Prob(\underline{M}_\infty \ge \ensuremath{\underline{b}} \mbox{ and } \overline{M}_\infty \le \ensuremath{\overline{b}}) =
\Prob(\underline{M}_\infty > \ensuremath{\underline{b}} \mbox{ and } \overline{M}_\infty < \ensuremath{\overline{b}}),
\]
for any $(\ensuremath{\overline{b}},\ensuremath{\underline{b}})$ except those excluded in the statement of the
theorem. We note that it is sufficient to show that $\Prob(\underline{M}_\infty
= \ensuremath{\underline{b}}) = \Prob(\overline{M}_\infty = \ensuremath{\overline{b}}) = 0$, and by
Proposition~\ref{prop:supatom} it is only possible to have an atom
in the law of the maximum or the minimum if the process stops at the
maximum with positive probability; we note however that the stopping
time $\tau_J$, due to properties of the Az\'ema-Yor embedding
precludes such behaviour except at the points $\ell_{\mu}, r_{\mu}$.
Considering now the Perkins construction, we note from
\eqref{eq:25} and the fact that the function ${\gamma_+}$ is
decreasing, that we will stop at $\ensuremath{\underline{b}}$ only if ${\gamma_+}(F(\overline{M}_t))
= \ensuremath{\underline{b}}$ and $M_t = \underline{M}_t = \ensuremath{\underline{b}}$. It follows from \eqref{eq:24}
that there is a range of values $(\ensuremath{\overline{b}}_*,\ensuremath{\overline{b}}^*)$ for which
${\gamma_+}(F(b)) = \ensuremath{\underline{b}}$, and consequently, we must have $h(b) =
\Prob(\underline{M}_\infty = \ensuremath{\underline{b}}, \overline{M}_\infty < b)$ increasing in $b$ as $b$ goes
from $\ensuremath{\overline{b}}_*$ to $\ensuremath{\overline{b}}^*$, with $h(\ensuremath{\overline{b}}_*) = \Prob(\underline{M}_\infty = \ensuremath{\underline{b}},
\overline{M}_\infty < \ensuremath{\overline{b}}_*)=0$ and $h(\ensuremath{\overline{b}}^*) = \Prob(\underline{M}_\infty = \ensuremath{\underline{b}},
\overline{M}_\infty < \ensuremath{\overline{b}}^*)=\mu(\{\ensuremath{\underline{b}}\})$.\footnote{In fact, as above, it
follows from e.g.{} \cite[Theorem~2.2]{Rogers:93} that the maximum
must have a strictly positive density with respect to Lebesgue
measure, and therefore that the function $h$ is strictly
increasing between the points $\ensuremath{\overline{b}}_*$ and $\ensuremath{\overline{b}}^*$.} Similar
results for the function $g$ also follow. \end{proof}
\section*{Conclusions} In this paper, we studied the possible joint distributions of $(\overline{M}_\infty,\underline{M}_\infty)$ given the law of $M_\infty$, and were able to obtain number of qualitative properties and sharp quantitative bounds. It follows from our results that the interaction between the maximum and minimum is highly non-trivial which makes the pair above much harder to study than $\overline{M}_\infty$ and $\underline{M}_\infty$ on their own. This is best seen in the case of Brownian motion where $\overline{B}_t$ has an easily accessible distribution while the description of the joint distribution of $(\underline{B}_t,\overline{B}_t)$ is much more involved. A further natural question arising from our work is to characterise the joint distributions of the joint distributions of the triple $(M_\infty,\overline{M}_\infty,\underline{M}_\infty)$. At present it is not clear to us if, and to what extent, a complete characterisation of the possible joint distributions of this triple, in the spirit of Rogers \cite{Rogers:93} and Vallois \cite{Vallois:93}, is feasible. It remains an open and challenging problem.
\end{document} | arXiv |
How are regression, the t-test, and the ANOVA all versions of the general linear model?
How are they all versions of the same basic statistical method?
regression self-study anova generalized-linear-model t-test
gung - Reinstate Monica
AmahabirsinghAmahabirsingh
$\begingroup$ related: Why is ANOVA taught / used as if it is a different research methodology compared to linear regression? $\endgroup$
– Haitao Du
$\begingroup$ related: R: Anova and Linear Regression $\endgroup$
$\begingroup$ related: Why is ANOVA equivalent to linear regression? $\endgroup$
Consider that they can all be written as a regression equation (perhaps with slightly differing interpretations than their traditional forms).
Regression: $$ Y=\beta_0 + \beta_1X_{\text{(continuous)}} + \varepsilon \\ \text{where }\varepsilon\sim\mathcal N(0, \sigma^2) $$
t-test: $$ Y=\beta_0 + \beta_1X_{\text{(dummy code)}} + \varepsilon \\ \text{where }\varepsilon\sim\mathcal N(0, \sigma^2) $$
ANOVA: $$ Y=\beta_0 + \beta_1X_{\text{(dummy code)}} + \varepsilon \\ \text{where }\varepsilon\sim\mathcal N(0, \sigma^2) $$
The prototypical regression is conceptualized with $X$ as a continuous variable. However, the only assumption that is actually made about $X$ is that it is a vector of known constants. It could be a continuous variable, but it could also be a dummy code (i.e., a vector of $0$'s & $1$'s that indicates whether an observation is a member of an indicated group--e.g., a treatment group). Thus, in the second equation, $X$ could be such a dummy code, and the p-value would be the same as that from a t-test in its more traditional form.
The meaning of the betas would differ here, though. In this case, $\beta_0$ would be the mean of the control group (for which the entries in the dummy variable would be $0$'s), and $\beta_1$ would be the difference between the mean of the treatment group and the mean of the control group.
Now, remember that it is perfectly reasonable to have / run an ANOVA with only two groups (although a t-test would be more common), and you have all three connected. If you prefer seeing how it would work if you had an ANOVA with 3 groups; it would be: $$ Y=\beta_0 + \beta_1X_{\text{(dummy code 1)}} + \beta_2X_{\text{(dummy code 2)}} + \varepsilon \\ \text{where }\varepsilon\sim\mathcal N(0, \sigma^2) $$ Note that when you have $g$ groups, you have $g-1$ dummy codes to represent them. The reference group (typically the control group) is indicated by having $0$'s for all dummy codes (in this case, both dummy code 1 & dummy code 2). In this case, you would not want to interpret the p-values of the t-tests for these betas that come with standard statistical output--they only indicate whether the indicated group differs from the control group when assessed in isolation. That is, these tests are not independent. Instead, you would want to assess whether the group means vary by constructing an ANOVA table and conducting an F-test. For what it's worth, the betas are interpreted just as with the t-test version described above: $\beta_0$ is the mean of the control / reference group, $\beta_1$ indicates the difference between the means of group 1 and the reference group, and $\beta_2$ indicates the difference between group 2 and the reference group.
In light of @whuber's comments below, these can also be represented via matrix equations:
$$ \bf Y=\bf X\boldsymbol\beta + \boldsymbol\varepsilon $$ Represented this way, $\bf Y$ & $\boldsymbol\varepsilon$ are vectors of length $N$, and $\boldsymbol\beta$ is a vector of length $p+1$. $\bf X$ is now a matrix with $N$ rows and $(p+1)$ columns. In a prototypical regression you have $p$ continuous $X$ variables and the intercept. Thus, your $\bf X$ matrix is composed of a series of column vectors side by side, one for each $X$ variable, with a column of $1$'s on the far left for the intercept.
If you are representing an ANOVA with $g$ groups in this way, remember that you would have $g-1$ dummy variables indicating the groups, with the reference group indicated by an observation having $0$'s in each dummy variable. As above, you would still have an intercept. Thus, $p=g-1$.
gung - Reinstate Monicagung - Reinstate Monica
$\begingroup$ The ANOVA equation would make sense as an ANOVA (and not a t-test) only if $\beta_1$ were interpreted as a vector and multiplied on the right. $\endgroup$
– whuber ♦
$\begingroup$ These aren't matrix equations; I rarely use those here, as many people don't read them. The 1st ANOVA represents an identical situation as the preceding t-test. I'm just pointing out that if you can run a 2-sample independent t-test, you can run the same data as an ANOVA (which many people should recognize / remember from their stats 101 class). I add another ANOVA version w/ 3 groups lower down to clarify that a 2-group situation isn't the only ANOVA case that can be understood as a regression; but the reg equation now looks different--I was trying to maintain a more explicit parallel above. $\endgroup$
– gung - Reinstate Monica
$\begingroup$ My point is that unless you do make it a matrix equation, your characterization of ANOVA is too limited to be useful: it is identical to your characterization of the t-test and so is more confusing than it is helpful. When you start introducing more groups, you suddenly change the equation, which may also be less than clear. Whether you want to use matrix notation is of course up to you, but in the interest of communicating well you should strive for consistency. $\endgroup$
$\begingroup$ Could you please explain a bit more on how you arrive from popular definition of t-test to the equation you have shown.Basically I can't figure out what is Y here (it could be naivity or less IQ for stats). However how to arrive from t = (y-x-u0)/s to this equation. $\endgroup$
– Gaurav Singhal
$\begingroup$ It doesn't, although this may be unfamiliar to you. $Y$ is continuous (& assumed conditionally normal) in all cases listed. There are no distributional assumptions about $X$, it can be continuous, dichotomous, or a multi-level categorical variable. $\endgroup$
They can all be written as particular cases of the general linear model.
The t-test is a two-sample case of ANOVA. If you square the t-test statistic you get the corresponding $F$ in the ANOVA.
An ANOVA model is basically just a regression model where the factor levels are represented by dummy (or indicator) variables.
So if the model for a t-test is a subset of the ANOVA model and ANOVA is a subset of the multiple regression model, regression itself (and other things besides regression) is a subset of the general linear model, which extends regression to a more general specification of the error term than the usual regression case (which is 'independent' and 'equal-variance'), and to multivariate $Y$.
Here's an example showing the equivalence of the ordinary (equal-variance) two sample-$t$ analysis and a hypothesis test in a regression model, done in R (the actual data looks to be paired, so this isn't really a suitable analysis):
> t.test(extra ~ group, var.equal=TRUE, data = sleep)
Two Sample t-test
data: extra by group
t = -1.8608, df = 18, p-value = 0.07919
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
-3.363874 0.203874
sample estimates:
mean in group 1 mean in group 2
0.75 2.33
Note the p-value of 0.079 above. Here's the one way anova:
> summary(aov(extra~group,sleep))
Df Sum Sq Mean Sq F value Pr(>F)
group 1 12.48 12.482 3.463 0.0792
Residuals 18 64.89 3.605
Now for the regression:
> summary(lm(extra ~ group, data = sleep))
(some output removed)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.7500 0.6004 1.249 0.2276
group2 1.5800 0.8491 1.861 0.0792 .
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.899 on 18 degrees of freedom
Multiple R-squared: 0.1613, Adjusted R-squared: 0.1147
F-statistic: 3.463 on 1 and 18 DF, p-value: 0.07919
Compare the p-value in the 'group2' row, and also the p-value for the F-test in the last row. For a two-tailed test, these are the same and both match the t-test result.
Further, the coefficient for 'group2' represents the difference in means for the two groups.
$\begingroup$ Having same p values in all 3 scenarios is magical and impressive, however if you could explain a bit more on how these p-values gets calculated, it would definitely make this answer more interesting. I don't know if showing p-value calculations will make it more useful as well, so that is something you could decide. $\endgroup$
$\begingroup$ @Gaurav The p-values are the same because you're testing the same hypothesis on the same model, just represented slightly differently. If you're interested in how some specific p-value is calculated, it would be a new question (it would not be an answer to the question here). You're free to ask such a question though try a search first since it may already have been answered. $\endgroup$
– Glen_b
$\begingroup$ Thanks @Glen_b, sorry for asking an obvious question and that too in not the best way. And you still answered my question - "same hypothesis on the same model (and/or data)". I did not give enough thoughts on how they are testing the same hypothesis. Thanks $\endgroup$
This answer that I posted earlier is somewhat relevant, but this question is somewhat different.
You might want to think about the differences and similarities between the following linear models: $$ \begin{bmatrix} Y_1 \\ \vdots \\ Y_n \end{bmatrix} = \begin{bmatrix} 1 & x_1 \\ 1 & x_2 \\ 1 & x_3 \\ \vdots & \vdots \\ 1 & x_n \end{bmatrix} \begin{bmatrix} \alpha_0 \\ \alpha_1 \end{bmatrix} + \begin{bmatrix} \varepsilon_1 \\ \vdots \\ \vdots \\ \varepsilon_n \end{bmatrix} $$ $$ \begin{bmatrix} Y_1 \\ \vdots \\ Y_n \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & & \vdots \\ 1 & 0 & 0 & \cdots & 0 \\ \hline 0 & 1 & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & & \vdots \\ 0 & 1 & 0 & \cdots & 0 \\ \hline 0 & 0 & 1 & \cdots & 0 \\ \vdots & & & & \vdots \\ \vdots & & & & \vdots \end{bmatrix} \begin{bmatrix} \alpha_0 \\ \vdots \\ \alpha_k \end{bmatrix} + \begin{bmatrix} \varepsilon_1 \\ \vdots \\ \vdots \\ \varepsilon_n \end{bmatrix} $$
Michael HardyMichael Hardy
$\begingroup$ Some description and comment to the questions would useful for the readers since now they have to guess where did they came from and how do they relate to the question... $\endgroup$
– Tim ♦
Anova is similar to a t-test for equality of means under the assumption of unknown but equal variances among treatments. This is because in ANOVA MSE is identical to pooled-variance used in t-test. There are other versions of t-test such as one for un-equal variances and pair-wise t-test. From this view, t-test can be more flexible.
pemfirpemfir
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Using a general linear model to perform z-test | CommonCrawl |
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Work-family-conflict in the context of the working conditions of university employees – comparison of professions
L. Jerg-Bretzke, M. Kempf, S. Walter, H. Traue, P. Beschoner
Journal: European Psychiatry / Volume 64 / Issue S1 / April 2021
Published online by Cambridge University Press: 13 August 2021, pp. S400-S401
Working conditions at universities are often considered precarious. Employees complain of fixed-term contracts and extensive unpaid overtime (Dorenkamp et al. 2016). Studies from various fields of work show that occupational groups with a high workload suffer particularly from a conflictual compatibility of work and family.
The aim of this study was to assess the WFC in the context of working conditions.
N=844 university employees (55% women, 41% men) were asked about the burden of work/life balance using Work-family-conflict (WFC) - Family-work-conflict (FWC) -Scales (Netemeyer 1996). The dichotomously formulated question on overtime worked was supplemented by a five-step scaled item on the burden of overtime. The correlation analyses were calculated according to Spearman.
Overtime performed by 83% of the total sample and 64% feel burdened by it. 95% of the scientists and physicians, 68% of the administrative staff, 63% of the service providers work overtime and 90% of the physicians and 72% of the scientists feel burdened by it. Significantly high correlations were found between the burden of overtime and the conflict of compatibility. The higher the burden of overtime, the higher the WFC and FWC. The highest correlation was found among physicians (r=.649), followed by scientists (r=.533), administration (r=.451), services (r= (total sample r=.562).
The additional work and strain caused by this, as well as the connections with the problem of compatibility, show need for action for employers regarding the working conditions of physicians and scientists. Especially with regard to reducing overtime and improving the compatibility of work and family.
CUSTOMER SPECIFIC COMPATIBILITY MATRICES FOR FUNCTIONAL INTEGRAL PRODUCT ARCHITECTURES
Design Methods
J. Siebrecht, G. Jacobs, C. Konrad, C. Wyrwich, W. Schäfer
Journal: Proceedings of the Design Society: DESIGN Conference / Volume 1 / May 2020
Published online by Cambridge University Press: 11 June 2020, pp. 1105-1114
Supplier of system components face the challenge of customer requirements influencing the property level functional integral product architectures. For this, solution approaches focusing on the re-use of pre-engineered part variants are not applicable. However, to generate a valid product structure, customer-specific properties have to fit modelled product knowledge. Therefore, the approach models a reference class structure and analysis compatibilities on the property level for customer specific inputs concerning explicit product knowledge and constraints.
5 - Compatibility Relation
Mark Pankov
Book: Wigner-Type Theorems for Hilbert Grassmannians
Two projections commute if and only if their images are compatible. Using combinatorial methods we describe compatibility preserving bijective transformations of Grassmannians. In some cases, these transformations form a class greater than the class of transformations induced by unitary and anti-unitary operators.
2 - Analysis of Strain
from Part I - Fundamentals of Solid Mechanics
Marko V. Lubarda, Vlado A. Lubarda, University of California, San Diego
Book: Intermediate Solid Mechanics
Published online: 16 December 2019
Print publication: 09 January 2020, pp 31-50
The components of the infinitesimal strain tensor are defined, which represent measures of the relative length changes (longitudinal strains or dilatations) and the angle changes (shear strains) at a considered material point with respect to the chosen coordinate axes. The principal strains (maximum and minimum dilatations) and the maximum shear strains are determined, as well as the areal and volumetric strains. The expressions for the strain components are derived in terms of the spatial gradients of the displacement components. The Saint-Venant compatibility equations are introduced which assure the existence of single-valued displacements associated with a given strain field. The matrix of local material rotations, which accompany the strain components in producing the displacement gradient matrix, is defined. The determination of the displacement components by integration of the strain components is discussed.
3 - Case Law Analysis
Rob van Gestel, Universiteit van Tilburg, The Netherlands, Jurgen de Poorter, Universiteit van Tilburg, The Netherlands
Book: In the Court We Trust
Print publication: 19 December 2019, pp 59-103
Our case law analysis reveals that and how it takes two to tango in terms of organizing a dialogue. From the side of the referring courts closed-questions may force the CJEU to yes or no answers. The same holds true for compatibility questions asking the Court where a national laws are in compliance with EU law while the procedure is not meant to "solve" these problems. A positive way to stimulate dialogue with the CJEU by the referring court could be to make use of the possibility to offer provisional answers to the questions being referred. However, this only works when the CJEU explicitly responds to these answers. The CJEU can also discourage dialogue by reformulating questions in a way that makes the legal problem become unrecognizable to the referring court without issuing a request to the referring court to clarify the questions first. With respect to compatibility questions, the CJEU sometimes almost seems to operate as an appellate court trying to protect citizen's right by taking over the responsibility of national courts. At the same time, though, the Court shows little interest in what happens with preliminary rulings in the aftermath of its decision.
COMPATIBILITY AND ATTAINABILITY OF MATRICES OF CORRELATION-BASED MEASURES OF CONCORDANCE
Marius Hofert, Takaaki Koike
Journal: ASTIN Bulletin: The Journal of the IAA / Volume 49 / Issue 3 / September 2019
Measures of concordance have been widely used in insurance and risk management to summarize nonlinear dependence among risks modeled by random variables, which Pearson's correlation coefficient cannot capture. However, popular measures of concordance, such as Spearman's rho and Blomqvist's beta, appear as classical correlations of transformed random variables. We characterize a whole class of such concordance measures arising from correlations of transformed random variables, which includes Spearman's rho, Blomqvist's beta and van der Waerden's coefficient as special cases. Compatibility and attainability of square matrices with entries given by such measures are studied—that is, whether a given square matrix of such measures of concordance can be realized for some random vector and how such a random vector can be constructed. Compatibility and attainability of block matrices and hierarchical matrices are also studied due to their practical importance in insurance and risk management. In particular, a subclass of attainable block Spearman's rho matrices is proposed to compensate for the drawback that Spearman's rho matrices are in general not attainable for dimensions larger than three. Another result concerns a novel analytical form of the Cholesky factor of block matrices which allows one, for example, to construct random vectors with given block matrices of van der Waerden's coefficient.
A national FFQ for the Netherlands (the FFQ-NL1.0): development and compatibility with existing Dutch FFQs
Simone JPM Eussen, Martien CJM van Dongen, Nicole EG Wijckmans, Saskia Meijboom, Henny AM Brants, Jeanne HM de Vries, H Bas Bueno-de-Mesquita, Anouk Geelen, Diewertje Sluik, Edith JM Feskens, Marga C Ocké, Pieter C Dagnelie
Journal: Public Health Nutrition / Volume 21 / Issue 12 / August 2018
In the Netherlands, various FFQs have been administered in large cohort studies, which hampers comparison and pooling of dietary data. The present study aimed to describe the development of a standardized Dutch FFQ, FFQ-NL1.0, and assess its compatibility with existing Dutch FFQs.
Dutch FFQTOOLTM was used to develop the FFQ-NL1.0 by selecting food items with the largest contributions to total intake and explained variance in intake of energy and thirty-nine nutrients in adults aged 25–69 years from the Dutch National Food Consumption Survey (DNFCS) 2007–2010. Compatibility with the Maastricht-FFQ, Wageningen-FFQ and EPICNL-FFQ was assessed by comparing the number of food items, the covered energy and nutrient intake, and the covered variance in intake.
FFQ-NL1.0 comprised 160 food items, v. 253, 183 and 154 food items for the Maastricht-FFQ, Wageningen-FFQ and EPICNL-FFQ, respectively. FFQ-NL1.0 covered ≥85 % of energy and all nutrients reported in the DNFCS. Covered variance in intake ranged from 57 to 99 % for energy and macronutrients, and from 45 to 93 % for micronutrients. Differences between FFQ-NL1.0 and the other FFQs in covered nutrient intake and covered variance in intake were <5 % for energy and all macronutrients. For micronutrients, differences between FFQ-NL and other FFQs in covered level of intake were <15 %, but differences in covered variance were much larger, the maximum difference being 36 %.
The FFQ-NL1.0 was compatible with other FFQs regarding energy and macronutrient intake. However, compatibility for covered variance of intake was limited for some of the micronutrients. If implemented in existing cohorts, it is advised to administer the old and the new FFQ in combination to derive calibration factors.
Morphology, mechanical and thermal properties of poly(lactic acid) (PLA)/natural rubber (NR) blends compatibilized by NR-graft-PLA
Phijittra Sookprasert, Napida Hinchiranan
Journal: Journal of Materials Research / Volume 32 / Issue 4 / 28 February 2017
Published online by Cambridge University Press: 06 February 2017, pp. 788-800
Natural rubber (NR) is expected to enhance impact strength of poly(lactic acid) (PLA). Because the polarity difference of NR and PLA leads PLA/NR blends having phase separation and poor mechanical properties, this research aimed to synthesize NR-graft-PLA (NR–PLA) via esterification of maleated NR (NR-MAH) with PLA. The role of NR–PLA used as a compatibilizer on mechanical and thermal properties of the PLA/NR blends was studied. Maximum grafted PLA level at 66.8% (w/w) was reached when NR-MAH was esterified with PLA [2/1 (w/w) PLA/NR-MAH] catalyzed by 0.05 M 4-dimethylaminopyridine at 140 °C. The addition of 5% (w/w) NR–PLA [36.6% (w/w) grafted PLA content] into PLA/NR blend [80/20 (w/w)] increased Izod impact strength of the neat PLA plate from 28.9 J/m to 62.7 J/m due to partial miscibility of blends attested by morphology analysis and Molau test. Hydrolytic degradation of PLA/NR blends with and without the addition of NR–PLA was also examined.
Computing Optimal Interfacial Structure of Modulated Phases
Equilibrium statistical mechanics
Statistical mechanics, structure of matter
Jie Xu, Chu Wang, An-Chang Shi, Pingwen Zhang
Journal: Communications in Computational Physics / Volume 21 / Issue 1 / January 2017
Published online by Cambridge University Press: 05 December 2016, pp. 1-15
We propose a general framework of computing interfacial structures between two modulated phases. Specifically we propose to use a computational box consisting of two half spaces, each occupied by a modulated phase with given position and orientation. The boundary conditions and basis functions are chosen to be commensurate with the bulk phases. We observe that the ordered nature of modulated structures stabilizes the interface, which enables us to obtain optimal interfacial structures by searching local minima of the free energy landscape. The framework is applied to the Landau-Brazovskii model to investigate interfaces between modulated phases with different relative positions and orientations. Several types of novel complex interfacial structures emerge from the calculations.
Ultrafiltration performance and fouling resistance of PVB/SPES blend membranes with different degree of sulfonation
Shuhong Jiang, Jun Wang, Jun Wu, Hongzhong Zhou, Chuanwei Jiang
Journal: Journal of Materials Research / Volume 30 / Issue 18 / 28 September 2015
Published online by Cambridge University Press: 18 August 2015, pp. 2688-2701
In the present study, we investigated the effects of different degree of sulfonation (DS) on the performance of the poly (vinyl butyral)/sulfonated polyethersulfone (PVB/SPES) blend membranes. The compatibility of the PVB/SPES blending system was characterized by shear viscosity and Fourier transform infrared attenuated total reflection, respectively. Results stated that all PVB/SPES blending systems were partially compatible. Contact angle, equilibrium water content, and x-ray photoelectron spectroscopy measurements were carried out to investigate the hydrophilicity of the PVB/SPES blend membranes. With increasing DS, the blend membranes became more hydrophilic. The pure water flux of the blend membranes increased with DS, while the rejection decreased due to microstructures of the PVB/SPES membranes. The mechanical properties of the PVB/SPES blend membranes increased slightly with DS. Fouling resistances of blend membranes evaluated by bovine serum albumin solution filtration revealed the PVB/SPES blend membranes with DS = 27% exhibited the superior antifouling properties.
THE NATURAL PARTIAL ORDER ON THE SEMIGROUP OF ALL TRANSFORMATIONS OF A SET THAT REFLECT AN EQUIVALENCE RELATION
Semigroups
LEI SUN, XIANGJUN XIN
Journal: Bulletin of the Australian Mathematical Society / Volume 88 / Issue 3 / December 2013
Published online by Cambridge University Press: 25 January 2013, pp. 359-368
Print publication: December 2013
Let ${ \mathcal{T} }_{X} $ be the full transformation semigroup on a set $X$ and $E$ be a nontrivial equivalence relation on $X$. Denote
$$\begin{eqnarray*}{T}_{\exists } (X)= \{ f\in { \mathcal{T} }_{X} : \forall x, y\in X, (f(x), f(y))\in E\Rightarrow (x, y)\in E\} ,\end{eqnarray*}$$
so that ${T}_{\exists } (X)$ is a subsemigroup of ${ \mathcal{T} }_{X} $. In this paper, we endow ${T}_{\exists } (X)$ with the natural partial order and investigate when two elements are related, then find elements which are compatible. Also, we characterise the minimal and maximal elements.
Benefits of fidelity: does host specialization impact nematode parasite life history and fecundity?
J. KOPRIVNIKAR, H. S. RANDHAWA
Journal: Parasitology / Volume 140 / Issue 5 / April 2013
The range of hosts used by a parasite is influenced by macro-evolutionary processes (host switching, host–parasite co-evolution), as well as 'encounter filters' and 'compatibility filters' at the micro-evolutionary level driven by host/parasite ecology and physiology. Host specialization is hypothesized to result in trade-offs with aspects of parasite life history (e.g. reproductive output), but these have not been well studied. We used previously published data to create models examining general relationships among host specificity and important aspects of life history and reproduction for nematodes parasitizing animals. Our results indicate no general trade-off between host specificity and the average pre-patent period (time to first reproduction), female size, egg size, or fecundity of these nematodes. However, female size was positively related to egg size, fecundity, and pre-patent period. Host compatibility may thus not be the primary determinant of specificity in these parasitic nematodes if there are few apparent trade-offs with reproduction, but rather, the encounter opportunities for new host species at the micro-evolutionary level, and other processes at the macro-evolutionary level (i.e. phylogeny). Because host specificity is recognized as a key factor determining the spread of parasitic diseases understanding factors limiting host use are essential to predict future changes in parasite range and occurrence.
Evaluation of the R&R and the compatibility index for non-independent measurements
A. Garcia-Benadí, E. Molino-Minero-Re, J. del Río-Fernández, A. Mànuel-Lázaro
Journal: International Journal of Metrology and Quality Engineering / Volume 4 / Issue 1 / 2013
Published online by Cambridge University Press: 05 June 2013, pp. 23-28
This paper describes the methodology of the compatibility criteria En and the methodology of repeatability and reproducibility (R&R) throught the average and range method. With this paper we will use the methodology R&R for the evaluation of the compatibility criteria between the staff of the laboratories, where independent measurements are not insured.
Microstructural Characterization of Consolidant Products for Historical Renders: An Innovative Nanostructured Lime Dispersion and a More Traditional Ethyl Silicate Limewater Solution
Giovanni Borsoi, Martha Tavares, Rosário Veiga, Antonio Santos Silva
Journal: Microscopy and Microanalysis / Volume 18 / Issue 5 / October 2012
Published online by Cambridge University Press: 15 October 2012, pp. 1181-1189
Print publication: October 2012
The conservation and durability of historical renders must be carried out through compatible techniques and materials. An important operation is the restitution of historical renders cohesion, turned friable by the loss of binder, usually due to physical and/or chemical actions. Surface consolidation is based on the use of materials with aggregating properties. This operation is reached usually through the application of organic or mineral consolidants, but inorganic consolidants are becoming preferred due to better compatibility and durability. In this article two mineral compatible consolidation products were studied: a commercial suspension of calcium hydroxide nanoparticles in propanol and a limewater dispersion of ethyl silicate. Microscopy (optical and scanning electron microscopy) and X-ray microanalyses of the consolidation products and of the consolidated mortar specimens were carried out. To assess the mechanical properties and product's efficacy, analyses of the compression, flexural strength, and superficial hardness were performed. Microscopy results show that limewater dispersion of ethyl silicate forms platelike silica gels, which can interfere in product penetration. Otherwise, nanolime particles permit homogeneous distribution and optimum penetration on the treated substrate, improving cementing action and the agglomeration process.
PARTIAL ORDERS ON SEMIGROUPS OF PARTIAL TRANSFORMATIONS WITH RESTRICTED RANGE
KRITSADA SANGKHANAN, JINTANA SANWONG
Journal: Bulletin of the Australian Mathematical Society / Volume 86 / Issue 1 / August 2012
Let X be any set and P(X) the set of all partial transformations defined on X, that is, all functions α:A→B where A,B are subsets of X. Then P(X) is a semigroup under composition. Let Y be a subset of X. Recently, Fernandes and Sanwong defined PT(X,Y )={α∈P(X):Xα⊆Y } and defined I(X,Y ) to be the set of all injective transformations in PT(X,Y ) . Hence PT(X,Y ) and I(X,Y ) are subsemigroups of P(X) . In this paper, we study properties of the so-called natural partial order ≤ on PT(X,Y ) and I(X,Y ) in terms of domains, images and kernels, compare ≤ with the subset order, characterise the meet and join of these two orders, then find elements of PT(X,Y ) and I(X,Y ) which are compatible. Also, the minimal and maximal elements are described.
Assessment of Radio Frequency Compatibility between Compass Phase II and Other GNSSs
Li Liu, Xingqun Zhan, Wei Liu, Mancang Niu
Journal: The Journal of Navigation / Volume 64 / Issue S1 / November 2011
Published online by Cambridge University Press: 14 October 2011, pp. S55-S72
As the technology of global navigation satellite system (GNSS) and augmentation systems are evolving rapidly, compatibility becomes a critical issue for system providers. By April 2011, China had successfully launched eight satellites of the Compass phase II (CP II) navigation system, which will provide positioning, navigation, timing and communication services to the Asia-Pacific region by the year 2012. Due to the limitations of available radio frequency bandwidths, it is important to assess the compatibility and to design signals based on the compatibility within these limited radio frequency bandwidths. This paper presents a modified compatibility assessment methodology, derived from the traditional methodologies that are based on the spectral separation coefficient (SSC) and the effective carrier-power-to-noise density ratio. The modified methodology takes into account additional factors including the Doppler offset, code tracking loop, and band-limiting, sampling and quantisation (BSQ) of the GNSS receiver. In the simulation section, the comprehensive compatibility assessment between CP II and other GNSSs, such as GPS, Galileo, Wide Area Augmentation System (WAAS) and European Geostationary Navigation Overlay Service (EGNOS) on L1 Band are carried out and presented with some new results. Simulation results reveal that CP II does not cause serious interference on GPS, Galileo, WAAS and EGNOS as the interference level is below the 0·25 dB threshold recommended by ITU.
Assessment of Radio Frequency Compatibility Relevant to the Galileo E1/E6 and Compass B1/B3 Bands
Wei Liu, Gang Du, Xingqun Zhan, Chuanrun Zhai
Journal: The Journal of Navigation / Volume 63 / Issue 3 / July 2010
Print publication: July 2010
The intersystem interference between Galileo and Compass, known as a radio frequency compatibility problem, has become a matter of great concern for the system providers and user communities. This paper firstly describes two fundamentally different methods to assess the Global Navigation Satellite System (GNSS) intersystem interference, by using different interference coefficients that are calculated for each combination of signals: the spectral separation coefficient (SSC) and code tracking spectral sensitivity coefficient (CT_SSC). And then a complete methodology combining the SSC and CT_SSC is presented. Real simulations are carried out to assess the interference effects where Galileo and Compass signals are sharing the same band (E1/B1 and E6/B3 bands) at every time and place on the Earth. Simulation results show that the effects of intersystem interference are significantly different by using these two methodologies. It is also shown that the Compass system leads to intersystem interference on Galileo but that the maximal values are lower than Galileo interference to Compass. The design and implementation of any new signal has to be conducted carefully in order for there to be radio frequency compatibility.
INTERNATIONAL LAW AND GEOPOLITICS: ONE OBJECT, CONFLICTING LEGITIMACIES?*
Alexander Orakhelashvili
Journal: Netherlands Yearbook of International Law / Volume 39 / December 2008
The disciplines of international law and geopolitics have evolved around the same object – the exercise of State power in space. But the interaction between geopolitical and legal categories has not been properly examined yet. Similar to international law, geopolitics focuses on certain, albeit not formally binding, laws that govern or explain the conduct of States in relation to space. There is room for the geopolitical laws reasoning to lead to outcomes that differ from those required under international legal obligations of States. In other cases, geopolitical laws and reasoning could actually explain why certain international legal rules and institutions are what they are. This contribution is the first attempt to study geopolitics and international law in parallel to each other. It is demonstrated that the evolution of geopolitical thinking, whether as part of a particular expansionist or containment agenda or as scientific approach, has constantly reflected on the categories of international law, and also has been used in practice by States to justify their particular conduct in defiance of international legal requirements. At the same time, international law has traditionally left to States the room for pursuing their geopolitical agenda without breaking the requirements of international law. It is here that the significance of geopolitical factors for international law becomes clear, as the allegedly lawful expansionist action by States can lead, and has repeatedly led, to reactions that involve breaches, and potentially damage the integrity, of international law. Despite geopolitical agenda being allegedly lawful, it still has to observe certain geopolitical laws in order to avoid broader negative repercussions both for that agenda and for international law.
Effects of laboratory culture on compatibility between snails and schistosomes
A. THERON, C. COUSTAU, A. ROGNON, S. GOURBIÈRE, M. S. BLOUIN
Journal: Parasitology / Volume 135 / Issue 10 / September 2008
The genetic control of compatibility between laboratory strains of schistosomes and their snail hosts has been studied intensively since the 1970s. These studies show (1) a bewildering array of genotype-by-genotype interactions – compatibility between one pair of strains rarely predicts compatibility with other strains, and (2) evidence for a variety of (sometimes conflicting) genetic mechanisms. Why do we observe such variable and conflicting results? One possibility is that it is partly an artifact of the use of laboratory strains that have been in culture for many years and are often inbred. Here we show that results of compatibility trials between snails and schistosomes – all derived from the same natural population – depend very much on whether one uses laboratory-cultured or field-collected individuals. Explanations include environmental effects of the lab on either host or parasite, and genetic changes in either host or parasite during laboratory culture. One intriguing possibility is that genetic bottlenecks during laboratory culture cause the random fixation of alleles at highly polymorphic loci that control the matched/mismatched status of hosts and parasites. We show that a simple model of phenotype matching could produce dose response curves that look very similar to empirical observations. Such a model would explain much of the genotype-by-genotype interaction in compatibility observed among strains.
NATURALLY ORDERED TRANSFORMATION SEMIGROUPS PRESERVING AN EQUIVALENCE
LEI SUN, HUISHENG PEI, ZHENGXING CHENG
Let 𝒯X be the full transformation semigroup on a set X and E be a nontrivial equivalence on X. Write then TE(X) is a subsemigroup of 𝒯X. In this paper, we endow TE(X) with the so-called natural order and determine when two elements of TE(X) are related under this order, then find out elements of TE(X) which are compatible with ≤ on TE(X). Also, the maximal and minimal elements and the covering elements are described. | CommonCrawl |
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